Properties Of Random Materials Research Articles

Imposing different boundary conditions for thermal computational homogenization problems with FFT‐ and tensor‐train‐based Green's operator methods

AbstractTo compute the effective properties of random heterogeneous materials, a number of different boundary conditions are used to define the apparent properties on cells of finite size. Typically, depending on the specific boundary condition, different numerical methods are used. The article at hand provides a unified framework for Lippmann–Schwinger solvers in thermal conductivity and Dirichlet (prescribed temperature), Neumann (prescribed normal heat flux) as well as periodic boundary conditions. We focus on the explicit jump finite‐difference discretization and discuss different techniques for computing the discrete Green's operator. These include Fourier‐type methods, that is, based on discrete sine, cosine and Fourier transformations, as well as low‐rank tensor methods, that is, the tensor‐train approximation, whose computational prowess was demonstrated recently. We use the developed computational technology to investigate a number of interesting random materials and assess different microstructure‐generation techniques in terms of their compatibility with the prescribed boundary conditions.

Study of thermal properties of TiCNO by Debye-Einstein model and Debye-Grüneisen model

The thermal properties of TiCNO are studied from the Debye-Einstein model based on the first-principle in combination with the special quasi-random structural model, meanwhile the derived result is compared with those obtained by the Debye-Grüneisen model. The present results show that Debye-Einstein model is excellent in predicting the thermal properties of multicomponent random structure materials due to consideration of the 3 n − 3 optical frequencies at the Γ point to represent the contributions of optic branches, although the Debye-Grüneisen models is also effective methods for predicting the thermal properties of multicomponent materials. The two methods show that when temperature rises, the thermal expansion coefficients of TiCNO first increase rapidly and then slowly, and Debye-Einstein predicts a stronger volume sensitivity to lower temperature due to the contribution of 3 n − 3 optic frequencies at Γ point. Although both models also predict a comparable temperature dependence of bulk modulus, the high temperature softening effect of TiCNO calculated by Debye-Einstein is gentler than that from Debye-Grüneisen, which demonstrates that the influence of optical branches cannot be negligible. With temperature increasing, the thermodynamic entropy predicted by both models increases due to the greater disorder degree of the system. Moreover, the thermodynamic entropy predicted by Debye-Einstein is larger owing to considering the optical frequencies at the Γ point. Due to the strong coupling between the acoustic and optical phonons lying in low frequency regime, both isobaric and isochoric heat capacities predicted by Debye-Einstein are larger than those predicted by Debye-Grüneisen at low temperatures. This work is a useful reference for predicting the thermal properties of multicomponent materials.

Risk-averse estimates of effective properties in heterogeneous elasticity

In this work, we propose a theoretical framework for computing pessimistic and optimistic estimates of effective properties in the case of heterogeneous elastic materials with uncertain microscopic elastic properties. We rely on a risk-averse measure widely used in finance called the conditional-value at risk (CVaR). The CVaR computes the conditional expectation of events occurring above a given risk level, thereby characterizing the extreme tails of the probability distribution of a random variable. In the context of elastic materials, we propose to use the CVaR on the elastic free energy to compute an optimistic estimate of the global stiffness for some confidence level α. Similarly, we also use the CVaR on the complementary elastic energy to compute a pessimistic estimate of the global stiffness. The obtained CVaR estimates benefit from a convex optimization formulation. The resulting material behavior is still elastic but not necessarily linear anymore. We discuss approximate formulations recovering a linear elastic behavior. We apply the proposed formulations to the micromechanical estimates of effective elastic properties of random heterogeneous materials.

