Keywords: Fiber-orientation tensor, fiber-reinforced composites, interpolation, process simulation, anisotropy

Abstract: We propose and assess a new decomposition-based interpolation method on fourth-order fiber-orientation tensors. This method can be used to change the resolution of discretized fields of fiber-orientation tensors, e.g., obtained from flow simulations or computer tomography, which are common in the context of short- and long-fiber–reinforced composites. The proposed interpolation method separates information on structure and orientation using a parametrization which is based on tensor components and a unique eigensystem. To identify this unique eigensystem of a given fourth-order fiber-orientation tensor in the absence of material symmetry, we propose a sign convention on tensor coefficients. We explicitly discuss challenges associated with material symmetries, e.g., non-distinct eigenvalues of the second-order fiber-orientation tensor and propose algorithms to obtain a unique set of parameters combined with a minimal number of eigensystems of a given fourth-order fiber-orientation tensor. As a side product, we specify for the first time, parametrizations and admissible parameter ranges of cubic, tetragonal, and trigonal fiber-orientation tensors. Visualizations in terms of truncated Fourier series, quartic plots, and tensor glyphs are compared.

Keywords: Cahn–Hilliard-type phase-field model; Elasto-plastic deformations; Hydride formation; Open metal hydrogen system; Thin film

Abstract: For the usage of intercalating material systems to store and convert energy of renewable sources, their phase stabilities need to be engineered to adjust to the desired operation conditions. This can, e.g., be achieved by miniaturization, leading to constraints that modify the systems thermodynamics. The experimental investigation of such systems is cumbersome, as experiments on nano-sized systems are time intensive. Numerical simulations based on chemo-mechanically coupled continuum models can serve as a tool helping to understand these systems and to study different effects of miniaturization. In this work we present a phase-field model for the example of open, constrained metal hydrogen thin film systems, that allows the prediction of the hydrogen intercalation and hydride formation. The model relies on a free energy density consisting of chemical, mechanical and interfacial parts. The first two contributions are based on measurements of the thermodynamics of open Niobium–Hydrogen thin films, that are chosen as a model. The interfacial contribution of Cahn–Hilliard-type introduces a phase-field description for both phases. To study the systems behavior a numerical implementation in the commercial Finite Element solver ABAQUS is presented. Numerical results are presented and compared to previously obtained experimental results on the open systems thermodynamics. We show, that the model is capable of reproducing experimentally observed behavior of thin films especially regarding the coexistence of α- and hydride-phase in thermodynamic equilibrium, where the equilibrium concentrations in both phases drastically differ from bulk values, and gradients in concentration and stresses result due to the interfacial constraint conditions.

Keywords: Continuum thermodynamics; Coupled problems; Finite element method; Thermo-chemo-mechanical model

Abstract: Physically coupled models are used in various research fields to solve problems concerning the interaction of solid materials with thermal, chemical or electrical boundary conditions. If beside the mechanical fields two or more additional fields (e.g. temperature and concentration) have to be taken into account, the determination of the impact on the mechanical fields (e.g. stress and yield strength) and the influence of the boundary conditions leads to an ambitious task. To deal with this issue, a thermo-chemo-mechanical model using a geometrically linear theory and a thermodynamically consistent derivation, is presented. The model is specified for linear elastic isotropic solid materials. A fully coupled set of partial differential equations is obtained. A Finite Element implementation using the User Element subroutine of ABAQUS is performed. A detailed description about the steps necessary to derive the corresponding element formulation is provided, thereby supporting the development of user-defined elements. The user-defined element is used for a series of simulations involving submodels with up to seven different combinations of active fields including thermo-chemical, thermo-mechanical and chemo-mechanical couplings. Three sets of boundary conditions are considered, leading to closed and open systems. This procedure, which is exemplified in one- and three-dimensional examples, enlightens strong and weak couplings and outlines the mechanical role on the interaction between the solid material and the chemical environment. Additionally, the implications of a geometrical linearization on the interpretation of the concentration is illustrated.

Keywords: Enthalpy relaxation; Fictive temperature; Glass transition; Heat capacity; Polystyrene

Abstract: The glass transition is a central physical phenomenon in polymeric materials. It is related with significant changes of the thermomechanical properties and pronounced process-dependent behavior. This work deals with process-dependent caloric behavior in the glass transition regime which is known as enthalpy relaxation. Based on Coleman-Noll procedure and the scheme of linear irreversible thermodynamics, a model of enthalpy relaxation is derived and analyzed. Therefore, an internal temperature is introduced as internal variable and the analogy with viscoelasticity is utilized. The non-equilibrium part of the entropy is identified as thermodynamic driving force. For a purely temperature-dependent relaxation time of the internal temperature, the qualitative behavior of the model is discussed. Furthermore, the model parameters are examplarily adapted for polystyrene. In a first step, rate-independent behavior of the heat capacity in glassy and equilibrium is approximated by linear functions. Subsequently, the parameters for the evolution equation of the internal temperature are identified based on the reduced heat capacity. By comparing two different temperature rates, a rate-dependency of the relaxation time is observed. Therefore, the model is extended by a rate-dependency function for the relaxation time where temperature- and rate-dependency are multiplicatively splitted.

Keywords: gradient crystal plasticity; laminate microstructures; micro mechanics; size effect

Abstract: Metallic materials may show an ultra-fine lamellar morphology leading to desirable macroscopic mechanical properties. In this paper, an analytical method for modeling the size-dependent mechanical behavior of material systems with lamellar microstructure is proposed. The main contribution of this manuscript is the combination of the exact elastic localization relations for a periodic laminate having inhomogeneous phases with a gradient plasticity constitutive model. This allows a new formulation of the equations governing the evolution of the plastic deformation as a closed system of equations. The yield functions form a system of integro-differential equations that can be solved analytically. This new framework allows to model the size-dependent mechanical behavior of multiphase lamellar materials where the subdomains are elastoplastic single crystals. This is demonstrated using a two-phase laminate. In addition, bounds for the qualitative distributions of the plastic slip and the back stress are derived using a dimensionless quantity. The kinematic hardening due to the back stress depends on the lamella width and the slip system orientation. The back stress vanishes not only for increasing lamella widths but also in slip systems where the slip plane normal is parallel to the lamination direction.

Keywords: elastoplasticity; harmonic decomposition; linear elasticity; Tensorial basis; texture

Abstract: In continuum mechanics, one regularly encounters higher-order tensors that require tensorial bases for theoretical or numerical calculations. By using results from (Formula presented.) representation theory, we present a method to derive tensorial harmonic bases which unifies existing approaches in one framework. Applying this convention to symmetric second-order tensors results in a second-order harmonic basis which, for example, simplifies the depiction and numerical handling of stiffness tensors for most common material symmetries and reduces the computational effort involved in implementations of incompressible elastoviscoplastic material laws. The same convention can be applied to texture analysis to yield tensorial texture coefficients for both polycrystals and fiber-reinforced composites. Rotations of higher-order tensors are particularly efficient in harmonic bases as both material and index symmetries can be exploited. While special attention is given here to the examples of small-strain material laws and texture analysis, the framework is entirely general and can be used to simplify calculations in other physical contexts involving tensors of arbitrary order and symmetry. A Python implementation of the harmonic basis convention defined in this work is available as a Supplementary Material.

Keywords: Maximum entropy method · Linear thermoelasticity · Heterogeneous materials · Diffraction · Residual stress analysis

Abstract: X-ray diffraction methods are an established technique to analyze residual stresses in polycrystalline materials. Using diffraction, lattice plane distances are measured, from which residual stresses can be calculated by using diffraction elastic constants which can be inferred from experimental measurements or calculated based on micromechanical model assumptions. We consider two different generalizations of existing micromechanical models for the case of texture-free, i.e. statistically isotropic, single-phase polycrystals. The first is based on the singular approximation method of classical micromechanics, from which existing Voigt, Reuss, Hashin-Shtrikman and self-consistent methods are recovered. The second approach, which is newly proposed in this work, is based on the micromechanical Maximum Entropy Method. Both approaches are applied to the problem of calculating diffraction elastic constants of texture-free cubic polycrystals and are found to be consistent with each other in that case. Full-field FFT simulations are used to validate the analytical models by simulating X-ray diffraction measurements of copper. In the simulative setting, many sources of experimental measurement error are not present, which results in a particularly accurate validation of theoretical bounds and approximations. The first core result of the paper is a formulation of diffraction elastic constants for texture-free polycrystals in terms of the macroscopically measurable effective shear modulus. These diffraction elastic constants can be adapted to the properties of a given material sample. The second core result is the validation of the Maximum Entropy Method for X-ray diffraction stress analysis of texture-free single-phase materials as a preliminary step before extending the method to textured and multi-phase materials.

Keywords: curved fibers; fused sequential addition and migration; Long fiber reinforced hybrid composites; microstructure generation; representative volume elements

Abstract: We present a fused sequential addition and migration (fSAM) algorithm to generate synthetic microstructures for long fiber reinforced hybrid composites. To incorporate the bending of long fibers, we model the fibers as polygonal chains and use an optimization framework to search for a non-overlapping fiber configuration with the desired properties. As microstructures of hybrid composites consist of materials with varying characteristics, e.g., different fiber orientation distributions or fiber volume fractions, the unit cell needs to be divided in subcells. Then, the objective function is required to account for the characteristics of the subcells individually. To ensure that the location of a fiber is restricted to its respective subcell, we apply box constraints during the optimization procedure. Depending on the materials and the manufacturing processes, the interfaces within hybrid composites show either strict separation or interlocking of the different materials. Hence, we enable selecting the box-constraint type to model the interface according to the desired composite. We provide a detailed discussion of the extensions of the original fSAM algorithm to generate hybrid composites, considering two alternative procedures to handle the box constraints during the optimization procedure. As the selection of the box-constraint type influences the synthetic microstructures, we investigate the shape of the interfacial areas and the computed effective stiffnesses. Last but not least, to validate the capability of the fSAM algorithm to generate representative microstructures, we compare the resulting mechanical behavior with experimental data.

Keywords: curvature control; curved fibers; fused sequential addition and migration; long fiber reinforced composites; microstructure generation; representative volume elements

Abstract: The degree of curvature of fiber inclusions in fiber-reinforced composites impacts the elastic properties of the composite, but numerical studies of this effect are limited by the lack of efficient microstructure generators with fiber curvature control. We propose an extension to the fused Sequential Addition and Migration algorithm (Int. J. Numer. Methods Eng., 125(22), e7573, 2024) to allow for the direct control of the average fiber curvature. To this end, we prescribe an averaged curvature parameter for the entire microstructure, while also constraining local curvature for every fiber to a maximum value. While local curvature is unambiguously defined, the choice of the averaged curvature parameter is not obvious. We introduce as a novel curvature measure the deviation from straight curve (DSC), which computes as the trace of the local second-order fiber orientation tensor's covariance for a single fiber. Averaging this over all fibers leads to a scalar averaged curvature parameter. Using this parameter, we define an optimization procedure for generating curved-fiber reinforced microstructures. We compare the microstructure generation capabilities of the proposed extension with the prior algorithm without curvature control and find that the extended algorithm is capable of realizing a significantly wider spectrum of curvatures. Finally, we study the influence of fiber curvature on elastic properties, and find that for an example composite of glass-fiber reinforced polypropylene, increased fiber curvature leads to a reduction of the Young's modulus by as much as 10%.

Keywords: Homogenization; Mean-field theory; Probabilistic micromechanics; Stress fluctuations; Unidirectional fibers

Abstract: Statistical fluctuations of local tensorial fields beyond the mean are relevant to predict localized failure or overall behavior of the inelastic composites. The expression for second moments of the local fields can be established using the Hill-Mandel condition. Complete estimation of the statistical fluctuations via second moments is usually ignored despite its significance. In Eshelby-based mean-field approaches, the second moments are evaluated through derivatives of Hill's Polarization tensor ℙo using a singular approximation. Typically, semi-analytical procedures using numerical integration are used to evaluate the derivatives of the polarization tensor Po. Here, new analytically derived explicit expressions are presented for calculating the derivatives, specifically for unidirectional fibrous composites with isotropic phases. Full-field homogenization using finite element is used to compute the statistical distribution of local fields (exact solution) for the class of random fibrous microstructures. The mean-field estimates are validated with the exact solution across different fiber volume fractions and aspect ratios. The results indicate that the fiber volume fraction significantly influences the fluctuation of stress tensor invariants, whereas the aspect ratio has minimal effect.

Keywords: (Visco-)plasticity; Automatic differentiation; Finite deformation; Finite elements; Lithium-ion battery; Numerical simulation

Abstract: As an anode material for lithium-ion batteries, amorphous silicon offers a significantly higher energy density than the graphite anodes currently used. Alloying reactions of lithium and silicon, however, induce large deformation and lead to volume changes up to 300%. We formulate a thermodynamically consistent continuum model for the chemo-elasto-plastic diffusion-deformation behavior of amorphous silicon and it’s alloy with lithium based on finite deformations. In this paper, two plasticity theories, i.e. a rate-independent theory with linear isotropic hardening and a rate-dependent one, are formulated to allow the evolution of plastic deformations and reduce occurring stresses. Using modern numerical techniques, such as higher order finite element methods as well as efficient space and time adaptive solution algorithms, the diffusion-deformation behavior resulting from both theories is compared. In order to further increase the computational efficiency, an automatic differentiation scheme is used, allowing for a significant speed up in assembling time as compared to an algorithmic linearization for the global finite element Newton scheme. Both plastic approaches lead to a more heterogeneous concentration distribution and to a change to tensile tangential Cauchy stresses at the particle surface at the end of one charging cycle. Different parameter studies show how an amplification of the plastic deformation is affected. Interestingly, an elliptical particle shows only plastic deformation at the smaller half axis. With the demonstrated efficiency of the applied methods, results after five charging cycles are also discussed and can provide indications for the performance of lithium-ion batteries in long term use.

Keywords: Cross-fluid; Effective viscosity; Fiber-reinforced composites; Non-Newtonian suspension; Supervised machine learning

Abstract: In this article, we combine a Fast Fourier Transform based computational approach and a supervised machine learning strategy to discover models for the anisotropic effective viscosity of shear-thinning fiber suspensions. Using the Fast Fourier Transform based computational approach, we study the effects of the fiber orientation state and the imposed macroscopic shear rate tensor on the effective viscosity for a broad range of shear rates of engineering process interest. We visualize the effective viscosity in three dimensions and find that the anisotropy of the effective viscosity and its shear rate dependence vary strongly with the fiber orientation state. Combining the results of this work with insights from literature, we formulate four requirements a model of the effective viscosity should satisfy for shear-thinning fiber suspensions with a Cross-type matrix fluid. Furthermore, we introduce four model candidates with differing numbers of parameters and different theoretical motivations, and use supervised machine learning techniques for non-convex optimization to identify parameter sets for the model candidates. By doing so, we leverage the flexibility of automatic differentiation and the robustness of gradient based, supervised machine learning. Finally, we identify the most suitable model by comparing the prediction accuracy of the model candidates on the fiber orientation triangle, and find that multiple models predict the anisotropic shear-thinning behavior to engineering accuracy over a broad range of shear rates.

Keywords: deep material network; effective viscosity; fiber-reinforced composites; infinite material contrast; non-Newtonian suspension; supervised machine learning

Abstract: We extend the laminate based framework of direct deep material networks (DMNs) to treat suspensions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks that are capable of treating incompressible fluid phases and infinite material contrast. In particular, we leverage existing results for linear elastic laminates to identify closed form expressions for the linear homogenization functions of two-phase layered emulsions. To treat infinite material contrast, we rely on the repeated layering of two-phase layered emulsions in the form of coated layered materials. We derive necessary and sufficient conditions which ensure that the effective properties of coated layered materials with incompressible phases are non-singular, even if one of the phases is rigid. With the derived homogenization blocks and non-singularity conditions at hand, we present a novel DMN architecture, which we name the flexible DMN (FDMN) architecture. We build and train FDMNs to predict the effective stress response of shear-thinning fiber suspensions with a Cross-type matrix material. For 31 fiber orientation states, six load cases, and over a wide range of shear rates relevant to engineering processes, the FDMNs achieve validation errors below 4.31% when compared to direct numerical simulations with fast-Fourier-transform based computational techniques. Compared to a conventional machine learning approach introduced previously by the consortium of authors, FDMNs offer better accuracy at an increased computational cost for the considered material and flow scenarios.

Keywords: flexoelectricity; multiscale methods; second-order homogenization; size-dependent effects

Abstract: Flexoelectricity shows promising applications for self-powered devices with its increased power density. This paper presents a second-order computational homogenization strategy for flexoelectric composite. The macro-micro scale transition, Hill–Mandel energy condition, periodic boundary conditions, and macroscopic constitutive tangents for the two-scale electromechanical coupling are investigated and considered in the homogenization formulation. The macrostructure and microstructure are discretized using (Formula presented.) triangular finite elements. The second-order multiscale solution scheme is implemented using ABAQUS with user subroutines. Finally, we present numerical examples including parametric analysis of a square plate with holes and the design of piezoelectric materials made of non-piezoelectric materials to demonstrate the numerical implementation and the size-dependent effects of flexoelectricity.

Keywords: micromechanics short fiber reinforced thermoplastics fatigue Material modeling

Abstract: This work covers the topic of micromechanical modeling of the elastic and fatigue behavior of short glass fiber reinforced thermoplastic composites (SFRTs). A novel algorithm extracts microstructural data from x-ray micro-computed tomography scans. Analytical homogenization schemes are studied and used to predict elastic properties, while a high cycle fatigue model using the Two-Step scheme incorporates damage variables for fibers and matrix, predicting fatigue limits.

Keywords: computational Micromechanics; Data-driven modelling; Datengetriebene Modellierung; Deep Material Networks; Fasersuspensionen; Fiber suspensions; Machine learning; Numerische Mikromechanik; Maschinelles Lernen

Abstract: We present a Fast-Fourier-Transform (FFT) based computational approach to computing the viscous stress response of rigid fibers suspended in a non-Newtonian medium. We identify closed-form models for the fiber suspension viscosity from data obtained with the FFT-based computational approach by leveraging supervised machine learning techniques. Furthermore, we present a novel Deep Material Network architecture capable of treating suspensions of rigid particles with high computational efficiency.

Keywords: Micromechanics; Maximum Entropy; Diffraction; Polycrystals; Texture

Abstract: In this work, we investigate various methods of calculating local stress and strain statistics of heterogeneous materials with particular focus on polycrystalline metals. The main method under consideration is the micromechanical Maximum Entropy Method, an approximation technique based on results from information theory, which predicts local stress and strains based on the macroscopic properties of a material without explicit descriptions of the microstructure.

Keywords: Thermoplastics; Viscoelasticity; Enthalpy relaxation; Time-temperature superposition

Abstract: This work presents the theory of linear thermoviscoelasticity. Mechanical and calorical part of the model are formulated by using an analogy. The corresponding parts of the model are applied regarding two specific problems. First, the modeling of the viscoelastic behavior of polyamide 6 under hydrothermal influences is considered. In addition, the process-dependent calorical behavior of polystyrene in the vicinity of the glass transition is modeled.

Keywords: Diffusion, Continuum theory; chemo-mechanically coupled; diffusion

Abstract: Investigating multi-physics problems using continuum models is increasingly important, e.g., the coupling of chemical and mechanical fields in the design of energy storage systems. To address real processes, validated models and efficient implementation schemes in commercial Finite-Element solvers are essential. In this work, we derive chemo-mechanically coupled continuum models. We adress hydrogen embrittlement, phase-separation in metal-hydrogen systems and water diffusion in polymers.

Keywords: Fiber suspensions; Fiber orientation; Mean-field homogenization; Anisotropic microstructure; Composites

Abstract: This work deals with the micromechanical modeling of the orientation evolution of short-fibers in suspensions with an emphasis on mean-field homogenization. At the beginning, the impact of the anisotropic viscosity of the suspension on the stress field under load after the fluid-solid transition is studied. The subsequent parts contain the investigation of novel closure approximations and the generalized micromechanical formulation of the orientation tensor evolution of arbitrary order.

Keywords: Deep material networks; Fatigue; Inverse parameter identification; Multi-scale modeling; Short-fiber composites

Abstract: A key challenge for the virtual characterization of components manufactured using short fiber-reinforced thermoplastics (SFRTs) is the inherent anisotropy which stems from the manufacturing process. To address this, a multi-scale approach is necessary, leveraging deep material networks (DMNs) as a micromechanical surrogate, for a one-stop solution when simulating SFRTs under highly nonlinear long-term load cases like creep and fatigue. Therefore, we extend the a priori fiber orientation tensor interpolation for quasi-static loading (Liu et al. in Intelligent multi-scale simulation based on process-guided composite database. arXiv:2003.09491, 2020; Gajek et al. in Comput Methods Appl Mech Eng 384:113,952, 2021; Meyer et al. in Compos Part B Eng 110,380, 2022) using DMNs with a posteriori approach. We also use the trained DMN framework to simulate the stiffness degradation under fatigue loading with a linear fatigue-damage law for the matrix. We evaluate the effectiveness of the interpolation approach for a variety of load classes using a dedicated fully coupled plasticity and creep model for the polymer matrix. The proposed methodology is validated through comparison with composite experiments, revealing the limitations of the linear fatigue-damage law. Therefore, we introduce a new power-law fatigue-damage model for the matrix in the micro-scale, leveraging the quasi-model-free nature of the DMN, i.e., it models the microstructure independent of the material models attached to the constituents of the microstructure. The DMN framework is shown to effectively extend material models and inversely identify model parameters based on composite experiments for all possible orientation states and variety of material models.

Keywords: Chemo-mechanical coupling; Flory–Huggins theory; Polyamide 6; Swelling; Viscoelasticity; Water diffusion

Abstract: Polyamides can absorb or desorb water from or to their surrounding environment. The impact of this process is significant as water molecules lead locally to a swelling and a coupling of diffusion and deformation behavior. To model these phenomena, a strongly coupled chemo-mechanical (or diffuso-mechanical) model is required, considering both local water concentration and the viscoelastic material behavior of polyamide. In the present work, we derive and apply such a model to polyamide 6. A diffusion equation describing changes in water concentration is coupled to the balance of linear momentum in polyamide 6. The interaction between deformation and concentration is derived from thermodynamic considerations by introducing a free energy consisting of a mechanical and a chemical part. The mechanical part describes a linear viscoelastic model and includes chemical strains due to the presence of water molecules. The chemical part builds upon the theory of Flory and Huggins, that takes into account changes in enthalpy and entropy of mixing due to the interaction of polymer and water molecules. The coupling of deformation to water concentration arises due to a dependency of the water flux on the hydrostatic stress inside the polyamide. We successfully apply the derived model in Finite-Element simulations to predict the drying of polyamide 6 specimens without any coupling to mechanical loads. In addition, we reproduce experimentally obtained data from relaxation measurements, where the drying of polyamide specimens leads to an increase in relaxation modulus.

Keywords: Critical temperature; Mechanical stress; Metal-hydrogen system; Phase stability; Thermodynamics

Abstract: Different elastic constraint conditions affect the phase stabilities of metal-hydrogen systems. This is studied considering the free energy density within a chemo-mechanically coupled approach with linear elastic deformations and homogeneous concentrations. Utilizing the palladium-hydrogen alloy as a model system, the effects of various mechanical constraints in 1D, 2D and 3D are investigated. These constraints change occurring mechanical deformations and strongly influence the systems chemical potential compared to the unconstrained system. With increasing dimensionality of the constraints, large compressive mechanical stresses occur, which destabilize the hydride phase. This yields a reduced critical temperature of hydride formation. Spinodal and equilibrium miscibility gaps of the system are deduced as a function of the boundary conditions. Notably, the critical temperature of the ideal palladium-hydrogen system with 2D constraints is predicted to be [Formula presented], revealing a driving force for hydride formation at room temperature even under these constraints.

Keywords: Critical temperature; Elasto-plastic deformation; Equilibrium concentrations; Metal-hydrogen system; Thin film

Abstract: Constraint conditions and elasto-plastic deformation alter phase stabilities of metal-hydrogen systems. Experiments on niobium-hydrogen thin films reveal, that elastically deforming films suppress hydride formation at [Formula presented], while elasto-plastically deforming films can form a hydride phase. Building upon a thermodynamic model studying the coupling of elastic deformations, constraint conditions and phase separation, elasto-plastic deformations are incorporated to investigate hydride formation. The stress state for each constraint condition for both elastically and elasto-plastically deforming Nb is specified. The monotony of the resulting chemical potential reveals hydride formation to be possible in elasto-plastically deforming niobium-hydrogen films, while it is suppressed by large stresses in elastically deforming films. Critical temperatures of hydride formation and miscibility gaps for both elastically and plastically deforming niobium are computed. The critical temperature is way below [Formula presented] in elastically deforming films, while it is close to that of a stress free system in strongly elasto-plastically deforming films.

Keywords: Fiber orientation, Fiber suspensions, Homogenization, Anisotropy, Composites, Short fibers

Abstract: Fiber orientation tensors are widely used to efficiently describe the fiber orientation evolution of anisotropic fiber suspensions during mold-filling. Real viscous fiber suspensions show fiber-induced anisotropic behavior. It is an open question how to correctly account for the anisotropic microstructure of fiber suspensions in general in the evolution equation of the fiber orientation tensors. In this study, a generalized evolution equation for the fiber orientation tensor of arbitrary even order is formulated, which takes into account the microstructural anisotropy of the fiber suspension. This formulation is based on a linear homogenization approach which allows the application of arbitrary mean-field models. The derived interaction term, representing the anisotropic environment of a single fiber, is discussed as a micromechanical convergence criterion of the underlying integral operator. It is shown and discussed in detail how the special cases of the Jeffery equation and the Folgar–Tucker equation follow from this general formulation. In addition, the generalized evolution equation is specified for selected mean-field models. In this context, an equation is presented to describe the evolution of the fiber orientation tensor depending on the spatial distribution of the fibers. For simple shear flow, all models describing the orientation dynamics are numerically investigated and compared. The results for the second-order orientation tensor as well as for a single fiber show that the well-known periodic behavior is present and strongly depends on the volume fraction and the chosen mean-field model. Considering the spatial distribution of the fibers has a significant effect on the orientation evolution and strongly affects the periodic reorientation.

Abstract: Polymer-based composites play a crucial role in lightweighting applications by providing resource-efficient semi-structural materials. Among these, polyamides, in particular polyamide 6 (PA 6), serve as widely used matrix materials in various technical applications, especially in fiber-reinforced composites. The mechanical properties and performance of PA 6 are significantly affected by environmental factors such as temperature and humidity. Previous research has investigated the influence of hydrothermal conditions on the mechanical properties of pure PA 6. This study aims to extend the preceding investigations with additional experimental tests. As a preliminary work to later investigate the influence of the fibers on the hydrothermal behavior, discontinuous long carbon fiber-reinforced PA 6 is considered for micromechanical modeling of the effective behavior. In this context, a mean-field homogenization approach based on the Hashin–Shtrikman principle is used. By means of a variable reference stiffness, an approach specifically tailored to the material system under consideration can be applied. The simulation data are compared with data from experiments under tension load indicating good agreements for a specific choice of the reference material.

Keywords: Closure approximation; Effective elastic stiffness; Fiber orientation tensors; Fiber-reinforced composite; Mean field homogenization; Virtual process chain

Abstract: Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.

Keywords: curved fibers, d’Alembert type constrained mechanical systems, fused sequential addition and migration, long fiber reinforced composites, microstructure generation, representative volume elements

Abstract: We introduce the fused sequential addition and migration (fSAM) algorithm for generating microstructures of fiber composites with long, flexible, nonoverlap- ping fibers and industrial volume fractions. The proposed algorithm is based on modeling the fibers as polygonal chains and enforcing, on the one hand, the nonoverlapping constraints by an optimization framework. The connectivity constraints, on the other hand, are treated via constrained mechanical systems of d’Alembert type. In case of straight, that is, nonflexible, fibers, the proposed algorithm reduces to the SAM (Comput. Mech., 59, 247–263, 2017) algorithm, a well-established method for generating short fiber-reinforced composites. We provide a detailed discussion of the equations governing the motion of a flex- ible fiber and discuss the efficient numerical treatment. We elaborate on the integration into an existing SAM code and explain the selection of the numeri- cal parameters. To capture the fiber length distributions of long fiber reinforced composites, we sample the fiber lengths from the Gamma distribution and intro- duce a strategy to incorporate extremely long fibers. We study the microstructure generation capabilities of the proposed algorithm. The computational examples demonstrate the superiority of the novel microstructure-generation technology over the state of the art, realizing large fiber aspect ratios (up to 2800) and high fiber volume fractions (up to 32% for an aspect ratio of 150) for experimentally measured fiber orientation tensors.

Keywords: Crystal plasticity theory; Mechanical jump conditions; Multiphase-field theory

Abstract: An implementation of the crystal plasticity theory in the context of the multiphase-field method provides a numerically efficient tracking of evolving grain boundaries, modeled as diffuse interfaces. In literature, several approaches exist for the implementation of the plastic material behavior within the diffuse interface, based on interpolation, homogenization, or the mechanical jump conditions. Among these, only the jump condition approach exhibits an intrinsic relationship to the sharp interface (SI) theory. Therefore, in the work at hand, the implementation of the crystal plasticity theory within the jump condition approach, referred to as phase-specific plastic fields approach (PSPFA), is discussed in detail. The PSPFA is compared to the interpolation approach, referred to as common plastic fields approach (CPFA), using three-dimensional benchmark simulations of a bicrystal set-up. The comparison reveals that the PSPFA and SI coincide convincingly regarding the accumulated plastic slip and the Mises stress. In contrast, a significant deviation of CPFA and SI is observed both quantitatively and qualitatively, not only within the diffuse interface region, but throughout the complete simulation domain. A variation of the interface width illustrates that this observation can be transferred to the normal components of the total strain, even for smaller interface widths. Consequently, a quantitative estimate of the plastic material behavior, which is crucial for the prediction of the dynamic behavior of grain boundaries, is only provided by the PSPFA. The application of the crystal plasticity in the context of PSPFA to more complex microstructures is illustrated with respect to a periodic honeycomb-structure and an octotuple.

Keywords: CAE chain; Co-molding simulation; LFT; Structural simulation; Mean-field homogenization

Abstract: A virtual process chain for compression molded long fiber-reinforced thermoplastic (LFT) composites with co-molded continuous fiber-reinforced thermoplastics (CoFRTP) consisting of a compression molding and structural simulation step is established. The compression molding simulation considers the three-dimensional initial fiber orientation distribution of the semi-finished LFT plastificate and applies the Moldflow rotary diffusion (MRD) model to predict the reorientation of fibers. The predicted fiber orientations are compared to experimental results obtained from micro computed tomography (µCT) scans. The mapping step from molding to structural simulation allows the transfer of higher order anisotropy. Challenges in homogenizing the effective elastic material behavior of the direct (D-) LFT are discussed. The structural simulation is validated by means of coupon-level fourpoint bending tests on a D-LFT tape sandwich. The predicted bending stiffness shows higher accuracy if the mapped fiber orientation data are considered.

Keywords: Ammonia; Hydrogen enrichment; Stagnation flow flame; Thermo-mechanical stress; Corrosion

Abstract: The steady laminar stagnation flow flame of NH3-H2-air gas mixture stabilized at a plane wall is numerically investigated. Its interaction with the wall with the consideration of heat loss is the focus of this work. The numerical study of the combustion system is performed by using the full chemical mechanism and detailed transport model including the differential diffusion and Soret effect. The simulation of the solid mechanics is based on the theory of isotropic linear thermo-elasticity. With the numerical simulation, it will be discussed how the wall material would change the flame stability in terms of extinction limit, and how the combustion system such as mixture composition, flame strain rate, and pressure would vary the thermo-mechanical stresses in the solid wall and the corrosive behavior at the surface of the wall.

Keywords: -

Abstract: A laminar stagnation flow flame stabilized at a plane wall is theoretically analyzed, coupled with a thermo-elasticity model in the wall. Mathematical models for both the flame and the wall will be proposed, and corresponding analytical solutions based on their dimensionless forms will be obtained. The mathematical analysis of this flame-solid interaction primarily focuses on the effect of the flame on combustion-induced thermo-mechanical stresses and the influence of wall material on flame properties such as flame temperature and stability against extinction. A sensitivity analysis is further performed to examine how thermo-mechanical stress inside the wall is affected by changes in other combustion conditions (e.g., mixture composition, flame stretch rate). This study offers valuable insights into understanding the interaction between flames and solids, a relationship crucial in engineering applications such as the interaction between combustion processes and blades within gas turbine machinery.

Keywords: Closure approximation; Effective elastic stiffness; Fiber orientation tensors; Fiber-reinforced composite; Mean field homogenization; Virtual process chain

Abstract: Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors.

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Abstract: In this contribution, selected results from [1–3] are presented in a compact and simplified way. In addition, the variety of fiber orientation tensors is used to determine a maximum deviation of the direction-dependent Young's modulus, which can arise if only second-order directional information is included in a specific meanfield homogenization. Focusing on the special case of planar fiber distributions, the variety of fiber orientation tensors identified in [1] is considered as a design space. This design space is completely explored for the orientation-averaging homogenization following [4], fixed material parameters and fixed fiber volume content. The possible directional dependence of the resulting effective stiffnesses is graphically presented using polar plots of the direction-dependent Young's modulus. These polar plots are arranged on two-dimensional slices within the parameter space of planar fourth-order fiber orientation tensors. This gives a complete representation of the influence of the orientation tensor on the anisotropic stiffness tensor. Consequences of closure approximations, i.e., restriction to second-order directional information, are demonstrated and motivate measurement of fourth-order fiber orientation tensors.

Keywords: Closure approximation; Effective elastic stiffness; Fiber-orientation tensor; Fiber-reinforced composite; Injection molding

Abstract: Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations.

