Effective Properties Of Heterogeneous Materials Research Articles

Fast construction of cluster interaction matrix for data-driven cluster-based reduced-order model and prediction of elastoplastic stress-strain curves and yield surface

The data-driven cluster-based reduced-order model (CROM) methods[47–50] are very effective for predicting the nonlinear effective properties of heterogeneous materials. The Cluster Interaction Matrix (CIM) in these methods can be regarded as a clustered discrete Green's function and is the critical part of CROM methods, which more and more scholars have realized. However, the explicit expression of Green's function is available only for homogeneous material. The CIM for heterogeneous materials needs to be calculated through numerical approaches, which face the challenge of computational efficiency. Therefore, this paper proposes an efficient method to construct the CIM of heterogeneous materials. By deriving the Green's function in Fourier space under the governing equation of FCA, the CIM of homogeneous materials that can provide the self-equilibrium stress field is calculated efficiently. Then, to avoid the effects of stress averaging on clusters, we augment the proposed cluster-element interaction matrix with the self-equilibrium stress basis vector of heterogeneous material without introducing additional computational cost. Based on the augmented cluster-element interaction matrix, the self-equilibrium stress space is improved, and the CIM of highly heterogeneous materials is calculated using the minimum complementary energy principle. The proposed algorithm significantly improves the offline computational accuracy of FCA, enabling efficient prediction of nonlinear effective properties and yield surface under complex load paths. Numerical examples demonstrate the efficiency and accuracy of the proposed method. In addition, we provide the relationship between the CIMs of a homogeneous material of FCA and SCA. Based on the proposed algorithm in this paper, it can be helpful for other CROM methods.

Effective properties of heterogeneous materials as functions of contrast between properties of constituents

One of the key difficulties in applications of the effective media theories to real materials is the complex and “irregular” character of microstructures. However, in cases of moderate contrast between properties of the constituents, the sensitivity of the effective properties to microgeometries is much lower; it vanishes in the first approximation. The latter actually remains applicable at substantial (up to 70%–100%) contrast in properties. The effective elastic and conductive properties are considered as functions of property contrast between constituents. Whereas in the isotropic cases the Voigt–Reuss bounds provide guidance, in anisotropic cases these issues are less obvious. Two origins of effective anisotropies are discussed: (A) anisotropy of material constants and (B) geometric factors: non-random orientations of non-spherical inclusions. To the first order in contrast, the factor (A) is dominant, and factor (B) can be ignored (effective properties are independent of microgeometries, such as inhomogeneity shapes); the importance of factor (B) increases sharply with increasing contrast. For matrix composites (continuous matrices containing inhomogeneities), an interesting interplay takes place between the moderate contrast assumption and the non-interaction assumption (NIA). It shows that, in cases of moderate contrast, the NIA remains accurate at substantial concentrations. An application to monitoring fatigue of austenitic steel, whereby property contrast between the martensite and austenite phases is moderate, is discussed. Results of the second order in contrast, that are limited to the ellipsoidal inhomogeneities, are also given.

Learning from 2D: Machine learning of 3D effective properties of heterogeneous materials based on 2D microstructure sections

Microstructure—property relationships are key to effective design of structural materials for advanced applications. Advances in computational methods enabled modeling microstructure-sensitive properties using 3D models (e.g., finite elements) based on microstructure representative volumes. 3D microstructure data required as input to these models are typically obtained from either 3D characterization experiments or digital reconstruction based on statistics from 2D microstructure images. In this work, we present machine learning (ML) approaches to modeling effective properties of heterogeneous materials directly from 2D microstructure sections. To this end, we consider statistical learning models based on spatial correlations and convolutional neural networks as two distinct ML strategies. In both strategies, models are trained on a dataset of synthetically generated 3D microstructures and their properties obtained from micromechanical 3D simulations. Upon training, the models predict properties from 2D microstructure sections. The advantage of the presented models is that they only need 2D sections, whose experimental acquisition is more accessible compared to 3D characterization. Furthermore, the present models do not require digital reconstruction of 3D microstructures.

Hybrid approach to predict the effective properties of heterogeneous materials using artificial neural networks and micromechanical models

AbstractIn this article, an investigation was carried out to verify hybrid models capabilities to predict the effective properties of heterogeneous materials. A hybrid model is developed by combining artificial neural networks and micromechanical modeling. The homogenization approach used in this study is mainly based on Eshelby's inclusion problem. The model, once trained on an Eshelby's tensors database, showed an excellent predictive capabilities of the effective mechanical behavior and local stresses in heterogeneous materials. The obtained results with are compared to numerical estimations which are often costly in terms of computational time. The results presented in this work show that the developed hybrid model can provide a significant computational time saving by a factor up to 2000 for phases while maintaining its accuracy and reliability.

