Minimum Representative Volume Element Size Research Articles

Fabricating and determining representative volume elements of two-dimensional random aggregate numerical model for asphalt concrete without damage

In order to optimize the fabricating method of representative volume element (RVE) for two-dimensional (2D) random aggregate numerical model of asphalt concrete, and determine the minimum size of RVE corresponding to the 2D random aggregate numerical model of typical graded asphalt concrete in the nondestructive state. The authors first proposed an RVE fabricating method that includes “aggregate region identification” algorithm and can place “partial aggregate”. Then, by using this method, 9 different sizes of RVEs corresponding to AC-13C, AC-20C and AC-25C asphalt concrete (AC-13C, AC-20C and AC-25C asphalt concrete are dense-graded asphalt mixtures, and their nominal maximum aggregate sizes are 13.2 mm, 19 mm and 26.5 mm respectively) were fabricated. Subsequently, the geometric characteristics (aggregate area proportion, aggregate angle, “density” homogeneity) of these RVEs were analyzed. After that, virtual dynamic modulus tests were performed on these RVEs. Finally, the results of geometric analysis and virtual dynamic modulus test were comprehensively analyzed. The results show that the “aggregate region identification” algorithm can effectively improve the aggregate placement efficiency. By placing “partial aggregate”, the RVEs with small-sizes that meet the requirements of aggregate area proportion in each particle size range can be fabricated. In the nondestructive state, the minimum sizes of 2D random aggregate numerical model’s RVE corresponding to AC-13C, AC-20C and AC-25C asphalt concrete are 20 mm, 30 mm and 40 mm respectively.

Multiscale stress and strain statistics in the deformation of polycrystalline alloys

Multiscale stresses and strains in polycrystalline metals are always inhomogeneous. In this study, a rate-independent crystal plasticity formulation was implemented for a cubic representative volume element (RVE) of an fcc polycrystal generated by 3D Delaunay tessellation. Multiple realizations were generated with crystallographic orientation permutations and different grain morphologies in order to investigate the statistical distribution of stress, elastic lattice strain and total strain at the macro-, meso- and micro-scale. Macroscopically, at 1.55% total strain (elasto-plastic deformation), the overall stress statistics among different RVEs were observed to follow a normal distribution, whose profile shape is affected by the parameters that describes the lognormal grain size distribution. On the mesoscale, the orientation-specific elastic strains were accurately reproduced via the use of diffraction post-processing and validated by neutron diffraction data for a polycrystalline alloy. Microscopically, the local elastic strains (and hence stresses) universally follow a normal distribution, while plastic strains follow a lognormal probability distribution. Reliable knowledge of the statistical distributions of stresses and strains give new guidance for the determination of the minimum RVE size. The above finding reveals the nature of stress and strain inhomogeneity at multiple scales and emphasizes the fact that the dispersion of local stress and strain is much larger than that of the macroscopic average. The statistical analysis of stress and strain distribution at multiple scales provide further rich insights into the connection between microstructure and mechanical properties under monotonic and cyclic loading.

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.

Microstructure measures and the minimum size of a representative volume element: 2D numerical study

In the paper, a numerical study of the size of a representative volume element (RVE) for both the heat flow as well as the linear elasticity problems is presented. A particular two-phase random microstructure is studied and the method is applied to the digital image of reconstructed 2D realization of random media. The minimum size of RVE is determined by the investigation of the convergence of apparent properties as the size of RVE is increasing. Then, two estimates of the minimum RVE size are proposed and it is shown that the estimates are in a good agreement with the results determined by the investigation of the convergence of apparent properties. The minimum size of RVE can be successfully predicted based only on the microstructure morphology. The statistical measures used in this work are: the two-point probability and the linealpath functions.