Keywords: Creep; Deep material networks; Inverse parameter identification; Multiscale model; Short fiber reinforced thermoplastics; Viscoplasticity

Abstract: In this work, we propose to use deep material networks (DMNs) as a surrogate model for full-field computational homogenization to inversely identify material parameters of constitutive inelastic models for short fiber-reinforced thermoplastics (SFRTs). Micromechanics offers an elegant way to obtain constitutive models of materials with complex microstructure, as these emerge naturally once an accurate geometrical representation of the microstructure and expressive material models of the constituents forming the material are known. Unfortunately, obtaining the latter is non-trivial, essentially for two reasons. For a start, experiments on pure samples may not accurately represent the conditions present in the composite. Moreover, the manufacturing process may alter the material behavior, and a subsequent modification is necessary. To avoid modeling the physics of the manufacturing process, one may identify the material models of the individual phases of the composite based on experiments on the composite. Unfortunately, this procedure requires conducting time-consuming simulations. In the work at hand, we use Deep Material Networks to replace full-field simulations, and to carry out an inverse parameter optimization of the matrix model in a SFRT. We are specifically concerned with the long-term creep response of SFRTs, which is particularly challenging to both experimental and simulation-based approaches due to the strong anisotropy and the long time scales involved. We propose a dedicated fully coupled plasticity and creep model for the polymer matrix and provide details on the experimental procedures.

Keywords: Micromechanical modeling; Morphology variation; Shear-lag modeling; Stationary creep

Abstract: Analytic shear-lag models enjoy great popularity for assessing and interpreting microstructure dependent stationary creep in fibrous metal composites, especially the formulation of Kelly-Street (Kelly and Street, 1972 [2]). Beyond the original model's scope, i.e. large aspect ratios of inclusions, predictions are highly inaccurate, which was recently pointed out by Wicht et al. (Wicht et al., 2022 [10]), by comparing model predictions to micromechanical Fast-Fourier-Transform-based simulations. In this study we therefore modify basic Kelly-Street model assumptions, concerning effective creep rate, stress transfer and inclusion spacing to arrive at a modified model with an extended scope. To validate the modified model, we benchmark the model against Fast-Fourier-Transform-based micromechanical simulations. We show, that the proposed modifications are successful in extending the model's scope to inclusions with small aspect ratios ≤20. Thus, the reformulated model is a powerful tool to describe and interpret microstructure dependent creep.

Keywords: Micromechanical modeling; Morphology variation; Shear-lag modeling; Stationary creep

Abstract: In this work, a physically based dislocation theory of plasticity is derived within an extended continuum mechanical context. An orientation-dependent grain boundary flow rule is introduced for the modelling of dislocation pile-up at grain boundaries and dislocation transmission through grain boundaries. With the conventional grain boundary modelling approach according to Gurtin (Gurtin. 2008 J. Mech. Phys. Solids 56, 640-662. (doi:10.1016/j.jmps.2007.05. 002)) the single-crystal consistency check for the limit case of adjacent grains that hold no misorientation is not satisfied. In order to overcome this modelling shortcoming, a slip system coupling based on a geometric measure of slip system compatibility is introduced. In order to investigate the grain boundary modelling approaches, the analytical solution of a three-phase periodic laminate is used to study the interactions of dislocations and grain boundaries within the gradient crystal plasticity framework. With the developed grain boundary model two grain boundary states, i.e. microhard and microcontrolled, are observed for misaligned grains. This allows the modelling of slip activation at grain boundaries based on the dislocation pile-up stress.

Keywords: digital image-based, microstructure characterization, Minkowski tensor, quadratic normal tensor

Abstract: For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean field homogenization or as target value for generating synthetic microstructures.

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Abstract: Deep material networks (DMN) are a data-driven homogenization approach that show great promise for accelerating concurrent two-scale simulations. As a salient feature, DMNs are solely identified by linear elastic precomputations on representative volume elements. After parameter identification, DMNs act as surrogates for full-field simulations of such volume elements with inelastic constituents. In this work, we investigate how the training on linear elastic data, i.e., how the choice of the loss function and the sampling of the training data, affects the accuracy of DMNs for inelastic constituents. We investigate linear viscoelasticity and derive a material-informed sampling procedure for generating the training data and a loss function tailored to the problem at hand. These ideas improve the accuracy of an identified DMN and allow for significantly reducing the number of samples to be generated and labeled.

Keywords: Damage; High cycle fatigue; Mean field homogenization; Micromechanics; Polyamide 6.6; Short fiber reinforced composites

Abstract: This paper presents a micromechanical high cycle fatigue damage model for short fiber reinforced composite materials. Because the damage processes within such materials are influenced strongly by their microstructure, we use a mean field homogenization framework to compute the macroscopic behavior of the composite as well as the microscopic stresses and strains in the constituent phases. This allows us to account for damage phenomena related to both the fiber phase as well as in the matrix material. Fiber and fiber-interface damage is modeled using a Tsai–Wu and Weibull-based approach and the progressive damage due to cyclic loading is described by a progressive matrix damage model. The latter includes a novel coupling term in which the matrix damage progression is linked to the damage state of the reinforcing fibers. A cycle-based numerical formulation is used to overcome the computational limits of such load cases in the time domain. While the approaches are in principle applicable to different types of fiber–matrix composite, a short glass fiber reinforced polyamide 6.6 is used as an example material, for which microstructural analyses as well as tensile and fatigue tests are reported. The model's capabilities with regard to complex fiber orientations and different fiber fractions are studied using different grades of this material class. Furthermore, the model is analyzed via benchmarks of the numerical schemes and by parameter sensitivity studies. The results show that the approach is capable of modeling the complex and microstructure-dependent fatigue damage and fatigue limits for the different material grades with limitations only becoming visible at very high fiber fractions.

Keywords: Anisotropy; Closure approximation; Composites; Fiber orientation; Folgar–Tucker equation; Homogenization

Abstract: A novel closure approximation method for fiber orientation tensors is proposed namely the fully symmetric implicit closure. Besides the full index symmetry, implicitly formulated closures based on the contraction condition fulfill the trace condition and the trace-preserving property of the Folgar–Tucker equation. As a first example, the fully symmetric implicit quadratic closure is considered as a simple modification of a recently proposed symmetric quadratic closure. It is shown that this closure can be realized by a fiber orientation distribution function. Secondly, the fully symmetric implicit hybrid closure is proposed as a counter-example of a closure not being based on a orientation distribution function. Both closures are compared against classical approximations in view of orientation evolution in a simple shear flow. Furthermore, the capability of predicting the effective viscous and elastic behavior of fiber suspensions and solid composites is investigated for a given fiber orientation state. The results show that the proposed implicit closures can be used to approximate the maximum entropy closure. Thereby, both the quadratic and the hybrid approach alleviate the high computational burden of the maximum entropy closure, as their evaluation requires solving a one-dimensional problem only. In addition, the predicted effective behavior based on the implicit closures shows an overall good agreement with predictions based on measured orientation data.

Keywords: Flow–fiber coupling; Homogenization; Micromechanics; Numerical stabilization; Short-fiber reinforced composites

Abstract: The anisotropic elastic properties of injection molded composites are fundamentally coupled to the flow of the fiber suspension during mold-filling. Regarding the modeling of mold-filling processes, both a decoupled and a flow–fiber coupled approach are possible. In the latter, the fiber-induced viscous anisotropy is considered in the computation of the flow field. This in turn influences the evolution of the fiber orientation compared to the decoupled case. This study investigates how flow–fiber coupling in mold-filling simulation affects the stress field in the solid composite under load based on the final elastic properties after fluid–solid transition. Furthermore, the effects of Newtonian and non-Newtonian polymer matrix behavior are investigated and compared. The entire process is modeled micromechanically unified based on mean-field homogenization, both for the fiber suspension and for the solid composite. Different numerical stabilization methods of the mold-filling simulation are discussed in detail. Short glass fibers with a typical aspect ratio of 20 and a volume fraction of 20% are considered, embedded in polypropylene matrix material. The results show that the flow–fiber coupling has a large effect on the fiber orientation tensor in the range of over ± 30% with respect to the decoupled simulation. As a consequence, the flow–fiber coupling affects the stress field in the solid composite under load in the range of over ± 10%. In addition, the predictions based on a non-Newtonian modeling of the matrix fluid differ significantly from the Newtonian setup and thus the necessity to consider the shear-thinning behavior is justified in a quantifiable manner.

Keywords: DMA; generalized Maxwell model; humidity influence; polyamide 6; Time-temperature superposition; viscoelasticity

Abstract: Polyamides serve as matrix material for fiber reinforced composites and are widely applied in many different engineering applications. In this context, they are exposed to various environmental influences ranging from temperature to humidity. Thus, the influence of these environmental conditions on the mechanical behavior and the associated implications on the performance of the material is of utmost importance. In this work, the thermoviscoelastic behavior of polyamide 6 (PA 6) for two equilibrium moisture contents is investigated. To this end, dynamic mechanical analysis tests with and without humidity control of the environmental chamber were performed. In terms of relaxation tests, the experimental results reveal drying effects and increased diffusion activities when the sample’s equilibrium moisture content differs from the ambient humidity level within the testing chamber. Temperature-frequency tests quantify the humidity-induced shift of the glass transition temperature. The linear generalized Maxwell model (GMM) and time-temperature superposition are used to analyze the hydrothermal effects on the linear viscoelastic material properties and the onset of mechanical nonlinearity. Based on these investigations and findings, insight is gained on the humidity influence on the material properties and the limitations of linear thermoviscoelastic modeling. Furthermore, the computational construction of master curves and the parameter identification for a generalized Maxwell model are described in detail.

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Abstract: Thermoplastic polyamides are used in many industrial areas due to their potential in lightweight applications. Polyamides serve as matrix material in fiber reinforced thermoplastics, for instance. The mechanical behavior of polyamides is characterized by pronounced viscoelastic effects that are strongly affected by environmental conditions like temperature or humidity. In this work, linear thermoviscoelastic behavior of polyamide 6 is considered. Viscoelastic behavior is modeled by the generalized Maxwell model while extended time-temperature superposition is used to model temperature dependency. A temperature-frequency sweep conducted by dynamic mechanical analysis serves as input for the model. By horizontal and vertical shifting, master curves of the loss factor, storage modulus, and loss modulus are obtained. Based on this, limitations of time-temperature superposition and linear thermoviscoelastic modeling are discussed. Furthermore, it is shown that the horizontal shifts can be well approximated by the Williams-Landel-Ferry equation for temperatures above and below the glass transition temperature.

Keywords: Micromechanics, Local fields, Second moments, Mean field, Random sphere assemblage

Abstract: In micromechanical approaches to failure prediction of inhomogeneous microstructures or homogenization of inelastic materials, local field fluctuations beyond the mean field are relevant, but not easily accessible except for resorting to full-field simulations. However, any analytical estimate of the effective strain energy density combined with the Hill–Mandel condition implies some information about the probability distributions of local fields such as stresses and strains. In this paper, analytical equations for the second moments of local fields are derived for particulate composites with isotropic phases. Explicit expressions are provided using the Mori–Tanaka approach, the Hashin–Shtrikman bounds and the singular approximation. The results are compared to full-field simulations of a random sphere assemblage. As the Hashin–Shtrikman covariance is isotropic, it is not well-suited for predicting the behavior under deviatoric loadings. The singular approximation and Mori–Tanaka methods yield sensible approximations for the second moment of matrix strain.

Keywords: Fiber orientation-length distribution; Fiber-reinforced composites; Microstructure generation

Abstract: We present an algorithm for generating short fiber-reinforced microstructures with almost planar fiber orientation. The orientation corrected shaking (OCS) method achieves a high accuracy regarding the volume fraction, fiber length distribution and fiber orientation state. Additionally, the algorithm is capable of generating microstructures for industrial materials, e.g., for a PA66GF35 material with a volume fraction of 19.3% and an aspect ratio of 33. For typical manufacturing processes, short fiber-reinforced composites show a mainly planar fiber arrangement. Therefore, we extend the two-step shaking algorithm of Li et al. [J. Ind. Text. 51(1), pp. 506–530, 2022] for a user-selected rectangular size of the unit cell and periodic boundary conditions. Additionally, the hidden closure structure of the algorithm is uncovered and a precise realization of the fiber orientation state achieved. We examine the representative volume element size for the OCS method, observing representative errors below 2% even for unit cells with edge lengths smaller than the mean fiber length. Additionally, the influence of different closure approximations on the stiffness is investigated. When applied to an industrial PA66GF35 material with sandwich structure, the OCS method demonstrates differences below 2% and 9% for the computed directional Young's moduli E1 and E2 compared to experimental data.

Keywords: Microstructure generation; Fiber-reinforced composites; Fiber orientation-length distribution

Abstract: We present an adaption of the Orientation Corrected Shaking (OCS) method, originally developed for discontinuous short fiber-reinforced polymers, for generating continuous discontinuous fiber-reinforced polymers. Due to the combination of continuous and discontinuous reinforced phases, the interface between the two layers needs to be accounted for in a realistic way. The material flow during compression molding of CoDiCoFRP leads to ply migration at the interface resulting in an interlinked region between the layers, which is observable in 3D imaging. To account for this type of interface, we adapt the OCS method to enforce soft constraints for the fibers, i.e., a fiber's midpoint but not the entire fiber is constrained to its respective layer. Hence, at the interface fibers may penetrate the opposite phase, providing a link between the phases. In a computational study, we generate CoDiCoFRP with the OCS method and investigate whether the selection of the fiber length distribution type for given volume-weighted mean m and standard deviation s influences the effective stiffness of the generated microstructures. For Weibull, Gamma and log-normal distribution the results almost coincide, with differences smaller than 0.5%, revealing that for these cases the statistical quantities m and s are the only important descriptors to model the fiber length distribution.

Keywords: Micro-mechanics; Numerical analysis; Statistical properties; Compression molding; Virtual process chain

Abstract: The manufacturing process of Sheet Molding Compound (SMC) influences the properties of a component in a non-deterministic fashion. To predict this influence on the mechanical performance, we develop a virtual process chain acting as a digital twin for SMC specimens from compounding to failure. More specifically, we inform a structural simulation with individual fields for orientation and volume fraction computed from a direct bundle simulation of the manufacturing process. The structural simulation employs an interpolated direct deep material network to upscale a tailored SMC damage model. We evaluate hundreds of virtual specimens and conduct a probabilistic analysis of the mechanical performance. We estimate the contribution to uncertainty originating from the process-induced inherent random microstructure and from varying initial SMC stack configurations. Our predicted results are in good agreement with experimental tensile tests and thermogravimetric analysis.

Keywords: Cross-fluid; Effective viscosity; FFT-based method; Fiber-reinforced composites; Non-Newtonian

Abstract: In this work, we investigate the fiber orientation dependent viscosity of fiber suspensions using a computational homogenization method. To improve computational prediction capabilities for the viscosity of fiber suspensions, we extend an existing, Fast Fourier Transform based computational approach for fiber suspensions with linear viscous, i.e., Newtonian, matrix behavior to nonlinear viscous matrix behavior. Specifically, a Cross-type shear-thinning rheology is assumed for the matrix fluid. We employ composite voxels to lower resolution requirements and find through resolution studies that the resolution error decreases for certain shear rates. Furthermore, we conduct a volume element study and find that the representative volume element sizes for engineering considerations in a specific Newtonian case and the investigated Cross-type case are highly similar. For shear rates of engineering process interest we visualize the effective suspension viscosity in three dimensions and study the effects of the fiber volume fraction and the imposed macroscopic shear rate tensor on the suspension viscosity. We find that the elongational viscosity and the degree of anisotropy of the suspension viscosity vary stronger with the shear rate for higher fiber volume fractions. In a comparison with an analytical mean-field model for the suspension viscosity, the deviations between the computational and analytical results turn out to be substantial.

Keywords: A-plane; Condition; Flame stability; Mechanical stress; Plane walls; Solids process; Stagnation flows; Strain-rates; Thermo-mechanical analysis; Thermo-mechanical stress

Abstract: While thermomechanical analyses in solid and combustion processes in the gas phase are intensively studied separately, their interaction is less investigated. On one hand, the combustion system can be affected by the solid as a result of, for example, the heat conduction. On the other hand, the high-temperature flame can induce the thermal load in the solid, which significantly affects the mechanical stress field in the material. This work focuses on the coupling between the solid and flame and the combustion-induced mechanical stress in the material. The influence of the solid on the flame structures and properties and also the influence of the flame on the thermomechanical behavior in the solid are investigated. A stagnation flow NH3-H2-air flame to a plane wall is considered as a representative model, which is simple but also realistic in many engineering conditions. It is mainly found that an increase of the flame strain rate leads to an increase of thermomechanical stress in the wall, and the system pressure improves the flame stability against extinction but enlarges the induced thermomechanical stress at the same time. Furthermore, it is observed that the hydrogen content in the gas mixture does not affect the thermomechanical stress in the wall if the flames with different hydrogen additions are imposed with the same strain rate. On the basis of various flame parameters, it is also shown that the solid would also flow plastically under certain conditions, such as high pressures. From the viewpoint of the wall, it is mainly shown that the flame stability against extinction can be improved using wall materials with larger heat conductivities.

Keywords: Characterization; forming; microstructural modeling; residual stress; Sheet metal; simulation; springback; strain

Abstract: In scientific studies, sheet metal is usually considered as a two-dimensional body. Thus, it is accepted that material properties are in most cases regarded homogeneous in thickness direction. However, a gradation of certain properties becomes apparent when going beyond the standard characterization methods for sheet metals, which can for example, influence the springback behavior and the thinning of the sheet after forming. Thus, the aim of this work is to further improve the prediction accuracy of springback after forming in simulations, by implementing several inhomogeneous properties over the sheet thickness in an existing material model. For this purpose, the entire procedure from the identification of the inhomogeneous properties for describing the gradation to the implementation in a numerical model and its validation by comparing experimental and simulated bending operations is carried out on a DC04 cold-forming steel in order to prove its influence on the springback behavior. It is shown that including graded material properties in simulations does indeed have an impact on the prediction quality of springback and that the information about inhomogeneous properties can be provided by existing characterization methods with a high local resolution like electron backscatter diffraction or X-ray stress analysis. In a further step, it was possible to validate the improvement in numerical accuracy by comparing the prediction of the springback angle from both the existing and the extended model with experimental bending results. Both the initial model as well as the model supplemented with the 3D properties provide a good prediction accuracy in the solution heat treated material state. For the predeformed material, however, the initial numerical model predicts a springback angle of about 13°, which deviates remarkably from the experimentally obtained mean value of about 17°. The extended model delivers a significantly improved accuracy in springback prediction in relation to the initial prediction (deviation of 4°) with a minor deviation of only about 0.8°, which proves the importance of considering the gradation of material properties in thickness direction for an overall higher dimensional accuracy of sheet metal products.

Keywords: Effective crack energy; FFT-based computational homogenization; Phase-filed fracture; Minkowski tensors; Fast marching methods

Abstract: Materials of industrial interest often show a complex microstructure which directly influences their macroscopic material behavior. For simulations on the component scale, multi-scale methods may exploit this microstructural information. This work is devoted to a multi-scale approach for brittle materials. Based on a homogenization result for free discontinuity problems, we present FFT-based methods to compute the effective crack energy of heterogeneous materials with complex microstructures.

Keywords: Polycrystalline metals; Fatigue; Micromechanical modeling; Laguerre tessellations; Texture coefficients optimization

Abstract: Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments.

Keywords: deep material networks; Two-scale simulations; Deep Material Networks; micromechanics; machine learning

Abstract: We investigate deep material networks (DMN). We lay the mathematical foundation of DMNs and present a novel DMN formulation, which is characterized by a reduced number of degrees of freedom. We present a efficient solution technique for nonlinear DMNs to accelerate complex two-scale simulations with minimal computational effort. A new interpolation technique is presented enabling the consideration of fluctuating microstructure characteristics in macroscopic simulations.

Keywords: Fiber orientation tensor; Fiber-reinforced composites; Mean field homogenization; Anisotropy; Elasticity

Abstract: Effective mechanical properties of fiber-reinforced composites strongly depend on the microstructure, including the fibers' orientation. Studying this dependency, we identify the variety of fiber orientation tensors up to fourth-order using irreducible tensors and material symmetry. The case of planar fiber orientation tensors, relevant for sheet molding compound, is presented completely. Consequences for the reconstruction of fiber distributions and mean field homogenization are presented.

Keywords: SMC composite; Thermomechanical properties; Microstructure-based macro modeling; Biaxial damage and failure experiments

Abstract: The aim of this work is to model and experimentally characterize the anisotropic material behavior of SMC composites on the macroscale with consideration of the microstructure. Temperature-dependent thermoelastic behavior and failure behavior are modeled and the corresponding material properties are determined experimentally. Additionally, experimental biaxial damage investigations are performed. A parameter identification merges modeling and experiments and validates the models.

Keywords: mechanics; continuum mechanics; mechanics of materials; linear elasticity; fiber orientation; tensors notation

Abstract: The Python package mechkit is a toolkit for researchers in the field of continuum mechanics and material modeling. mechkit contains methods and operators for elementary tasks concerning tensor algebra and tensor notation, e.g., linear mapping, base transformations and active as well as passive transformations of tensors. In the context of engineering applications in three spatial dimensions, strains and stresses in solids are usually described by second-order tensors. In linear elastictiy, mappings between stresses and strains are important. To this end, the methods in mechkit are focussed on second- and fourth-order tensors. Main motivations can thus be found in the research concerning linear elasticity (Bertram & Glüge, 2015), (Mandel, 1965), (Fedorov, 1968), (Mehrabadi & Cowin, 1990), (Thomson, 1856), (Cowin & Mehrabadi, 1992), (Rychlewski, 2000), (Spencer, 1970), (Böhlke & Brüggemann, 2001), (Brannon, 2018) and the description of microstructures of fiber-reinforced composite materials (Bauer & Böhlke, 2021), (Ken-Ichi, 1984), (Advani & Tucker III, 1987). The implementations in mechkit aim at usability, seek to provide understandable source code, and do not put major emphasis on performance. Furthermore, the implementations follow, as directly as possible, the notation and formulation in the respective primary scientific sources. A redundant implementation of identical operations based on different sources is aimed at, for validation reasons.

Keywords: Anisotropy; Fiber orientation tensor; fiber-reinforced composites; harmonic decomposition

Abstract: Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given.

Keywords: Anisotropy; Closure approximation; Elasticity; Fiber orientation tensor; Fiber reinforced composites; Mean field homogenization; Small strain

Abstract: A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young's modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed.

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Keywords: Computational homogenization; Creep loading; Deep material networks; Multiscale methods; Short fiber-reinforced thermoplastics

Abstract: Deep material networks (DMNs) are a recent multiscale technology which enable running concurrent multiscale simulations on industrial scale with the help of powerful surrogate models for the micromechanical problem. Classically, the parameters of the DMNs are identified based on linear elastic precomputations. Once the parameters are identified, DMNs may process inelastic material models and were shown to reproduce micromechanical full-field simulations with the original microstructure to high accuracy. The work at hand was motivated by creep loading of thermoplastic components with fiber reinforcement. In this context, multiple scales appear, both in space (due to the reinforcements) and in time (short- and long-term effects). We demonstrate by computational examples that the classical training strategy based on linear elastic precomputations is not guaranteed to produce DMNs whose long-term creep response accurately matches high-fidelity computations. As a remedy, we propose an inelastically informed early stopping strategy for the offline training of the DMNs. Moreover, we introduce a novel strategy based on a surrogate material model, which shares the principal nonlinear effects with the true model but is significantly less expensive to evaluate. For the problem at hand, this strategy enables saving significant time during the parameter identification process. We demonstrate that the novel strategy provides DMNs which reliably generalize to creep loading.

Keywords: Computational homogenization; Concurrent two-scale simulations; Deep material networks; Hierarchical laminates; Short fiber reinforced polyamide; Thermomechanical composites

Abstract: We extend the FE-DMN method to fully coupled thermomechanical two-scale simulations of composite materials. In particular, every Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN). Such a DMN serves as a high-fidelity surrogate model for full-field solutions on the microscopic scale of inelastic, non-isothermal constituents. Building on the homogenization framework of Chatzigeorgiou et al. (Int J Plast 81:18–39, 2016), we extend the framework of DMNs to thermomechanical composites by incorporating the two-way thermomechanical coupling, i.e., the coupling from the macroscopic onto the microscopic scale and vice versa, into the framework. We provide details on the efficient implementation of our approach as a user-material subroutine (UMAT). We validate our approach on the microscopic scale and show that DMNs predict the effective stress, the effective dissipation and the change of the macroscopic absolute temperature with high accuracy. After validation, we demonstrate the capabilities of our approach on a concurrent thermomechanical two-scale simulation on the macroscopic component scale.

Keywords: Crystal plasticity; Elastic anisotropy; Fast Fourier transformation; Grain boundaries; Maximum entropy method; Spatial stress distribution

Abstract: The onset of nonlinear effects in metals, such as plasticity and damage, is strongly influenced by the heterogeneous stress distribution at the grain level. This work is devoted to studying the local stress distribution in fcc polycrystals using FFT-based solvers. In particular, we focus on the distribution of shear stresses resolved in the slip systems as the critical driving force for plastic deformations. Specific grain orientations with respect to load direction are investigated in the linear elastic regime and at incipient plastic deformations based on a large ensemble of microstructures. The elastic anisotropy of the single crystal is found to have a crucial influence on the scatter of the stress distribution, whereas the Young's modulus in the respective crystal direction governs the mean stress in the grain. It is further demonstrated that, for higher anisotropy, the shear stresses deviate from the normal distribution and are better approximated by a log-normal fit. Comparing the full-field simulations to the Maximum Entropy Method (MEM), reveals that the MEM provides an excellent prediction up to the second statistical moment in the linear elastic range. In a study on the spatial distribution of shear stresses, the grain boundary is identified as a region of pronounced stress fluctuations and as the starting point of yielding during the elastic-plastic transition.

Keywords: Continuum damage mechanics; Failure; Anisotropy; Multiscale; Bayesian optimization; Sheet molding compound composite

Abstract: We present a holistic multiscale approach for constructing anisotropic criteria describing the macroscopic failure of sheet molding compound composites based on full-field simulations of microscale damage evolution. We use an anisotropic damage model on the microscale that directly operates on the compliance tensor, captures matrix and bundle damage via dedicated stress-based damage criteria and allows for a comparison of simulation and experimental results. To identify the damage material-parameters used in the non-linear and time-consuming full-field simulations, we utilize Bayesian optimization with Gaussian processes. We validate the full-field predictions on the microscale and subsequently identify macroscopic failure criteria based on distributions taken from experimental findings. We propose failure surfaces in stress space and stiffness-reduction triggered failure surfaces to cover both a structural analysis and a design process perspective.

Abstract: This work is devoted to anisotropic continuum-damage mechanics in the quasi-static, isothermal, small-strain setting. We propose a framework for anisotropic damage evolution based on the compliance tensor as primary damage variable, in the context of generalized standard models for dissipative solids. Based on the observation that the Hookean strain energy density of linear elasticity is jointly convex in the strain and the compliance tensor, we design thermodynamically consistent anisotropic damage models that satisfy Wulfinghoff’s damage-growth criterion and feature a convex free energy. The latter property permits obtaining mesh-independent results on component scale without the necessity of introducing gradients of the damage field. We introduce the concepts of stress-extraction tensors and damage-hardening functions, implicitly describing a rigorous damage-analogue of yield surfaces in elastoplasticity. These damage surfaces may be combined in a modular fashion and give rise to complex damage-degradation behavior. We discuss how to efficiently integrate Biot’s equation implicitly, and show how to design specific stress-extraction tensors and damage-hardening functions based on Puck’s anisotropic failure criteria. Last but not least we demonstrate the versatility of our proposed model and the efficiency of the integration procedure for a variety of examples of interest.

Keywords: Effective anisotroy; Fiber reinforced composites; Fiber suspensions; Mean-field modeling; Micromechanics; Short fibers

Abstract: Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited.

Keywords: Homogenization; Localization; Maximum Entropy Method; Micromechanics; Polycrystals

Abstract: The prediction of local field statistics from effective properties is an open problem in the field of micromechanics. Partial information on the local field statistics is accessible from homogenization assumptions. In particular, exact phase-wise second moments of stresses can be calculated analytically from the effective strain energy density. In recent years, full-field calculations have become efficient enough to sample large ensembles of microstructures in the plastic regime (e.g. Gehrig et. al [4]). In the present work, the maximum entropy method known from statistical thermodynamics is used to estimate first and second moments of local stresses from known eigenstrain distributions. The simple and refined formulations of the maximum entropy method proposed by Kreher and Pompe [9] are considered. While the simple method yields satisfactory results for a large amount of material classes (cf. Krause and Böhlke [7]), we prove that it does not respect the linearity of the eigenstrain problem. We further show that neither method corresponds to the exact second moments of stresses known from the effective strain energy density. By incorporating additional information, we find an improved maximum entropy method. As an example, we analyze stress fluctuations in polycristalline titanium. For the exact analytical solution and the maximum entropy methods, we use the singular approximation and the Hashin-Shtrikman bounds. For comparison, we numerically approximate full-field statistics using an FFT approach. In all methods, the stress fluctuations caused by the anisotropy of the single crystal strongly influence the elastic-plastic transition.

Keywords: Crystal plasticity; Crystallographic texture; Microstructure modeling; Multiscale methods; Polycrystalline microstructures

Abstract: This work is concerned with synthetic microstructure models of polycrystalline materials. Once a representation of the microstructure is generated, the individual grains need to be furnished with suitable crystal orientations, matching a specific crystal orientation distribution. We introduce a novel method for this task, which permits to prescribe the orientations based on tensorial Fourier coefficients. This compact representation gives rise to the texture coefficient optimization for prescribing orientations method, enabling the determination of representative orientations for digital polycrystalline microstructures. We compare the proposed method to established and dedicated algorithms in terms of the linear elastic as well as the non-linear plastic behavior of a polycrystalline material.

Keywords: Bayesian optimization; Crystal plasticity; Multiscale method; Parameter identification

Abstract: In this work, we advocate using Bayesian techniques for inversely identifying material parameters for multiscale crystal plasticity models. Multiscale approaches for modeling polycrystalline materials may significantly reduce the effort necessary for characterizing such material models experimentally, in particular when a large number of cycles is considered, as typical for fatigue applications. Even when appropriate microstructures and microscopic material models are identified, calibrating the individual parameters of the model to some experimental data is necessary for industrial use, and the task is formidable as even a single simulation run is time consuming (although less expensive than a corresponding experiment). For solving this problem, we investigate Gaussian process based Bayesian optimization, which iteratively builds up and improves a surrogate model of the objective function, at the same time accounting for uncertainties encountered during the optimization process. We describe the approach in detail, calibrating the material parameters of a high-strength steel as an application. We demonstrate that the proposed method improves upon comparable approaches based on an evolutionary algorithm and performing derivative-free methods.

Keywords: Anisotropy; Characteristic lengths; Higher-rank constitutive tensors; Strain gradient elasticity; Voigt average

Abstract: In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

Keywords: Cauchy continuum; Crystal plasticity theory; Extended continua; Purely mechanical; Vanishing dissipation

Abstract: The concept of a purely mechanical theory is revisited within the framework of continuum thermodynamics. It is based on three assumptions: (i) the additive split of the specific free energy into a mechanical contribution not depending on the temperature and a contribution that depends on the temperature only, (ii) a homogeneous and stationary temperature field, and (iii) a negligible heat supply. The consequences of these three assumptions on the dissipation inequality in form of the Clausius–Duhem inequality are discussed. It is shown, that these three assumption yield a vanishing dissipation. This result is valid for the classical Cauchy–Boltzmann continuum as well as for extended continua. Many plasticity theories consider the first two assumptions tacitly, however the role of the heat supply is not discussed. Here, the consequences of a vanishing dissipation with respect to a local crystal plasticity theory are discussed, exemplarily. The paper does not favor a purely mechanical theory but highlights the implications of such an approach.

Keywords: Embossing; Reforming; Residual stresses; Sheet metal material

Abstract: In sheet metal forming, combination of embossing and reforming allows the mechanical properties of sheet metal materials to be specifically improved. Here, local property modification is achieved by the residual stresses induced as a result of the one-sided embossing process followed by a reforming step. The residual stresses induced in this specific way can lead to a significantly increase in the fatigue strength of processed sheet metal components. However, in order to ensure this kind of component optimization in continuous operation, the induced stresses have to be homogeneous. In this respect, the main objective of the study reported about in this paper was to identify a forming strategy, consisting of the process steps embossing and reforming, that generates preferably homogeneous residual stress distributions into sheet metal blanks. For this, numerical and experimental investigations were carried out with samples of the stainless steel (X6Cr17) having a thickness of 1.5 mm. It was found that embossing and reforming, integrated into a conventional forming process, is a novel approach to specifically induce very localized homogeneous compressive residual stresses in sheet metal materials. This eliminates the need for costly post-processing by means of surface treatment.

Keywords: Cauchy continuum; Crystal plasticity theory; Extended continua; Purely mechanical; Vanishing dissipation

Abstract: Directionally solidified eutectics of NiAl matrix and fibrous refractory metals, like Mo, can form cellular mesostructures with significant fiber misalignment and changing fiber volume fraction, for example, when being solidified at high growth rates or when increased solidification intervals are present in the alloys. In order to reveal the deteriorating impact of the mesostructure, i.e., the volume fraction and aspect ratio of the well-aligned cells, on the creep response of such cellular eutectics, we rely on scale-bridging numerical simulations, using the level-set framework by Sonon et al. [1] for microstructure generation and FFT-based solvers for computing the creep response. Our results indicate, firstly, that the fraction of properly aligned regions in cellular NiAl composites is lower than estimated in earlier experimental studies, due to the existence of degenerated regions surrounding the well-aligned cell interiors. Secondly, studying the influence of the cell aspect ratio shows that the apparent stress exponent of the composite is very sensitive with respect to this parameter, providing a possible explanation for the large scatter of experimentally determined stress exponents in previous studies. A comparison of the numerical simulations to a linear rule of mixtures and the frequently applied analytical Kelly-Street model illustrates that both fail to accurately describe the magnitude of minimum creep rates in the investigated ranges of volume fractions and aspect ratios. The heterogeneity of the strain-rate field on the mesoscale is identified as the primary error source, demonstrating that either numerical simulations or more sophisticated analytical models are required for reliably predicting for the creep response of cellular materials.