Efficient prediction of the effective nonlinear properties of porous material by FEM-Cluster based Analysis (FCA)

The present paper extends the FEM-Cluster based Analysis (FCA) method to the efficient prediction of effective properties of porous materials in the nonlinear range. Different from the traditional methods of homogenization or mean field theory, this new approach does not fill the pore with low modulus materials which causes numerical inaccuracy and convergence difficulties as reported in several literature. With no material filling in pores, the average strain theorem and the Hill–Mandel condition which is the basis of the Representative Volume Element (RVE) method for the prediction of effective properties of heterogeneous materials is almost impossible to implement. Furthermore, because of the lack of pore strain information, the compatibility condition of cluster strains cannot be applied in the incremental analysis of reduced order model in the nonlinear stage. In the present study we have overcome these difficulties by several approaches. First, it is found that the energy equivalence principle provides a more suitable basis for effective properties prediction of porous materials. Second, the self-equilibrium cluster stress field obtained from the interaction matrix and the cluster-based minimum complementary energy in FCA can be successfully applied to the incremental analysis of reduced order model in the nonlinear stage instead of the strain compatibility condition. However, due to the errors introduced by averaging stress and strain over each cluster, the interaction matrix becomes non-singular and the self-equilibrium properties of the cluster stress field generated from the interaction matrix become very poor. Fortunately, the most exciting discovery of the present study is that we have found a way to recover the singularity of the interaction matrix by minimizing its rank on the premise that the elastic energy norm of RVE remains unchanged within the given tolerance. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed approach.

Elliptic functions and lattice sums for effective properties of heterogeneous materials

Effective properties of fiber-reinforced composites can be estimated by applying the asymptotic homogenization method. Analytical solutions are possible for infinite long circular fibers based on the elliptic quasi-periodic Weierstrass Zeta function. This process leads to numerical convergences issues related to lattice sums calculations. The lattice sums original series converge slowly, which make the calculation difficult. This problem needs to be addressed because effective properties are highly sensitive to these values. Therefore, a systematic review and analysis for the lattice sums are a necessity. In the present work, the Eisenstein–Rayleigh lattices sums are reviewed and numerically implemented for fiber-reinforced composites with parallelogram unit periodic cell whose fibers are centered, or not, at the coordinate origin. Numerical values are reported and compared with available data in the literature obtaining good agreements. In this work, new Eisenstein–Rayleigh lattice sums are obtained that are easy to implement and a set of tables with numerical values are given.

Nanoscale Effect Investigation for Effective Bulk Modulus of Particulate Polymer Nanocomposites Using Micromechanical Framework

This paper investigates the nanoscale effect on the effective bulk modulus of nanoparticle‐reinforced polymer. An interface‐based model is introduced in this work to study the nanoscale effects on the effective properties of heterogeneous materials. That interface model is able to capture discontinuity of mechanical fields across the surface between the nanoparticle and matrix. A generalized self‐consistent scheme is then employed to determine the effective bulk modulus. It has been seen from the results that, in a certain range of limits, the influence of nanoscale effects on effective properties of heterogeneous materials is significant and needs to be taken into account. In particular, when the nanoparticle radius is smaller than 10 nm, the value of effective bulk modulus significantly increases when the characteristic size of nanofillers decreases. Besides, it is seen that the harder the inclusion, the smaller the nanoscale influence effects on the overall behaviors of composite materials. Finally, parametric studies in terms of surface strength and filler’s volume fractions are investigated and discussed, together with a comparison between the proposed model and other contributions in the literature.

A unified methodology for calculation of compliance and stiffness contribution tensors of inhomogeneities of arbitrary 2D and 3D shapes embedded in isotropic matrix – open access software.

Property contribution tensors constitute basic building blocks for evaluation of effective properties of heterogeneous materials. Most of the existing results, however, are obtained for inhomogeneities of simple shapes, like ellipsoidal. With this paper we introduce open access software that allows one to calculate the components of compliance and stiffness contribution tensors of inhomogeneities of any shape. The software uses mesh free method based on a class of Gaussian approximating functions. Here we present details of the method and illustrate it by several examples. The software can be downloaded from the website of NMSU Center for Micromechanics https://centerformichromechanics.nmsu.edu.