Ultra-efficient reconstruction of 3D microstructure and distribution of properties of random heterogeneous materials containing multiple phases

Ultra-fast 3D material microstructure reconstruction and quantitative structure-property mapping are crucial components of integrated computational material engineering (ICME). It is particularly challenging for modeling random heterogeneous materials such as alloys, composites, polymers, porous media, and granular matters, which exhibit strong randomness and variations of their material properties due to the hierarchical uncertainties associated with their complex microstructure at different length scales. An explicit mixture random field (MRF) model is proposed to characterize and reconstruct multi-phase stochastic material property and microstructure simultaneously. The proposed method is shown to have ultra-high computational efficiency and only requires minimal imaging and property input data. The material property field is modeled by this strongly non-Gaussian random field, which is generated by a nonlinear mapping from the underlying Gaussian random field explicitly. The corresponding microstructure is represented by discrete phase indicators obtained from the material property field. A decomposed Karhunen–Loève (K-L) expansion method is used to dramatically reduce the computational costs and memory requirement for high-dimensional and high-resolution generations. The feasibility and superior efficiency are demonstrated by reconstructing various materials, from 2D to 3D, bi-phase to multi-phase, isotropic to anisotropic materials. The results show that it only takes approximately seconds to reconstruct a 3D material with the resolution of 500×500×500 pixels. The elegant expression for explicit microstructure-property mapping with uncertainty quantification capability can be directly incorporated into the ICME framework for material design and optimization.

A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network

This work addresses the inverse identification of apparent elastic properties of random heterogeneous materials using machine learning based on artificial neural networks. The proposed neural network-based identification method requires the construction of a database from which an artificial neural network can be trained to learn the nonlinear relationship between the hyperparameters of a prior stochastic model of the random compliance field and some relevant quantities of interest of an ad hoc multiscale computational model. An initial database made up with input and target data is first generated from the computational model, from which a processed database is deduced by conditioning the input data with respect to the target data using the nonparametric statistics. Two- and three-layer feedforward artificial neural networks are then trained from each of the initial and processed databases to construct an algebraic representation of the nonlinear mapping between the hyperparameters (network outputs) and the quantities of interest (network inputs). The performances of the trained artificial neural networks are analyzed in terms of mean squared error, linear regression fit and probability distribution between network outputs and targets for both databases. An ad hoc probabilistic model of the input random vector is finally proposed in order to take into account uncertainties on the network input and to perform a robustness analysis of the network output with respect to the input uncertainties level. The capability of the proposed neural network-based identification method to efficiently solve the underlying statistical inverse problem is illustrated through two numerical examples developed within the framework of 2D plane stress linear elasticity, namely a first validation example on synthetic data obtained through computational simulations and a second application example on real experimental data obtained through a physical experiment monitored by digital image correlation on a real heterogeneous biological material (beef cortical bone).

A framework for optimal microstructural design of random heterogeneous materials

An efficient micromechanical methodology is developed for tailoring elastic and thermal properties of random heterogeneous materials. The methodology involves three steps: (i) statistical reconstruction based on the two-point correlation function (TPCF), (ii) thermomechanical homogenization, and (iii) optimization. The method relies on the tailoring the state of anisotropy of the microstructure to achieve desired directional effective properties. This capability is demonstrated by designing heterogeneous microstructures with optimal elastic modulus and thermal conductivity in perpendicular directions. The study aims at presenting optimal computational algorithms for the reconstruction, homogenization and optimization steps. Several studies study are presented to demonstrate the feasibility of obtaining desired thermomechanical properties with a minimal set of design variables. These case studies highlight the advantages and limitations of the method, along with its computational cost.

Effective Medium Theory for the Elastic Properties of Composite Materials with Various Percolation Thresholds.

It is discussed that the classical effective medium theory for the elastic properties of random heterogeneous materials is not congruous with the effective medium theory for the electrical conductivity. In particular, when describing the elastic and electro-conductive properties of a strongly inhomogeneous two-phase composite material, the steep rise of effective parameters occurs at different concentrations. To achieve the logical concordance between the cross-property relations, a modification of the effective medium theory of the elastic properties is introduced. It is shown that the qualitative conclusions of the theory do not change, while a possibility of describing a broader class of composite materials with various percolation thresholds arises. It is determined under what conditions there is an elasticity theory analogue of the Dykhne formula for the effective conductivity. The theoretical results are supported by known experiments and show improvement over the existing approach. The introduction of the theory with the variable percolation threshold paves the way for describing the magnetorheological properties of magnetoactive elastomers. A similar approach has been recently used for the description of magneto-dielectric and magnetic properties.