Keywords: Fourier transform analysis; Fourier transform rheology on solid materials; nonlinear Schapery model; nonlinear viscoelasticity; thermoplastics

Abstract: The nonlinear mechanical behavior of viscoelastic materials is modeled based on the Schapery integral model containing internal variables. In this context, a new approach for the strain-dependent material properties is introduced considering a one-dimensional formulation with strain-dependent nonlinear functions for an oscillatory load case. In addition to the viscoelastic storage and loss modulus, the higher order harmonic oscillations in the stress response are computed and compared to experimental data from Fourier transform rheology of Polyamide 6 (PA6). The comparison reveals a good agreement between the predictions of the nonlinear model and the experimental data for the higher harmonic intensity (Formula presented.).

Keywords: Micromechanics; homogenization; computational mechanics; nonlinear optimization; thermomechanics

Abstract: The mechanical behavior of many applied materials arises from their microstructure. Thus, to aid the design, development and industrialization of new materials, robust computational homogenization methods are indispensable. The present thesis is devoted to investigating and developing FFT-based micromechanics solvers for efficiently computing the (thermo)mechanical response of nonlinear composite materials with complex microstructures.

Keywords: SMC; Microstructure; Modeling; Damage; Composite

Abstract: We introduce an algorithm that allows for a fast generation of SMC composite microstructures. An exact closure approximation and a quasi-random orientation sampling ensure high fidelity. Furthermore, we present a modular framework for anisotropic damage evolution. Our concept of extraction tensors and damage-hardening functions enables the description of complex damage-degradation. In addition, we propose a holistic multiscale approach for constructing anisotropic failure criteria.

Keywords: : Gradient Crystal Plasticity; Extended Continuum Theory; Continuum Dislocation Theory; Grain Boundary Modeling; Finite Element Method

Abstract: A physically-based dislocation theory of plasticity is derived within an extended continuum mechanical context. Thermodynamically consistent flow rules at the grain boundaries are derived. With an analytical solution of a three-phase periodic laminate, dislocation pile-up at grain boundaries and dislocation transmission through the grain boundaries are investigated. For the finite element implementations, numerically efficient approaches are introduced based on accumulated field variables.

Keywords: Effective viscosity; FFT-based method; Fiber-reinforced composites; Staggered grid; Thermoplastic melt

Abstract: Understanding the effective viscosity of fiber-filled polymer melts is essential for predicting the local fiber orientation of injection molded short-fiber reinforced components. To circumvent the intrinsic difficulties of experimentally determining the strongly anisotropic viscosity of such particle-reinforced melts, an FFT-based computational homogenization method for computing the effective quasi-static viscosity of a particle-suspended Newton fluid is introduced, based on a dual formulation. The dependence of the effective viscosity on microstructural parameters, like fiber volume fraction, fiber aspect ratio and fiber orientation are investigated by computational experiments on representative volume elements, and the results are compared to an analytical model for the effective viscosity. Based on the computational findings, a closed-form approximation for the effective viscosity is provided.

Keywords: -

Abstract: Recently, mathematically well-defined homogenization results for the Francfort-Marigo fracture model were established. To solve the resulting cell formula, efficient computational methods were developed and improvements on solver and discretization techniques were investigated. We discuss an approach for solving the governing cell formula based on a rewriting as a second order cone problem, a specific normal form for optimization problems. For such a formulation, potent high-accuracy optimization solvers are available. We demonstrate our approach on heterogeneous two-dimensional microstructures.

Keywords: Author keywords: Anisotropy; Artificial neural networks; Data-driven modeling; Finite hyperelasticity; Machine learning; Metamaterials

Abstract: This work investigates the capabilities of anisotropic theory-based, purely data-driven and hybrid approaches to model the homogenized constitutive behavior of cubic lattice metamaterials exhibiting large deformations and buckling phenomena. The effective material behavior is assumed as hyperelastic, anisotropic and finite deformations are considered. A highly flexible analytical approach proposed by Itskov (Int J Numer Methods Eng 50(8): 1777–1799, 2001) is taken into account, which ensures material objectivity and fulfillment of the material symmetry group conditions. Then, two non-intrusive data-driven approaches are proposed, which are built upon artificial neural networks and formulated such that they also fulfill the objectivity and material symmetry conditions. Finally, a hybrid approach combing the approach of Itskov (Int J Numer Methods Eng 50(8): 1777–1799, 2001) with artificial neural networks is formulated. Here, all four models are calibrated with simulation data of the homogenization of two cubic lattice metamaterials at finite deformations. The data-driven models are able to reproduce the calibration data very well and reproduce the manifestation of lattice instabilities. Furthermore, they achieve superior accuracy over the analytical model also in additional test scenarios. The introduced hyperelastic models are formulated as general as possible, such that they can not only be used for lattice structures, but for any anisotropic hyperelastic material. Further, access to the complete simulation data is provided through the public repository https://github.com/CPShub/sim-data.

Keywords: Author keywords: Computational homogenization; Deep material networks; Laminates; Micromechanics; Multiscale simulation; Short fiber reinforced composites

Abstract: In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by linear elastic precomputations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents. We discuss how to extend direct DMNs to account for varying fiber orientation, and propose a simplified sampling strategy which significantly speeds up the training process. To enable concurrent multiscale simulations, evaluating the DMNs efficiently is crucial. We discuss dedicated techniques for exploiting sparsity and high-performance linear algebra modules, and demonstrate the power of the proposed approach on an injection molded quadcopter frame as a benchmark component. Indeed, the DMN is capable of accelerating two-scale simulations significantly, providing possible speed-ups of several magnitudes.

Keywords: -

Abstract: Deep material networks (DMN) are a promising piece of technology for accelerating concurrent multiscale simulations. DMNs are identified by linear elastic pre-computations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents. The offline training phase is independent of the online evaluation, such that a pre-trained DMN may be applied for varying material behavior of the constituents. In this contribution, we investigate a two-scale component simulation of industrial complexity accelerated by DMNs. To this end, a DMN is solved implicitly at every Gauss point to include the microstructure information into the macro simulation.

Keywords: Author keywords: FFT; Mean field homogenization; Micromechanics; Polyamide 6.6; Short fiber reinforced composites; Short glass fiber reinforced thermoplastics; Stress localization

Abstract: This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class.

Keywords: Author keywords: Anisotropic property; Asymptotic solutions; Elastic composites; Fiber-fiber interactions; Light-weight constructions; Numerical implementation; Orientation distributions; Orientation tensor

Abstract: Anisotropic fiber-reinforced composites are used in lightweight construction, which is of great industrial relevance. During mold filling of fiber suspensions, the microstructural evolution of the local fiber arrangement and orientation distribution is determined by the local velocity gradient. Based on the Folgar-Tucker equation, which describes the evolution of the second-order fiber orientation tensor in terms of the velocity gradient, the present study addresses selected states of deformation rates that can locally occur in complex flow fields. For such homogeneous flows, exact solutions for the asymptotic fiber orientation states are derived and discussed based on the quadratic closure. In contrast to the existing literature, the derived exact solutions take into account the fiber-fiber interaction. The analysis of the asymptotic solutions relying upon the common quadratic closure shows disadvantages with respect to the predicted material symmetry, namely, the anisotropy is overestimated for strong fiber-fiber interaction. This motivates us to suggest a novel normalized fully symmetric quadratic closure. Two versions of this new closure are derived regarding the prediction of anisotropic properties and the fiber orientation evolution. The fiber orientation states determined with the new closure approach show an improved prediction of anisotropy in both effective viscous and elastic composite behaviors. In addition, the symmetrized quadratic closure has a simple structure that reduces the effort in numerical implementation compared to more elaborated closure schemes.

Keywords: Author keywords: Anisotropic constitutive law; Anisotropic elastic properties; Closure approximation; Fiber orientation distribution; Fiber volume fractions; Flow-induced anisotropy; Mold-filling simulation; Short-fiber-reinforced composites

Abstract: The present work discusses the impact of the back coupling of the fiber orientation distribution on the base flow and on the fiber orientation itself during mold filling simulations. Flows through a channel and over a backward-facing step are investigated. Different closure approximations are considered for modeling the flow-induced evolution of anisotropy. The results corresponding to the decoupled approach, in which the effect of fibers on local fluid properties is neglected, build the basis of comparison. The modeling is limited to a laminar, incompressible, and isothermal flow of a fiber suspension consisting of rigid short fibers suspended in an isotropic Newtonian matrix fluid. A linear, anisotropic constitutive law is used in combination with a uniform fiber volume fraction of 10% and an aspect ratio of 10. To evaluate the impact of back coupling and of different closure methods in view of the manufactured solid composite the resulting anisotropic elastic properties are investigated based on the Mori–Tanaka method combined with an orientation average scheme. Relative to the range [0, 1] the pointwise difference in fiber orientation between the decoupled and the coupled approach is found to be ± 5 % in the channel and ± 30 % in the backward-facing step, respectively. The viscosity and the elasticity tensor show both significant flow-induced anisotropies as well as a strong dependence on closure and coupling.

Keywords: Author keywords: CO2 emissions; electricity; fossil-based energy; Leibniz's fundamental theorem of calculus; non-fossil-based energy

Abstract: The substitution of energy based on fossil fuels in different sectors like household or traffic by electric energy saves CO2 of this specific sector due to decreased fossil fuel consumption. An important quantity is the additional CO2 emission (Formula presented.) due to an increased electric power demand (Formula presented.) for the average electricity power demand (Formula presented.). Commonly, the formula (Formula presented.) is used (called simplified formula), where (Formula presented.) represents mean average CO2 footprint. It is shown in the present manuscript, that the simplified formula may underestimate the CO2 footprint significantly if the average CO2 footprint depends on the average electricity power demand, which is the case for most of mixed partly renewable and partly non-renewable electric energy systems. Therefore, the real CO2 emissions would outmatch those according to simplified easily by factor 2 in reality depending on the status of the electricity system. In order to establish a more precise calculation of the CO2 footprint, the general formula (Formula presented.) which is exact and contains the simplified formula as a special case, is derived in this article. The simplified formula requires an additional term that takes into account the change of the mean average CO2 footprint (Formula presented.) depending on the electricity power demand.

Keywords: Author keywords: Analytical calculation; Crystallographic textures; Field simulation; Heterogeneous materials; Mechanical behavior; Polycrystalline phasis; Stochastic evaluations; Stress and strain distribution

Abstract: Austenite–ferrite duplex steels generally consist of two differently textured polycrystalline phases with different glide mechanisms. For estimating the effective mechanical behavior of heterogeneous materials, there exist well established approaches, two of which are the classes of mean-field and full-field methods. In this work, the local fields resulting from these different approaches are compared using analytical calculations and full-field simulations. Duplex steels of various textures measured using X-ray diffraction are considered. Special emphasis is given to the influence of the crystallographic texture on the stress and strain distributions.

Keywords: Author keywords: Beam element; Four-point-bending experiment; Mean field approach; Residual stresses

Abstract: The resulting shapes in production processes of metal components are strongly influenced by deformation induced residual stresses. Dual-phase steels are commonly used for industrial application of, e.g., forged or deep-drawn structural parts. This is due to their ability to handle high plastic deformations, while retaining desired stiffness for the products. In order to influence the resulting shape as well as component characteristics positively it is important to predict the distribution of phase-specific residual stresses which occur on the microscale of the material. In this contribution a comparative study is presented, where two approaches for the numerical simulation of residual stresses are applied. On the one hand a numerically efficient mean field theory is used to estimate on the grain level the total strain, the plastic strains and the eigenstrains based on macroscopic stress, strain and stiffness data. An alternative ansatz relies on a Taylor approximation for the grain level strains. Both approaches are applied to the corrosion-resistant duplex steel X2CrNiMoN22-5-3 (1.4462), which consists of a ferritic and an austenitic phase with the same volume fraction. Mean field and Taylor approximation strategies are implemented for usage in three dimensional solid finite element analysis and a geometrically exact Euler–Bernoulli beam for the simulation of a four-point-bending test. The predicted residual stresses are compared to experimental data from bending experiments for the phase-specific residual stresses/strains which have been determined by neutron diffraction over the bending height of the specimen.

Keywords: Author keywords: Beam element; Four-point-bending experiment; Mean field approach; Residual stresses

Abstract: Polymer matrix exhibit linear viscoelasticity during processing of composites. The viscoelastic homogenization problem in the time-domain play a vital role in the virtual process simulations. In this paper, full-field simulations using finite element (FE) are carried out for different regular periodic microstructures of unidirectional (UD) fiber reinforced polymer (FRP) (i.e., short, and long fiber) composite and particulate composites. The incremental variational based mean-field homogenization (MFH) as proposed by Lahellec and Suquet (Lahellec and Suquet, 2007a) is used here for comparing with the full-field solutions. Two different elastic bounds-based homogenization methods are coupled with this scheme. Implementation is validated with benchmark problems of particulate composites. These are then compared with the solutions of different particulate microstructures (face centered cubic (FCC), body centered cubic (BCC) and simple cubic (SC)). Mean-field response is closer to FCC arrangement. Additionally, FCC and BCC arrangement indicated nearly the same response with different local field statistics. Relaxation in the effective modulus of UD FRP composites obtained from the MFH compared well with the full-field solution as a function of direction and time for hexagonal arrangement of long fiber and staggered arrangement of ellipsoidal short fibers. The effect of different aspect ratio, volume fraction and contrast in properties on the macroscopic response is additionally investigated. Glass and carbon fiber with an isotropic approximation is used for studying the effect of contrast. Double inclusion MFH scheme coupled with the incremental variational approach indicated better comparisons with the full-field solution of UD FRP composites.

Keywords: Author keywords: Deep-drawing process; Efficient predictions; Finite element modelling; Mean-field homogenizations; Micro residual stress; Numerical predictions; Simulation

Abstract: Residual stress development in deep drawing processes is investigated based on cylindrical cups made of duplex stainless steel sheet. Using a two-scale approach combining finite element modelling with a mean field homogenization scheme the macro residual stresses as well as the phase-specific micro residual stresses regarding the phases ferrite and austenite are calculated for steel X2CrNiN23‑4 for various drawing depths. The simulation approach allows for the numerical efficient prediction of the macro and phase-specific micro residual stress in every integration point of the entire component. The simulation results are validated by means of X‑ray diffraction residual stress analysis applied to a deep-drawn cup manufactured using corresponding process parameters. The results clearly indicate that the fast simulation approach is well suited for the numerical prediction of residual stresses induced by deep drawing for the two-phase duplex steel; the numerical results are in good agreement with the experimental data. Regarding the investigated process, a significant influence of the drawing depth, in particular on the evolution of the residual stress distribution in drawing direction, is observed. Considering the appropriate phase-specific strain hardening, the two-scale approach is also well suited for the prediction of phase specific residual stresses on the component level.

Keywords: Author keywords: Computed tomography; Hashin-Shtrikman two-step method; Mean-Field homogenization; Mechanical testing; Orientation analysis; Sheet molding compound

Abstract: Fiber reinforced polymers, for example sheet molding compounds (SMC), have gained significant importance as lightweight materials. In this work, a SMC composite based on unsaturated polyester-polyurethane hybrid resin (UPPH) reinforced with discontinuous long glass fibers is considered. The mechanical behavior of this composite is highly governed by its microstructure. Based on samples extracted from compression molded sheets, the underlying microstructure is characterized by means of X-ray computed tomography (µCT) scans. Tensile tests are performed to investigate and characterize the elastic properties of this composite. A mean-field method is presented to approximate the effective elastic behavior. In this context, the tensile tests performed serve as validation data for the simulation results. The effective Young's modulus is in good agreement with the experimentally obtained data with a deviation less than 15%.

Keywords: Author keywords: Anderson acceleration; computational homogenization; directionally solidified eutectics; FFT-based method; fiber-reinforced composites; metal-matrix composites

Abstract: Classical solution methods in fast Fourier transform-based computational micromechanics operate on, either, compatible strain fields or equilibrated stress fields. By contrast, polarization schemes are primal-dual methods whose iterates are neither compatible nor equilibrated. Recently, it was demonstrated that polarization schemes may outperform the classical methods. Unfortunately, their computational power critically depends on a judicious choice of numerical parameters. In this work, we investigate the extension of polarization methods by Anderson acceleration and demonstrate that this combination leads to robust and fast general-purpose solvers for computational micromechanics. We discuss the (theoretically) optimum parameter choice for polarization methods, describe how Anderson acceleration fits into the picture, and exhibit the characteristics of the newly designed methods for problems of industrial scale and interest.

Keywords: Author keywords: composites; FFT-based micromechanics; homogenization; thermomechanical coupling; viscoelasticity

Abstract: Thermomechanical couplings are present in many materials and should therefore be considered in multiscale approaches. Specific cases of thermomechanical behavior are the isothermal and the adiabatic regime, in which the behavior of real materials differs. Based on the consistent asymptotic homogenization framework for thermomechanically coupled generalized standard materials, the present work is devoted to computing the effective thermomechanical behavior of composite materials in the context of fast Fourier transform (FFT)-based micromechanics. Exploiting the homogeneity of the temperature on the microscale, we develop a fast implicit staggered solution scheme for the coupled problem, which is compatible to existing strain-based micromechanics solvers. Due to its implicit formulation, the algorithm permits large time steps for computations involving strong thermomechanical coupling. We investigate the performance of modern FFT-based algorithms combined with the proposed thermomechanical solution strategy. In this context, the Barzilai–Borwein method is identified as particularly efficient, inducing only a small overhead compared with the traditional isothermal setting. We demonstrate the effectiveness of the presented approach for short-fiber reinforced composites with viscoelastic matrix behavior.

Keywords: Author keywords: sharp interface; martensite; phase transformation; kinetic

Abstract: Martensite forms under rapid cooling of austenitic grains accompanied by a change of the crystal lattice. Large deformations are induced which lead to plastic dislocations. In this work a transformation model based on the sharp interface theory, set in a finite strain context is developed. Crystal plasticity effects, the kinetic of the singular surface as well as a simple model of the inheritance from austenite dislocations into martensite are accounted for.

Keywords: Author keywords: Anisotropic yield function; crystallographic texture; orientation distribution function; plastic anisotropy; tensorial texture coefficients

Abstract: A micro-mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group-theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises - Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro-mechanically motivated generalization of the von Mises - Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension-compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface.

Keywords: Author keywords: Brittle fracture; FFT; Fiber reinforced composites; Heavy-ball method; Micromechanics; Phase-field fracture

Abstract: We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memory-efficient FFT-based solution techniques, and identify the heavy ball scheme as particularly powerful. We discuss the memory-efficient implementation and present demonstrative numerical examples.

Keywords: Author keywords: Computational homogenization; Deep material network; Fiber reinforced polyamide; J2-elasto-plasticity; Laminate; Metal matrix composite; Micromechanics; Volterra series

Abstract: We investigate deep material networks (DMNs), recently introduced by Liu et al. [Comput. Method Appl. M., vol. 345, pp. 1138–1168, 2019], from the viewpoint of classical micromechanics at small strains. We aim to establish the basic micromechanical principles of deep material networks, shed light on the characteristics of the building blocks and introduce a simple, robust and fast solution technique for inelastic deep material networks. In their original formulation, DMNs are solely trained by linear elastic data, but applied to nonlinear and inelastic problems with astonishing accuracy. We clarify this phenomenon theoretically by showing that, to first order in the strain rate, the effective inelastic behavior of composite materials is determined by linear elastic localization. Our argumentation applies to arbitrary microstructures comprising nonlinear generalized standard materials at small strains. The main technical tool is a Volterra series approximation of the stress of a generalized standard material, which we adapt from nonlinear dynamical systems theory. Next, we establish that deep material networks inherit thermodynamic consistency and stress-strain monotonicity from their phases. These properties root in the definition of the DMN as a tree of hierarchical laminates and contrast with other applications of neural networks to the approximation of material laws, where consistency and monotonicity typically cannot be guaranteed far away from the training set. Last but not least, we introduce rotation-free DMNs with arbitrary directions of lamination and exploit a novel formulation, uniting the implementation of DMNs of arbitrary tree topology and multi-phase laminates, and apply our insights to microstructures of industrial complexity.

Keywords: Author keywords: Computational homogenization; Exact closure; FFT-based micromechanics; Representative volume element; Sheet molding compound

Abstract: Sheet molding compound (SMC) composites combine high lightweight potential with excellent formability and are frequently used in industrial applications. To reduce safety factors in dimensioning SMC parts, the influence of processing parameters and stochastic variation of microstructural and physical properties needs to be quantified accurately. Taking into account the inherent three-scale structure of SMC, we improve the microstructure generator of Chen et al. [Compos. Struct. 188, pp. 25–38, 2018] in various respects. Firstly, we consistently rely upon state-of-the-art closure approximations for the fourth order fiber orientation tensor. More precisely, we show that for a planar fiber orientation state, there is an explicit formula for the fast exact closure approximation of Montgomery-Smith et al. [J. Fluid Mech. 680, pp. 321–335, 2011]. Secondly, we exploit the use of quasi-random numbers in sampling the fiber orientation distribution, leading to dramatic improvements in accuracy compared to pseudo-random Monte Carlo sampling. Last but not least, we rely upon fast Fourier transform based methods for rapid computational homogenization. With these methodological improvements at hand, we thoroughly investigate the influence of the mechanical and microstructural parameters on the effective elastic properties of SMC composites, and compare the results to direct numerical simulations on large scale digital volume images and mean-field estimates.

Keywords: Author keywords: Duplex steel; homogenisation; metal forming; residual stresses; two-phase materials; two-scale modelling

Abstract: For modern engineering applications, the prediction of residual stresses is a starting-point for the subsequent optimisation of the design with regard to the process induced stresses on the macro and the grain scale. In this paper, a new approach for the numerically efficient prediction of phase-specific residual stresses (residual stresses of second kind) in two-phase materials is presented. The proposed model allows for a two-scale simulation of complex forming processes, solely based on the phase-specific constitutive equations. This enables the calculation of the average stresses and strains in each phase during the entire process and the prediction of the macroscopic material response. The numerical results are compared to experimental diffraction-based data of a bending bar and a deep drawn cup and are aligned better compared to existing models.

Keywords: Author keywords: Dynamic mechanical analysis; effective thermoelastic properties; fiber-reinforced polymer composite; Mean-field homogenization; orientation average; two-step Hashin–Shtrikman method

Abstract: Fiber-reinforced polymers contribute significantly to weight-reducing components for various industrial applications. A discontinuous glass fiber-reinforced thermoset resin is considered which is produced by the sheet molding compound (SMC) process. Related to the production process, the samples considered in this work exhibit an anisotropic fiber orientation distribution which highly affects the thermomechanical properties. The thermoviscoelastic material behavior of three selected samples is characterized by means of dynamic mechanical analysis. These tests show the temperature-dependent elastic modulus and the glass transition of the composite. Measurements of the thermal expansion of the SMC composite provide data on the coefficient of thermal expansion (CTE). These experimental investigations provide data for the thermoelastic material modeling. Aiming at the prediction of the effective thermal and mechanical properties, a Hashin–Shtrikman-based homogenization method is presented. Based on an eigenstrain formulation, the effective Young’s modulus and CTE are computed in two steps. Moreover, the mean-field method is given in dependence of a variable reference stiffness allowing to tailor the approach to the material system. The influence of this variable reference stiffness on the effective quantities as well as the predicted behavior is analyzed with respect to the experiments. The presented numerical results are in good agreement with the experimental data

Keywords: Author keywords: Heterogeneous materials; Homogenisation; Linear thermoelasticity; Maximum entropy method; Statistical second moments

Abstract: Mean-field methods are a common procedure for characterizing random heterogeneous materials. However, they typically provide only mean stresses and strains, which do not always allow predictions of failure in the phases since exact localization of these stresses and strains requires exact microscopic knowledge of the microstructures involved, which is generally not available. In this work, the maximum entropy method pioneered by Kreher and Pompe (Internal Stresses in Heterogeneous Solids, Physical Research, vol. 9, 1989) is used for estimating one-point probability distributions of local stresses and strains for various classes of materials without requiring microstructural information beyond the volume fractions. This approach yields analytical formulae for mean values and variances of stresses or strains of general heterogeneous linear thermoelastic materials as well as various special cases of this material class. Of these, the formulae for discrete-phase materials and the formulae for polycrystals in terms of their orientation distribution functions are novel. To illustrate the theory, a parametric study based on Al-Al2O3 composites is performed. Polycrystalline copper is considered as an additional example. Through comparison with full-field simulations, the method is found to be particularly suited for polycrystals and materials with elastic contrasts of up to 5. We see that, for increasing contrast, the dependence of our estimates on the particular microstructures is increasing, as well.

Keywords: Author keywords: Gradient solvers; Grain size distributions; Laguerre tessellations; Polycrystalline materials

Abstract: Ideas from the mathematical theory of optimal transport have recently been transferred to the micromechanics of polycrystalline materials, leading to fast methods for generating polycrystalline microstructures with grains of prescribed volume fraction in terms of centroidal Laguerre tessellations. In this work, we improve the state of the art solvers. For a given set of seeds and corresponding volume fractions summing to unity, there is a set of Laguerre weights such that the corresponding Laguerre tessellation realizes the prescribed volume fractions exactly. Furthermore, the Laguerre weights are unique up to a constant and can be determined by solving a convex optimization problem. However, whenever the optimization algorithm encounters a weight vector leading to an empty cell, the optimization problem is no longer locally strictly convex. To account for the latter, backtracking strategies are typically employed. We show that modern gradient-based optimization algorithms devoid of backtracking, like the Malitsky–Mishchenko method and the Barzilai–Borwein scheme easily overcome the described difficulty, leading to a significant speed-up compared to more traditional solvers. Furthermore, for computing centroidal Laguerre tessellations of prescribed volume fraction, we propose an Anderson-accelerated version of Lloyd's algorithm, and show, by numerical experiments, that it consistently reduces the run-time. We demonstrate the capabilities of our proposed methods for generating microstructures of polycrystalline materials with prescribed grain size distribution.

Keywords: Author keywords: Gradient solvers; Grain size distributions; Laguerre tessellations; Polycrystalline materials

Abstract: The paper focuses on the evaluation of the effective properties of linear viscoelastic composites with a periodic structure, containing long cylindrical fibers of circular cross-section and for two different cell arrangements: square and hexagonal unit cells. For this purpose, we use the two-scale asymptotic homogenization method (AHM) and a numerical homogenization method (NHM). Based on the correspondence principle, the local functions and the relaxation overall properties are obtained in explicit form by the AHM using the Prony series. Additionally, the NHM is established for a three-dimensional representative cell, and the problem is solved under appropriate boundary conditions, by using the Finite Element Method. The numerical results obtained by the AHM and NHM are compared and verified with other theoretical approaches. The comparisons show a good agreement and a benchmark for further experimental and theoretical investigations.

Keywords: Author keywords: Boundary conditions; Extended continuum; Grain boundaries; Invariance considerations; Slip gradient crystal plasticity theory

Abstract: The presence of grain boundaries significantly influences the overall mechanical behavior of materials with an underlying crystalline microstructure. They act as an obstacle against the movement of dislocations and, thus, significantly contribute to size effects as for example the Hall–Petch effect. Hence, the thermodynamically consistent modeling of the behavior of the plastic slip at a grain boundary is of utmost interest. To this end, balance equations at a grain boundary are derived from an extended energy balance by means of invariance considerations. The grain boundary is considered as a material singular surface with own internal and kinetic energy as well as energy supply. Consequently, the balances at the grain boundary depend on its mean curvature. The framework presented is applied to a small strain slip gradient crystal plasticity theory, regarding single slip. Accounting for the derived balance equations, thermodynamically consistent flow rules for the plastic slip at the grain boundary are obtained by exploitation of the Coleman–Noll procedure. In this context, a classification of flow rules for the plastic slip at the grain boundary is provided. Finally, the distribution of the plastic slip is presented regarding a three-phase laminate material with two elastoplastic phases and one elastic phase. The two elastoplastic phases represent the grains of a bicrystal. A grain boundary effect is discussed which is based on a variation of the internal length scale in one of the two elastoplastic phases.

Keywords: Author keywords: Additional field equations; Conservation of micro-inertia; Extended energy balance; Green–Naghdi–Rivlin theorem; Micro-stress

Abstract: Gradient plasticity theories are of utmost importance for accounting for size effects in metals, especially on the grain scale. Today, there are several methods used to derive the governing equations for the additional degrees of freedom in gradient plasticity theories. Here, the equivalence between an extended principle of virtual power and an extended energy balance is shown. The energy balance of a Boltzmann continuum is supplemented by contributions based on a scalar-valued degree of freedom. It is considered to be invariant with respect to a change of observer. This yields unambiguously the existence of a corresponding micro-stress vector, which is presumed from the outset in the context of an extended principle of virtual power. A thermodynamically consistent nonlocal evolution equation for the additional, scalar-valued degree of freedom is obtained by evaluation of the dissipation inequality in terms of the Clausius–Duhem inequality. Partitioning the nonlocal flow rule yields a partial differential equation, often referred to as micro-force balance. The approach presented is applied to derive a slip gradient crystal plasticity theory regarding single slip. Finally, the distribution of the plastic slip is exemplified with respect to a laminate material consisting of an elastic and an elastoplastic phase.

Keywords: Author keywords: Asymptotic homogenization method; Effective complex permittivity; Interface/interphase; Multi-phase fiber-reinforced composites; Transport problems

Abstract: The two-scale asymptotic homogenization method is used to find closed-form formulas for effective properties of periodic multi-phase fiber-reinforced composites where constituents have complex-valued transport properties and parallelogram unit cells. An antiplane problem relevant to linear elasticity is formulated in the frame of transport properties. The application of the method leads to the need of solving some local problems whose solution is found using potential theory and shear effective coefficients are explicitly obtained for n-phase fiber-reinforced composites. Simple formulae are explicitly given for three- and four-phase fiber-reinforced composites. The broad applicability, accuracy and generality of this model is determined through comparison with other methods reported in the literature in relation to shear elastic moduli and several transport problems of multi-phase fiber-reinforced composites in their realm, such as conductivity in a biological context and permittivity leading to gain and loss enhancement of dielectrics. Also, the example of gain enhancement of inertial mass density is looked into. Good agreement with other theoretical approaches is obtained. The formulas may be useful as benchmarks for checking experimental and numerical results.

Keywords: Author keywords: Duplex stainless steel; Load partitioning; Mean-field homogenisation; Micro residual stresses

Abstract: Multi-phase materials often times consist of constituents with high contrasts in phase-specific mechanical properties. Here, even after homogeneous plastic deformation phase-specific residual stresses develop that may affect the components behaviour in service. For numerical simulation of phase-specific residual stresses, knowledge of the particular phase-specific strain hardening behaviour is essential. In this study, the strain hardening of ferrite and austenite in cold rolled duplex stainless steel of type X2CrNiN23-4 is investigated. By means of X-ray diffraction, the phase-specific load partitioning and residual stress evolution are analysed for uniaxial load application in three directions within the sheets plane, taking into account the sheet metals phase specific anisotropy. In order to assess the necessity for experimental determination of anisotropic phase specific behaviour, the strain hardening parameters, derived from only one loading direction, are implemented in a mean-field approach for prediction of phase-specific stresses. A simplified simulation approach is applied that only considers macroscopic plastic anisotropy and results are compared to experimental findings. For all investigated loading directions, it was observed that austenite is the high-strength phase. This load partitioning behaviour was confirmed by the evolution of phase-specific residual stresses as a result of uniaxial elasto-plastic loading. With the simplified and fast numerical approach, satisfying results for prediction of anisotropic phase-specific (residual) stresses are obtained.

Keywords: Author keywords: Computational homogenization; Crystal elasto-viscoplasticity; FFT; Micromechanics; Polycrystalline materials

Abstract: Computational homogenization schemes based on the fast Fourier transform (FFT) enable studying the effective micromechanical behavior of polycrystalline microstructures with complex morphology. In the conventional strain-based setting, evaluating the single crystal elasto-viscoplastic constitutive law involves solving a non-linear system of equations which dominates overall runtime. Evaluating the inverse material law is much less costly in the small-strain context, because the flow rule is an explicit function of the stress. We revisit the primal and dual formulation of the unit cell problem of computational homogenization and use state of the art FFT-based algorithms for its solution. Performance and convergence behavior of the different solvers are investigated for a polycrystal and a fibrous microstructure of a directionally solidified eutectic.

Keywords: Author keywords: BFGS; composites; crystal viscoplasticity; FFT-based micromechanics; homogenization; Quasi-Newton methods

Abstract: This work is devoted to investigating the computational power of Quasi-Newton methods in the context of fast Fourier transform (FFT)-based computational micromechanics. We revisit FFT-based Newton-Krylov solvers as well as modern Quasi-Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, one of the most powerful Quasi-Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi-Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT-based context, we promote a Dong-type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi-)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast.

Keywords: Extended continuum; Invariance considerations; Slip gradient crystal plasticity; Grain boundaries; Gradient stress

Abstract: An overview of different methods for the derivation of extended continuum models is given. A gradient plasticity theory is established in the context of small deformations and single slip by considering the invariance of an extended energy balance with respect to Euclidean transformations, where the plastic slip is considered as an additional degree of freedom. Thermodynamically consistent flow rules at the grain boundary are derived. The theory is applied to a two- and a three-phase laminate.

Keywords: Model reduction; Conjugate heat transfer; Engine heat transfer; Transient simulation; Thermal management

Abstract: In this work, the dynamic behavior of a lean combustion race engine is investigated with regard to cycle-averaged component temperatures, heat fluxes and piston failure under high cycle fatigue loading. A special focus lies on the influence of various engine settings, like ignition time and the airfuel ratio. For the introduced stationary, as well as transient, temperature calculation method, in cylinder pressure curves are measured and processed statistically by probability density functions.