Homogenization of the average thermo-elastoplastic properties of particle reinforced metal matrix composites: The minimum representative volume element size

The average thermo-elastoplastic properties of particle reinforced metal matrix composites (PRMMC) including the average coefficient of thermal expansion (CTE), Young’s modulus, Poisson’s ratio and isotropic hardening function are investigated. Computational homogenization method based on 3D realistic microstructures (RMs) is employed. Unit cell microstructure (UCM) based model and analytical models are also employed for comparison. As an illustration, 17vol.%SiCp (3μm)/2124Al composite is studied. Compared to RMs, UCM underestimates the average CTE and Poisson’s ratio, while it overestimates the average Young’s modulus and isotropic hardening function. The minimum representative volume element (RVE) size for determining the average CTE, Young’s modulus and Poisson’ ratio is δ=15, δ=20 and δ=20, respectively, where δ is the size ratio of microstructure model, which is defined by the ratio of the side length of the RVE to the nominal mean radius of reinforcement. The minimum size of RVE for estimating average isotropic hardening function of plastic deformation is dependent on both the temperature and the plastic deformation condition.

The Minimum Representative Volume Element Size for Coefficient of Thermal Expansion of Particle Reinforced Metal Matrix Composites

A 3D realistic microstructure based computational homogenization model is proposed, in order to determine the temperature dependent effective coefficient of thermal expansion of particle reinforced metal matrix composites The model employed three-dimensional realistic microstructures with different domain sizes, where particles had random shape, sharp edges and were randomly distributed. The unit cell microstructure based model and classical analytical models were also presented for comparison. As an illustration of the model, a 17% vol. SiCpreinforced 2124Al composite was investigated. Its minimum RVE size is found to beδ= 15, whereδis called the size ratio and defined by the ratio between the side length of microstructure and the mean particle radius.

A micromechanics-based nonlocal constitutive equation and minimum RVE size estimates for random elastic composites containing aligned spheroidal heterogeneities

An explicit micromechanics-based nonlocal constitutive equation is derived that relates the ensemble averages of stress and strain for macroscopically transversely isotropic elastic composites consisting of a matrix containing a random distribution of aligned impenetrable spheroidal inclusions or voids. The analysis builds on and generalizes previous models developed by the authors [Drugan, W.J., Willis, J.R., 1996. J. Mech. Phys. Solids 48,1359–1387; Monetto, I., Drugan, W.J., 2004. J. Mech. Phys. Solids 52, 359–393) wherein elastic composites containing spherical and randomly oriented spheroid-shaped inclusions were analyzed. One advance involves derivation of a closed-form statistical characterization of the aligned spheroidal microstructure that is valid for the full physically attainable range of inclusion volume fractions. This was accomplished by applying a simple scale transformation to a sensible model of the microstructure, without the use of rigid “security spheres”. An explicit micromechanics-based nonlocal constitutive equation is derived by employing this statistical characterization. This constitutive equation is then employed to explore the nonlocal effects of both shape and orientation of inclusions on the macroscopic response of the composite. Finally, quantitative estimates of the characteristic internal length of the random material are derived for the extreme cases of spheroidal voids and rigid particles; this length is indicative of the minimum representative volume element (RVE) size needed for a standard local composite constitutive equation sensibly to characterize the composite's overall response. Among the results, it is found that this minimum volume element size must be substantially larger for accurate description of composites containing aligned stiff spheroids as compared with those containing randomly oriented spheroids or spherical particles.

Stochastic homogenization of random elastic multi-phase composites and size quantification of representative volume element