FEM-Cluster based reduction method for efficient numerical prediction of effective properties of heterogeneous material in nonlinear range

A novel FEM-Cluster based reduced order method or FEM-Cluster based Analysis method (FCA) which enables efficient numerical prediction of effective properties of heterogeneous material in nonlinear range is proposed. The cluster concept initially presented in the work by WK Liu et al. is introduced and extended to derive a full FEM multi-scale formulation of the Representative Unit Cell (RUC) to circumvent the heavy burden due to huge computational efforts required for a direct numerical simulation (DNS) of the high-fidelity RUC. The proposed FEM-Cluster based reduced order method is formulated in a consistent framework of finite element method. The offline clustering process with construction of the cluster-interaction matrix derived under the assumption of the linear elasticity is carried out by the devised FE procedure of RUC. The online elasto-plastic process is performed by the incremental non-linear FE analysis using the constant cluster-interaction matrix, which plays a role in the present work conceptually similar to the initial elastic modular matrix used in the ”initial stiffness method” for the traditional incremental elasto-plastic analysis. Accurate and efficient numerical prediction of effective properties of heterogeneous material in nonlinear range are developed in a consistent way. The performances of the proposed reduced order model and its numerical implementation are studied and demonstrated. Several numerical examples show its efficiency and applicability.

Modelling of thermal expansion of single- and two-phase ceramic polycrystals utilising synthetic 3D microstructures

Determining effective properties of heterogeneous materials is interesting not only in the theoretical aspect but also in the practical one – engineering or technological. Therefore, steps allowing to obtain values of the effective coefficients of thermal expansion of single- and two-phase ceramic polycrystals, were described in the hereby work. At first, based on scanning electron microscope (SEM) images of materials used in the research, the basic microstructure parameters were determined. Then, by means of the Scheil-Schwartz-Saltykov method, the distributions of equivalent grain diameters (3D) were established. The obtained information on grains morphology were utilised in the reconstruction of three-dimensional microstructures of analysed polycrystals. Reconstructions were performed by means of the Laguerre tessellation. In the next step, on the basis of the proposed criterion, numbers of independent realisations – necessary to obtain effective coefficients of thermal expansion – were determined. Finally, the planned simulations (Finite Element Analysis – FEA) were performed and the obtained effective values were compared with the dilatometric measurements results. The path, presented in this work, leading from two-dimensional microstructure images via synthetic three-dimensional models of polycrystals to FEA models, allowed to obtain the effective coefficients of thermal expansion within the measurement error of experimental values.

Determination of the RVE size for polycrystal metals to predict monotonic and cyclic elastoplastic behavior: Statistical and numerical approach with new criteria

Abstract The determination of the size of the Representative Volume Element “RVE” is extremely important to predict mechanical and physical effective properties of heterogeneous materials. In the present work, the optimal size of RVE for polycrystal materials, which is made–up of grains having a polyhedron Voronoi shape, is investigated for the prediction of an elastoplastic behavior using the crystal plasticity model. Ten realizations with a different number of grains ranging from 10 to 250 have been used in the numerical simulation. The homogenization technique combined with the Finite Element Method (FEM) is used to obtain the macroscopic strain and stress. Several new criteria are introduced as indicators for estimating the optimal RVE's size. Their formulation is based on the standard deviation with respect to the average value of different mechanical quantities. These criteria ensure that, for an optimal RVE size, the response of the crystalline aggregate is not sensitive to the orientations evolution and grains' dimensions. It turns out that, for the cyclic behavior, the influence of the loading mode is very important, in particular the asymmetrical stress mode. The influence of the value of strain or stress used to perform different simulation tests on the RVE size has been analyzed. The simulation results are in good agreement with the experimental results from the literature.

Multi-scale genome modeling for predicting fracture strength of silicon carbide ceramics

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general purpose micromechanics approach capable of predicting the effective properties of heterogeneous materials and recovering the local fields. In this study, a novel micromechanics approach has been developed enabling VAMUCH to homogenize heterogeneous microstructure and predict its crack formation through a multi-scale materials genome model. A variational form for homogenization is formulated in combination with a cohesive zone model. The weak form of the problem is derived using an asymptotic method, discretized using finite element formulations, and implemented into VAMUCH. The advantages of the present approach are demonstrated through homogenizing silicon carbide ceramics and predicting its fracture strength. Both the elastic properties and fracture strength can be predicted in a computationally efficient manner using this approach compared with the multi-scale finite element model.

A new multiscale XFEM for the elastic properties evaluation of heterogeneous materials

In the present work, extended finite element method (XFEM) in conjunction with multiscale finite element method (MsFEM) is employed to model heterogeneous materials i.e. matrix having particles and/or voids. This approach, named as multiscale extended finite element method (MsXFEM), is used to determine the effective elastic properties of heterogeneous materials. The elastic properties are calculated by analyzing a representative volume element (RVE) under periodic boundary conditions. Three cases of particles and voids in the matrix i.e. matrix with hard particles, matrix with voids, and matrix with voids and hard particles both, are considered for analysis. The particles and voids are randomly distributed in the matrix. A strain energy based homogenization is carried out numerically for reinforcement volume fractions up to 50%. RVEs containing large number of discontinuities are analyzed by the proposed MsXFEM. The evaluated effective properties of the heterogeneous material are compared with the results available in the literature. These simulations show that the use of MsXFEM leads to a significant reduction in the CPU time as compared with the standard XFEM.