Keywords: Creep A; Directional solidification; Finite elements C; Gradient plasticity; IGFEM

Abstract: Interfaces can act as dislocation obstacles, sinks or dislocation sources and, therefore, influence strongly the mechanical properties of metals. To consider these effects at high temperature creep, a three-dimensional gradient crystal plasticity model is introduced. The interaction between dislocations and the fiber-matrix interface is included by an interface flow rule, which accounts for the gradient stresses and the normal component of the Cauchy stress on the interface. Motivated by the interface-enriched generalized finite element method (IGFEM), continuous shape functions allowing for weak discontinuities are introduced. These shape functions are used to evaluate the interface flow rule at sharp interfaces and are validated by comparing numerical simulation results of a laminate for single slip with an analytical solution. To investigate the slip transfer/activation at the interface, the directionally solidified NiAl-9Mo composite is modeled as regular fibrous microstructure. The simulated creep curves agree well with experimentally measured ones. It is found that the stress dependency of the interface flow rule is necessary to reproduce the well known composite's Norton behavior. The simulations reveal that the creep behavior of the composite is mainly controlled by the fibers and the interface properties. Finally, the specific shape of the creep curve could be explained.

Keywords: -

Abstract: Discontinuous long fiber-reinforced polymer structures with local continuous fiber reinforcements represent an important class of lightweight materials with broad design possibilities and diverse technical applications, e.g., in vehicle construction. However, in contrast to continuous fiber-reinforced composites, extensively used in the aircraft industry, there is still a lack of integrated and experimentally proven concepts for manufacture, modeling, and dimensioning of combinations of discontinuously and continuously reinforced polymer structures. This is partly ascribed to the complexity of the manufacturing processes of discontinuously reinforced polymers, with heterogeneous, anisotropic, and nonlinear material and structural properties, but also to the resulting bonding problem of both material types. This book addresses these issues, including both continuous and discontinuous fiber processing strategies. Specific design strategies for advanced composite reinforcement are provided, with an integrated and holistic approach taken for composites material selection, product design, and mechanical properties. Characterization, simulation, technology, design, future research, and implementation directions are also included. Especially in the field of application of three-dimensional load-bearing structures, this book provides an excellent foundation for the enhancement of scientific methods and the education of engineers who need an interdisciplinary understanding of process and material techniques, as well as simulation and product development methods.

Keywords: -

Abstract: The description of material failure as an energy minimization problem, i.e., the Francfort–Marigo model, has been studied widely in recent years. The approximation of the crack surface as a phase field, i.e., smeared interface, enjoys great popularity, as it allows describing fracture as a set of partial differential equations. In numerical homogenization, FFT‐based solution methods have been established over the past two decades. Their purpose is to compute the overall response of a heterogeneous microstruture w.r.t. a macroscopic loading and can be applied to a variety of nonlinear materials. The benefits lie in a fast implementation and the possibility to use image data like CT‐scans as input without further need for meshing. Based on the results of the master thesis of the first author, we investigate phase field crack propagation on heterogeneous microstructures using FFT‐based solvers.

Keywords: Homogenization; Materials design; Polycrystals; Texture

Abstract: The Hashin-Shtrikman bounds accounting for eigenfields are represented in terms of tensorial texture coefficients for arbitrarily anisotropic materials and arbitrarily textured polycrystals. This requires a short review of the Hashin-Shtrikman bounds with eigenfields, an investigation of the polarization field determined by the stationarity condition and, finally, the analysis of the resulting expressions of the Hashin-Shtrikman bound of the effective potential. The resulting expressions are given naturally in terms of symmetric second-order tensors and minor and major symmetric fourth-order tensors. These properties induce, based on the tensorial Fourier expansion of the crystallite orientation distribution function, a dependency of all Hashin-Shtrikman properties in terms of solely the second- and the fourth-order texture coefficients. This is a new result, which is not self-evident, since an alternative formulation of the polarization field would alter the implied algebraic properties of the Hashin-Shtrikman functional. The results obtained by the polarization field, determined through the stationarity condition of the Hashin-Shtrikman functional, are discussed and demonstrated with an example for linear thermoelasticity in which bounds for elastic and thermoelastic properties are illustrated.

Keywords: -

Abstract: The Hashin-Shtrikman bounds accounting for eigenfields are represented in terms of tensorial texture coefficients for arbitrarily anisotropic materials and arbitrarily textured polycrystals. This requires a short review of the Hashin-Shtrikman bounds with eigenfields, an investigation of the polarization field determined by the stationarity condition and, finally, the analysis of the resulting expressions of the Hashin-Shtrikman bound of the effective potential. The resulting expressions are given naturally in terms of symmetric second-order tensors and minor and major symmetric fourth-order tensors. These properties induce, based on the tensorial Fourier expansion of the crystallite orientation distribution function, a dependency of all Hashin-Shtrikman properties in terms of solely the second- and the fourth-order texture coefficients. This is a new result, which is not self-evident, since an alternative formulation of the polarization field would alter the implied algebraic properties of the Hashin-Shtrikman functional. The results obtained by the polarization field, determined through the stationarity condition of the Hashin-Shtrikman functional, are discussed and demonstrated with an example for linear thermoelasticity in which bounds for elastic and thermoelastic properties are illustrated.

Keywords: Discontinuous reinforcement; Computational modeling; Damage mechanics; Compression molding

Abstract: A virtual process chain for sheet molding compound (SMC) composites is established and validated by means of experimental investigations on a demonstrator structure. The flow in the compression molding step is simulated via a Coupled-Eulerian-Lagrangian approach using an anisotropic non-Newtonian fluid flow model. Evolution of the fiber orientation distribution (FOD) is described by Jeffery's equation. The predicted FOD is mapped to structural simulations employing a neutral data format. A mean-field anisotropic damage model is used to predict the damage evolution in the demonstrator. Simulated FOD at the end of the compression molding is validated by computer tomography. Structural simulations are validated by means of a cyclic four-point bending test on the demonstrator. The predicted results show increased accuracy with the experiments by transferring FOD data within the virtual process chain. Critical points of high damage concentrations leading to failure agree with the experimental observations.

Keywords: Injection molding; Non-destructive testing; Polymer-matrix composites (PMCs); Short fiber composites; Single fiber segmentation

Abstract: As a result of the injection molding process, short glass fiber reinforced thermoplastics (SFRT) exhibit complex microstructures with fibers of different length and orientation as well as spatially varying fiber volume fractions. Their mechanical and functional performance can therefore only be predicted and ensured based on precise knowledge of said microstructure. To that end, x-ray micro-computed tomography (μCT) is commonly employed, with analysis software most-often yielding the fiber orientation tensors but lacking more detailed results. In order to increase the quality of the microstructural data, more accurate analysis techniques are necessary that include single fiber information for detailed correlations of microstructural parameters. In this work, a novel algorithm is suggested that relies on an iterative single fiber segmentation and merging procedure to obtain the fiber characteristics: orientation, location, radius and length. The algorithm is implemented in an efficient manner in Python, making heavy use of parallelization techniques. It is then applied to μCT scans and artificially generated 3D data of short glass fiber reinforced polyamide 6.6 with fiber mass fractions of 35%. The algorithm has been validated using reference data, commercial software and experimental fiber length data from an incinerated specimen and was shown to be both robust and accurate.

Keywords: Conjugate heat transfer; Engine heat transfer; Lumped element model; Model reduction; Monte Carlo simulation; Thermal management

Abstract: Reduction of heat and friction losses is a proven approach to increase the engine efficiency. Therefore, and due to a stabilized, robust combustion, a specific adjustment of component temperatures is desirable in highly transient conditions. In this paper, a turbocharged SI engine is investigated numerically concerning the potential regulation of temperatures, including heat fluxes, only by controlling the water mass flow rate. Using two independent models, a simplified lumped capacity model and a detailed three-dimensional CFD-CHT simulation, an efficient, two-stage calculation method is suggested for an optimized determination of control strategies and their parameters. This complements existing published works which usually control more than one parameter, but use one model. Different control strategies, like feed forward or feedback controllers, are proposed and compared. In addition, a more holistic approach is presented performing a Monte Carlo simulation which evaluates temperatures, as well as hydraulic pumping losses. Using expedient control strategies and parameters, it could be shown that the engine temperatures can be effectively regulated within a wide range. The two different models show partly similar results, and the efficient, two-stage optimization method has proven its worth. However, there are some significant differences between simplified and detailed modelling which are worth mentioning.

Keywords: Conjugate heat transfer; Engine heat transfer; Lumped element model; Model reduction; Monte Carlo simulation; Thermal management

Abstract: A calculation method for engine temperatures is presented. Special focus is placed on the transient and scattering boundary conditions within the combustion chamber, including fired and coasting conditions, as well as the dynamic heat transfer of the water jacket. Model reduction is achieved with dimensional analysis and the application of probability density functions, which allows for a timescale separation. Stationary in-cylinder pressure measurements are used as input values and, according to the transient behavior, modified with an own part-load model. A turbocharged spark ignition race engine is equipped with 70 thermocouples at various positions in proximity to the combustion chamber. Differentiating from already published works, the method deals with the transient engine behavior during a race lap, which undergoes a frequency range of 0.1–1 Hz. This includes engine speed build-ups under gear changes, torque variations, or the transition from fired to coasting conditions. Different thermal behaviors of various measuring positions can be simulated successfully. Additionally, cylinder individual temperature effects resulting from an unsymmetrical ignition sequence and different volumetric efficiencies with unequal residual gas can be predicted. Up to a few percent, the energy balance of the water jacket is fulfilled and variations of water inlet temperatures can be simulated accurately enough.

Keywords: -

Abstract: Biaxial tensile tests allow the investigation of damage evolution in sheet molding compound under biaxial stress states. This is of particular interest due to the different damage phenomena in composite materials. A key challenge is to find a suitable specimen design, because typical cruciform specimens fail in the arms before damage occurs in the area of interest which is the area of the biaxial stress state in the center area of the specimen. A specimen was found which enables the observation of anisotropic stiffness degradation which is one phenomenon of damage. In this proceedings the results of the experiments are presented.

Keywords: Effective anisotropic properties; Materials design; Orientation average; Polycrystals; Texture; Texture coefficients

Abstract: The present work generalizes the results of Böhlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Keywords: High-strength steel; Hot stamping; Phase transformation; Thermomechanical simulation; Two-scale method

Abstract: The hot stamping process of sheet metals into a w-shape profile is considered in experiments and numerical simulation using the FEM tool ABAQUS. The influence of the processing parameters of the hot stamping process on the final shape and the residual stress state are investigated. Tensile tests of thermomechanically treated specimens serve as a starting point for the modeling of the thermomechanical behavior of the steel grade 22MnB5. Based on the thermo-micromechanical (TMM) model for small deformations suggested by Neumann and Böhlke (2016), the updated Lagrangian formulation is used to render the finite deformation process. The model captures in a two-scale framework the temperature dependent elasto-plastic effect of the different phases, the phase transformation effect, and the phase transformation accompanying transformation strain occurring during the quenching of the austenitized sheet metal. Additionally, a model for the transformation induced plasticity (TRIP) effect in the context of the two-scale approach is suggested. The homogenization and localization of the thermomechanical properties is performed with a HashinöShtrikman (HS) type homogenization scheme. In this work, a reduced formulation for isotropic elastic behavior of the different phases is found reducing the amount of numerical costs. Furthermore, a self-consistent scheme based on the HS type method is introduced. Finally, the suggested TMM model is validated by considering the time-temperature-transformation- (TTT) and continuous-cooling-time-curves (CCT), dilatation tests under external load, and the hot stamping of w-shape profiles. The TMM model is compared to both experimental results and a phenomenological reference model usually used for the simulation of hot stamping processes. It is found that the suggested TMM model fits the final shape and the residual stress state of hot stamped parts better than the phenomenological reference model.

Keywords: Computational homogenization; Douglas–Rachford splitting; Elasto-viscoplasticity; FFT

Abstract: We revisit the polarization-based schemes introduced to FFT-based computational homogenization by Eyre–Milton, Michel–Moulinec–Suquet and Monchiet–Bonnet. When applied to nonlinear problems, these polarization-based methods suffer from two handicaps. Firstly, the optimal choice of algorithmic parameters is only known for the linear elastic case. Secondly, in its original version each iteration of the polarization scheme requires solving a nonlinear system of equations for each voxel. We overcome both difficulties for small-strain elastic–viscoplastic materials. In particular, we show how to avoid solving the nonlinear system. As a byproduct, we identify a computationally efficient convergence criterion enabling a fair comparison to gradient-based solvers (like the basic scheme). The convergence behavior of the polarization schemes is compared to the basic scheme of Moulinec–Suquet and fast gradient methods, based on numerical demonstrations.

Keywords: Deep drawing; Mean-field homogenization; Residual stresses; Two-scale simulation

Abstract: The final geometry and fatigue behavior of deep drawn components in service is strongly influenced by deformation-induced residual stresses. For multi-phase materials, besides macro residual stresses (first kind), phase-specific residual stresses (second kind) occur on the microscale of the material. In order to influence the component characteristics positively it is important to predict the distribution of the residual stresses on both scales. A two-scale simulation and measurement approach is presented which allows for an efficient determination and validation of the phase-specific residual stresses. Finite-element simulations are performed to predict the deformation-induced macro residual stresses. A numerically efficient mean-field homogenization is used to estimate the total strain, the plastic strain and the eigenstrain on the grain level based on macroscopic stress, strain and stiffness data. The simulated residual stresses are compared to experimental data. Macro residual stresses are determined by means of incremental hole drilling method, whereas phase-specific residual stresses are analyzed with use of X-ray diffraction according to the sin 2ψ method. The simulation and measurement approaches are applied to a representative deep-drawing process for the lean duplex stainless steel X 2 CrNiN 23 - 4 , which consists of a ferritic and an austenitic phase both with the same volume fraction. The results indicate that the proposed two-scale simulation approach is well suited for the prediction of phase-specific residual stresses after a deep drawing process of lean duplex steel.

Keywords: Long fiber-reinforced composite; Mean- and full-field homogenization; Thermoelastic properties; Dynamic mechanical analysis

Abstract: A discontinuous fiber-reinforced thermoset material produced by the Sheet Molding Compound process is investigated. Due to the process-related fiber orientation distribution, a composite with an anisotropic microstructure is created which crucially affects the mechanical properties. The central objectives are the modeling of the thermoelastic behavior of the composite accounting for the underlying microstructure, and the experimental characterization of the pure resin and the composite material.

Keywords: Finite element simulation; Creep; Gradient plasticity; Directional solidification

Abstract: A single-crystal plasticity model as well as a gradient crystal plasticity model are used to describe the creep behavior of directionally solidified NiAl based eutectic alloys. To consider the transition from theoretical to bulk strength, a hardening model was introduced to describe the strength of the reinforcing phases. Moreover, to account for microstructural changes due to material flux, a coupled diffusional-mechanical simulation model was introduced.

Keywords: Composites; Damage; DMA; Fiber reinforced polymers; LFT; Viscoelasticity

Abstract: This paper describes an experimental technique based on dynamic mechanical analysis (DMA) that is shown to be useful in separating nonlinear viscoelastic effects from the characterization of irreversible stress-induced damage responsible for stiffness reduction in fiber-reinforced polymer composites. In this work, we characterize the damage evolution of polypropylene (PP) reinforced with long, discontinuous glass fibers (GF) produced by the Direct Long-Fiber Thermoplastic/Compression Molding (D-LFT/CM) process. The experimental technique is comprised of three phases: 1) dynamic stabilization at low load to measure the time-dependent storage and loss moduli, followed by 2) a frequency sweep to provide rate-dependent viscoelastic properties and 3) a quasi-static application of a peak load. These three phases are repeated with the peak load of Phase 3 increased in each iteration. Experimental results for D-LFT/CM PP/GF30 presented here show a direct correlation between peak load and the irreversible stiffness reduction. Furthermore, the stabilization phase following peak load application is shown to lead to a stiffness recovery of up to 40% due to nonlinear viscoelastic recovery.

Keywords: -

Abstract: This contribution investigates the thermodynamic consistency of a meanfield damage model published by the authors [1]. The model is based on an empirical description of the fiber orientation distribution function (FODF). Due to a combination of matrix damage and fiber‐matrix interface debonding, the model is able to predict the inhomogeneous and anisotropic evolution of the overall behavior of the considered Sheet Molding Compound (SMC) composite.

Keywords: Sheet molding compound (SMC) composites; multiscale modeling; damage; interface characterization

Abstract: This contribution shortly introduces the anisotropic, micromechanical damage model for sheet molding compound (SMC) composites presented in the authors’ previous publication [1]. As the considered material is a thermoset matrix reinforced with long (≈25 mm) glass fibers, the leading damage mechanisms are matrix micro-cracking and fiber-matrix interface debonding. Those mechanisms are modeled on the microscale and within a Mori-Tanaka homogenization framework. The model can account for arbitrary fiber orientation distributions. Matrix damage is considered as an isotropic stiffness degradation. Interface debonding is modeled via a Weibull interface strength distribution and the inhomogeneous stress distribution on the lateral fiber surface. Hereby, three independent parameters are introduced, that describe the interface strength and damage behavior, respectively. Due to the high non-linearity of the model, the influence of these parameters is not entirely clear. Therefore, the focus of this contribution lies on the variation and discussion of the above mentioned interface parameters.

Keywords: Conjugate heat transfer; Engine heat transfer; Model reduction; Scattering boundary conditions; Variation model

Abstract: The focus of this paper is the fast determination of thermal boundary conditions in engine combustion chambers. In contrast with many other studies, only cycle integrated quantities like the induced torque are needed as input variables, which means that no crank angle resolved in-cylinder pressure data are required. Changes in the engine mapping like variations in ignition time and boost pressure or various lambda strategies are studied concerning component temperatures, and not to crank angle resolved heat fluxes, as it was often the case in previous published works. It is demonstrated that variations of cycle averaged solid temperatures can be predicted with the proposed identification method for thermal boundary conditions. The limit of the model for highly non-uniform pressure changes, as it is the case in ignition time variations, is well discussed. A variety of thermal boundary conditions is tested within a CFD-CHT simulation in order to get component temperatures. The new calculation algorithm combines proven models according to Woschni with a statistical method, which takes pressure fluctuations into account. Probability density functions and realisations of chosen random variables, like heat transfer coefficients, are transformed according to different engine operating conditions. For model validation, engine temperature measurements are conducted.

Keywords: Similarity mechanics; Buckingham Pi-Theorem; Conjugate heat transfer; Engine heat transfer; Scattering boundary conditions

Abstract: A three-dimensional CFD–CHT simulation method is presented and validated with a turbocharged single cylinder SI engine. Various ignition time and lambda strategies as well as variations of boost pressure are investigated with regard to cycle averaged component temperatures. This complements existing published works which experimentally studied crank angle resolved heat fluxes or temperature swings rather than averaged temperatures. Cyclical fluctuations in the pressure curves were measured and processed statistically using probability density functions for the heat transfer coefficient and the cylinder gas temperature. The corresponding joint probability density function considers their strong correlation. The interpretation as random variables enables a time-scale separation with a low-pass filter function. The thermomechanical problem of heat transfer is addressed with simplified models according to Woschni, Eichelberg, and Hohenberg. Previous investigations primarily focused on their predictive quality of instantaneous in-cylinder heat fluxes. In this paper, their effect on cycle averaged component temperatures is investigated and the corresponding different sensitivities to specific engine settings are presented and compared with measurements. It is shown that, by choosing the right model, the suggested simulation approach is an alternative to prevailing experimental methods in temperature analysis: all thermodynamic variations examined are in good agreement with theoretical predictions.

Keywords: Dynamic mechanical analysis; homogenization; parameter identification; polypropylene; unsaturated polyester-polyurethane hybrid resin; viscoelasticity

Abstract: Fiber-reinforced polymer composites have gained significant importance in engineering applications and are widely used as structural components. The optimal choice of combinations of the type of polymer matrix and the fiber content depend on the required applications. In view of this, polypropylene (PP), a thermoplastic, reinforced with 30 wt.% of short glass fibers (PP-GF30) is considered for this study. Additionally, a rather new composite system based on unsaturated polyester-polyurethane hybrid resin (UPPH) is introduced. This thermoset-based material is reinforced with 41 wt.% of long discontinuous glass fibers (UPPH-GF41). Dynamic mechanical analysis (DMA) is performed on the pure polymer and the composite samples to experimentally characterize the temperature- and frequency-dependent material behavior. It is observed that the stiffness of the materials is highly temperature-dependent. PP exhibits a nonlinear viscoelastic behavior in the temperature range of possible applications whereas the UPPH material shows a less pronounced behavior. The resulting experimental data not only give general information on the material behavior of the composite subjected to a temperature and frequency load but also provide input as well as validation data for the developed material modeling methods. In an industrial environment, homogenized elastic material properties are desirable for analysis of the structural components at room temperature conditions. A mean field homogenization method is introduced to estimate the effective elastic material properties on a macroscopic scale. This method is formulated explicitly in terms of orientation tensors of second- and fourth-order, describing quantitatively the microstructure of the corresponding composite. Additionally, the thermoviscoelastic material behavior for the considered temperature range is modeled by the free volume concept. In this context, the material parameters for the pure PP and UPPH material are identified. The calculated simulation results are compared with the experimental data entailing good agreements.

Keywords: Sheet molding compound (SMC); biaxial tensile testing; cruciform specimen design; unidirectional reinforcements; stiffness degradation; finite element simulation

Abstract: This proceedings paper presents the investigation of different cruciform specimen designs for the characterization of Sheet Molding Compounds under biaxial loading. Biaxial tensile tests allow the investigation of damage evolution under multiaxial stress states, which is particularly interesting due to the different damage phenomena in composite materials. A key challenge is to find a suitable specimen shape, because typical cruciform specimens fail in the arms before damage occurs in the area of interest which is the area of the biaxial stress state in the center region of the specimen. For all of the introduced designs the stiffness degradation is analyzed more in detail and compared to that of a uniaxial bone specimen. For the best performing specimen which is reinforced by unidirectional reinforced tapes on the arms, the strain field is analyzed by finite element simulations, taking into account the mechanical properties of the different layers of the specimen. Especially in the center area and at critical points, strain concentrations and non-symmetrical strain distributions are analyzed and evaluated.

Keywords: Discontinuous fibers; sheet molding compound (SMC); biaxial tensile testing; cruciform specimen design; unidirectional reinforcements

Abstract: This paper presents an investigation of different cruciform specimen designs for the characterization of sheet molding compound (SMC) under biaxial loading. The considered material is a discontinuous glass fiber reinforced thermoset. We define various (material-specific) requirements for an optimal specimen design. One key challenge represents the achievement of a high strain level in the center region of the cruciform specimen in order to observe damage, at the same time prevention of premature failure in the clamped specimen arms. Starting from the ISO norm for sheet metals, we introduce design variations, including two concepts to reinforce the specimens’ arms. An experimental evaluation includes two different loading scenarios, uniaxial tension and equi-biaxial tension. The best fit in terms of the defined optimality criteria, is a specimen manufactured in a layup with unidirectional reinforcing outer layers where a gentle milling process exposed the pure SMC in the center region of the specimen. This specimen performed superior for all considered loading conditions, for instance, in the uniaxial loading scenario, the average strain in the center region reached 87% of the failure strain in a uniaxial tensile bone specimen.

Keywords: Inverse Parameter Identification, Biaxial Tensile Tests, Sheet Molding Compound (SMC), Fiber Orientation Distribution, Fiber Orientation Tensor (FOT)

Abstract: Discontinuous fiber reinforced composites offer great advantages for high-volume lightweight components. The characterization of their process-dependent, macro-heterogeneous, anisotropic mechanical behavior presents, however, a challenge to composite material science. Biaxial tensile tests allow for the loading of various stress states on the specimen. The inhomogeneous stress and strain fields require an inverse parameter identification. Previous biaxial tensile tests in the elastic range showed fluctuations in the elastic properties within one specimen. Micro CT scans suggested that some of these fluctuations derive from an inhomogeneous fiber orientation distribution. The identification of a generally inhomogeneous stiffness leads, however, to an ill-posed problem which does not allow for a unique solution. We introduce the assumption of linearity between the stiffness tensor and the fiber orientation distribution. This simplification reduces the problem size to five degrees of freedom per specimen which do not depend on fiber orientation distribution. Four of these parameters are identifiable and are determined in a Gauss-Newton type optimization procedure.

Keywords: Damage mechanics; Debonding; Interfacial strength; Multiscale modeling; Polymer-matrix composites (PMCs)

Abstract: This paper presents an anisotropic, micromechanical damage model for sheet molding compound (SMC) composites, that is a thermoset matrix reinforced with long (≈25mm) glass fibers. The model captures the dominant damage mechanisms – matrix damage and fiber-matrix interface debonding – in a Mori-Tanaka homogenization framework. Matrix damage is modeled as a phase-averaged isotropic stiffness degradation. The interface damage is governed by an equivalent interface stress on the lateral fiber surface. Hereby, the inhomogeneous stress distribution in the fiber-matrix interface is taken into account in the definition of the equivalent stress. A Weibull distribution for the interface strength is assumed. The model can account for an anisotropic distribution and evolution of load-carrying fibers with intact interfaces. The model is validated by means of tensile tests on unsaturated polyester polyurethane hybrid and epoxy resin systems with different glass fiber contents (23-50vol.%). The model yields satisfyingly accurate predictions under uniaxial and biaxial stress states.

Keywords: -

Abstract: The mechanical behavior of polymers and other glass forming materials is strongly time‐ and temperature‐dependent. Based on the rheological Poynting‐Thomson Model and the Adam‐Gibbs equation, a shift function is developed which relates the time constants of a material to temperature. The behavior of this function is compared to the established Williams‐Landel‐Ferry and Arrhenius equations in the glassy and rubbery temperature regimes.

Keywords: composite; damage modeling; parameter identification; biaxial tensile testing; specimen design

Abstract: The focus of this work lies on the microstructure-based modeling and characterization of a discontinuous fiber-reinforced thermoset in the form of sheet molding compound (SMC). A microstructure-based parameter identification scheme for SMC with an inhomogeneous fiber orientation distribution is introduced. Different cruciform specimen designs, including two concepts to reinforce the specimens' arms are evaluated. Additionally, a micromechanical mean-field damage model for the SMC is introduced.

Keywords: Homogenization; Materials design; Theoretical bounds; Polycrystals; Multiphase materials

Abstract: This work approaches the fields of homogenization and of materials design for the linear and nonlinear mechanical properties with prescribed properties-profile. The set of achievable properties is bounded by the zeroth-order bounds (which are material specific), the first-order bounds (containing volume fractions of the phases) and the second-order Hashin-Shtrikman bounds with eigenfields in terms of tensorial texture coefficients for arbitrarily anisotropic textured materials.

Keywords: Anisotropic viscosity, Homogenization, Jeffery’s equation

Abstract: Background The commonly used flow models for Fiber Reinforced Polymers (FRPs) often neglect the flow-induced anisotropy of the suspension, but with increasing fiber volume fraction, this plays an important role. There exist already some models which count on this effect. They are, however, phenomenological and need a fitted model parameter. In this paper, a micromechanically-based constitutive law is proposed which considers the flow-induced anisotropic viscosity of the fiber suspension. Methods The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber suspension which depends on the actual fiber orientation distribution. Assuming incompressible material behaviour, a homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described based on Jeffery’s equation. The reorientation of the fibers is modeled in two ways: by describing them with fiber orientation vectors, and by fiber orientation tensors. A numerical implementation of the introduced model is applied to representative flow modes. ResultsThe predicted effective stress values depending on the actual fiber orientation distribution through the anisotropic viscosity are analyzed in transient and stationary flow cases. In the case of the assumed incompressibility, they show similar effective viscous material behaviour as the results obtained by the use of the Dinh-Armstrong constitutive law. Conclusions The introduced model is a possible way to describe the flow-induced anisotropic viscosity of a fluid-fiber suspension based on the mean field theory.

Keywords: -

Abstract: This presentation gives a short review on the Fiber-Orientation-Evolution models commonly used in commercial softwares. The basis of the considered models is Jeffery’s equation from 1922 which describes the motion of a single ellipsoidal particle in a Newtonian fluid. The later models extend Jeffery’s equation for many fiber system, with the use of Fiber-Orientation-Tensors. Models are discussed up to the o-iARD-RPR model (Tseng 2016), and compared considering representative flow modes.

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Abstract: The implementation of novel material models in the microscale gives a deeper understanding of inner and intercrystalline effects of crystalline materials. For future works, this allows more precise predictions of macroscale models. Here, we present a finite gradient crystal plasticity theory which preserves the single crystal slip kinematics. However, the model is restricted to one gradient-stress, associated with the gradient of the accumulated plastic slip, in order to account for long range dislocation interactions in a physically simplified, numerically efficient approach. In order to model the interaction of dislocations with and their transfer through grain boundaries, a grain boundary yield condition is introduced. The grain boundary flow rule is evaluated at sharp interfaces using discontinuous trial functions in the finite element implementation, thereby allowing for a discontinuous distribution of the accumulated plastic slip. Simulations of crystal aggregates are performed under different loading conditions which reproduce well the size dependence of the yield strength. An analytical solution for a one-dimensional single slip supports the numerical results.

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Abstract: Micro Abstract We present a two-scale anisotropic damage model that captures matrix damage and fiber-matrix interface debonding. Based on the fiber orientation distribution and a Weibull probability distribution of the interface strength, the damage evolution on the microscale is determined. Within this work focus lies on the comparison of different matrix damage evolution models. To predict the macroscopic behavior, a mean field homogenization with the Mori-Tanaka method based on orientation tensors of second and fourth order is applied.

Keywords: Anisotropic damage model; associated damage evolution; glass-fiber-reinforced polymer composite; vinylester matrix material

Abstract: Beyond a certain loading threshold, glass-fiber-reinforced polymer composites with vinylester matrix can show considerably nonlinear deformation behavior due to damage of the matrix material. To model such nonlinear deformation properties, we follow the approach of a fully anisotropic damage model suggested by Govindjee et al. (Govindjee S, Kay GJ and Simo JC. Anisotropic modelling and numerical simulation of brittle damage in concrete. International Journal for Numerical Methods in Engineering 1995; 38(21): 3611-3633). In addition, the inter-fiber fracture criterion introduced by Puck (Puck A. Festigkeitsanalyse von Faser-Matrix-Laminaten: Modelle für die Praxis. München, Germany: Hanser Fachbuchverlag, 1996) is used for the damage function that defines the damage initiation and the material strength. A particular procedure is chosen to identify the material properties to validate the model: three glass-fiber-reinforced polymer laminates with different layups are tested in tension under different loading orientations, assuming laminate theory to be reasonably well fulfilled. Finally, further experiments are compared with corresponding simulation results to demonstrate the performance of the model.

Keywords: -

Abstract: Based on two artificial microstructures representing a long fiber reinforced thermoset material, the effective linear elastic material properties are calculated by both a mean and a full field homogenization method. Concerning the mean field method, the effective elastic material properties are approximated using the homogenization scheme by Mori and Tanaka, formulated explicitly in terms of orientation averages. This allows to use orienation tensors of 2nd and 4th order describing the orientation information on the micro level. The full field method is based on the fast Fourier transformation (FFT), for which the effective material properties are determined by volume averaging. The comparison between both methods show good agreements, the deviations are in the range between 2% and 12%.

Keywords: Composites; Effective anisotropic properties; Materials design; Orientation average; Texture

Abstract: It is shown in the present work that for linear elastic multiphase materials the orientation average of a stiffness tensor can be given in a closed form if central model functions are taken into consideration for the description of the material orientation distribution. This holds for arbitrary numbers of material constituents and for arbitrary degree of anisotropy of any of the constituents. The first-order bounds of Voigt and Reuss, the geometric average of Aleksandrov and Aisenberg and, specifically, the Hashin–Shtrikman bounds are given in closed forms depending on the single crystal stiffnesses of the material constituents, on their volume fractions and, for elasticity, on a central orientation and two scalar parameters per central model function. The central orientation and the two scalar parameters, referred to as texture eigenvalues in the present work, reflect the complete influence of the corresponding central model function on the elastic properties. The set of all admissible texture eigenvalues is derived. The closed form expressions can be used, e.g., for efficient and low dimensional homogenization procedures taking the crystallographic texture into account or, e.g., for the determination of estimates for unknown single crystal behavior. It is further shown, that central crystallite orientation distributions exist which conserve the anisotropy of the single crystal behavior but invert the direction of the anisotropy. The closed form expressions documented in the present work are used for an exemplary materials design problem in order to determine favorable volume fractions and texture eigenvalues for the orientation distributions of considered constituents for given prescribed anisotropic properties.

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Abstract: Based on a slip gradient crystal plasticity theory, cf. [1], an analytical, one dimensional solution for the plastic slip is discussed with respect to a homogeneous stress state. Multislip is considered in a bicrystal with two slip systems in each grain. The orientation of the grains is accounted for by the choice of a free energy based on the grain boundary Burgers tensor, cf. [2].

Keywords: Dislocation-density based modeling; Dual-phase steel; Long-range stresses; Mean-field modeling

Abstract: Numerical simulations based on phase-field methods are indispensable in order to investigate interesting and important phenomena in the evolution of microstructures. Microscopic phase transitions are highly affected by mechanical driving forces and therefore the accurate calculation of the stresses in the transition region is essential. We present a method for stress calculations within the phase-field framework, which satisfies the mechanical jump conditions corresponding to sharp interfaces, although the sharp interface is represented as a volumetric region using the phase-field approach. This model is formulated for finite deformations, is independent of constitutive laws, and allows using any type of phase inherent inelastic strains.