By using effective properties derived from homogenization techniques, a heterogeneous medium is transformed into a homogenous one, and the number of the degrees of freedom (DOF) can therefore be significantly reduced. This dimension reduction approach has been a most widely used computational approach for efficient simulation of heterogeneous materials. The success of such a deterministic Representative Volume Element (RVE) practice critically relies on satisfaction of the assumption on scale decoupling, i.e. the size of heterogeneity is significantly smaller than the size of finite elements or RVE. The condition however has been incautiously or unknowingly ignored in quite a number of engineering practices. One typical example is finite element modeling of concrete materials where the element size is often chosen to be comparable to or even smaller than the size of aggregates. In computational modeling of small scale systems such as NEMS and MEMS, the scale decoupling assumption is commonly invalid. For such circumstances, the use of deterministic homogenization to reduce DOF results in large errors and new methodology using stochastic homogenization and non-deterministic material constants in finite elements should be considered. This paper is aimed to quantify the size effect of RVE and therefore provide estimates for the threshold of scale separation. The generalized variational principles developed in Xu [Xu, X.F., submitted for publication. Generalized variational principles for uncertainty quantification of boundary value problems of random heterogeneous materials] are adapted to stochastic homogenization problems, which results in size-dependent energy bounds and Hashin–Shtrikman (HS) bounds. The difference between the size dependent stochastic bounds and the classical deterministic bounds demonstrates the size effect of RVE. Numerical characterization suggests that for three-dimensional problems the size threshold of scale-decoupling, or equivalently the minimum size of deterministic RVE, is approximately between 10 and 20 times of “correlation diameter” based on a range of accuracy criteria between 1% and 0.1%. The threshold for two-dimensional problems is approximately between 15 and 50 times of correlation diameter for the same criteria, which shows stronger size effect than the 3-D cases.

Generalized self-consistent estimation of the apparent isotropic elastic moduli and minimum representative volume element size of heterogeneous media

Under investigation is a heterogeneous material consisting of an elastic homogeneous isotropic matrix in which layered elastic isotropic inclusions or pores are embedded. The generalized self-consistent model (GSCM) is extended so as to be capable of estimating the apparent elastic properties of a finite-size specimen smaller than a representative volume element (RVE). The kinematical or static apparent shear modulus is determined as a root of a cubic polynomial equation instead of a quadratic polynomial equation as in the classical GSCM of Christensen and Lo [Christensen, R.M., Lo, K.H., 1979. Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330]. It turns out that the extended GSCM establishes a link between the composite sphere assemblage model (CSAM) of Hashin [Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150] and the classical GSCM. Demanding that the normalized distance between the kinematical and static apparent moduli of a finite-size specimen be smaller than a certain tolerance, the minimum RVE size is estimated in a closed form.

Effects of grain sizes, shapes, and distribution on minimum sizes of representative volume elements of cubic polycrystals

In the literature the concept of representative volume element (RVE) was introduced to correlate the effective or macroscopic properties of materials with the properties of the microscopic constituents and microscopic structures of the materials. However, to date little quantitative knowledge is available about minimum RVE sizes of various engineering materials. In our recent paper [J. Mech. Phys. Solids 50 (2002) 881], a new definition of minimum RVE size was introduced based on the concept of nominal modulus. Numerical experiments using the finite element method (FEM) were then carried out for determining the minimum RVE sizes of more than 500 cubic polycrystals in the plane stress problem, under the assumption that all grains in a polycrystal have the same square shape––called the simple polycrystal model. The major finding is that the minimum RVE sizes for effective elastic moduli have a roughly linear dependence on crystal anisotropy degrees. The present paper takes into account the effect of grain sizes, shapes, and distribution on the minimum RVE sizes for real cubic polycrystals that are formed by crystallization processes. Similar roughly linear dependence is found again, with the slope about 19% lower than that in the simple polycrystal model. This finding is interesting and useful because numerical experiments on minimum RVE sizes for a large number of crystals are quite time-consuming and the simple polycrystal model reduces significantly the FEM pre- and post-processing works. This should be particularly true in numerically testing minimum RVE sizes for three-dimensional polycrystals and for nonelastic properties in future works. With a maximum relative error 5%, all the polycrystals tested have a minimum RVE size of 16 or less times the grain size.