Porosity Effects on Mechanical Properties of 3D Random Fibrous Materials at Elevated Temperatures

In this study, we prepare the specimens of three-dimensional random fibrous (3D RF) material along its through-the-thickness (TTT) and in-plane (IP) directions. The experimental tests of tensile and compressive properties as well as fracture toughness of 3D RF material are performed at elevated temperatures. Then, the porosity (83%, 87% and 89%) and temperature dependence of the tensile and compressive strength, elastic modulus, fracture toughness and fracture surface energy of the 3D RF materials for both the TTT and IP directions are analyzed. From the results of the tensile strength and elastic modulus versus material porosities at various temperatures, we find that tensile strength and elastic modulus for the TTT direction are more sensitive to the porosity, but not for the IP direction. Fracture toughness increases firstly and then decreases at a certain critical temperature. Such critical temperature is found to be the lowest for the porosity of 83%. On the other hand, at below 1073 K, the temperature-dependent fracture surface energies with three porosities for the TTT direction show similar variation trends.

The Choice of Representative Volumes in the Approximation of Effective Properties of Random Materials

The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: a sample of the random material is chosen—the representative volume—and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well; for example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in materials science, Le Bris, Legoll and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.

On modeling of carbon nanotubes reinforced materials and on influence of carbon nanotubes spatial distribution on mechanical behavior of structural elements

A recently developed approximate approach for modeling of carbon nanotubes and for evaluation of the effective properties of carbon nanotubes reinforced materials is applied to analysis of square plate with hole under uniaxial tension. It is assumed, that the plate is made of polymer with randomly distributed and randomly oriented carbon nanotubes. Distributions of carbon nanotubes volume fraction is varied over the surface of the plate, with the constraint that the total volume of nanotubes in the structure is the same, and the effects of that on stress concentration is evaluated. Such special variation volume fraction makes the carbon nanotubes reinforced polymer a functionally graded material.In evaluation of the local effective properties the Gutrin–Murdoch material surface model is applied to a hollow cylinder of finite thickness and combined with the notion of energy equivalence to replace carbon nanotube with an equivalent solid cylinder. The interphase, if any, can be modeled as a spring layer and subsequently added to obtain another equivalent solid cylinder. The material reinforced with such solid cylinders is then evaluated using the method of conditional moments, a stochastic homogenization approach proposed earlier for determination of the effective properties of random materials.

A Bayesian statistics based investigation of binder hardening’s influence on the effective strength of particulate reinforced metal matrix composites (PRMMC)

In order to understand how hardening of the binder phase in particulate reinforced metal matrix composites (PRMMC) influences the effective strength, we present in this work a numerical framework consisting of the direct method (DM) and statistical models. Using this approach we created a large number of statistically equivalent representative volume element (SERVE) models to represent an exemplary PRMMC material WC-20 Wt.% Co and predicted its effective strengths using DM. After the global strength was calculated from each SERVE sample all derived data are interpreted by Bayesian network and diagnostic testing. By doing so the relationship between material strength and few selected characteristics have been clarified. The study shows the formulated approach as a novel means for investigating how the overall mechanical properties of random heterogeneous materials react to a certain constituent. Meanwhile, the study also demonstrates how statistical models, in particular the Bayesian network, can be used as a powerful supplement to micromechanical models for result analysis and knowledge discovery.

Computational analysis of particle reinforced viscoelastic polymer nanocomposites – statistical study of representative volume element

The concept of representative volume element (RVE) is widely used to determine the effective material properties of random heterogeneous materials. In the present work, the RVE is investigated for the viscoelastic response of particle-reinforced polymer nanocomposites in the frequency domain. The smallest RVE size and the minimum number of realizations at a given volume size for both structural and mechanical properties are determined for a given precision using the concept of margin of error. It is concluded that using the mean of many realizations of a small RVE instead of a single large RVE can retain the desired precision of a result with much lower computational cost (up to three orders of magnitude reduced computation time) for the property of interest. Both the smallest RVE size and the minimum number of realizations for a microstructure with higher volume fraction (VF) are larger compared to those of one with lower VF at the same desired precision. Similarly, a clustered structure is shown to require a larger minimum RVE size as well as a larger number of realizations at a given volume size compared to the well-dispersed microstructures.