Keywords: Finite deformations; Mechanical jump conditions; Multiphase-field; Phase-field

Abstract: Numerical simulations based on phase-field methods are indispensable in order to investigate interesting and important phenomena in the evolution of microstructures. Microscopic phase transitions are highly affected by mechanical driving forces and therefore the accurate calculation of the stresses in the transition region is essential. We present a method for stress calculations within the phase-field framework, which satisfies the mechanical jump conditions corresponding to sharp interfaces, although the sharp interface is represented as a volumetric region using the phase-field approach. This model is formulated for finite deformations, is independent of constitutive laws, and allows using any type of phase inherent inelastic strains.

Keywords: Hot Stamping; Hot Working; Phase Transformation; Semi-Analytical Homogenization; Two-Scale Modeling

Abstract: Hot stamping is a hot drawing process which takes advantage of the polymorphic steel behavior to produce parts with a good strength-to-weight ratio. For the simulation of the hot stamping process, a nonlinear two-scale thermomechanical model is suggested and implemented into the FE tool ABAQUS. Phase transformation and transformation induced plasticity effects are taken into account. The simulation results regarding the final shape and residual stresses are compared to experimental findings.

Keywords: DMA; Fiber reinforced composites; polypropylene; damage; Mori-Tanaka

Abstract: Dynamical mechanical analysis is used to investigate the temperature dependent, anisotropic visco-elastic stiffness, damage induced stiffness degradation as well as physical aging of polypropylene and the anisotropic compression molded long fiber reinforced polypropylene. Based on micro-computer tomography data, the approximations of the linear elastic properties using the Mori-Tanaka method are presented.

Keywords: Creep; Crystal plasticity; Directional solidification; Finite element method; Nickel-aluminide

Abstract: Directional solidification (DS) of near-stoichiometric NiAl with a reinforcing refractory metal is of great interest for high-temperature structural applications. A three-dimensional model based on single-crystal plasticity theory is introduced for the description of the creep behavior of DS NiAl-9Mo (at.%) with a well-aligned fibrous microstructure. A hardening model considering also the transition from theoretical to bulk strength is motivated. To evaluate the model, NiAl-9Mo samples were directionally solidified using various growth rates. With the DS samples, creep experiments were performed at 900 °C and 1000°C for different applied stresses. The model reproduces correctly a change of the applied stress, the temperature as well as a change in the fiber diameter. It is found that creep of the composite is mainly controlled by the plastic behavior of the fibers. A closer insight into the interactions between the fiber and the matrix is obtained by the simulation. Finally, by revealing the impact of each phase on the composite's behavior, the shape of the creep curve could be explained.

Keywords: -

Abstract: Depending on the process parameters, the directional solidification (DS) of eutectic alloys leads to a fibrous or lamellar microstructure. A physically motivated creep model which was evaluated for a DS-eutectic with a fibrous microstructure is applied to a DS-eutectic with a lamellar microstructure. Creep curves are simulated and compared to experimentally measured ones. It is shown, that the simulation is in good agreement with the experiment.

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Abstract: Common cruciform specimen for biaxial tensile testing of sheet moulding compound, take damage and finally fail in uniaxially loaded areas. When using these specimen, an observation of damage initialization and failure in biaxially loaded areas is, therefore, not possible. In this paper, a parametric shape optimization is described to find a more suitable specimen shape. The parametrization of the specimen is presented. Objective functions are introduced to measure the appropriateness of specimen. A weighted summation transfers the constraint multiobjective optimization problem into a constraint scalar-valued problem. Findings of experiments suggest that a specimen shape with straight, non-tapering arms and slits along the arms is reasonable.

Keywords: -

Abstract: A small-deformation strain gradient plasticity (GP) model for single-crystals has been proposed in [1], including a grain boundary (GB) yield condition without hardening. It has been extended by a hardening term for the GBs after a comparison to discrete dislocation dynamics (DDD) results in [2]. Differences between the strain gradients of the GP results and the DDD results motivate the consideration of a non-quadratic defect energy [3] in the GP model. It is shown that the gradients in the GP model can be improved using an exponent different from two. Remaining discrepancies in the strain profiles, compared to the DDD results, are attributed to the neglect of the individual gradients of plastic slip and due to the lack of a mechanism for the misorientation-dependent elastic interactions of dislocations across GBs [4] in the GP model.

Keywords: Deformation behavior; Finite element simulations; Gradient plasticity theory; Loading condition; Mechanical response; Micro-structural; Microstructural variation; Uniaxial tensions

Abstract: When Chen et al. (Acta Mater 87:78–85, 2015) investigated the deformation behavior of oligocrystalline gold microwires with varying diameters in both uniaxial tension and torsion, contrary size effects were observed for the different load cases. In accompanying microstructural studies it was found that the microwires of different thicknesses reveal distinctive differences in grain size and texture, respectively. As a consequence, a significant influence of these microstructural variations on the determined size effects was assumed. However, within the frame of their work, a direct confirmation could only be presented for the effect of the grain size. In the present work, the size-dependent mechanical response of the microwires is modeled with a gradient plasticity theory. By finite element simulations of simplified grain aggregates, the influence of the texture on the size effects is investigated under both loading conditions. It is shown that the experimentally observed contrary size effects can only be reproduced when taking into account the individual textures of the microwires of different thicknesses within the modeling.

Keywords: Defect energy; Equivalent plastic strain; Gradient plasticity; Plastic strain gradients

Abstract: A single-crystal gradient plasticity model is presented that includes a power-law type defect energy depending on the gradient of an equivalent plastic strain. Numerical regularization for the case of vanishing gradients is employed in the finite element discretization of the theory. Three exemplary choices of the defect energy exponent are compared in finite element simulations of elastic-plastic tricrystals under tensile loading. The influence of the power-law exponent is discussed related to the distribution of gradients and in regard to size effects. In addition, an analytical solution is presented for the single slip case supporting the numerical results. The influence of the power-law exponent is contrasted to the influence of the normalization constant.

Keywords: Computational simulation; Crystal plasticity models; Experimental investigations; Grain boundary orientation; Single slip; Slip system; Slip transmissions; Theoretical framework

Abstract: A comprehensive overview is given of the literature on slip transmission criteria for grain boundaries in metals, with a focus on slip system and grain boundary orientation. Much of this extensive literature has been informed by experimental investigations. The use of geometric criteria in continuum crystal plasticity models is discussed. The theoretical framework of Gurtin (J Mech Phys Solids 56:640–662, 2008) is reviewed for the single slip case. This highlights the connections to slip transmission criteria from the literature that are not discussed in the work itself. Different geometric criteria are compared for the single slip case with regard to their prediction of slip transmission. Perspectives on additional criteria, investigated in experiments and used in computational simulations, are given.

Keywords: -

Abstract: The commonly used flow models for fiber reinforced polymers often neglect the flow induced mechanical anisotropy of the suspension. With an increasing fiber volume fraction, this plays, however, an important role. There are some models which count on this effect, they are, however, phenomenological and require a fitted model parameter. In this paper, a micromechanically based constitutive law is proposed which considers the flow induced anisotropic viscosity of the fiber suspension. The introduced viscosity tensor can handle arbitrary anisotropy of the fluid-fiber mixture depending on the actual fiber orientation distribution. A homogenization method for unidirectional structures in contribution with orientation averaging is used to determine the effective viscosity tensor. The motion of rigid ellipsoidal fibers induced by the flow of the matrix material is described by Jeffery's equation. A numerical implementation of the introduced model is applied to representative flow modes. The calculated stress values are analyzed in transient and stationary flow cases. They show a less pronounced anisotropic viscous behaviour in every investigated case compared to the results obtained by the use of the Dinh-Armstrong constitutive law. The reason for the qualitative difference is that the presented model depends on the complete orientation information, while the other one is linear in the fourth-order fiber orientation tensor.

Keywords: -

Abstract: An equivalent plastic strain gradient theory [1] is extended to account for finite strains. The presented model considers discontinuities of the equivalent plastic strain at grain boundaries by using enriched trial functions in the Finite-Element discretization. As a consequence, a grain boundary flow rule is introduced, depending on both the mean value and the jump of the equivalent plastic strain.

Keywords: -

Abstract: The effective elastic material properties of short fiber reinforced polypropylen are determined by means of the self-consistent (SC) method and the interaction direct derivative (IDD) method. In order to account for thermoelastic effective material properties, a Hashin-Shtrikman (HS) based two-step homogenization method with variable reference stiffness is used. The influence of the reference stiffness, dependent on a scalar parameter is investigated. Information on the microstructure are derived by computed tomography scans (µCT) and considered within the homogenization schemes. Thermomechanical properties of a long fiber reinforced polymer (LFRP) and a short fiber reinforced polymer (SFRP) are obtained by means of dynamic mechanical analysis (DMA). Simulation results for SFRP are compared to experimental results.

Keywords: Dynamic mechanical analysis; Homogenization; Long fiber reinforced composite; Short fiber reinforced composite; μCT data

Abstract: In order to predict the elastic material properties of a short fiber reinforced composite, the self-consistent scheme and the interaction direct derivative method are used. Furthermore, thermoelastic effective material properties are calculated by the Hashin-Shtrikman two-step method using a reference stiffness that is variable, dependent on a scalar parameter. Within the homogenization methods, information on the microstructure obtained by μCT scans are considered. Using dynamic mechanical analysis (DMA), the material properties of long fiber reinforced polymers (LFRP) and short fiber reinforced polymers (SFRP) are investigated by tension tests under thermal load. The homogenized material properties of SFRP are compared with experimental results obtained by DMA.

Keywords: Effective anisotropic linear elastic properties; Material data bases; Materials design; Optimal bounds

Abstract: Zeroth-order bounds of elastic properties have been discussed by Kröner (1977) and by Nadeau and Ferrari (2001). These bounds enclose the effective linear elastic properties of multiphase materials constituted of materials with arbitrary symmetry and of an arbitrary number of phases by using solely the material constants of the single materials. Nadeau and Ferrari showed that these bounds are isotropic tensors and presented an algorithm for the determination of the upper and the lower zeroth-order bound. It is shown in this paper that a problem arises for the lower bound, since the algorithm presented in Nadeau and Ferrari (2001), results in a negative compression modulus and/or shear modulus although the considered stiffness is positive definite. A simple analytic example for this undesirable property is given, together with a short Mathematica® code of the algorithm. In the present work, the definition of the lower bound by Nadeau and Ferrari is modified, thereby assuring its positive definiteness. The Mathematica® code of the corrected algorithm is also given. Furthermore, new bounds for non-diagonal components are derived, which give information of, in principle, accessible values for non-diagonal stiffness components using the zeroth-order bounds of the present work. The practical application of zeroth-order bounds for local and online material data bases of stiffness tensors is presented, in order to accelerate purposes in materials design through efficient materials screening.

Keywords: Anisotropy; Mechanical properties; Multiscale modeling; Orientation tensors; Short-fiber composites

Abstract: The main objective of this work is to give an answer to the question: is it sufficient to consider only the second-order fiber orientation tensor as microstructure variable describing the orientation distribution of short-fiber reinforced composites (SFRCs) in the prediction of effective elastic properties? This question is addressed in the context of SFRCs on the one hand with an overall transversal symmetric orientation distribution of fibers and, hence, effective transversally isotropic properties, and on the other hand with experimentally determined microstructure data using micro-computed tomography. Applying the maximum entropy principle, it is shown, how the fiber orientation distribution function (FODF) can be estimated by relying on the second and/or the fourth-order orientation tensor, only. Both estimates are used within the self-consistent and the interaction direct derivate approach to calculate the effective linear elastic properties. It is shown, that the predicted stiffness tensors significantly depend on the estimation of the FODF. Relative deviations of up to 20% in terms of stiffnesses and up to 46% in terms of Young's modulus are observed. For the experimentally determined microstructure, small deviations of up to 4.3% are found.

Keywords: Fiber reinforced composites; Injection-molded specimens; Micro computer tomographies; Microstructural information; Orientation distributions; Segmentation algorithms; Self-consistent method; Short-fiber-reinforced composites

Abstract: Mechanical properties of short-fiber reinforced composites are crucially influenced by their microstructure. The microstructure itself is mainly governed by the manufacturing process like injection or compression molding. The main contribution of this paper lies in the homogenization of linear elastic properties using experimental microstructural information. For this purpose, the microstructure of injection-molded specimens made of polypropylene reinforced with 30wt.% of short glass fibers are analyzed through micro-computer tomography (1/4CT) measurements. Applying a recently developed segmentation algorithm, the spatial position, the orientation distribution and the length of the fibers are determined. This data is evaluated in terms of orientation tensors and length distribution, and is used within three mean field approaches: a self-consistent homogenization method, the interaction direct derivative estimate, which is based on the three-phase model, and a two-step bounding method. All methods account for the orientation, the length and the diameter distribution. The numerical results are compared to experimental tensile tests.

Keywords: Finite elements; Microstructures; Phase transformation; Semi-analytical homogenization; Thermomechanical processes

Abstract: Thermomechanical treatment of steel plays an important role in production, e.g., hot stamping. To capture the complex polymorphic material behavior of steel under thermal and mechanical loads, the usual approach in literature is to use a macroscopic phenomenological ansatz. In contrast, the aim of this work is to establish a more physically based two-scale model taking the material behavior of the different phases on the micro scale into account. The scale transition from macro scale to micro scale is performed by a nonlinear localization method of Hashin-Shtrikman type, which is based on a phase-wise constant stress polarization. The thermomechanical constitutive law is determined by both the phase transformation on micro scale and the temperature-dependent mechanical behavior of each phase. The mechanical behavior is based on a thermodynamical approach, i.e. introduction of a Helmholtz free energy for each phase and use of potential relations gained from the Clausius-Duhem (CD) inequality. The temperature-affected diffusionless evolution of the microstructure is modeled by both a Koistinen-Marburger (KM) consistent rate law and a suggested nonlinear extension of the KM model. The suggested transformation model leads to a better agreement with experimental results compared to the usually used approaches. The Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is chosen to describe the diffusion driven phase transformation. After a parameter identification of the steel 42CrMo4 and a validation of the thermomechanical coupling, the influence of the latent heat, the transformation strain occurring during phase transformation, and the additional parameter arising in the homogenization scheme on the macroscopic behavior is investigated.

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Keywords: Engineering uncontrolled terms; Ab initio simulations; EBSD data; Finite element solution; Hashin-Shtrikman bounds; Mean-field homogenizations; Statistical characterization; Thermo-mechanical behaviors; Thermoelastic properties

Abstract: The present work focuses on the determination of the effective thermoelastic properties and the statistical characterization of stress fluctuations in silicon nitride's local phases. For that purpose, full field finite element solutions have been considered, based on 3D electron backscatter diffraction (EBSD) data of silicon nitride. A second-order mean field homogenization scheme, consisting in Hashin-Shtrikman bounds, has been also considered. Ab-initio simulations have been performed in order to determine the temperature-dependent elastic properties of the local phases. The isotropic material microstructure has been checked based on both experimental results and full field solutions. The effective thermoelastic properties have been assessed with the newly obtained experimental results. The stress fluctuations within silicon nitride's local phases have been examined under mechanical and thermal loadings. It has been shown that the amorphous phase is the most vulnerable to fracture and to micro-cracks initiation.

Keywords: Gradient plasticity; Single-crystal plasticity; Grain boundary influence; Microwires; Finite element method

Abstract: In experiments on metallic microwires, size effects occur as a result of the interaction of dislocations with, e.g., grain boundaries. In continuum theories this behavior can be approximated using gradient plasticity. A numerically efficient geometrically linear gradient plasticity theory is developed considering the grain boundaries and implemented with finite elements. Simulations are performed for several metals in comparison to experiments and discrete dislocation dynamics simulations.

Keywords: micromechanics; multiscale; crystal plasticity; homogenization; sheet metal forming

Abstract: The deep drawing of an aluminum alloy used in the packaging industry for the beverage can manufacturing process is investigated. In this work, the effective constitutive behavior is based on a crystal plasticity model in combination with a non-linear Hashin-Shtrikman type homogenization scheme in which a reference stiffness controls the stress and strain fluctuations. The simulation results are compared to experiments in terms of deep drawing earing profiles, texture evolution, and localization.

Keywords: dual-phase steel; material modelling; two-scale model; dislocation mechanism; composite

Abstract: Dual-phase steels exhibit good mechanical properties due to a microstructure of strong martensitic inclusions embedded in a ductile ferritic matrix. This work presents a two-scale model for the underlying work-hardening effects; such as the distinctly different hardening rates observed for high-strength dual-phase steels. The model is based on geometrically necessary dislocations and comprises the average microstructural morphology as well as a direct interaction between the constituents.

Keywords: fiber-reinforced composites; micromechanical modeling; semi-analytical homogenization; elastic properties; orientation tensors

Abstract: This work is focused on the prediction of elastic behavior of short-fiber reinforced composites by mean-field homogenization methods, which account for experimentally determined and artificially constructed microstructure data in discrete and averaged form. The predictions are compared with experimental measurements and a full-field voxel-based approach. It is investigated, whether the second-order orientation tensor delivers a sufficient microstructure description for such predictions.

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Abstract: The directional solidification (DS) of the eutectic alloy NiAl-9Mo (at. %) leads to a formation of single-crystal molybdenum-rich fibers embedded in a NiAl matrix and has promising high-temperature properties. Material models describing each phase are necessary to be able to predict the materials behavior of the in-situ composite under several conditions. Using a one-dimensional Voigt-Taylor model based on a phenomenological approach, the creep behavior is simulated and the results are compared to experimental data.

Keywords: Discrete Dislocation Dynamics; Grain Boundary Hardening; Grain Boundary Yielding; Strain Gradient Plasticity

Abstract: The gradient crystal plasticity framework of Wulfinghoff et al. (Wulfinghoff et al. 2013 Int. J. Plasticity 51, 33-46. (doi:10.1016/j.ijplas.2013.07.001)), incorporating an equivalent plastic strain yeq and grain boundary (GB) yielding, is extended with GB hardening. By comparison to averaged results from many discrete dislocation dynamics (DDD) simulations of an aluminium-Type tricrystal under tensile loading, the new hardening parameter of the continuum model is calibrated. Although the GBs in the discrete simulations are impenetrable, an infinite GB yield strength, corresponding to microhard GB conditions, is not applicable in the continuum model. A combination of a finite GB yield strength with an isotropic bulk Voce hardening relation alone also fails to model the plastic strain profiles obtained by DDD. Instead, a finite GB yield strength in combination with GB hardening depending on the equivalent plastic strain at the GBs is shown to give a better agreement to DDD results. The differences in the plastic strain profiles obtained in DDD simulations by using different orientations of the central grain could not be captured. This indicates that the misorientationdependent elastic interaction of dislocations reaching over the GBs should also be included in the continuum model.

Keywords: Dynamic mechanical analysis (DMA); Fiber orientation; Long fiber reinforced composites; Multiscale modeling

Abstract: The entire simulation process for long fiber reinforced thermoplastics is examined to determine the effective mechanical properties which are influenced by the microstructural fiber orientation state. Therefore, flow and fiber orientation simulations are conducted and the obtained fiber orientation tensors are used in two-scale structural simulations. The fiber orientation distributions as well as the mechanical properties are compared with micro-computed tomography data and results from threepoint bending tests performed by dynamical mechanical analysis (DMA), respectively. The validated results show that prediction of the essential mechanical properties is possible with the applied combinated methods and that the knowledge of the fiber orientation and its gradients is of crucial importance for the entire simulation process.

Keywords: Dynamic mechanical analysis; Long fiber reinforced composites; Multiscale modeling

Abstract: The mechanical properties of long fiber reinforced thermoplastics (LFT), which highly depend on the fiber orientation induced through manufacturing on a direct LFT line, are predicted for compression molded rectangular plates. Therefore, three two-scale structural mechanical simulation schemes are applied and discussed: a two-step approach, the Mori-Tanaka scheme and the self-consistent method. Fiber orientation tensors based on measured micro computed tomography data of selected samples as well as on filling simulations are used for the determination of mechanical properties, as e.g. the storage modulus. The results have been compared with dynamic mechanical analysis measurements of tensile specimens. The influence of the initial strand position on the effective mechanical properties of the plate and the variation of those are examined.

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Abstract: In order to predict the effective material properties of a short-fiber reinforced polymer (SFRP), homogenization of elastic properties with the self-consistent (SC) scheme and the interaction direct derivative (IDD) method is performed by means of µCT data describing the microstructure of the composite material. Using dynamic mechanical analysis (DMA), the material properties of both, polypropylene and fiber reinforced polypropylene are investigated by tensile tests under thermal load. The measured storage modulus of the matrix material is used as input parameter for the homogenization scheme. The effective properties of SFRP are compared to experimental results from DMA.

Keywords: Aluminum alloys; Band nucleation; Band propagation; Extended Estrin-McCormick model; PLC effect

Abstract: The purpose of the current work is the experimental investigation and modeling of Portevin-Le Châtelier (PLC) band development in AlMg3 aluminum alloys, in particular in AA5754. The experimental investigation employs both mechanical and thermal (infrared) radiation measurement methods. The former involve force, displacement, and strain measurements using strain gauges. The latter employ a high-speed infrared camera to capture PLC band trajectories and evolution. In addition, the critical strain for band nucleation, as well as band characteristics such as velocity, type, and strain state, have been determined. To model the experimental results, a modification of the Estrin-McCormick model (e.g., Estrin and McCormick, 1991) due to Böhlke et al. (2009) is employed. In this modification, the saturation value of the Cottrell-Bilby-Louat (CBL) contribution to the effective flow stress due to dynamic strain aging is not constant, but rather is assumed to be linearly dependent on the accumulated effective inelastic strain. This model is incorporated into a finite-element model for the experimental specimens constructed with the help of convergence and mesh-sensitivity studies. Using selected mechanical test data, model parameters for AA5754 are then identified. With the identified model, a number of comparisons between experimental results and model predictions are carried out for validation purposes. Attention is focused here in particular on aspects of PLC band nucleation and propagation. Nucleation behavior is studied in particular with respect to stress gradients. It was found that the stress gradient is not the main trigger for PLC band nucleation, and is less relevant with increasing strain. The comparison of experimental and simulation results for spatio-temporal strain patterns and stress distribution shows that strong stress drops are correlated with the start of longitudinal propagation of a fully evolved PLC band. Small oscillations between large stress drops during propagation of type B bands indicate the consecutive nucleation, propagation and disappearing of these bands. The transition region from normal to inverse behavior (decreasing strain rate) denotes the transition from type A to type B. A transition with increasing strain is observed both experimentally and in the simulation. For lower strain rates and higher strains, type C bands are observed as well. Generally speaking, the band velocity decreases with decreasing strain rate and increasing strain. On the other hand, the band strain is found to increase with increasing strain, and only slightly with increasing strain rate. Simulation and experimental results, generally, show very good agreement.

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Abstract: For a chosen combination of materials for a multiphase polycrystalline composite, a crystallite orientation distribution function for each phase is formulated as a superposition of analytic central model von Mises-Fisher functions. The chosen central model functions allow the analytic integration of orientation averages of arbitrarily anisotropic fourth-order tensors. Based on this result, the first-order bounds of Voigt and Reuss and the geometric average of the stiffness can be expressed explicitly in closed forms for arbitrarily anisotropic polycrystalline materials and number of phases depending on the volume fractions and the influence of the crystallographic texture of each constituent. These expressions can then be used for the materials design aiming the determination of volume fractions and orientation distribution of selected materials for prescribed effective properties.

Keywords: Cubic crystal aggregates; Hashin–Shtrikman bounds; Materials design; Tensorial texture coefficients

Abstract: For polycrystals made of cubic materials like copper, aluminum, iron and other metals and ceramics, the macroscopic elastic behavior can be bounded using minimum energy principles. Böhlke and Lobos (Acta Mater. 67:324–334, 2014) have shown that not only the Voigt and the Reuss bound but also the Hashin–Shtrikman bounds can be represented explicitly depending on the texture in form of the fourth-order texture coefficient. Considering the inequalities due to these bounds, the texture can be enclosed independently of the specific cubic material parameters. This implies domains for the texture parameters. Materials design is defined as the identification of materials and microstructures such that the effective constitutive properties correspond best to a prescribed properties profile. The design space is proposed to be constituted by the material design space and microstructure design space, delivering a total of twelve scalar design variables in the present model for linear elasticity of cubic crystal aggregates. Based on analytical results, materials design is established as an algorithm following Adams et al. (Microstructure Sensitive Design for Performance Optimization, 2013). In the present work, the scheme consists of four steps: (i) material selection, (ii) homogenization scheme, (iii) properties closure, and (iv) microstructure optimization. As an example, Young’s modulus of a polycrystal is designed with respect to four prescribed directions for a macroscopical orthotropic sample symmetry. For the orthotropic texture domain, a mathematically equivalent parametrization is derived in order to facilitate the constrained numerical optimizations.

Keywords: Fast Fourier transformation; Full field simulation; Interaction direct derivative; Mean field homogenization; Self-consistent method; Short-fiber reinforced composites

Abstract: The main contribution of this work lies in the detailed comparison of the predictions of linear elastic properties of mean field homogenization approaches and full field, voxel-based homogenization methods for short-fiber reinforced materials. In the former case, the self-consistent, the interaction direct derivative and a two-step-bounding approach, applying the Hashin-Shtrikman bounds, are used. In the latter case, the boundary value problem for representative volume elements is solved using fast Fourier transformation. Model microstructures with unidirectional aligned and two misaligned fiber configurations are considered exemplarily. Fiber volume fractions of 13%, 17% and 21% and phase contrasts of 44, 100 and 1000 in the elastic moduli have been taken into account. The different homogenization schemes are compared by means of effective directional dependent Young's modulus. This detailed comparison shows that mean field and full field solutions deliver similar results for moderate phase contrasts and volume fractions. Especially in the range of realistic phase contrasts like 44 for a composite of polypropylene and glass, the mean field approaches pose reliable alternatives for full field solution. Large phase contrasts result in relative deviations of up to 68%.

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Abstract: An equivalent plastic strain gradient theory, cf. [1], is extended by a GB flow rule, that accounts for the misorientation of adjacent grains. In contrast to previous works, the grain boundary (GB) resistance against plastic flow is not treated as a constant parameter. Therefore, in the work at hand, the grain boundary dislocation density (GBD), cf. [2], is used as a measure for geometrical mismatch between all glide system orientations of two adjacent grains.

Keywords: Crystal plasticity; Dual-phase steel; Mean-field modeling; Micromechanical modeling; Representative volume element; Work hardening

Abstract: The mechanical behavior of automotive dual-phase steel (DP) is modeled by two different approaches: with a full-field representative volume element (RVE) and with a mean-field model. In the first part of this work, the full-field RVE is constituted by a crystal plasticity-based ferrite matrix with von Mises-type martensite inclusions. To isolate the martensite influence, the full-field DP results were compared to a full-field comparison RVE. In the comparison RVE, all martensite inclusions were replaced by a phase that exhibits the average ferrite behavior. A higher relative martensite grain boundary coverage facilitates an increased average dislocation density after quenching. However, for uniaxial deformations above ∼10%, the grain size-dependent relation reverses and exhibits slowed-down hardening. In the second part, we incorporate the main findings from the full-field simulations into a nonlinear mean-field model of Hashin–Shtrikman type. The dislocation density production parameter and the saturated dislocation density are modeled based on grain size and martensite coverage. The comparison of both approaches shows good agreement for both the overall and constituent averaged behavior.

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Abstract: Modeling the damage of brittle materials is of great importance considering a variety of structural components. Prominent examples are high strength engineering ceramics. The present work is concerned with silicon nitride, a material with increasing importance in industrial applications. Using EBSD data allows to map the microstructure of polycrystallines and provides a tool to examine the grains, grain boundaries and their respective orientation. Based on EBSD data, micromechanical simulations of silicon nitride was performed using a micromechanical thermoelastic model by Wippler (2012). The micromechanical model includes damage for the different constituents of silicon nitride and the respective interface. As a result, the effective material properties of silicon nitride were determined. In the sense of a hierarchical model structure the effective properties were applied to macromechanical, phenomenological damage models for monotonous and cyclic loading. It is, thereby, essential to capture the general anisotropic nature of damage. For this purpose, an anisotropic smeared crack model has been adopted which was originally formulated by Govindjee et al. (1995). This model is extended to include crack-closure based on a geometrical approximation proposed by Ortiz (1985). The monotonous damage model, hereby, uses a damage evolution equation proposed by Govindjee et al. (1995). The cyclic damage model is based on an effective crack evolution for slow crack growth in silicon nitride (Lube et al., 2007) which was reinterpreted in terms of damage. The effect of damage on the effective material properties is shown by making use of the directional dependent Young’s modulus. Furthermore the application of the cyclic damage model to a four point bending test of silicon nitride as well as the application of the monotonous model to a three-dimensional thrust bearing are discussed.

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Abstract: In the present article we discuss the characterization of the mechanical properties of long fiber reinforced thermoplastics (LFT) and sheet molding compounds (SMC) with biaxial tensile tests and the inverse parameter identification. The full 3D strain field is measured via digital image correlation (DIC). The anisotropic viscoelastic material properties are identified through inverse modelling by comparison of the heterogeneous experimental and simulated strain fields. A Gauss-Newton type algorithm is used to identify the optimal parameter set [1].

Keywords: Biaxial tensile testing; Fiber reinforced composites; Homogenization; Interaction direct derivative estimate; Inverse parameter identification

Abstract: The present article deals with the characterization of the mechanical properties of long fiber reinforced thermoplastics (LFT), the resulting inverse parameter identification and homogenization based on fiber orientation distribution. To gain a better insight into this inhomogenous and anisotropic material behavior, biaxial tensile tests on cruciform specimen are performed. The full 3D strain field is measured via digital image correlation (DIC). The observed results including inelastic effects are discussed in detail for uniaxial and biaxial loading paths. The anisotropic material properties are identified through inverse modeling by comparison of the heterogeneous experimental and simulated strain fields. A Gauss-Newton type algorithm is used to identify the optimal parameter set [12]. Hereby the polymer composite is assumed to behave as a viscoelastic Maxwell material, to take rate-dependent effects into account. The identified linear elastic material parameters are compared to homogenized properties based on fiber orientation distributions obtained by micro-computed tomography. Meanfield approaches such as the interaction direct derivative estimate [20] and a Mori-Tanaka scheme are used to determine the macroscopic material properties.

Keywords: Elasticity; Heterogeneous systems; Interfacial excess energy; Jump-conditions; Microstructure evolution; Phase-field

Abstract: The present article deals with the characterization of the mechanical properties of long fiber reinforced thermoplastics (LFT), the resulting inverse parameter identification and homogenization based on fiber orientation distribution. To gain a better insight into this inhomogenous and anisotropic material behavior, biaxial tensile tests on cruciform specimen are performed. The full 3D strain field is measured via digital image correlation (DIC). The observed results including inelastic effects are discussed in detail for uniaxial and biaxial loading paths. The anisotropic material properties are identified through inverse modeling by comparison of the heterogeneous experimental and simulated strain fields. A Gauss-Newton type algorithm is used to identify the optimal parameter set [12]. Hereby the polymer composite is assumed to behave as a viscoelastic Maxwell material, to take rate-dependent effects into account. The identified linear elastic material parameters are compared to homogenized properties based on fiber orientation distributions obtained by micro-computed tomography. Meanfield approaches such as the interaction direct derivative estimate [20] and a Mori-Tanaka scheme are used to determine the macroscopic material properties.

Keywords: Phase-field, Multiphase-field, Elasto-plasticity, Crack propagation

Abstract: A small strain plasticity model, based on the principles of continuum mechanics, is incorporated into a phase-field model for heterogeneous microstructures in polycrystalline and multiphase material systems (Nestler et al., Phys Rev 71:1–6, 2005). Thereby, the displacement field is computed by solving the local momentum balance dynamically (Spatschek et al., Phys Rev 75:1–14, 2007) using the finite difference method on a staggered grid. The elastic contribution is expressed as the linear approximation according to the Cauchy stress tensor. In order to calculate the plastic strain, the Prandtl–Reuss model is implemented consisting of an associated flow rule in combination with the von Mises yield criterion and a linear isotropic hardening approximation. Simulations are performed illustrating the evolution of the stress and plastic strain using a radial return mapping algorithm for single phase system and two phase microstructures. As an example for interface evolution coupling with elasto-plastic effects, we present crack propagation simulations in ductile material.

Keywords: Gradient plasticity; Logarithmic defect energy; Non-smooth defect energy; Regularization; Sub-differential

Abstract: Abstract Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress-strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress-strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.

Keywords: Crystal plasticity; Dislocations; Gradient plasticity; Numerical algorithms; Particulate reinforced material

Abstract: This work aims at the formulation of a gradient crystal plasticity model which incorporates some of the latest developments in continuum dislocation theory and is, at the same time, well-suited for a three-dimensional numerical implementation. Specifically, a classical continuum crystal plasticity framework is extended by taking into account continuous dislocation density and curvature field variables which evolve according to partial differential equations (Hochrainer et al., 2014; Ebrahimi et al., 2014). These account for dislocation transport and curvature-induced line-length production and have been derived from a higher-dimensional continuum dislocation theory. The dislocation density information is used to model work hardening as a consequence of dislocation entanglement. A composite microstructure is simulated consisting of a soft elasto-plastic matrix and hard elastic inclusions. The particles are assumed to act as obstacles to dislocation motion, leading to pile-ups forming at the matrix-inclusion interface. This effect is modeled using gradient plasticity with a simplified equivalent plastic strain gradient approach (Wulfinghoff et al., 2013) which is used here in order to allow for an efficient numerical treatment of the three-dimensional numerical model. A regularized logarithmic energy is applied which is intended to approximate the higher order gradient stress of the statistical theory of Groma et al. (2003).