A Quantitative study of minimum sizes of representative volume elements of cubic polycrystals—numerical experiments

The concept of representative volume element (RVE) plays a key role in correlating the properties of microscopically heterogeneous materials with those of their macroscopically homogenized ones. However, up to now little quantitative knowledge is available about RVE scales or sizes of various engineering materials, which have been becoming a necessity due to the rapid development of, for instance, microelectromechanical systems. A new and convenient definition of the minimum RVE size is introduced. Then more than 500 kinds of cubic polycrystalline material in the planar stress state are numerically tested. The major finding from these numerical experiments is that the RVE size for the effective shear modulus (as well as the Young's modulus) depends roughly linearly upon the anisotropy degree of the single crystal, while the effective area modulus does not. For the latter observation a theoretical proof is also given. With a maximum relative error 5%, all the materials tested (with one exception) have a minimal RVE size of 20 or less times as large as the grain size.

Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites

A generalization of the Hashin–Shtrikman variational formulation to random composites, due to J.R. Willis, is employed to derive micromechanics-based variational estimates for a higher-order nonlocal constitutive equation relating the ensemble averages of stress and strain, for a class of random linear elastic composite materials. We analyze two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), within a formulation accounting for one- and two-point probabilities, and derive an explicit nonlocal constitutive equation that includes terms up through the fourth gradient of average strain. The analysis is carried out first for an arbitrary comparison medium. Then, a new approach is outlined and applied which employs the nonlocal correction to determine the optimal choice of comparison medium, and hence the optimal effective modulus tensor (as well as the optimal tensor coefficients of the nonlocal terms) for the amount of statistical information employed. The new higher order analysis provides a highly accurate nonlocal constitutive equation, valid down to quite small volume size scales and to rather strong variations of average strain with position. Among several applications illustrated, it permits accurate analytical assessment of the remarkably small predictions derived by Drugan and Willis (1996. Journal of the Mechanics and Physics of Solids 44, 497–524) of the minimum representative volume element (RVE) size needed for accuracy of the standard constant-effective-modulus macroscopic constitutive equation for elastic matrix-inclusion composites that have spherical inclusions/voids. It also affords an analytical assessment of the improved (i.e., reduced) minimum RVE size scale, compared to a standard constant-effective-modulus constitutive equation, to which the leading-order nonlocal constitutive equation derived by Drugan and Willis applies. This improvement is shown to be dramatic in some example cases.

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

Comments on rarity and chaos, Thomas D. Rogers, math. biosci. 72:13-18 (1984)

By using effective properties derived from homogenization techniques, a heterogeneous medium is transformed into a homogenous one, and the number of the degrees of freedom (DOF) can therefore be significantly reduced. This dimension reduction approach has been a most widely used computational approach for efficient simulation of heterogeneous materials. The success of such a deterministic Representative Volume Element (RVE) practice critically relies on satisfaction of the assumption on scale decoupling, i.e. the size of heterogeneity is significantly smaller than the size of finite elements or RVE. The condition however has been incautiously or unknowingly ignored in quite a number of engineering practices. One typical example is finite element modeling of concrete materials where the element size is often chosen to be comparable to or even smaller than the size of aggregates. In computational modeling of small scale systems such as NEMS and MEMS, the scale decoupling assumption is commonly invalid. For such circumstances, the use of deterministic homogenization to reduce DOF results in large errors and new methodology using stochastic homogenization and non-deterministic material constants in finite elements should be considered. This paper is aimed to quantify the size effect of RVE and therefore provide estimates for the threshold of scale separation. The generalized variational principles developed in Xu [Xu, X.F., submitted for publication. Generalized variational principles for uncertainty quantification of boundary value problems of random heterogeneous materials] are adapted to stochastic homogenization problems, which results in size-dependent energy bounds and Hashin–Shtrikman (HS) bounds. The difference between the size dependent stochastic bounds and the classical deterministic bounds demonstrates the size effect of RVE. Numerical characterization suggests that for three-dimensional problems the size threshold of scale-decoupling, or equivalently the minimum size of deterministic RVE, is approximately between 10 and 20 times of “correlation diameter” based on a range of accuracy criteria between 1% and 0.1%. The threshold for two-dimensional problems is approximately between 15 and 50 times of correlation diameter for the same criteria, which shows stronger size effect than the 3-D cases.