A stochastic multiscale method for thermo-mechanical analysis arising from random porous materials with interior surface radiation

A stochastic multiscale computational method for predicting the thermo-mechanical coupling properties of random porous materials is proposed. Here, the thermal radiation effect at microscale of the material consisting of randomly distributed pores is considered, which has an important impact on the macroscopic temperature and stress fields. A novel unified homogenization procedure, based on two-scale asymptotic expressions, has been established. The higher-order multiscale formulations for solving the dynamic thermo-mechanical coupling problem with the inertia term, coupling term and radiation term are given successively. Then, the statistical prediction algorithm based on the multiscale method is described in detail. Finally, numerical examples for porous materials with varying probability distribution models are calculated by the proposed algorithm, and compared with the data by finite element method with very fine meshes. The comparison shows that the stochastic multiscale computational method developed in this paper is useful for determination of the thermo-mechanical properties of porous materials and demonstrates its potential applications in engineering and technology.

Assessment of the effect of microstructural uncertainty on the macroscopic properties of random composite materials

The linking of microstructural uncertainty with the random variation in the response of heterogeneous structures at the macroscale is particularly important in the framework of the stochastic finite element method. In this work, the effect of uncertainty in the constituent material properties and the geometry of the microstructure, on the macroscopic properties of composite materials is assessed through computational homogenization. Based on Hill–Mandel homogeneity condition, the homogenization procedure utilizes the excellent synergy of the extended finite element method and the Monte Carlo simulation. In this way, the computation of the statistical characteristics of the homogenized elasticity tensor of random composite materials reinforced with arbitrarily shaped inclusions is performed in a computationally efficient manner. The effect of stochastic variation in the elastic properties of the constituents as well as the effect of inclusion shape on the statistical characteristics of the homogenized elasticity tensor is assessed through probabilistic sensitivity analysis. A comparison is performed with regard to the relative influence of material and geometrical uncertainty which are considered separately. More realistic results are obtained by considering simultaneously material and geometrical uncertainty in the microstructural modeling of composite materials. The results can be further exploited in the stochastic finite element analysis of composite structures where material properties with random characteristics obtained by the presented multiscale homogenization procedure will be assigned to each finite element.

Elasticity of Random Multiphase Materials: Percolation of the Stiffness Tensor

Topology and percolation effects play an important role in heterogeneous materials, but have rarely been studied for higher-order tensor properties. We explore the effective elastic properties of random multiphase materials using a combination of continuum computational simulations and analytical theories. The effective shear and bulk moduli of a class of symmetric-cell random composites with high phase contrasts are determined, and reveal shortcomings of classical homogenization theories in predicting elastic properties of percolating systems. The effective shear modulus exhibits typical percolation behavior, but with its percolation threshold shifting with the contrast in phase bulk moduli. On the contrary, the effective bulk modulus does not exhibit intrinsic percolation but does show an apparent or extrinsic percolation transition due to cross effects between shear and bulk moduli. We also propose an empirical approach for bridging percolation and homogenization theories and predicting the effective shear and bulk moduli in a manner consistent with the simulations.

Effective dynamic properties of random nanoporous materials with consideration of surface effects

The frequency-dependent dynamic effective properties (velocity, attenuation and elastic modulus) of nanoporous materials with randomly positioned spherical cavities are studied. The surface energy theory proposed by Huang and Wang (Handbook of micromechanics and nanomechanics. Pan Stanford Publishing Pte Ltd., Singapore, pp 303-348, 2013) is used to derive the non-traditional boundary condition on the surface of the nanocavity and further the scattering matrix of a single nanocavity. Then, the total wave field is obtained by considering the multiple scattering processes among the dispersive spherical cavities. The configuration average of the total wave field results in the coherent waves or the averaged waves. The numerical examples of the effective speed, the effective attenuation of the coherent waves and associated effective dynamic modulus of a nanoporous material are given and shown graphically. The effects of surface elasticity and the residual surface tension are both considered. Moreover, the influences of the non-symmetric parts of in-plane surface stress and the out-of-plane parts of the surface stress are also discussed based on the numerical examples.