Keywords: Crystal plasticity; Deformation patterns; Equivalent plastic strain; Laue microdiffraction; Microtorsion; Strain gradients

Abstract: Abstract In order to investigate an almost pure extrinsic size effect we propose an experimental approach to investigate the deformation structure within single crystalline cross-sections of twisted bamboo-structured Au microwires. The cross-sections of individual <100> oriented grains of 25 μm thick Au microwires have been characterized by Laue microdiffraction. The diffraction data were used to calculate the misorientation of each data point with respect to the neutral fiber in the center of the cross-section as well as the kernel average misorientation to map the global and local deformation structure as function of the imposed maximum plastic shear strain. The study is accompanied by crystal plasticity simulations which yield the equivalent plastic strain distributions in the cross-section of the wire. The global deformation structures are directly related to the activated slip systems, resulting from the real orientations of the investigated grains. When averaging the degree of deformation along ring segments, an almost continuous but non-linear increase of misorientation from the center toward the surface is observed, reflecting the overall strain gradient imposed by torsion. For the local deformation structure, pronounced and graded deformation traces are observed which often pass over the neutral fiber of the twisted wire and which are obviously reflecting domains of high geometrically necessary dislocations content.

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Abstract: The applicability of a strain gradient crystal plasticity model to size effects observed on microwires in torsion experiments is discussed. Finite Element simulations of simplified cylindrical grain aggregations are presented and the resulting overall mechanical response is compared to experimental data for gold from the literature.

Keywords: Cubic crystal aggregates; Effective linear elastic properties; Hashin-Shtrikman bounds; Materials design; Tensorial texture coefficients

Abstract: The Hashin-Shtrikman bounds of aggregates of cubic crystals are explicitly represented in terms of tensorial texture coefficients. The formula is valid for arbitrary crystallographic textures and isotropic two-point statistics. The isotropy of the two-point statistics implies that the grain shape is isotropic on average. The new explicit representation has the advantage that the set of energetically admissible crystallographic textures and corresponding effective linear elastic properties can be directly determined and analyzed based on minimum principles of the elastic strain energy density. It is shown that all energetically admissible textures with maximum anisotropy have an effective elastic behavior with cubic sample symmetry. Furthermore, it is proven that there exist texture states without maximum anisotropy which have the extreme elastic properties peculiar to states with maximum anisotropy. This is an important result for the design of elastic material properties.

Keywords: grain size effects; homogenization; phase-transformation; transformation induced plasticity; two-scale modeling

Abstract: A nonlinear homogenization scheme is applied to consider two classes of micromechanical problems, i.e. to estimate the grain size dispersion influence and to model phase transformation effects in steels. The macroscopic material behavior is described purely based on micro-mechanical constitutive models. The coarse graining procedure contains the classical bounds, i.e. mixture theories, as special cases and determines the macroscopic stress and the evolution of the microstructural variables. In the context of the grain size dependent flow behavior of polycrystals, a log-normal distributed grain size is assumed together with a grain size dependent local plastic behavior. The numerical results are well approximated by a simple analytical expression. It is found that grain size dispersion leads to a decrease of the material strength, especially for small mean diameters around one micron. In the context of phase transformation phenomena, the thermo-mechanically strongly coupled Greenwood-Johnson effect is considered. Here, temperature driven phase transformation under external applied stress below the yield stress is modeled that leads to plastic strains. The main result of the present work is, that based on the suggested nonlinear homogenization technique of Hashin-Shtrikman type a numerically effective modeling approach is given, that can be applied at the Gauss point level of structural finite element simulations. In contrast to FE2 schemes, full three-dimensional microstructures can be modeled effectively with standard computers. The simulation results presented in this work are in good agreement with the experimental data. A nonlinear homogenization scheme is applied to model grain size and phase transformation effects. The macroscopic material behavior is described only based on micro-mechanical constitutive models. Regarding the grain size dependent flow behavior of polycrystals, a log-normal distributed grain size is approximated by an analytical expression. In the context of phase transformation phenomena, the thermo-mechanically strongly coupled Greenwood-Johnson effect is modeled based on the micro-mechanical approach.

Keywords: elasto-plasticity; large deformations; phase field; spectral solver

Abstract: Most solid-state phase transformations are accompanied by large deformations, stemming either from external load, transformation strains or plasticity. The consideration of such large deformations will affect the numerical treatment of such transformations. In this paper, we present a new scheme to embed large deformations in an explicit phase-field scheme and its implementation in the open-source framework OpenPhase. The suggested scheme combines the advantages of a spectral solver to calculate the mechanical boundary value problem in a small strain limit and an advection procedure to transport field variables over the calculation grid. Since the developed approach should be used for various sets of problems, e.g. simulations of thermodynamically driven phase transformations, the mechanic formulation is kept general. However, to ensure compatibility with phase-field methods using the concept of diffuse interface, the latter is treated with special care in the present work.

Keywords: Composite materials; FFT; Finite deformations; Lippmann–Schwinger equation

Abstract: In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton–Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton’s method using linear fixed point solvers and roughly 100 times less iterations than the nonlinear fixed point method. However, the Newton–Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant–Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

Keywords: Quality Control; Lightweight; SMC Production

Abstract: The use of sheet molding compounds (SMC) in diverse applications requires different specific material properties for each type of finished parts. These material properties have to be assured by a reliable quality control, which does not only have to be performed for the prefabricated SMC itself but also during the production process of the semi-finished material. This is of high importance because quality fluctuations and defects can already occur during the production of the semi-finished SMC. This results in high scrap rates as well as machine failure and can additionally cause further problems in the following process steps. Hence, an inline quality control can help to establish objective quality criteria for semi-finished SMC and can enable controlled and stable production processes. Therefore, this paper deals with quality assurance in the production process of semi-finished sheet molding compounds. Air entrapping and fiber distribution are identified as two parameters that influence the quality of the semi-finished product significantly. In addition, the early detection of a pending carrier foil failure can help to establish a stable process. The focus of this paper lies on how various, individually adapted metrology systems can be used for the detection of the respective characteristics and integrated into the production process of the semi-finished SMC. In particular, optical systems, such as area scan cameras and laser stripe sensors as well as thermographic sensors are discussed and possibilities for application-related sensor data evaluation are shown. This helps to reduce the scrap rates of parts and to establish a further automated production process.

Keywords: Coherent domains; Elastic properties; Homogenization; Image processing; Pyrolytic carbon

Abstract: On the nanoscale, the microstructure of pyrolytic carbon (PyC) is constituted by an ensemble of graphene planes, which manifest themselves as lattice fringes in high-resolution transmission electron microscope images. This microstructure can be also considered by an aggregate of so-called coherent domains consisting of stacks of graphene planes with a common unit normal vector. In order to homogenize the elastic behavior of PyC on the micro-level, image processing techniques are used to detect the coherent domains. Subsequently, the domain orientation distribution function (DODF) is modeled by means of a von Mises-Fisher distribution. The main objective of the present paper is to estimate the elastic properties of the coherent domains of the PyC microstructure. Moreover, the Hashin-Shtrikman bounds for the elastic properties can be determined by taking into account the DODF and by applying a nonlinear averaging procedure of the spatially dependent deviations of the local elastic properties. The elastic properties of the coherent domains are estimated by an inverse parameter identification of the Hashin-Shtrikman homogenization method by using effective elastic properties. The latter ones have been obtained based on an Fourier-based image processing algorithm and the orientation distribution function of the graphene planes in a recent paper (Böhlke et al. in Z Angew Math Mech, 2012).

Keywords: -

Abstract: For crystal aggregates, the orientation distribution of single crystals affects the anisotropic linear elastic properties. In the singular approximation for cubic materials, this influence is reflected by a fourth-order texture coefficient. From this approximation, the statistical bounds of Voigt, Reuss and Hashin-Shtrikman, and an isotropically self-consistent singular approximation can be obtained. Here, an approximation is called isotropically self-consistent, if, for a vanishing texture, it results in the isotropic self-consistent approximation. The isotropically self-consistent singular approximation has the following advantages: i) it lies between the bounds of Voigt, Reuss and Hashin-Shtrikman, ii) it offers a useful approximation of the effective material behavior of textured anisotropic polycrystals, and iii) it can be used for material design purposes tailoring anisotropic properties mainly depending on the crystallographic texture.

Keywords: Fiber orientation distribution; Fiber-reinforced composite; Homogenization; Maximum entropy method; Self-consistence method; Two-scale modeling

Abstract: The main contribution of this work is the prediction of the linear elastic properties of short fiber-reinforced composites (SFRCs) and the estimation of the fiber orientation distribution function (FODF) based only on fiber orientation tensors of second order. A SFRC consisting of polypropylene with 30wt% of short glass fibers is considered exemplary. In the first part, two micromechanically based mean field methods are applied to approximate the linear elastic properties of the composite. Both presented methods, the self-consistence method and a two-step bounding approach, are able to consider segmented microstructure date from, e.g., micro computer tomography measurements. In the second part, it is shown, how the FODF can be estimated without closure approximations using the maximum entropy method (MEM).

Keywords: -

Abstract: An incremental homogenization scheme for the prediction of elastic properties of composites is reviewed. Similar to the differential scheme, the inclusions are included step-by-step. This approach accounts for high volume fractions of inclusion of different shape and elastic properties. A numerical example for a composite consisting of a polymeric matrix, glass fibers and voids is shown. The fiber distribution is chosen equivalently to a distribution in an injection molded short-fiber reinforced composite. The volume fraction of the voids is varied.

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Abstract: This article discusses a conceptual problem of gradient plasticity theories including a scalar plastic field variable, the Laplacian of which enters the yield criterion. Since the plastic field variable is assumed a scalar, it contains no information on the direction of the plastic flow. The work at hand demonstrates that even arbitrarily small perturbations can determine the direction of the plastic deformation in many formulations, which is considered an unphysical behavior. In this sense, the solution is not stable with respect to the boundary conditions.

Keywords: Dislocations; plasticity; crystals; size effect; hardening

Abstract: This book is a contribution to the further development of gradient plasticity. Several open questions are addressed, where the efficient numerical implementation is particularly focused on. The book inspects an equivalent plastic strain gradient plasticity theory and a grain boundary yield model. Experiments can successfully be reproduced. The hardening model is based on dislocation densities evolving according to partial differential equations taking into account dislocation transport.

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Abstract: A one-dimensional strain gradient plasticity model is reviewed in the context of Finite Element Method (FEM) solvability. It is found that under certain conditions specified the resulting system of two coupled partial differential equations in space and time leads to a symmetric global stiffness matrix. This is achieved upon standard weak form approach and yields a favorable system for an efficient numerical solution.

Keywords: Homogenization; Image processing; Pyrolytic carbon

Abstract: In this work, the linear elastic material properties of differently textured variants of pyrolytic carbon are homogenized from the submicro- to the micro-scale. In high resolution transmission electron microscope (HRTEM) lattice fringe images, the microstructure of pyrolytic carbon manifests itself in terms of projections of graphene layers. According to their orientation distribution, different textures of pyrolytic carbon have been classified. Assuming a von Mises-Fisher distribution for the spatial orientation of single graphene layers, the orientation distribution function of the projected layers in the image plane is analytically found to be a modified Struve function. For each pyrolytic carbon texture, Maximum-likelihood estimates for the mean orientation and the concentration parameter of the von Mises-Fisher distribution are obtained numerically. Hereby, Fourier transformation and appropriate filters are used to determine the probabilities for discrete orientations of the graphene layers directly from HRTEM images. First- and second-order bounds of the linear elastic properties of pyrolytic carbon of the different textures are computed. Elastic constants of graphite and pyrolytic graphite have been used for modeling the elastic behavior of the graphene layers within a continuum mechanical setting. Due to the high anisotropy of all analyzed textures of pyrolytic carbon, the differences even between the second-order bounds are quite large. In this work, the linear elastic material properties of differently textured variants of pyrolytic carbon are homogenized from the submicro- to the micro-scale. Assuming a von Mises-Fisher distribution for the spatial orientation of single graphene layers, the orientation distribution function of the projected layers in the image plane is analytically found to be a modified Struve function. For each pyrolytic carbon texture, Maximum-likelihood estimates for the mean orientation and the concentration parameter of the von Mises-Fisher distribution are obtained numerically. Hereby, Fourier transformation and appropriate filters are used to determine the probabilities for discrete orientations of the graphene layers directly from HRTEM images. First- and second-order bounds of the linear elastic properties of pyrolytic carbon of the different textures are computed.

Keywords: Failure probability; Hashin-Shtrikman bounds; Second-order estimates; Weakest link approach

Abstract: For brittle materials in engineering, it is well known that the defect structure governs the macroscopic strength. However, there is a class of materials [34] where the failure events occur on the micro-structure level with characteristic length smaller than the characteristic length of the defect. In this case, a mean field approach in which the stress fluctuations are estimated by a second-order moment approach in combination with a quantification of failure probabilities may give useful insight to the level of failure. Weibull's type failure probabilities are introduced in this work without reference to a defect microstructure.

Keywords: Computational homogenization; Material of Green type; Plastic compressibility; Porous materials; Three-scale homogenization

Abstract: The constitutive response of porous materials is investigated computationally. For the solid phase elasto- plastic behavior of Green type is considered, i.e. an isotropic compressible yield criterion is assumed. A wide range of material parameters and porosities from 0.1 to 30 % are investigated by means of FEM simulations of periodic ensembles of spherical pores. The dilatation of the pores and of the compressible matrix are evaluated. It is found that a large part of the total dilatation is due to plastic volume changes of the solid phase. The asymptotic stress states of the simulations are compared to analytical predictions by Shen et al. (Comput Mater Sci 62:189-194, 2012). Based on the computational data, an effective constitutive law is proposed and verified by means of additional computations. A three-scale homogenization procedure for double porous materials is proposed that depends only on the micro- and mesoscale porosity and the yield stress of the solid phase.

Keywords: A. Particle-reinforced composites; B. Non-linear behavior; C. Complex moduli; C. Multiscale modeling; Nonuniform Transformation Field Analysis (NTFA)

Abstract: A reduced basis order-reduction based homogenization method for the constitutive behavior of viscoelastic composites is presented. The approach is based on a modification of the Nonuniform Transformation Field Analysis (NTFA) introduced by Michel and Suquet [1]. A new evolution law for the set of macroscopic internal variables of the homogenized material is derived based on dissipative considerations. The theoretical aspects of the method are extensively discussed including an explicit algorithm for the computation of the anisotropic complex moduli of the material. Details concerning the numerical implementation and a selection of representative numerical examples are provided. The method accurately captures the anisotropic transient mechanical response of composites for various loading conditions while offering numerical savings in the order of 106-1010 with respect to both, CPU time and memory requirements.

Keywords: clustering techniques; crystallite orientation distribution function; data reduction; orientation space; tessellation methods

Abstract: Experimental techniques [e.g. electron backscatter diffraction (EBSD)] yield detailed crystallographic information on the grain scale. In both two- and three-dimensional applications of EBSD, large data sets in the range of 10 5-109 single-crystal orientations are obtained. With regard to the precise but efficient micromechanical computation of the polycrystalline material response, small representative sets of crystallographic orientation data are required. This paper describes two methods to systematically reduce experimentally measured orientation data. Inspired by the work of Gao, Przybyla & Adams [Metall. Mater. Trans. A (2006), 37, 2379-2387], who used a tessellation of the orientation space in order to compute correlation functions, one method in this work uses a similar procedure to partition the orientation space into boxes, but with the aim of extracting the mean orientation of the data points of each box. The second method to reduce crystallographic texture data is based on a clustering technique. It is shown that, in terms of representativity of the reduced data, both methods deliver equally good results. While the clustering technique is computationally more costly, it works particularly well when the measured data set shows pronounced clusters in the orientation space. The quality of the results and the performance of the tessellation method are independent of the examined data set.

Keywords: -

Abstract: Since the microstructure of short fiber reinforced composites is inhomogeneous, the application of micromechanical models is useful, that take into account their characteristics like the fiber orientation and the aspect ratio of fibers. Two different methods are considered in this work: A two-step approach is utilized to get approximately the upper and lower bounds of the elastic properties. Furthermore, an approximation for the elastic properties is calculated by the self-consistence method. Both methods use discretely microstructural information including the length, the diameter and the orientation of each single fiber.

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Abstract: For the description of the phase transformation effect TRIP (transformation induced plasticity), a thermo-micro-mechanical model is introduced. The scale transition is realized by a Hashin-Shtrikman type homogenization scheme. The macroscopical behavior is purely based on micro-mechanical, thermo-mechanical constitutive models. The model takes into account the thermal strains arising during cooling, and the Greenwood-Johnson effect. The Greenwood-Johnson effect describes the TRIP strain developing during the growth of a product phase in the parent phase. The TRIP strain is determined by a micromechanical model gained by an extended Leblond ansatz. For the diffusion driven phase transformation, the JMAK model is incorporated into the homogenization scheme. The diffusionless transformation is described by a rate law of Koistinen-Marburger type.

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Abstract: The deformation in polycrystals is often heterogenous, e.g. due to grain size dependent hardening. In a semi-analytical representative volume element (RVE), a log-normal distributed grain size is assumed together with a grain size dependent local plastic behavior. The numerical results are well approximated by a simple analytical expression. The effect of the homogenization comparison stiffness on the transient behaviour is explained using a simplified localization equation.

Keywords: -

Abstract: The mechanical behaviour of αFe-Cu composites is numerically investigated for large plastic deformations under simple tension and compression by three-dimensional (3D) finite element (FE) simulations, where an elasto-viscoplastic material model is applied. Seven types of the aforementioned polycrystals are systematically studied in order to reveal the effects of local events on the global behaviour, in particular, the role of the bcc-fcc-grain interaction. Compared to the axisymmetric 3D model taking the real microstructure as the cross-section in Schneider et al. (2010), the current work uses periodic boundary conditions (PBCs) and a Poisson-Voronoi microstructure to simulate the flow behaviour, the stress in each phase, the crystallographic texture, and the local strain distribution of the Fe-Cu polycrystals. In particular, the crystallographic texture evolution and its dependence upon the phase distribution have been investigated. A quantitative study is performed for the mean value of the local strain in both phases, where a good agreement with the experimental result is shown for the Fe17-Cu83 composite under tension. Furthermore, a comparison is performed between the numerical results presented here and those in Schneider et al. (2010) which uses the same material model for two types of the above mentioned seven polycrystals.

Keywords: Crystallographic texture; Deep drawing; Earing; Finite element simulation; Generic process modeling; Statistical learning

Abstract: Advanced process controls require information about the system state, the effect of control quantities on state transitions, and the related effort to optimally execute a process step. This information has to be acquired and processed in real time based on models for state monitoring, state transition, and cost. A method is presented on how to derive such real-time state monitoring (observer) models from sophisticated material behavior models. By means of micromechanical modeling, the material behavior is captured, which is verified by experimental findings. Numerical simulations based on the material behavior model deliver the data to extract a specific state monitoring model, which relates online measured data to the process state via statistical learning. For this purpose, the parameters of a generic model are fitted to the simulation data. The resulting process observers have compressed the comprehensive data to essential characteristics. This is realized by using regression in combination with dimension reduction. The proposed approach is applied to the deep drawing of DC04 steel for a proof of concept.

Keywords: Fracture toughness; High-strength ceramics; Homogenization; Micromechanical modeling; R-curve

Abstract: Silicon nitride is often used, when high fracture toughness and strength is needed. For a safe and economic structural design with this material, a prediction of its resistance against thermal and mechanical loads is important. The finite element method together with a continuum damage mechanics model allows for such calculations. The parameters of the suggested model have been adjusted to three-dimensional micromechanical finite element simulations, which include models for the microstructure, the thermoelasticity and the fracture. The material model is used for four-point bend test simulations. The results are compared to recent experiments.

Keywords: -

Abstract: A viscoplastic strain gradient crystal plasticity theory based on the gradient of the equivalent plastic strain ∇γeq is proposed. A grain boundary yield condition is introduced. The microstructural explanation of the Hall-Petch effect, accounting for notch-like stress concentrations at the grain boundary as a result of discrete slip bands, is reviewed. Periodic tensile test FEM simulation results illustrate the prediction of the numerical model.

Keywords: Crystal plasticity; Elastic visco-plastic material; Grain boundary; Size effect; Yield condition

Abstract: A strain gradient crystal plasticity theory is presented that accounts for the resistance of grain boundaries against plastic flow based on an interface yield condition. This theory incorporates the previously presented numerically efficient visco-plastic treatment by the gradient of an equivalent plastic strain γeq in Wulfinghoff and Böhlke (2012). The finite element implementation is discussed and the three-dimensional numerical model is fitted to experimental data of polycrystalline copper micro-tensile tests. The size dependent yield strength is reproduced notably well.

Keywords: Equivalent plastic strain; Gradient plasticity; Power law

Abstract: A gradient crystal plasticity theory is presented including a defect energy based on the gradient of an equivalent plastic strain measure. Preserving the single crystal slip kinematics, the gradient hardening contribution models dislocation long range interactions adding a back-stress term to the flow rule, similar to other gradient crystal plasticity theories. Owing to a reduced number of four nodal degrees of freedom the finite element implementation can handle systems consisting of an increased number of grains (compared to other theories) without elaborate or costly computer systems. Emphasis is put on the enhancement of the power law material subroutine. The associated implicit Euler scheme is optimized based on an improved starting value for the Newton scheme. Three-dimensional simulations illustrate that the proposed algorithm facilitates significantly larger time steps compared to the standard Newton scheme.

Keywords: homogenization; inelastic properties; polycrystalline materials; mean-field approach

Abstract: This work is dedicated to the numerically efficient simulation of the material response of polycrystalline aggregates. Therefore, crystal plasticity is combined with a new non-linear homogenization scheme, which is based on piecewise constant stress polarizations with respect to a homogeneous reference medium and corresponds to a generalization of the Hashin-Shtrikman scheme. This mean field approach accounts for the one- and two-point statistics of the microstructure.

Keywords: Computational homogenization; Ductile porous metals; Finite element method; Plasticity; Representativeness

Abstract: The effective material response of ductile metals containing spherical pores at volume fractions between 0.1% and 30% is investigated on a computational basis. Periodic hard core models of spherical voids are used in a Monte Carlo type finite element study. The objective of the study is the investigation of the pressure dependency of the deviatoric limit stress of the three-dimensional microstructures. In order to characterize the underlying morphology, the statistical properties of the unit cells are evaluated. The representativeness of the computational results is investigated. With respect to the local material response, the inelastic deformations within the unit cell are analyzed and compared for different types of boundary conditions. The computational results are related to existing analytical models and an extension of the Gurson-Tvergaard-Needleman model is proposed to overcome the observed discrepancies. The new model has only one additional parameter, and is found to efficiently predict the pressure dependency of the limit stress for all examined porosities.

Keywords: Anisotropic viscoplasticity; Earing behavior; Finite element method; Texture components; Texture induced plastic anisotropy

Abstract: Aluminum sheets used for beverage cans show a significant anisotropic plastic material behavior in sheet metal forming operations. In a deep drawing process of cups this anisotropy leads to a non-uniform height, i.e., an earing profile. The prediction of this earing profiles is important for the optimization of the forming process. In most cases the earing behavior cannot be predicted precisely based on phenomenological material models. In the presented work a micromechanical, texture-based model is used to simulate the first two steps (cupping and redrawing) of a can forming process. The predictions of the earing profile after each step are compared to experimental data. The mechanical modeling is done with a large strain elastic visco-plastic crystal plasticity material model with Norton type flow rule for each crystal. The response of the polycrystal is approximated by a Taylor type homogenization scheme. The simulations are carried out in the framework of the finite element method. The shape of the earing profile from the finite element simulation is compared to experimental profiles.

Keywords: Homogenization; Rolling simulation; Texture prediction

Abstract: This work deals with comparing the prediction of the development of rolling textures by using a homogenization method that is based on a homogeneous reference material. The proposed homogenization scheme, assuming constant stress polarisations in each phase, has in a natural way the potential to model the transition between Taylor- and Sachs-type textures. Therefore, the stiffness of the homogeneous reference material has to be varied between infinitely stiff and infinitely compliant. In the present study, texture evolution during rolling is simulated, showing that the application of different comparison materials in the homogenization scheme leads to the development of different main texture characteristics (Cube, Cu, Bs, Goss) in the orientation distribution function. For efficiently carrying out the rolling simulations using the proposed method, the measured texture information of the bulk aluminum sample is representatively reduced by using a partitioning technique of the orientation space.

Keywords: Locally compact groups; Maximum entropy; Moment problem; Pseudo-Haar property; Texture analysis

Abstract: The estimation of the crystallite orientation distribution function based on the leading texture coefficients can be rephrased as a maximum entropy moment problem. In this paper, we prove the solvability of these moment problems under quite general assumptions on the moment functions which carries over to general locally compact and σ-compact Hausdorff topological groups.

Keywords: DC04; Deep drawing; EBSD; FEM; Micro-mechanical model

Abstract: In this work, a ferritic stainless steel (DC04) is investigated numerically on the micro and the macro scale. Single crystal orientations extracted from 2D electron backscatter diffraction (EBSD) data are reduced by using two methodically different methods. Based on these reduced data sets, a two-scale Taylor type model is applied at the integration points of the finite elements to simulate a deep drawing process. The earing profiles of the simulated cups are compared to the experimental data. In addition, the influence of the variation of the sheet holder force on the earing profile of the drawn cup for different friction coefficients is discussed. It is found that incorporating experimental microstructural information in the two-scale simulations, realistic earing profiles are obtained. With regard to efficient computation, a coarse FE model is used, resulting in a larger average cup height than the experimental slope. Nevertheless, reducing the friction coefficient in the model, the slope of the cup height agrees very well with the experiments which is expected to be obtainable for a finer FE mesh and the experimental friction coefficient as well.

Keywords: C. Si3N4; C. Toughness and toughening; E. Cutting tools; E. Structural applications; Grain size and shape distribution

Abstract: Silicon nitride is used for demanding tasks due to its high stiffness, strength and, especially, its high fracture toughness. Examples include cutting tools, forming rolls and ball bearings. The microstructure is characterized by elongated β-Si3N4 grains of different size and shape, which lead to the increased fracture toughness. Consequently, the paper will present an algorithm for the generation of three-dimensional and periodic silicon nitride-like microstructures, which will be used for micromechanical finite element simulations. The structure generation algorithm enhances the sequential adsorption technique with growth of particles and steric hindrance, which are motivated by experimental results. Results of the structure generator, such as the pseudo-time evolution and its statistical geometric distributions are presented and compared to literature data. With the finite element simulations, using a periodic unit cell, a validation of the model with literature values for Young's modulus and Poisson's ratio was possible.

Keywords: Finite element simulation; Fracture; Fracture toughness; Homogenization; R-curve; Silicon nitride

Abstract: The fracture toughness of silicon nitride ceramics is closely related to its microstructure, which is characterized by a bimodal distribution of partially large and elongated grains. In the first stage of fracture, these grains act as elastic bridges, which avoid a quick crack propagation. In order to improve the understanding of this R-curve effect, a unit cell approach for multiscale finite element simulation has been chosen. For the examination of the morphology influence on fracture stress and toughness unit cells with different mean aspect ratios have been utilized for the fracture simulations. As results, on the one hand effective stress-strain curves have been applied for the computation of the effective R-curves. On the other hand, these results have been compared with observations of the stress and strain fields on the local level. Findings on the local level support the concept of elastic bridging grains.

Keywords: Crystal plasticity; Finite elements; Strain gradient plasticity

Abstract: We propose a visco-plastic strain gradient plasticity theory for single crystals. The gradient enhancement is based on an equivalent plastic strain measure. Two physically equivalent variational settings for the problem are discussed: a direct formulation and an alternative version with an additional micromorphic-like field variable, which is coupled to the equivalent plastic strain by a Lagrange multiplier. The alternative formulation implies a significant reduction of nodal degrees of freedom. The local algorithm and element stiffness matrices of the finite-element discretization are discussed. Numerical examples illustrate the advantages of the alternative formulation in three-dimensional simulations of oligo-crystals. By means of the suggested formulation, complex boundary value problems of the proposed plastic strain gradient theory can be solved numerically very efficiently.

Keywords: Continuum dislocation dynamics; Dislocation based plasticity; Dislocation continuum theory; Dislocation transport; Gradient plasticity

Abstract: We present three-dimensional Finite Element simulation results of a visco-plastic continuum single crystal theory including a continuum dislocation dynamics model which accounts for the dislocation transport and line length production. One particular feature of the theory is that it accounts for all dislocations and does not a priori differentiate between geometrically necessary and statistically stored part. The approach comprises standard short and long range dislocation interaction models. The predicted dislocation density evolution is quantitatively consistent with the computed geometrically necessary dislocation density.

Keywords: finite element simulation; fracture; toughness; structure

Abstract: Silicon nitride is used for challening applications like cutting inserts or forming rolls. The extreme strength and toughness of the material is achieved by an interaction between the microstructure and fracture behaviour on the microlevel. In order to understand these mechanisms, detailed unit cells have been defined and used for the determination of the effective fracture properties. The results have been used for the implementation of an effective continuum damage mechanics model.

Keywords: Kristallographische Textur; Tensorielle Texturkoeffizienten; Plastische Anisotropie; Nichtlineare Schranken erster Ordnung

Abstract: In dieser Arbeit wird eine Formulierung der starr-viskoplastischen Fließpotentiale kubisch-flächenzentrierter Kristalle durch eine tensorielle Fourier-Reihenentwicklung mit Texturkoeffizienten entwickelt. Aus der Einkristallformulierung werden die makroskopischen Fließpotentiale durch die Berechnung der elementaren Schranken, sowohl im Spannungs- als auch im Dehnratenraum näherungsweise bestimmt. Numerische Beispiele demonstrieren die Güte der texturbasierten Approximation.

Keywords: -

Abstract: The microstructure of Pyrolytic Carbon (PyC) can be described as an aggregate of coherent domains with different orientations. The domains exhibit an approximately a hexagonal material symmetry. In order to apply statistical homogenization schemes which predict the thermomechanical properties in terms of the microstructural characteristics, a quantitative description of PyC microstructures based on the domain orientation distribution function (DODF) is introduced in this work. The von-Mises Fisher distribution is applied for modeling the orientation of the unit normal vectors of the graphene planes. The respective probabilty density function depends on the so-called concentration parameter κ. The parameter κ is estimated by the set of domain orientations, which can be obtained by applying the image processing technique. This procedure facilitates the estimate of the 2nd order bounds (Hashin-Shtrikman upper and lower bounds) based on the real microstructure.

Keywords: -

Abstract: In fibre reinforced materials, the interface toughness has a significant influence on strength and damage resistance. Especially in discontinuous fibre reinforced composites, where high densities of fibre ends are apparent, it has been shown that additives which improve the interfacial toughness can increase the effective strength of the materials [1]. Due to the fact that interface properties are strongly dependent on the manufacturing process, only experimental techniques providing the possibility to take these influences into account, are promising for an integrated material characterization.

Keywords: 3D pores of irregular shape; Carbon/carbon composites; Nanotextured matrix

Abstract: A multiscale modeling approach is utilized to evaluate the contribution of irregularly shaped three-dimensional pores to the overall elastic properties of carbon/carbon composites. The degree of anisotropy of a carbon matrix depends on nanotexture, which is defined by manufacturing conditions. Elastic properties of the matrix are predicted assuming a Fisher distribution of orientations of graphene planes with respect to the pyrolytic carbon deposition direction. X-ray computed microtomography is employed to identify pores in a sample of carbon/carbon composite. The pores have highly irregular shapes so that micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable. Thus, the cavity compliance tensor of an individual pore is found numerically by finite element method, and then used in a micromechanical modeling procedure. Examples of pores in isotropic and transversely isotropic pyrolytic carbon matrices are considered. The accuracy of pore approximation by ellipsoidal shapes is evaluated.

Keywords: -

Abstract: The homogenization of physically nonlinear composite materials with anisotropic morphology is investigated using the nonuniform transformation field analysis (NTFA) first introduced by [1, 2]. In this contribution a three-dimensional finite element implementation (see [3]) of the NTFA is used for the homogenization of composite with morphological anisotropy (see also [4]). The main focus is on the application to structural problems with spatially varying orientation of near-spherical and needle-shaped particle reinforcements.

Keywords: Metal-matrix composites (MMCs); Non-linear behavior; Anisotropy; Finite element analysis (FEA); Multiscale modeling

Abstract: The effective properties of metal matrix composites with particulate reinforcement are investigated using the nonuniform transformation field analysis (NTFA) developed by Michel and Suquet [30]. In particular the effect of the particle morphology on the effective mechanical response is examined in detail. For that, an existing periodic three-dimensional mesh generation technique for particulate composites is extended to allow for anisotropic morphologies. It is shown that the effects induced by the anisotropic particles can be captured by the NTFA. Additionally, the load partitioning between reinforcement and matrix material is investigated and a good agreement to full-field computations is attained with the NTFA.

Keywords: Finite element method; Metal matrix composites; Particle reinforced composites; Periodic mesh generation; Thermoelastic homogenization; Voronoi tessellation

Abstract: A method for the generation of three-dimensional model microstructures resembling particle reinforced composites is developed based on the periodic Voronoi tessellation. The algorithm allows for the generation of arbitrary particle volume fractions and produces periodic geometries based on the erosion procedure suggested by Christoffersen (1983). A technique for the creation of high quality periodic spatial discretizations of the particle systems for application with the finite element method is described in detail. The developed procedure is extensively applied to metal ceramic composites (Al-SiC p) at volume fractions ranging from 10 to 80%. The elastic and thermo-elastic material properties are investigated and the effect of higher statistical moments (see, e.g., Torquato, 2002), i.e. of the particle shape and relative position, is evaluated in terms of constraint point sets used in the generation of the random microstructures.

