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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. [ABSTRACT FROM AUTHOR]

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. [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]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [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]

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 (μCT) 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. [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]

Computational models based on the phase-field method typically operate on a mesoscopic length scale and resolve structural changes of the material and furthermore provide valuable information about microstructure and mechanical property relations. An accurate calculation of the stresses and mechanical energy at the transition region is therefore indispensable. We derive a quantitative phase-field elasticity model based on force balance and Hadamard jump conditions at the interface. Comparing the simulated stress profiles calculated with Voigt/Taylor (Annalen der Physik 274(12):573, ), Reuss/Sachs (Z Angew Math Mech 9:49, ) and the proposed model with the theoretically predicted stress fields in a plate with a round inclusion under hydrostatic tension, we show the quantitative characteristics of the model. In order to validate the elastic contribution to the driving force for phase transition, we demonstrate the absence of excess energy, calculated by Durga et al. (Model Simul Mater Sci Eng 21(5):055018, ), in a one-dimensional equilibrium condition of serial and parallel material chains. To validate the driving force for systems with curved transition regions, we relate simulations to the Gibbs-Thompson equilibrium condition (Johnson and Alexander, J Appl Phys 59(8):2735, ). [ABSTRACT FROM AUTHOR]

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]

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. [ABSTRACT FROM AUTHOR]

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

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

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

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. [Copyright &y& Elsevier]

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. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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. • Memory-efficient FFT-based solution methods for implicit phase field damage. • A novel fast gradient method, staggered and memory-efficient solver. • Demonstration of advantages compared to semi-explicit approaches. • Application to fiber reinforced composite microstructures of large scale. [ABSTRACT FROM AUTHOR]

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 E 1 and E 2 compared to experimental data. [ABSTRACT FROM AUTHOR]

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. • Fast method for computing polycrystalline microstructures for given grain-size distribution. • Laguerre-based convex optimization solver based on the Barzilai–Borwein method. • Anderson-accelerated Lloyd algorithm for centering. • Application to polycrystalline materials with prescribed grain size distribution. [ABSTRACT FROM AUTHOR]

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 −30.2GPa in tension and 18.8GPa in compression. The out-of-plane compressive modulus is 5.2GPa and the Poisson’s ratios are ν 12 =0.35 (compression), ν 23 =0.16 (tension), 0.22 (compression), and ν 21 =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. [Copyright &y& Elsevier]