Prediction of effective properties for random heterogeneous materials with extrapolation

This work presents an efficient method to predict effective thermo-mechanical properties of random heterogeneous materials, with the purpose of reducing the cost of computation. The method is applied to morphologically realistic microstructures generated by a modified random morphology description functions algorithm. To achieve this purpose, firstly the convergence rates of the approximate effective material properties by homogenization with increasing cell domain size are numerically investigated, and it shows first-order accuracy of the approximate effective material properties. Secondly, Richardson extrapolation technique is introduced to obtain more accurate approximations of effective material properties by using some smaller cell domains, without the need for computations in larger cell domain. Numerical examples demonstrate that the method significantly saves computation time and computer memory while improving the accuracy of the effective material properties.

Evaluation of a simple finite element method for the calculation of effective electrical conductivity of compression moulded polymer–graphite composites

Finite element (FE) techniques can be used for the calculation of the effective properties of random heterogeneous materials, the required input simply consisting of phase properties and representative three-dimensional models of material microstructure. This approach has been widely exploited in recent years, although limited by the considerable amount of computational power required to obtain statistically accurate results. By using simple microstructural models of compression moulded polymer–graphite composites and a FE code modified for execution on graphical processing units, we show that reliable predictions of electrical properties for these materials can now be obtained in a reasonable computational time and with acceptable accuracy and precision. By using an approach based on design of experiments, we also perform a set of simulations aimed at determining the microstructural details which are most significant for the effective properties of these materials.

New Approach to Bounding Effective Properties of Random Heterogeneous Materials

A new approach to bounding effective properties of random heterogeneous materials is developed in this paper. The expectation material properties are computed for a stochastic unit cell to represent the random heterogeneous materials. This stochastic unit cell is then used to compute the upper bound and lower bound of effective properties using the variational asymptotic method for unit cell homogenization, a recently developed general-purpose micromechanics code. In comparison with other approaches, the present approach has two main advantages. 1) The numerical implementation is not restricted to statically homogeneous and isotropic microstructures. 2) Both the lower bound and upper bound converge to the effective properties of a deterministic unit cell with decreasing randomness of the stochastic unit cell. To illustrate the application of this new approach, various examples are analyzed and compared with existing theories.

Computing effective properties of random heterogeneous materials on heterogeneous parallel processors

In recent decades, finite element (FE) techniques have been extensively used for predicting effective properties of random heterogeneous materials. In the case of very complex microstructures, the choice of numerical methods for the solution of this problem can offer some advantages over classical analytical approaches, and it allows the use of digital images obtained from real material samples (e.g., using computed tomography). On the other hand, having a large number of elements is often necessary for properly describing complex microstructures, ultimately leading to extremely time-consuming computations and high memory requirements. With the final objective of reducing these limitations, we improved an existing freely available FE code for the computation of effective conductivity (electrical and thermal) of microstructure digital models. To allow execution on hardware combining multi-core CPUs and a GPU, we first translated the original algorithm from Fortran to C, and we subdivided it into software components. Then, we enhanced the C version of the algorithm for parallel processing with heterogeneous processors. With the goal of maximizing the obtained performances and limiting resource consumption, we utilized a software architecture based on stream processing, event-driven scheduling, and dynamic load balancing. The parallel processing version of the algorithm has been validated using a simple microstructure consisting of a single sphere located at the centre of a cubic box, yielding consistent results. Finally, the code was used for the calculation of the effective thermal conductivity of a digital model of a real sample (a ceramic foam obtained using X-ray computed tomography). On a computer equipped with dual hexa-core Intel Xeon X5670 processors and an NVIDIA Tesla C2050, the parallel application version features near to linear speed-up progression when using only the CPU cores. It executes more than 20 times faster when additionally using the GPU.