Keywords: Fracture; Grain boundaries; Microstructure; R-curve; Toughness

Abstract: Crack-bridging mechanisms can provide substantial increases in toughness coupled with strength in ceramics. Herein, we describe the various bridging mechanisms, their toughening contributions and how they are affected by microstructure in silicon nitride ceramics, which are a classic example where both high strength and toughness are achieved. Crack growth resistance curves (R-curves) for seven different silicon nitrides doped with various metal oxides and with different microstructures were measured, and bridging stress distributions were calculated for each. Based on an analysis of those results combined with results of the relative interfacial toughness of two of the ceramics, a new mechanistic theory for the evolution of the R-curve is proposed. The initial steep rise in toughness is attributed to the formation of elastic bridges that experience no debonding. This mechanism has not previously been recognized in the literature and the high strength of these materials (up to 1 GPa) is here attributed primarily to that mechanism. As those bridges begin to fracture and the mechanism becomes saturated, a change in the R-curve slope is observed and the more traditional mechanisms of partially debonded elastic and fully debonded frictional bridges dominate the continuing rise in toughness. Furthermore, the saturation of each mechanism is associated with a change in slope of the R-curve. The results of this study provide a fundamental insight into how to optimize silicon nitride microstructures for high strength and toughness.

Keywords: Anisotropic elastic constants; Compressive moduli; Graphene sheets; Micro-structural; Out-of-plane anisotropy; Pyrolytic carbon; Standard strain; Tension compressions

Abstract: We used standard strain gage methods to measure the anisotropic elastic constants for a sample of pyrolytic carbon. The in-plane elastic modulus exhibits significant tension-compression anisotropy -30.2 GPa in tension and 18.8 GPa in compression. The out-of-plane compressive modulus is 5.2 GPa and the Poisson's ratios are ν12 = 0.35 (compression), ν23 = 0.16 (tension), 0.22 (compression), and ν21 = 0.97 (tension). The tension-compression anisotropy is attributed to buckling, puckering, or bending of the graphene sheets in compression versus simple stretching in tension. The in-plane to out-of-plane anisotropy is expected from the microstructural anisotropy.

Keywords: deformation texture; homogenization; polycrystalline materials; rolling

Abstract: The estimation of the texture in a sheet metal induced by rolling is an important issue for the accurate description of forming operations as, e.g., deep drawing. This work deals with comparing the prediction of the development of rolling textures in aluminum sheets based on different homogenization schemes. The crystallite orientation distribution function (codf) is evaluated by a class of homogenization methods based on a so-called comparison material and is compared to the widely used Taylor-type prediction. It is demonstrated that using the model based on the comparison material, the particular choice of the latter strongly influences the intensity distribution of the codf and also the location of the obtained β-fiber. The proposed homogenization method gives much better results for the reproduction of the codf than the Taylor-type model. When qualitatively comparing the computational results to experimental data, the location of the maxima in the codf generated by the rolling process are satisfactorily reproduced.

Keywords: Crystal plasticity; Homogenization; Rolling; Texture evolution

Abstract: To efficiently use homogenization methods, it is necessary to reduce the incorporated number of texture components used for the simulations. In this paper, a method to create a low-dimensional description of the initial texture is presented, which applies a partitioning technique of the orientation space into equal-volume boxes. By this scheme, the orientation information of, e.g., EBSD measurements (electron backscatter diffraction) of the initial texture can be reduced to a variable number of significant texture components. Thus, there is no need for any previous processing of raw EBSD data, e.g., removing measurement errors or clustering the pixelwise data to grains. A study is performed to evaluate the quality of the reduced data sets by varying the number of texture components. The low-dimensional texture representation can be used as input for texture-based homogenization schemes, e.g., to simulate sheet metal forming operations. As an example, the reduced data sets are used for simulations of a rolling process of an aluminum sample using the Taylor-type homogenization technique. The influence of the dimensionality reduction of the orientation input data is examined by comparing the predicted rolling textures. Conclusions are derived concerning the convergence of the obtained crystallite orientation distribution function and computational efficiency.

Keywords: -

Abstract: A crystal plasticity model and a homogenization method are used to analyze the local and global mechanical behavior of a ferritic stainless steel. In the first step the material constants are determined based on tensile tests and used to simulate the local deformation behavior on the grain scale in the second step. For that 2D EBSD data are discretized by finite elements. The computed local grain reorientations of three different BCC slip systems are compared to experimental data at the state of 20% elongation.

Keywords: -

Abstract: A crystal plasticity model and a homogenization method are used to analyze the local and global mechanical behavior of a ferritic stainless steel. In the first step the material constants are determined based on tensile tests and used to simulate the local deformation behavior on the grain scale in the second step. For that 2D EBSD data are discretized by finite elements. The computed local grain reorientations of three different BCC slip systems are compared to experimental data at the state of 20% elongation.

Keywords: crystal plasticity; ferritic steel DC04; micro-pillar tests; Taylor type model

Abstract: In this work, a ferritic stainless steel (DC04) is investigated in the following three steps. First, we use micropillar compression test data for the identification of a large strain single crystal plasticity model. In the second step the model is verified based on Electron Backscatter Diffraction (EBSD) measurements in a small specimen subjected to a large strain uniaxial tensile test. The two-dimensional EBSD data have been discretized by finite elements and subjected to homogeneous displacement boundary conditions for the second step. Finally, we apply a two-scale Taylor type model at the integration points of the finite elements to simulate the deep drawing process based on initial texture data. The texture data required for the specification of the two-scale model is determined based on the aforementioned EBSD data and by using a texture component method simultaneously to improve the computation time. The finite element simulations were performed with differently textured sheet metals and compared with experiment.

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Abstract: Intergranular cracking due to delamination of grain interfaces along with the development of bridging grains is the most important mechanism for the high fracture toughness of silicon nitride. In this line, an interface behavior, which is extending the Coulomb friction concept into the tensile domain has been implemented into a thermodynamical consistent frame work of Helmholtz free energy and dissipation. The model is used to describe the fracture process in a simple model geometry with a β-Si3N4 grain embedded into a precracked matrix of oxynitride glass. The material model considers the thermoelastic anisotropy of the grain and the thermal residual stresses, which evolve during the cooling of the model from the glass transition temperature to room temperature.

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Abstract: Intergranular cracking due to delamination of grain interfaces along with the development of bridging grains is the most important mechanism for the high fracture toughness of silicon nitride. In this line, an interface behavior, which is extending the Coulomb friction concept into the tensile domain has been implemented into a thermodynamical consistent frame work of Helmholtz free energy and dissipation. The model is used to describe the fracture process in a simple model geometry with a β-Si3N4 grain embedded into a precracked matrix of oxynitride glass. The material model considers the thermoelastic anisotropy of the grain and the thermal residual stresses, which evolve during the cooling of the model from the glass transition temperature to room temperature.

Keywords: Ceramics; Elastic behavior; Finite element analysis; Micromechanical modeling; Residual stresses

Abstract: The main contribution of this paper is the numerical homogenization of the effective thermoelastic properties of silicon nitride based on the temperature-dependent thermoelastic data of the two phases. Silicon nitride consists of two phases: rod-like β-Si3N4 grains (approximately 90 vol.%) and the glassy phase formed by the sintering additives. Due to its microstructure, silicon nitride has a high crack and thermal shock resistance. The homogenization is performed with the finite element method, where a new type of periodic boundary conditions is used that is applicable to non-conforming finite element meshes. The periodic microstructures considered are generated by means of a statistical microstructure generator. The temperature-dependent material properties on the microscale are experimental results taken from the literature. The effective properties obtained are compared to experimental data from the literature and to new experiments. The numerically determined local fields are discussed and different measures of triaxiality are examined.

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Abstract: The higher-dimensional kinematical dislocation continuum theory of Hochrainer (2006) and a homogenized version thereof by Hochrainer, Zaiser and Gumbsch (2010) are reviewed. A three-dimensional Finite Element solution of the homogenized theory is presented, interpreted physically and compared to the results of the Geometrically Necessary Dislocations theory.

Keywords: computational homogenization; microstructure; composite materials; order reduction method; mesh generation

Abstract: Engineering materials show a pronounced heterogeneity on a smaller scale that influences the macroscopic constitutive behavior. Algorithms for the periodic discretization of microstructures are presented. These are used within the Nonuniform Transformation Field Analysis (NTFA) which is an order reduction based nonlinear homogenization method with micro-mechanical background. Theoretical and numerical aspects of the method are discussed and its computational efficiency is validated.

Keywords: Apparent properties; Crystallographic texture; Homogenization; Microcomponents; Singular approximation

Abstract: The statistics of Young's modulus of microspecimens subjected to a tensile load is examined experimentally, numerically and analytically. The material considered is Stabilor®G, a dental alloy, mainly consisting of gold. Tensile tests with microspecimens produced by vacuum pressure casting have been performed using a self-designed universal micro-testing machine. The numerical approach is based on the finite element method with the microstructure being described by a periodic Voronoi tessellation with randomly oriented grains. The analytical approach uses an explicit formulation of the singular approximation in terms of texture coefficients and is valid for arbitrary sample symmetries. It is found that the finite element approach and the analytical approach reproduce the experimental findings. Furthermore, it is shown that the mean value and the standard deviation of Young's modulus of microspecimens made of cubic crystals subjected to tensile loads can be described by a unified scaling law. The results imply that the statistics of apparent properties of cubic crystal aggregates can be determined using anisotropic effective medium approximations.

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Abstract: The submicrostructure of Pyrolytic Carbon (PyC) can be described as an aggregate of approximately coherent domains with different orientations. The domains themselves exhibit a hexagonal material symmetry. In order to apply statistical homogenization schemes [1] which predict the thermomechanical properties in terms of the microstructural characteristics, a quantitative description of PyC microstructures based on experimental data is a precondition. In the study, a new efficient image segmentation technique is applied which extracts from micrographs the one- and two-point correlation functions of domain orientation which are necessary for computing bounds of the thermo-elastic properties of PyC. The segmentation technique, which segments the micrograph into regions with approximately coherent orientations, uses the distribution of Local Binary Patterns (LBPs) [4] for determining the similarity of adjacent image regions. This region-based algorithm yields a first coarse image segmentation.

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Abstract: In this paper, the first-order bounds, the geometric mean, the singular approximation and the self-consistent estimate of the linear elastic properties of pyrolytic carbon (PyC) are determined numerically. The texture, i.e. the orientation distribution of the normal direction of the graphene planes, is modeled by a Fisher distribution on the unit sphere. Fisher distributions depend only on one scalar concentration parameter. It is shown in detail how the effective elasticities of PyC can be estimated based on the one concentration parameter which describes the scatter width of the orientation distribution. The numerical predictions of the different bounds and estimates are compared.

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Abstract: The effective properties of fiber reinforced materials depend strongly on the properties of the interface between the phases. In the particular case of glass fiber reinforced thermoplastics, this bonding is inherently weak and needs to be improved, e.g., by addition of compatibilyzers. Reliable experimental setups are needed in order to investigate the effects of such additives on the interface properties. In this contribution, the sensitivity of push out experiments with respect to the microstructure and test setup are examined in order to find such an experimental configuration.

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Abstract: In this paper we derive an analytical expression for the inverse of the Jacobian occuring in the implicit time integration procedure for a large class of generalized standard materials. The resulting expression is easy to construct and implement into arbitrary programming languages.

Keywords: Finite element method; Metal matrix composite; Nonuniform transformation field analysis; Numerical multiscale method; Solids

Abstract: In this paper aspects of the nonuniform transformation field analysis (NTFA) introduced by Michel and Suquet (Int. J. Solids Struct. 2003; 40:6937-6955) are investigated for materials with three-dimensional microtopology. A novel implementation of the NTFA into the finite element method (FEM) is described in detail, whereas the NTFA was originally used in combination with the fast Fourier transformation (FFT). In particular, the discrete equivalents of the averaging operators required for the preprocessing steps and an algorithm for the implicit time integration and linearization of the constitutive equations of the homogenized material are provided. To the authors knowledge this is the first implementation of the method for three-dimensional problems. Further, an alternative mode identification strategy is proposed with the aim of small computational cost in combination with good efficiency. The new identification strategy is applied to three-dimensional metal matrix composites in order to investigate its effective non-linear behaviour. The homogenized material model is implemented into ABAQUS/STANDARD. Numerical examples at integration point level and in terms of structural problems highlight the efficiency of the method for three-dimensional problems.

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Abstract: Aspects of the numerical solution of the systems of equations resulting from the computational homogenization of unit cell problems using the finite element method are discussed. Different kinematic boundary conditions and solution techniques are examined and compared, both, theoretically and numerically. It is found that the combination of boundary conditions and solver significantly influences the computational cost in terms of memory and cpu time. Examples for model and real world problems are presented.

Keywords: Crystal plasticity; Magnesium; Nonconvex strain energy; Pseudoelasticity; Twinning; {101̄2}<1̄011>

Abstract: A pseudoelastic model for the simulation of deformation twinning on the microscale is developed and coupled with a crystal plasticity model for crystallographic slip. The material parameters are adopted to {101̄2}<1̄011gt; twinning and basal glide in a magnesium alloy. Special attention is drawn to the energy invariance of conjugate twin systems that emerges when twinning is treated elastically. The model is tested in three characteristic FE simulations, namely a simple shear test parallel and inclined to a twin system and an elongation test of a notched band. The slip-twin interaction is studied, as well as the practical implications of the strain energy invariance. Some characteristics of twinning could be reproduced. The most important observations are that the load drop at the twin nucleation, the cusp shape of the twin tip in the absence of slip and the kink patterns that evolve in slip-twin interaction could be simulated.

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Abstract: The estimation of the texture in a sheet metal induced by rolling is an important issue for the accurate description of forming operations as, e.g., deep drawing. In this paper, we discuss the influence of the number of grains in the polycrystal on the prediction of the pole figures as well as on the α and β fiber in the orientation distribution function. For this study, we use the Taylor model as homogenization method where it is assumed that the macroscopic deformation of the polycrystal is identically transferred to each single crystal.

Keywords: Crystallographic texture; Morphological texture; Self-consistent homogenization

Abstract: In this contribution, the prediction of the self-consistent homogenization method with regard to the effective material response of cubic crystal aggregates is analyzed and compared to results from full field simulations. The influence of the crystalline orientation distribution but also the effect of the grain shape on the macroscopic elastic response of sheet metals is especially emphasized.

Keywords: Cardiovascular blood flow; Partitioned structural extension; Strongly coupled interaction

Abstract: We present a 3D code-coupling approach which has been specialized towards cardiovascular blood flow. For the first time, the prescribed geometry movement of the cardiovascular flow model KaHMo (Karlsruhe Heart Model) has been replaced by a myocardial composite model. Deformation is driven by fluid forces and myocardial response, i.e., both its contractile and constitutive behavior. Whereas the arbitrary Lagrangian-Eulerian formulation (ALE) of the Navier-Stokes equations is discretized by finite volumes (FVM), the solid mechanical finite elasticity equations are discretized by a finite element (FEM) approach. Taking advantage of specialized numerical solution strategies for non-matching fluid and solid domain meshes, an iterative data-exchange guarantees the interface equilibrium of the underlying governing equations. The focus of this work is on left-ventricular fluid-structure interaction based on patient-specific magnetic resonance imaging datasets. Multi-physical phenomena are described by temporal visualization and characteristic FSI numbers. The results gained show flow patterns that are in good agreement with previous observations. A deeper understanding of cavity deformation, blood flow, and their vital interaction can help to improve surgical treatment and clinical therapy planning.

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Abstract: A conserved phase-field model is proposed to investigate the effect of interfacial energy on the morphological evolution of the pyrolytic carbon deposit in chemical vapor deposition. The equilibrium geometry of carbon deposit islands is analytically predicted, of which the contact angle was controlled through the boundary conditions of the phase-field parameter at the substrate surface according to the Young-Laplace equation. Simulations of deposit growth are carried out for single and multi island nucleation. It is clarified that the island morphology depends on the magnitude of the interface energy. It is also observed that high interface energy results in large island size fluctuation.

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Abstract: For the optimization of process chains in sheet metal forming it is required to accurately describe each partial process of the chain, e.g. rolling, press hardening and deep drawing. The prediction of the thickness distribution and the residual stresses in the blank has to be of high reliability, since the subsequent behavior of the semi-finished product in the following subprocesses strongly depends on the process history. Therefore, high-quality simulations have to be carried out which incorporate real microstructural data [1,2,3]. In this contribution, the ferritic steel DC04 is analyzed. A finite strain crystal plasticity model is used, for the application of which micro pillar compression tests were carried out experimentally and numerically to identify the material parameters of DC04. For the validation of the model, a two-dimensional EBSD data set has been discretized by finite elements and subjected to homogeneous displacement boundary conditions describing a large strain uniaxial tensile test. The results have been compared to experimental measurements of the specimen after the tensile test. Furthermore, a deep drawing process is simulated, which is based on a two-scale Taylor-type model at the integration points of the finite elements. At each integration point, the initial texture data given by the aforementioned EBSD measurements is assigned to the model. By applying this method, we predict the earing profiles of differently textured sheet metals.

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Abstract: Material parameter identification of the carbon/carbon composites was provided using numerical implementation of semi-analytical methods and FE-calculations. The distribution of the fibers and pores obtained from microstructural studies is used as input for homogenization schemes for the determination of the effective elastic constants. The predictions are compared to experimental results.

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Keywords: Anisotropy; Crystallographic texture; Elasto-viscoplastic material model; Heterogeneity; Morphology; RVE; Strain field; Two-phase polycrystals

Abstract: Two-phase composites, which consist of polycrystalline α-iron and copper particles, are studied mechanically under large plastic deformation. Due to the significant difference of the yield stress in the iron and the copper phase in which the slip system geometry is also dissimilar, a high heterogeneity and anisotropy characterize the plastic deformation behaviour. In this work, an elasto-viscoplastic material model is applied in finite element simulations, whereas the macroscopic material behaviour is established based on constitutive equations of the single crystal. Due to the natural spatial character of the slip system mechanisms of crystal plasticity, the numerical calculation must be performed fully 3D. However, since it is hardly possible to determine the grain geometry of a real material in 3D without destroying the sample by slicing or the like, real 2D cross-sections have been modelled and extended to the third dimension in an axisymmetric way producing an annular pattern, which comes closer to reality than a 2D structure. Numerical predictions include the grain deformation behaviour, the flow behaviour, the crystallographic texture, and the local strain in Fe-Cu composites. In particular, a quantitative study is performed for the mean value of the local strain in both phases, which shows a good agreement with the experimental result for the Fe17-Cu83 composite under tension.

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Abstract: The dissipation of rigid-viscoplastic material behaviour is related to the stress flow potential and its Legendre-Fenchel conjugated strain-rate flow potential. In this paper micromechanically based approximation of these flow potentials, expressed in terms of the deviatoric stresses and strain-rates, is derived from texture measurements. The dual potentials for single face-centered cubic crystals are approximated in terms of Fourier expansion with tensorial texture coefficients. The macroscopic flow stress and strain-rate potentials for an aggregate of polycrystals can be deduced from those of single crystal under the assumption of either homogeneous strain-rate, or homogeneous stress field from the codf measurements.

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Abstract: High performance ceramics have found their way into many highly challenging engineering tasks. For example silicon nitride is one of the best choices, if a material for demanding applications like metal forming and cutting is required. Due to the brittle nature of these hard and strong materials it is useful to know about thermal residual stresses, which can arise during the sintering process. In order to gain insight into the material behaviour, a single grain inclusion is exposed to thermal loads. Due to thermal mismatch, it undergoes a residual stress and strain field. The geometry of the model and the material data are motivated by the properties of silicon nitride. The stress fields are analyzed by three different measures for stress triaxiality.

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Abstract: The kinematic relationship between classical single crystal kinematics and geometrically necessary dislocations will be clarified by a demonstrative example. The starting point for the dynamics is the contribution of geometrically necessary dislocations to the free energy, which leads to a kinematic hardening law. A simple two-dimensional shear example will be discussed.

Keywords: Dynamic strain aging; Jerky flow; Non-linear finite element method; Normal and inverse behavior; Plastic instability

Abstract: In this work we investigate the plastic instabilities associated with the Portevin-Le Chatelier (PLC) effect in Al alloy 2024. A semi-phenomenological approach is taken. A simple geometrically non-linear elastic-viscoplastic constitutive model is proposed for simulation of material response under various applied strain rates. Using the model we determine numerically the relation between the critical strain for the onset of discontinuous yielding and the applied strain rate. The results obtained are in very good quantitative agreement with the available experimental data (collected from tests at room temperature) and cover both the normal and the inverse behavior of the critical strain. The simulations are performed using non-linear finite element method. Additional verification of the proposed constitutive framework was carried out using statistical analysis of the simulated stress-time series. A transition from a non-linear chaotic regime to self-organized critical behavior of the localized strain bands were predicted in terms of the temporal two-point correlation function of the stress-time series. Finally we investigated the influence of different factors, such as the geometry of the specimen, its orientation with respect to the rolling direction and loading conditions (strain rate), on the type of PLC instabilities and the critical conditions for their onset.

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Abstract: The elastic properties of heterogeneous materials without defects depend on both the constitutive properties of the constituents and the microstructural characteristics. In this paper the first-order bounds of aggregates of domains with hexagonal material symmetry are determined in terms of tensorial texture coefficients. The predictive capability of these bounds is compared to higher-order bounds for pyrolytic carbon.

Keywords: Crystallographic texture; Homogenization; Morphologic texture; Voronoi tesselation

Abstract: The influence of an anisotropic distribution of crystal orientations and an anisotropic average grain shape is analysed using finite element simulations. By the numerical approach, which is based on a statistical volume element with periodic microstructure and periodic boundary conditions, the influence of the crystallographic and the morphologic texture can be separated by combining (an)isotropic orientation distributions with (an)isotropic grain morphologies.

Keywords: C/C composites, Threshold parameters, Damage accumulation, Structural integrity, FFT

Abstract: Carbon-carbon (C/C) composites with different densities (1.8, 1.35, 0.8 g/cm 3), produced by chemical vapor infiltration (CVI) were tested mechanically under quasi-static loading in bending mode of uniform and notched beams. Acoustic emission (AE) technique was used to track the mechanical threshold parameters as well as to characterize the damage build-up profile to frac-ture. In both states of stress (uniform and tri-axial), threshold values detected by AE activity in-dicated the damage onset. The sensitivity of the AE method to the density changes was apparent by variations of the threshold values. Decreasing the density from 1.8 to 0.8 g/cm 3 decreases the thresholds values (σ th , K Ith) from 25 to 2 MPa and from 0.8 to 0.1 MPa·m 1/2 , respectively. Three stages in damage evolution to fracture were observed: Stage I, with no AE activity, Stage II, gradual/linear growth in AE counts up to an abrupt jump and Stage III with exponential increase in AE counts. Similarity in profile and threshold value were found between the cumulative AE counts vs. strain data and the crack density vs. strain predicted by micro mechanical model, indi-cating the importance of using AE in monitoring the damage evolution in composites with regard to structural integrity aspect. Wave analysis using fast Fourier transform (FFT) and short-time fast Fourier transform (ST-FFT) points out four possible failure micro-mechanisms: multilayer cracking, breaking of fiber bundles, interfacial matrix de-bonding and micro-crack growth. Breaking of fiber bundles was found to be the major damage mechanism for the low density C/C composite.

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Abstract: C/C composites with different densities, produced by chemical vapor infiltration (CVI) are tested mechanically under quasi-static loading and tracked simultaneously with Acoustic Emission (AE) technique. Threshold mechanical values varying with composite density are detected by AE activity which indicates the damage onset. Three stages in damage accumulation profile up to fracture are found in terms of AE response. Similarity in profile and threshold value are found between the cumulative AE counts vs. strain and crack density vs. strain, predicted by the micro mechanical model, indicating the importance of monitoring structural integrity by the AE method. Wave's analysis points out on four possible failure mechanisms: multilayer cracking, breaking of fiber bundles, interfacial matrix de-bonding and micro-crack growth.

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Abstract: In this paper we derive an analytical expression for the inverse of the Jacobian occuring in the implicit time integration procedure for a large class of generalized standard materials. The resulting expression is easy to construct and implement into arbitrary programming languages.

Keywords: Homogenization, Mesh generation, Metal Matrix Composites (MMC), Nonuniform Transformation Field Analysis (NTFA)

Abstract: The inelastic material properties of Metal Matrix Composites (MMC) with particulate reinforcement are investigated. In order to be able to investigate a variety of unit cells a method for the generation and spatial discretization of random model mi-crostructures is presented. The Nonuniform Transformation Field Analysis (NTFA) is employed to investigate the properties of the microheterogeneous material with physical non-linearity. The coefficients of the NTFA are determined from full-field simulations us-ing the Finite Element Method (FEM) on the microscopic scale with consideration of the exact geometry. The homogenized material model is implemented into ABAQUS STAN-DARD. Numerical examples highlight the efficiency of the method.

Keywords: Cubic crystal symmetry; Finite element method; Periodic mesh generation; Polycrystalline aggregate; Voronoi tessellation

Abstract: In this paper a method for the generation of three-dimensional periodic meshes for the numerical simulation of polycrystalline aggregates is presented. The mesh construction is based on Voronoi and Hardcore Voronoi tessellations of random point seeds. Special emphasis is paid on the periodicity of the mesh topologies which leads to favorable numerical properties for the determination of effective properties using unit cells. The mesh generation algorithm is able to produce high quality meshes at low computational costs. Based on unit cell simulations with different but statistically equivalent microstructures, the effective linear elastic properties of polycrystals consisting of grains with a cubic symmetry are determined. The numerical results are compared with first-, third- and fifth-order bounds and experimental data. Numerical simulations show the efficiency of the proposed homogenization technique.

Keywords: homogenization, mesh generation, metal matrix composites (MMC), nonuniform transformation field analysis (NTFA)

Abstract: The inelastic material properties of Metal Matrix Composites with particulate reinforcement are investigated. A method for the generation and spatial discretization of a class of model microstructures is presented. The Nonuniform Transformation Field Analysis is employed to examine the microheterogeneous material. Numerical examples are presented.

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Abstract: The application of homogenization methods to compute the macroscopic material response of metal matrix composites is a possibility to save memory and computation time in comparison to full field simulations. This paper deals with a method to extend the self-consistent scheme from linear elasticity theory to nonlinear problems. The idea is to approximate the nonlinear problem by an incrementally linear one. Since time discretization of the deformation process implies a certain linearization, we use the algorithmic consistent tangent operator of the composite for defining the linear comparison material in each time step. This is in contrast to classical incremental self-consistent approaches which use continuum tangent or secant operators.

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Abstract: This paper describes the combination of the finite element method and the incremental self­consistent scheme, which is an extension of the self­consistent method from linear elasticity theory to nonlinear problems. The incremental self­consistent method is implemented by a user subroutine UMAT into the commercial finite element software ABAQUS, so as to apply a homogenization procedure at each integration point of a macroscopic finite element model for solving a given boundary value problem. As a result, the microstructure can be taken into account in macroscopic simulations. Compared to the FE²­approach, where a separate full field simulation is carried out on the microscale, the large memory requirements and the computational effort can be avoided by using the presented procedure.

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Abstract: A study was conducted to predict the elastic properties of polycrystalline microcomponents by numerical homogenization. The microcomponents considered in the study were made of Stabilor®G, an alloy consisting of gold with a cubic crystal symmetry. The investigated microtensile specimens were produced by vacuum pressure diecasting with a muffle temperature of Tm=700°C. The finite element program ABAQUS was used to carry out simulations of uniaxial tensile tests on polycrystalline microspecimens made up of cubic single crystals. The model of the specimens were discretized by a regular mesh with 39000 linear hexahedral elements. The simulations showed a decreasing scatter band of Young's modulus with an increasing number of grains. The mean value and standard deviation of Young's modulus were investigated to understand its behavior in microspecimens.

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Abstract: Carbon/carbon composites produced by chemical vapour infiltration consist of carbon fibers embedded in a matrix of pyrolytic carbon with anisotropic mechanical properties. Microscopic studies show that the production process facilitates formation of a matrix consisting of cylindrically shaped pyrolytic carbon layers. The matrix layers may have different textures, which induce different mechanical properties in the axial, radial and circumferential directions. By modifying the production process parameters, it is possible to control the order, approximate width and degree of texture of the layers. Depending on the infiltration conditions pores with different geometry, size and orientations are formed between fibers with pyrolytic carbon coating. One of the goals of the present study is the microstructure characterization and the statistical description of the matrix texture, the fibers orientation distribution and the porosity. Furthermore, a micromechanical modeling using homogenization methods of the material on different length scales is performed. Correlation between calculated and experimentally obtained material properties is also discussed.

Keywords: carbon/carbon composites, effective material properties, homogenization, micromechanics, microstructure modeling

Abstract: Three-dimensional structural information obtained by X-ray computed tomography on carbon/carbon laminates is used as input for a mechanical model describing the elastic behavior of the composite. The model is based on a homogenization procedure consisting of three sequential steps covering the fiber and matrix interaction, the pores, and the laminate structure.

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Abstract: In this paper the application of a crystal plasticity model for body-centered cubic crystals in the simulation of a sheet metal forming process is discussed. The material model parameters are identified by a combination of a texture approximation procedure and a conventional parameter identification scheme. In the application of a cup drawing process the model shows an improvement of the strain and earing prediction as well as the qualitative springback results in comparison with a conventional phenomenological model.

Keywords: Crystal plasticity; Crystallographic texture; Flow potentials

Abstract: In this paper, an effective micromechanically based flow potential for face-centered cubic crystals is proposed. In a first step, the single crystal flow potential is represented in terms of texture coefficients of the crystallite orientation distribution function. An optimization problem on SO(3) is formulated and solved in order to identify the best approximation of the single crystal flow potential. In a second step, assuming homogeneous stress or strain rate fields, the flow potential of the corresponding polycrystal is deduced explicitly from the tensorial representation of the single crystal potential. In contrast to phenomenological yield criteria, the advantage of such an approach lies in the fact that the macroscopic yield function can be estimated directly based on a single texture measurement and the critical resolved shear stress of the single crystal.

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Abstract: In this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd-order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed.

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Abstract: In metal forming processes, the accuracy of their simulations depends on the ability of the constitutive model to describe the relevant features of the material. For the inclusion of the texture-induced anisotropy, micro-macro models are favourable. However, the numerical effort must be drastically reduced for practical applications. A reduction of the number of crystallites on the macroscale will, unfortunately, result in an overestimation of the anisotropy. In this paper, three different methods are suggested which lead to a reduction of the numerical effort, for each of which this overestimation has been avoided by different means.

Keywords: Crystallographic texture; Deformation-induced anisotropy; Macroscopic yield condition; Metal forming; Polycrystalline material

Abstract: In the present paper a two-scale approach for the description of anisotropics in sheet metals is introduced, which combines the advantages of a macroscopic and a microscopic modeling. While the elastic law, the flow rule, and the hardening rule are formulated on the macroscale, the anisotropy is taken into account in terms of a micro-mechanically defined 4th-order texture coefficient. The texture coefficient specifies the anisotropic part of the elasticity tensor and the quadratic yield condition. The evolution of the texture coefficients is described by a rigid-viscoplastic Taylor type model. The advantage of the suggested model compared to the classical v. Mises-Hill model is first that macroscopic anisotropy parameters can be identified based on a texture measurement, and second that the anisotropy of the elastic and the plastic behavior is generally path-dependent and that this path-dependence is related to a micro-mechanical deformation mechanism. An explicit modeling of the plastic spin is circumvented by the aforementioned micro-mechanical approach. The model is implemented into the FE code ABAQUS and applied to the simulation of the deep drawing process of aluminum.

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Abstract: The effective linear elastic properties of silicon nitride (Si3N4) are estimated based on first–, third–, and fifth–order bounds of the strain energy density. This specific type of material is a mixture of two linear elastic materials with different material symmetries. The β-Si3N4 grains have a hexagonal symmetry with significant amount of anisotropy, whereas the glassy phase is approximately isotropic. The results are as follows: i) The fifth–order upper and lower bounds are almost identical. Therefore, these bounds are sufficient for estimating the effective elastic properties. ii) For fixed elastic constants of the hexagonal β-Si3N4 grains, the effective properties of Si3N4 are determined as a function of properties of the glassy phase and its volume fraction. The corresponding diagrams allow for the inverse identification of the elastic properties of the glassy phase.

Keywords: Crystallographic Texture; Elastic Anisotropy; Microcomponents; Polycrystals

Abstract: This work deals with the elastic properties of polycrystalline microcomponents made of Gold. Finite element calculations with ABAQUS are carried out so as to identify the characteristic parameters of the distribution of Young's modulus. In the finite element model, the microstructure of the microspecimens is represented by a periodic Voronoi tessellation and a uniform distribution of single crystal orientations on SO(3). Experimental values of the mean value and the standard deviation of Young's modulus are compared to predictions of finite element simulations and to the Voigt and the Reuss bound as well as the Hashin-Shtrikman bounds.

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Abstract: During deformations of an Al-Mg alloy (AA5754) dynamic strain aging occurs in a certain range of temperatures and strain-rates. A manifestation of this phenomenon, usually referred to as the Portevin-Le Chatelier (PLC) effect, consists of the occurrence of strain localisation bands accompanied by discontinuous yielding. The PLC effect is due to dynamic dislocation-solute interactions and results in negative strain-rate sensitivity of the flow stress. The PLC effect is detrimental to the surface quality of sheet metals and also affects the ductility of the material. Since the appearance of the PLC effect strongly depends on the tri-axiality of the stress state, three-dimensional finite element simulations are necessary in order to optimise metal forming operations. We present a geometrically non-linear material model which reproduces the main features of the PLC effect. The material parameters are identified by experimental data from tensile tests. Special emphasis is put on the prediction of the critical strain for the onset of the PLC effect and the statistical characteristics of the stress drop distribution.