Nonlinear effective properties of heterogeneous materials with ellipsoidal microstructure

The analysis and the synthesis of the nonlinear effective response of particulate composite materials are of great importance for developing new systems such as nonlinear elastic and electromagnetic metamaterials, nonlinear waveguides, nonlinear magnetoelectric devices and photonic or phononic crystals. Typically, classical homogenization schemes take into account the shape of the inhomogenieties but neglect the spatial correlation among them, a crucial feature for the above applications. In this paper we develop a nonlinear homogenization technique for dispersions of nonlinear particles in a linear matrix, which is able to take account of spatial correlation by means of the so-called ellipsoidal microstructure. While the linear result corresponds to the well known Ponte Castañeda–Willis estimate, we propose new formulae for the second and third order nonlinear behavior. We finally show applications to the nonlinear elastic Landau coefficients and to the nonlinear hypersusceptibility of transport processes.

Numerical Version of Self-Consistent Scheme for Heterogeneous Materials

The self-consistent scheme (Budiansky, 1965) is one of the analytical methods to predict the effective properties of heterogeneous material. However, it has been noticed that this scheme cannot take the micro-structural information into the estimation. The full scale numerical simulation considering elaborately the micro-morphological characteristics, under certain conditions, needs too much computational resources (memory and time). A numerical version of self-consistent scheme based on finite element analysis is proposed to overcome the above drawbacks by combining the characteristics of numerical simulation and self-consistent scheme. As a demonstration, such scheme is applied to estimate the effective mechanical properties of bi-continuous composite in which all the constituent phases are on the equal status. The prediction is in good agreement with the full scale numerical simulation while the requirement on computational resource is reduced dramatically.

A statistical descriptor based volume-integral micromechanics model of heterogeneous material with arbitrary inclusion shape

A continuing challenge in computational materials design is developing a model to link the microstructure of a material to its material properties in both an accurate and computationally efficient manner. In this paper, such a model is developed which uses image-based data from characterization studies combined with a newly developed self-consistent volume-integral micromechanics model (SVIM) for linear elastic material. It is observed that SVIM is able to capture the effective stress/strain distribution inside the inclusion, as well as effects of volume fraction and nearest inclusion distance on the effective properties of heterogeneous material. More importantly, SVIM can be applied to inclusions with arbitrary shape through discretizing the inclusion domain. For both 2-dimensional and 3-dimensional problems with circular and spherical inhomogeneities, SVIM's capability of predicting effective elastic properties is validated against experiments and direct numerical simulations using the finite element method. Finally, the effect of inclusion shape is predicted by SVIM.

On the possibility to represent effective properties of a material with inhomogeneities in terms of concentration parameters

It is argued that representation of the effective properties of heterogeneous materials in terms of a single parameter of concentration of inhomogeneities (such as volume fraction or crack density) is generally impossible for mixtures of inhomogeneities of diverse shapes; the exceptions – when this is possible – are identified. Application of such parameters to microstructures of “irregular” microgeometries may lead to various inconsistencies. Instead, the effective constants in such cases can be represented in terms of a “tracking” parameter for a specific process of evolution of a microstructural pattern. These issues are discussed in the context of the elastic and conductive properties.

A micromechanics approach to homogenizing elasto-viscoplastic heterogeneous materials

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it.

Shielding Effectiveness of Composite Materials: Effect of Inclusion Shape

The use of composite materials for electromagnetic shielding applications contributes to the effort of structure lightening in aerospace industry. In these materials the strong interaction between the electromagnetic field and the microstructure makes the standard numerical tools difficult to implement. Indeed these methods would involve an excessive number of degrees of freedom to describe details of the microstructure. An efficient way to overcome this problem is the use of homogenization techniques providing the effective properties of heterogeneous materials. These effective properties can then be introduced in standard numerical tools to estimate the behavior of shielding enclosures. A recent paper proposes an extension to microwave frequencies of quasistatic homogenization methods. It introduces a characteristic length for the microstructure in the case of a square array of circular 2-D conductive phases embedded in a dielectric matrix. In this paper, a method to identify this length parameter is proposed for random microstructures.

Modeling Anisotropic Multiphase Heterogeneous Materials via Directional Correlation Functions: Simulations and Experimental Verification

The effective properties of heterogeneous materials critically depend on their complex microstructure. In this article, by successfully reconstructing the three-dimensional microstructure of a multiphase alloy with orientated anisotropic inclusions from two-dimensional slices, we show that such materials can be modeled by the two-point correlation functions of the inclusion phases along the three orthogonal characteristic directions of the inclusions. The reconstructions are compared to the actual microstructure obtained from high-resolution X-ray tomography experiments by quantifying certain directional cluster statistics associated with the microstructures.