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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 ▪, revealing a driving force for hydride formation at room temperature even under these constraints. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 ▪, 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 ▪ in elastically deforming films, while it is close to that of a stress free system in strongly elasto-plastically deforming films. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 ΔF(D¯,ΔD) due to an increased electric power demand ΔD for the average electricity power demand D¯. Commonly, the formula ΔF(D¯,ΔD)≈M(D¯)ΔD is used (called simplified formula), where M(D¯) 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 ΔF(D¯,ΔD)=D¯ΔM(D¯,ΔD)+ΔDM(D¯+ΔD) 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 ΔM depending on the electricity power demand. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • The evolution equation of orientation tensors of arbitrary even order is derived. • Fiber-induced anisotropic viscosity effects are covered in the orientation evolution. • Anisotropic effects are shown to be a micromechanical convergence criterion. • The chosen mean-field model greatly influences the fiber orientation dynamics. • The spatial fiber distribution strongly affects the periodic fiber reorientation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. Graphical abstract (Left) Set of texture coefficients of second- and fourth-order for polycrystals of hexagonal materials with macroscopic hexagonal symmetry. The blue region depictes the region described by the Frobenius norm of the textures coefficients bounded by unity. The orange region describes the region of all possible texture coefficients of second- and fourth-order. The texture coefficients for the single crystal are marked by the blue point, while the red point marks the texture coefficients for a uniform distribution.(Right) Properties-closure for a linear thermoelastic polycrystal of a hexagonal material with macroscopic hexagonal symmetry. The bounds of Voigt, Reuss and Hashin-Shtrikman have been evaluated with the representations presented in this work using the set of all possible texture coefficients depicted in orange in the left graphic. The bounds for the effective stiffness component C_1111 and for the effective thermal expansion coefficient beta_11 have been investigated. The properties-closure for these properties based on the Voigt and Reuss region (first-order bounds) delivers the green region. The properties-closure based on the Hashin-Shtrikman bounds (second-order bounds) delivers the red region. A propertiesprofile with accepted tolerances is depicted by the black point. The Hashin-Shtrikman bounds help material scientists in order to check if aimed properties-profile are reachable or not by polycrystals of considered materials. Image 1 [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • Fiber orientation distributions of sheet molding compound are approximately planar. • Orientation averaging mean field: Complete orientation dependence for the planar case. • Explicit linear representation of the Advani–Tucker orientation average. • New numeric formulation of the Advani–Tucker orientation average for the planar case. • Orientation averaging in stiffness or compliance domain affects generalized bulk modulus. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

Highlights: [•] Failure prediction in silicon nitride is based on a probabilistic approach. [•] The weakest link model accounts for volume defects and interface damage. [•] A sensitivity analysis is carried out for two loading modes. [Copyright &y& Elsevier]

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. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

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. • A purely mechanical continuum theory , defined by assumptions A1–5, is dissipation-free. • Heat conduction is, thus, not present and, consequently, the Taylor–Quinney effect is not reproducible. • Internal variables evolve without dissipation; a hysteresis-free material behavior is obtained. • A purely mechanical theory provides only a rate-independent crystal plasticity. • Since the dissipation vanishes, the derived crystal plasticity corresponds to a total strain theory. [ABSTRACT FROM AUTHOR]

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 105-109 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. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Obituary, in memoriam of Jürgen Zierep (1929-2021, Founding Editor of ACTA MECHANICA), accompanied by the previously unpublished note: "Mein Weg zu neuen naturwissenschaftlichen Erkenntnissen" by Jürgen Zierep: Professor em. Bei mir löst das durchaus keinen Triumph aus, denn ich weiß, dass der wissenschaftliche Erfolg manchmal mit einer Seelenanspannung verknüpft ist, die oft an die Grenze der Belastbarkeit reicht. Die Gleichungen oder das physikalische Modell werden I so i vereinfacht, dass die physikalisch wesentlichen Effekte noch enthalten sind, dabei aber die Gleichungen lösbar werden. Ich habe als heranwachsender junger Wissenschaftler das große Glück gehabt, das Vertrauen bedeutender akademischer Lehrer, die gleichzeitig hervorragende Wissenschaftler waren, zu genießen. [Extracted from the article]

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]. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]