Keywords: Layered structures; Microstructure; Fracture; Damage mechanics; Acoustic emission

Abstract: Three composites materials, Glare 2 fiber metal laminates, graphite/epoxy (Gr/Ep) and carbon/carbon (C/C) have been tested mechanically under quasi-static loading in uniaxial and bending modes using uniform and notched specimens. Acoustic emission (AE) technique was utilized in tracking the damage accumulation profile during loading up to fracture in terms of AE counts rate and cumulative. In addition wavelet transforms was used to process AE signals in order to obtain both frequencies and time information on the main failure mechanism and the sequential events during fracture process. This was supported by light and electron microscopies characterizations. The mechanical and acoustical responses were examined with respect to orientation and temperature effects for the Glare 2, exposure temperature effect for the Gr/Ep and porosity degree effect for the C/C. Manifestly, the AE results demonstrate different damage build-up profiles and point to a transition in failure micro-mechanisms with respect to the influence of each parameter (temperature, orientation, and density) on the specific composite tested. For the Gr/Ep, the wavelet transforms indicate the sequence of events in the fracture process, from fiber breaks followed by debonding and ending with matrix cracking. In some cases, it has been observed that the damage accumulation profile in terms of AE resembled the dependency of the crack density versus the strain predicted by a micro-mechanical damage model.

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Abstract: C/C composites with different porosities, produced by chemical vapor infiltration have been mechanically tested under quasi-static loading in bending modes using uniform and notched specimens. The acoustic emission (AE) method was used to monitor the damage accumulation profile during loading up to fracture, supported by optical and scanning electron microscope characterization. Three stages in the damage buildup up to fracture were observed: Stage I, with no AE activity, Stage II, gradual growth in AE counts up to an abrupt jump and Stage III, sharp increases in AE counts. Moreover, the similarity in the profile between the cumulative AE counts vs. strain data and the predicted crack density vs. strain by the micro mechanical model suggested for interlaminar cracking, indicates the importance of AE in monitoring the damage evolution in composites in terms of AE counts. Fast Fourier transform analysis of the AE waves revealed three characteristic frequencies in Stage III, which is a sign of three main micro-mechanisms of failure which control the failure progress: fiber fracture, debonding and matrix cracking seem to be the active mechanisms.

Keywords: composites, non-destructive evaluation, porosity, fracture, FE-Method

Abstract: Micro Computed Tomography (CT) can be applied for three-dimensional characterization of structural features like pores in a non-destructive way. The resolution of the volumetric data depends upon the size of the specimen, its x-ray absorption coefficient and the tomography system used. With a commercial desktop CT system we achieved a Voxel size of 14.7 m on a carbon/carbon composite specimen, which was further loaded until fracture in a four point cyclic bending test. Based on this investigation we present a methodology for brittle materials to determine the initial pore distribution (pores down to a size of about 50 m), the three dimensional stress-state and the fracture surface corresponding to the non-deformed microstructure.

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Abstract: To avoid time–consuming computations, several analytical approaches can be used to estimate the elastic properties of polycrystalline aggregates. In this paper we compute some well–known bounds and estimates for the effective shear modulus of aggregates of cubic crystals and compare them with results from finite element simulations using a representative volume element (RVE). It is shown that among the evaluated approaches, the singular approximation results in the best agreement with the RVE simulations for the examined materials.

Keywords: Carbon/carbon materials; Failure; Fibrous composites; Microstructure; Numerical modeling; Stress concentration

Abstract: Carbon/carbon composites produced by chemical vapor infiltration consist of carbon fibers embedded in a matrix of pyrolytic carbon with anisotropic mechanical properties. The matrix around fibers consists of cylindrically shaped pyrolytic carbon layers of coating, which may have different textures with different mechanical properties in the axial, radial and circumferential directions. The goal of the present numerical study is to investigate the influence of the coating microstructure on stress concentrations and possible modes of failure in the carbon composite. Numerical modeling was performed on two length scales. First, the material properties of the differently textured pyrolytic carbon layers were determined on the nanometer scale using methodology based on the Eshelby theory for continuously distributed inclusions. Then, the obtained material parameters for each layer were used as input for the finite element models on the micrometer scale. The numerical simulations were conducted for three basic loading scenarios: uniaxial tension, shear loading and hydrostatic compression. The calculated stress distributions show zones of maximum stress concentrations and provide information on the possible failure regions for each material under all considered loading cases. The numerical results demonstrate good correspondence with experimentally identified failure regions.

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Abstract: Two-phase composites, which consist of spherical polycrystalline α-iron and copper particles, are studied mechanically under large plastic deformation. Such polycrystals are produced from mixtures of iron and copper powders by powder metallurgy. Due to the significant difference of the yield stress in the iron and the copper phase in which the slip system geometry is also dissimilar, a high heterogeneity and anisotropy characterize the plastic deformation behaviour. In such bcc-fcc polycrystals, the harder phase shows higher stresses while the softer phase undergoes a larger deformation. To successfully predict the mechanisms of the plastic deformation for a certain grain, effects of the major factors should be taken into account like, e.g., the microscopic interaction, the influence of neighbouring grains, the phase volume fraction, the morphology, and the initial crystallographic texture. In this work, an elasto-viscoplastic material model is applied in axisymmetric finite element simulations, whereas the macroscopic material behaviour is established based on constitutive equations of the single crystal. In the simulations, real two-dimensional microstructures are selected as cross-sections in the axisymmetric model. The material parameters are identified from the experimental data in compression tests. Numerical predictions include the flow behaviour, the crystallographic texture, and the local strain in Fe-Cu composites. In particular, a quantitative study is performed for the mean value of the local strain in both phases, which shows a good agreement with the experimental result for the Fe17-Cu83 composite under tension. Numerical predictions and experimental measurements are compared for the flow behaviour and the texture in both Fe17-Cu83 and Fe50-Cu50 composites.

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Abstract: In this paper the effect of the texture development during the deep drawing of a ferritic steel is studied with a reduced crystal plasticity model. This model has been developed for the application in the forming and springback simulation for industrial applications. Based on a specific optimisation scheme, the number of crystals used at each integration point of the finite elements is reduced to less than 100. Even with such a low number of crystals, it is possible to predict the qualitative development of the texture during the deep drawing process.

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Abstract: This paper proposes a new micromechanically motivated yield criterion expressed in terms of the leading tensorial texture coefficient of the crystallite orientation distribution function. The central finding is that although only a 4th–order anisotropy tensor occurs in the expression for the yield locus, the ansatz provides significantly better results in comparison to the quadratic Hill condition. The improvement is due to cubic and quartic tensorial contributions in terms of the stress direction.

Keywords: Bcc crystals; Crystal plasticity; Deep drawing simulation; Rate-independent model; Springback simulation

Abstract: In this paper the application of a crystal plasticity model for body-centered cubic crystals in the simulation of a sheet metal forming process is discussed. The material model parameters are identified by a combination of a texture approximation procedure and a conventional parameter identification scheme. In the application of a cup drawing process the model shows an improvement of the strain and earing prediction as well as the qualitative springback results in comparison with a conventional phenomenological model.

Keywords: Generalized linear elastic laws; Generalized strain measures; Rank 1 convexity

Abstract: The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229-242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162-172. Pergamon, Oxford, New York (1964)] such as the St.Venant-Kirchhoff law or the Hencky law. The stored energy function of such materials has the form W̃(F) = W(α) := 1/2 σi=1 3 f(alpha;i)2 + β σ 1≤i<j≤3 f(αi) f (αj), where f: (0, ∞) → ℝ is a function satisfying f(1) = 0,f' (1) = 1, β ∈ ℝ, and α 1, α 2, α 3 are the singular values of the deformation gradient F. Two general situations are determined under which W̃ is not rank 1 convex: (a) if (simultaneously) the Hessian of W at α is positive definite, β ≠ 0, and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any m ∈ ℝ. No hypotheses about the range of f are necessary.

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Abstract: The texture evolution and the Swift effect in NiAl under torsion at 727 degrees C are studied by finite element simulations for two different initial textures. The material behaviour is modelled by an elastic-viscoplastic Taylor model. In order to overcome the well-known shortcomings of Taylor's approach, the texture evolution is also investigated by a representative volume element (RVE) with periodic boundary conditions and a compatible microstructure at the opposite faces of the RVE. Such a representative volume element takes into account the grain morphology and the grain interaction. The numerical results are compared with experimental data. It is shown that the modelling of a finite element based RVE leads to a better prediction of the final textures. However, the texture evolution path is not accounted for correctly. The simulated Swift effect depends much more on the initial orientation distribution than observed in experiment. Deviations between simulation and experiment may be due to continuous dynamic recrystallization.

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Abstract: In many Al, Cu, Fe and Ni based alloys jerky flow associated with the Portevin-Le Châtelier (PLC) effect occurs within a specific range of temperatures and strain-rates. This effect which reduces the ductility and the surface quality of sheet materials is caused by dynamic interaction of mobile dislocations with solute atoms [1]. A three-dimensional continuum mechanical model of this effect, also known as dynamic strain ageing, is presented. Furthermore, the predictions of the three-dimensional model are compared with experimental data. Special emphasis is put on the determination of the instability region in the strain and strain-rate space [2].

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Abstract: The modeling of coupled thermo-plasticity includes the consideration of self-heating which produces significant effects in many metal-forming applications. While it has become given practice to use a constant fraction of the plastic power as heat production term in computations, experiments show considerably varying Taylor-Quinney factors. The presented micromechanically motivated model describes non-constant and non-uniform latent energy storage effects. The Taylor-Quinney factor has been computed for different isotropic hardening laws for illustration.

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Abstract: There exist material models that incorporate mechanical twinning in a homogenized sense, or consider specific aspects, like grain refinement or texture evolution. However, since the RVE-technique became a standard method, it is possible to obtain more detailed predictions based on micromechanical models. In this work, an approach based on a nonconvex elastic potential and the corresponding results of FE calculations are presented.

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Abstract: In order to develop highly stressable microcomponents for various applications, the structural behavior of such parts has to be examined with respect to an accurate prediction of the elastic and plastic characteristics. When simulating microparts, the major challenge is to capture the significant material heterogeneities, which are the result of the small geometrical dimensions of the components being of the same order of magnitude as the grain size. For this reason, the concept of effective properties fails and apparent properties have to be considered. In this paper we examine the elastic properties of microspecimens made of gold. Therefore, various finite element simulations have been evaluated statistically in order to identify the characteristic parameters of the distribution. Polycrystals are modeled as a periodic Voronoi tesselation with a uniform distribution of crystal orientations.

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Abstract: For describing the influence of multi-phase materials with microstructure on different length scales as well as the evolution of phase changes under thermomechanical loading, an energetic model is developed. Relying on the incremental formulation of the energetic model, a new solution procedure for the coupled thermomechanical problem is proposed. This model can be applied to describe e.g. the macroscopic response of carbon fibre reinforced carbon.

Keywords: Continuum damage mechanics; Damage accumulation; High-cycle fatigue

Abstract: An overview on modeling of high-cycle fatigue is given and experimental findings of the damage accumulation are discussed. Finally we sketch an isotropic constitutive model for the description of the damage accumulation due to high-cycle fatigue.

Keywords: Carbon; Chemical vapor infiltration; Composites, Failure analysis; Thermal properties

Abstract: Carbon/carbon composites produced by chemical vapor infiltration consist of carbon fibers embedded in a pyrolytic carbon matrix with a cylindrically layered structure at the microscale. Each coating layer has a different texture and different mechanical properties that depend on temperature. Stress distributions in such carbon matrices subjected to thermal loading and their possible failure scenarios are analyzed. A two-scale numerical model is developed. At the nanoscale, material properties of each layer are determined using a methodology based on the Eshelby's theory for continuously distributed inclusions. The resulting material parameters for each layer are then used in the finite element modeling at the microscale. Calculations are conducted for composites with different matrix structures for several cases of thermal loading. Calculated stress distributions show zones of maximal stress concentration and provide information on possible failure regions which correspond well with experimentally identified failure regions.

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Abstract: During deformation of an Al-Mg alloy (AA5754) dynamic strain aging occurs in a certain range of temperatures and strainrates. An extreme manifestation of this phenomenon, usually referred to as the Portevin-Le Chatelier (PLC) effect, consists in the occurrence of strain localisation bands accompanied with discontinuous yielding. The PLC effect stems from dynamic dislocation-solute interactions and results in negative strain-rate sensitivity of the flow stress. The PLC effect is detrimental to the surface quality of sheet metals and also affects the ductility of the material. Since the appearance of the effect strongly depends on the triaxiality of the stress state, three-dimensional finite element simulations are necessary in order to optimize metal forming operations. We present a geometrically nonlinear material model which reproduces the main features of the PLC effect. The material parameters were identified based on experimental data from tensile tests. Special emphasis was put on the critical strain for the onset of PLC effect, εc, and the statistical characteristics of the stress drop distribution.

Keywords: Crystallographic texture; One-point correlation function of crystal orientations; Orientation distribution function; Quadratic programming; Texture components

Abstract: This paper considers the problem of approximating a given crystallite orientation distribution function (codf) by a set of texture components. Problems of this type arise for example if the codf has to be reconstructed from discrete orientations or if one looks for a physical interpretation of the codf. The same problem is encountered if crystallographic texture based constitutive models have to be specified. The equivalence of these tasks to a mixed integer quadratic programming problem (MIQP) - a standard but challenging problem in optimization theory - is shown. Special emphasis is given to the generation of a class of approximations with an increasing number of texture components. Furthermore, the constraints resulting from the non-negativity, the normalization, and the symmetry of the codf are analyzed. Finally, a set of approximations of three different experimental textures determined with this solution scheme is presented and discussed. Based on these hierarchical solutions, the engineer can decide in what detail the microstructure is considered.

Keywords: Crystallographic texture; Elastic-viscoplastic Taylor model; Taylor model

Abstract: In this paper two different texture-dependent material models based on the Taylor assumption are discussed and applied to the simulation of deep drawing operations of aluminium. From the numerical point of view, large-scale FE computations based on the Taylor model are very time-intensive and storage-consuming if the crystallographic texture is approximated by several hundred discrete crystals. Furthermore, the Taylor model in its standard form, which is based on discrete crystal orientations, has the disadvantage that the anisotropy is significantly overestimated if only a small number of crystal orientations are used. We quantitatively analyse this overestimation of anisotropy and suggest two Taylor-type models which allow us to reduce the sharpness of the crystallite orientation distribution function related to single crystals or texture components. One model is an elastic-viscoplastic Taylor model based on discrete orientations. The sharpness is reduced by modelling the isotropic background texture by an isotropic material law. The other model is a rigid-viscoplastic material one, which is based on continuous model functions on the orientation space. This model allows for a direct incorporation of the scattering around an ideal texture component since the model contains the half-width as a microstructural parameter which can be biased. These models are used to compute yield stresses, R values and earing profiles. The predictions are compared with experimental data.

Keywords: Electro Chemical Machining, Gap Pressure, Micro Machining, ECM/EDM with Oscillating Tool Electrode

Abstract: The optimal design of Electro Chemical Machining (ECM) processes is of significant technological importance. In the present work ECM processes with an oscillating tool electrode are considered. It is motivated by the fact that the workpiece electrode may suffer damage or may even fail if the applied load due to hydrodynamic forces is too large. A simple formula is developed for the reaction force acting on the tool electrode. The formula depends on the geometrical, the material and the process parameters. The predictions of the mechanical model are compared with experimental results.

Keywords: Anisotropic viscoplasticity; Crystallite orientation distribution function; Crystallographic texture; Evolution equation of texture coefficients; Maximum entropy method; Tensorial Fourier expansion

Abstract: The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (codf). The evolution equation of each coefficient depends on the complete codf and the lattice spin, which is a constitutive quantity. For the solution of the differential equation based on a finite number of coefficients, the codf has to be estimated. This estimate is obtained here by the maximum entropy method. By this approach the texture evolution can be described by modeling some low-order Fourier coefficients. It will be shown that such a low-dimensional approach yields a reasonable description of the texture evolution and of mechanical properties like the Taylor factor.

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Abstract: Skrotzki et al. [4] performed free end torsion tests on polycrystaline NiAl at 1000 to 1300 °K, where the texture evolution and the Swift effect have been measured. A material model is formulated and implemented to simulate the torsion tests and compared our results to the experimental findings.

Keywords: Crystallite orientation distribution function; Crystallographic texture; Maximum entropy method; Tensorial Fourier expansion

Abstract: The problem of estimating the crystallite orientation distribution function (codf) based on the leading texture coefficients is considered. Problems of such a type are called moment problems, which are well known in statistical mechanics and other areas of science. It is shown how the maximum entropy method can be applied to estimate the codf. Special emphasis is given to a coordinate-free formulation of the problem. The codf is represented by a tensorial Fourier series. The equations, which have to be solved for the estimate of the distribution function, are derived for all tensor ranks of the Fourier coefficients. As a numerical example, a model codf is estimated based on a set of discrete crystal orientations given by a full-constrained Taylor type texture simulation.

Keywords: Anisotropic Plasticity, Crystallographic Texture, Metal Forming

Abstract: -

Keywords: Anisotropic viscoplasticity; Crystallographic orientation distribution function; Crystallographic texture; Texture components; Upper bound model

Abstract: There are many crystallographic textures which can be approximated by a small number of texture components [see, e.g., Int. J. Mech. Sci. 31(7) (1989) 549]. In some cases, such texture components can be described by central distributions. Central distributions are characterized by a mean orientation and a half width. The classical Taylor model for viscoplastic polycrystals assumes that a discrete set of single crystals deforms homogeneously. If the viscoplastic version of the Taylor model is numerically implemented then the crystallite orientation distribution function (codf) is usually discretized by a set of Dirac distributions, where each of the Dirac distributions represents a single crystal. Due to the specific discretization of the codf this approach requires usually a large number of discrete crystal orientations even if the texture can be described by a small number of texture components. In the present work, we consider face-centered cubic (fcc) polycrystals and compare the classical upper bound model with an approach based on texture components. The texture components are modeled by Mises-Fischer distributions, which are central distributions. The stress of the polycrystal is obtained by a numerical integration of the single crystal stress state over the orientation space.

Keywords: anisotropic viscoplasticity, crystallite orientation distribution function, earing behavior, finite element method, texture components, texture induced plastic anisotropy

Abstract: In the present work the texture component model developed in [2] is used to predict the plastic anisotropy of a textured aluminum sheet in terms of the yield stress, the R value, and the earing profile resulting from a cup drawing operation. The model which is formulated in a viscoplastic setting, has been implemented in the commercial finite element code ABAQUS. The stress deviator is computed directly from the crystallite orientation distribution function which is approximated by a set of Gauss type model functions. The model is particularly suitable for the description of crystallographic textures which can be described by a small number of texture components. The numerical results are compared with experimental data and numerical results by Lege et al. [8].

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Abstract: The Voigt bound of the stress potential of face-centered cubic (fcc) polycrystals without texture is numerically determined for all types of strain rate states. The numerical findings reveal the dependence of the stress potential on both the second and the third principal invariant of the strain rate deviator. The dependence on the determinant vanishes only for a linear viscoplastic behavior. Due to the dependence of the stress potential on the third principal invariant, the determinant, the viscoplastic flow is generally nonproportional to the stress deviator. A simple analytical expression is found, which reproduces the numerical findings over the full range of strain rate sensitivities.

Keywords: maximum entropy method, tensorial Fourier expansion

Abstract: The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (CODF). The evolution equation of each coefficient depends on the complete CODF and the lattice spin, which is a constitutive quantity. Hence, for a solution of the differential equation for a finite number of coefficients, the CODF has to be estimated. This estimate is obtained here based on the maximum entropy method. By this approach the texture evolution can be described by modeling some low-order Fourier coefficients. It will be shown that such a low-dimensional approach yields a reasonable description of the texture evolution and of mechanical properties like the Taylor factor.

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Abstract: We present a new approach for modeling the texture induced anisotropy that is based on a tensorial Fourier expansion of the crystallite orientation distribution function (CODF). The general evolution equations for the Fourier coefficients are derived. By this approach the texture evolution can be described by modeling some low order Fourier coefficients and by estimating the higher order coefficients based on the maximum entropy method. It is shown that such a low dimensional approach of the CODF yields a reasonable description of the texture evolution and represents a versatile alternative to classical mixture theories (Taylor, Sachs).

Keywords: anisotropic viscoplasticity, crystallite orientation distribution function, earing behavior, finite element method, texture components, texture induced plastic anisotropy

Abstract: In the present work the texture component model developed in [2] is used to predict the plastic anisotropy of a textured aluminum sheet in terms of the yield stress, the R value, and the earing profile resulting from a cup drawing operation. The model which is formulated in a viscoplastic setting, has been implemented in the commercial finite element code ABAQUS. The stress deviator is computed directly from the crystallite orientation distribution function which is approximated by a set of Gauss type model functions. The model is particularly suitable for the description of crystallographic textures which can be described by a small number of texture components. The numerical results are compared with experimental data and numerical results by Lege et al. [8].

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Abstract: Geometrically nonlinear models have to be employed if large deformation problems are to be computed. The talk addresses the effect of geometric nonlinearities on strain hardening of aluminum under severe plastic deformation. As a ’test object’ for the current case study, the model with two internal variables due to Estrin and Kubin [5] is taken. A three-dimensional, geometrically nonlinear elasto-viscoplastic version of the model is presented. The model describes the rate-dependent viscoplastic flow in metals in terms of two evolving scalar quantities, namely the average densities of mobile and forest dislocations. It accounts for the strain hardening behavior over a wide range of strain rates and contains only such material parameters that can be interpreted microscopically. The consistency of the model with the second law of thermodynamics is demonstrated. Differences between the geometrically linear and nonlinear versions of the model are discussed with regard to large shear deformations of polycrystalline aluminum. The deformation behavior of aluminum during equal channel angular pressing (ECAP) is simulated using the finite element method. It is shown that the numerical simulations reproduce the observed strain pattern adequately. As a ’by-product’, the distribution of dislocation density (and hence the hardness) in the workpiece that underwent ECAP is obtained.

Keywords: Anisotropic material; Constitutive behavior; Crystal plasticity; Elastic–viscoplastic material; Finite strain; Polycrystalline material

Abstract: The asymptotic values of elastic anisotropy, induced in OFHC copper by uniaxial tension and compression, are determined by a phenomenological model and compared with the predictions of Taylor type texture simulations. The evolution equation for the texture-dependent anisotropic part of the effective elasticity tensor consists of two parts. The first term depends only on inelastic rate of deformation and represents the driving term for the evolving anisotropy. The second term governs the saturation of anisotropy and is linear in the anisotropic portion of the effective elasticity tensor. In the present paper it is shown that such an evolution equation implies a reasonable prediction of the asymptotic elastic anisotropy for uniaxial tension and uniaxial compression.

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Abstract: e consider a tensorial Fourier expansion of the crystallite orientation distribution function for aggregates of cubic crystals. The coefficient of rank four is determined explicitly. It is shown that this coefficient governs the anisotropic bounds of the linear elastic behavior of polycrystals. Recently, a phenomenological model has been proposed [4, 6], that describes the evolving elastic anisotropy in copper. The model also reproduces the cyclic Swift effect, which is due to a texture induced plastic anisotropy [7]. In the model the anisotropic part of the effective elastic stiffness tensor is used to define a texture dependent quadratic yield function. The present paper gives a new interpretation of the aforementioned model in terms of an approximation of the crystallographic texture by means of a 4th-order coefficient of the tensorial Fourier expansion of the crystallite orientation distribution function.

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Abstract: The Reuss bound of the strain rate potential of face-centered cubic polycrystals without texture is numerically determined for all types of stress states. The numerical results indicate a strong dependence of the potential on the determinant of the stress deviator. This dependence implies that the viscoplastic flow is not proportional to the stress deviator. A simple analytical expression is found, which reproduces the numerical findings over a wide range of strain rate sensitivities.

Keywords: Anisotropic material; Constitutive behaviour; Finite strain; inhomogeneous material; Polycrystalline material

Abstract: The main purpose of this work is to develop a phenomenological model, which accounts for the evolution of the elastic and plastic properties of fcc polycrystals due to a crystallographic texture development and predicts the axial effects in torsion experiments. The anisotropic portion of the effective elasticity tensor is modeled by a growth law. The flow rule depends on the anisotropic part of the elasticity tensor. The normalized anisotropic part of the effective elasticity tensor is equal to the 4th-order coefficient of a tensorial Fourier expansion of the crystal orientation distribution function. Hence, the evolution of elastic and viscoplastic properties is modeled by an evolution equation for the 4th-order moment tensor of the orientation distribution function of an aggregate of cubic crystals. It is shown that the model is able to predict the plastic anisotropy that leads to the monotonic and cyclic Swift effect. The predictions are compared to those of the Taylor-Lin polycrystal model and to experimental data. In contrast to other phenomenological models proposed in the literature, the present model predicts the axial effects even if the initial state of the material is isotropic.

Keywords: Anisotropic Materials, Constitutive Behavior, Crystallographic Texture Evolution, Finite Strain, lnhomogeneous Material, Polycrystalline Materials

Abstract: A phenomenological model for the texture induced elastic anisotropy was applied to estimate the plastic anisotropy of the polycrystal. The anisotropic portion of the elasticity tensor was used to formulate an anisotropic norm in terms of the stress deviator, which specifies the flow rule. It was shown that the model predicts the monotonic and cyclic Swift effect.

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Abstract: We consider a tensorial Fourier expansion of the crystallite orientation distribution function for aggregates of cubic crystals. The coefficient of rank four is determined explicitly. It is shown that this coefficient governs the anisotropic bounds of the linear elastic behavior of polycrystals. Recently, a phenomenological model has been proposed [4, 6], that describes the evolving elastic anisotropy in copper. The model also reproduces the cyclic Swift effect, which is due to a texture induced plastic anisotropy [7]. In the model the anisotropic part of the effective elastic stiffness tensor is used to define a texture dependent quadratic yield function. The present paper gives a new interpretation of the aforementioned model in terms of an approximation of the crystallographic texture by means of a 4th-order coefficient of the tensorial Fourier expansion of the crystallite orientation distribution function.

Keywords: Crystallographic texture development; Elastic anisotropy; FCC polycrystals; Finite plastic deformations; Hooke's law; Induced anisotropy; Plastic anisotropy

Abstract: Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro-macro simulations based on the Taylor-Lin model and experimental data.

Keywords: Microstructures; Anisotropic material; Elastic material; Inhomogeneous material; Polycrystalline material

Abstract: The determination of the effective elastic properties of aggregates of crystalline grains with or without texture is a long standing problem. Presently, such averages are investigated and their anisotropy is quantified by an anisotropy tensor. The harmonic decomposition of fourth-order tensors is applied to both the elasticity tensors of single crystals with cubic symmetry as well as to the effective elasticity tensors of aggregates of cubic single crystals. It is shown that the anisotropic parts of the different estimates of the effective stiffnesses are irreducible. A set of four discrete crystal orientations is presented, which ensures the isotropy of the effective elastic properties, i.e., a vanishing of the harmonic parts of different averages.

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Abstract: The anisotropic linear elastic behavior of single crystals can be described equivalently by a 4th-order elasticity tensor or two functions E(d) and K(d). These functions represent Young's modulus and a generalized bulk modulus as functions of the tensile direction d in a tension test. In the present paper three-and two-dimensional graphical representations of Young's modulus and the generalized bulk modulus are given for single crystals belonging to one of the following symmetry groups: monoclinic, rhombic, trigonal, tetragonal, hexagonal, and cubic symmetry.

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Abstract: Aggregates of cubic linear elastic crystals with arbitrary crystal orientation distribution are considered with respect to their effective elastic properties. It is shown that the effective elastic strain energy corresponding to a geometric averaging is bounded by the energies according to Voigt and Reuss estimates.

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Abstract: The effective elastic properties of polycrystals can vary significantly with their crystallographic texture [7]. Since a correlation of elastic and plastic properties has been proven (see [8] and references therein), a phenomenological modeling of the crystallographic texture induced elastic anisotropy is of importance in the context of both elasticity and plasticity. In the present paper an evolution equation for the effective elasticity tensors of aggregates of cubic crystals is specified by means of the theory of isotropic tensor functions. It is shown that constraints forced by the elastic symmetry on the micro scale simplify the phenomenological equations significantly.

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Abstract: The deformation induced mechanical anisotropy of polycrystalline metals is caused by crystallographic or morphological textures, i.e., non-uniform orientation distributions of crystals, grain boundaries, or grain edges. In the present work the anisotropy of the elastic behavior is considered, which is assumed to be affected by the crystallographic texture alone. We consider aggregates of crystals with a cubic symmetry which are subjected to large deformations at room temperature. The changes in the elastic anisotropy of textured polycrystalline materials are of interest for rather different applications. Examples are the spring-back analysis for metal forming processes and the investigation of wave propagation phenomena in geological materials. Furthermore, experiments and theoretical considerations show that the anisotropy of the plastic behavior of metals is correlated with the elastic anisotropy. Therefore, the anisotropy of the plastic behavior can be analyzed by means of the elastic anisotropy. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of polycrystals due to an evolving texture. The phenomenological model allows the determination of the elasticity tensor as a functional of the deformation process. A mechanical continuum theory based on internal variables is applied. The model is geometrically non-linear, fulfills the principle of objectivity identically, and can be used for the description of finite inelastic deformations. The transition from an elastically isotropic initial state to a path-dependent final anisotropic state is investigated for polycrystalline copper. The predictions of the model are compared with micro-macro simulations based on a Taylor-type model and experimental data. The anisotropic part of the elasticity tensor is used to formulate an anisotropic flow rule. It is shown that this anisotropic flow rule predicts the plastic anisotropy of rolled sheets and the cyclic Swift effect which can be observed in torsion experiments.

Keywords: anisotropy, cubic symmetry, discrete orientation distribution, invariant subspace, isotropy, linear elasticity, polycrystals, Reuss average, special orthogonal group, Voigt average

Abstract: e consider a model for the elastic behavior of a polyerystalline material based on volume averages. In this case the effective elastic properties depend only on the distribution of the grain orientations. The aggregate is assumed to consist of a finite number of grains each of which behaves elastically like a cubic single crystal. The material parameters are fixed over the grains. An important problem is to find discrete orientation distributions (DODs) which are isotropic, i.e., whose Voigt and Reuss averages of the grain stiffness tensors are isotropic. We succeed in finding isotropic DODs for any even number of grains N≥4 and uniform volume fractions of the grains. Also, N = 4 is shown to be the minimum number of grains for an isotropic DOD.

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Abstract: Initially isotropic aggregates of crystalline grains show an anisotropy of both their inelastic and elastic behavior induced by the crystallographic texture when submitted to large inelastic deformations. The elastic anisotropy, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for many materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of aggregates of cubic crystals. The anisotropy of the flow rule is coupled with the elastic anisotropy which is motivated by experimental findings and theoretical considerations.

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Abstract: The paper discusses a special aspect of averages approximating the elastic properties of polycrystalline aggregates with a uniform orientation distribution. It is shown that the well known isotropic estimations by VOIGT and REUSS and a recently suggested geometric mean of the local elasticity tensors minimize their distances to the single crystal elasticity tensors as well as to the corresponding volume averages. In this analysis, only the influence of the grain orientation distribution on the macroscopic response is considered.

Keywords: Finite inelasticity; Induced anisotropy; Texture development

Abstract: The texture induced elastic anisotropy of poly crystalline copper is studied for different finite deformation paths. As a homogenization technique, the Taylor-Lin model is used, which is a first-order theory and incorporates only a volume-fraction representation of microstructure. In order to identify the anisotropy of the macroscopic elastic behaviour, approximations of the macroscopic strain energy density corresponding to different symmetry groups are determined and compared.

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Abstract: We formulate a tensorial and a scalar quantity describing the deviation from the isotropic elastic state for polycrystals. Furthermore, an isotropy condition for discrete sets of cubic single crystals is expressed in terms of crystal orientations.

Keywords: Finite plasticity; Texture development

Abstract: Aspects of the homogenization problem for polycrystalline materials using finite element based representative volume elements (RVE) are discussed. Special attention is paid to the boundary conditions and the representation of initially isotropic states with limited numbers of grains. The RVE is used for simulations of shear tests with aluminium specimens as reported by Williams [O.W. Williams, Shear textures in copper, brass, aluminium, iron and zirconium, Trans. Met. Soc. AIME 224 (1962) 129–139.]. On the grain level, slip system theory, combining anisotropic elasticity with a viscoplastic flow rule, is used. The resulting texture is represented in the Rodrigues space and by pole figures. Initial and subsequent yield surfaces are investigated using different yield criteria.

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Abstract: In many metal-forming technologies, polycrystalline materials undergo large inelastic deformations which induce an evolution of the elastic and inelastic anisotropy. In the present work, a numerical example concerning deformation induced anisotropy is discussed. Special respect is given to the separation of the influence of texture evolution and accumulated residual stresses.

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Abstract: The development of texture including lattice rotations under large inelastic deformations is determined by the Represenative-Volume-Element-technique (RVE). The numerical results show the deformation induced anisotorpy due to eigenstresses and lattice rotations of the initially isotropic polycrystal.

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Abstract: The dependence of the macro stress on the macro deformation of the polycrystalline aggregate is simulated by using a representative volume element. The texture development and its influence on the elastic response and on yield loci, defined by different criteria, are discussed.

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Abstract: Es wird das mechanische Verhalten einer polykristallinen Aluminiumprobe bei einer einfachenScherung bis zu einer Scherzahl von 2.2 numerisch simuliert. Das Homogenisierungsproblem wird an einem repräsentativen Volumenelement (RVE) mittels der Finite Elemente Methode gelöst. Es werden die makroskopischen elastischen Eigenschaften und das elastisch-inelastische Übergangsverhalten in Form von Fließortkurven bestimmt. Die sich durch die Scherverformung ergebende Textur wird mit einem Experiment verglichen.

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Abstract: Die kinetische Energie eines einzelnen Starrkörperkreisels kann entweder als Funktion der kontravarianten Winkelgeschwindigkeitskomponenten oder als Funktion der kovarianten Winkelgeschwindigkeitsprojektionen ausgedrückt werden. In der vorliegenden Arbeit ist unter Verwendung von Eulerwinkeln als Koordinaten gezeigt, wie die eine Fassung zu den kovarianten Drehmomentprojektionen in Gestalt der bekannten Lagrangeschen Gleichungen führt, während die andere zu kontravarianten Drehmomentkomponenten führt.