Elsevier

Computational Materials Science

Volume 63, October 2012, Pages 91-104
Computational Materials Science

Comparison of spherical and cubical statistical volume elements with respect to convergence, anisotropy, and localization behavior

https://doi.org/10.1016/j.commatsci.2012.05.063Get rights and content

Abstract

The statistical volume element (SVE) technique is commonly used for the estimation of the effective properties of a micro-structured material. Mostly, cubical SVEs with periodic boundary conditions are employed, which result in a better convergence, compared to the uniform boundary conditions. In this work, the possibility of using spherical SVEs is discussed, since their use promises a reduction of the influence of the boundary, and thus a more efficient estimation of the effective material properties. We discuss the applicability of boundary conditions which are similar to the periodic boundary conditions to spherical SVEs. Then we assess the convergence (subject 1) of spherical and cubical SVEs to the effective material behavior for the uniform and periodic boundary conditions, focusing on the elastic and plastic properties of a macroscopically isotropic matrix-inclusion material. It is shown that the spherical SVEs perform indeed better than the cubical SVEs. Also, unlike the spherical SVEs, the cubical SVEs with periodic boundary conditions induce a spurious anisotropy (subject 2), which is quantified for the effective elastic properties. Finally, we examine the effect of the periodicity frame on the localization behavior (subject 3) of cubical SVE, since cubical SVE with periodic boundary conditions are commonly used to estimate macroscale material failure. It is demonstrated that the orientation of the periodicity frame affects the overall SVE response significantly. The latter is not observed for spherical SVE.

Highlights

► We carry out a direct comparison of spherical and cubical RVE. ► We observed a better convergence for spherical RVE, due to the better surface to volume ratio. ► The results indicate a spurious anisotropy induced by a cubical RVE shape, which is not the case for spherical RVE. ► This anisotropy is severe if localization of RVE is considered.

Introduction

The microscale structure of a material can have a considerable effect on the material properties as perceived on the macroscale. Common examples are polycrystals, which may exhibit a crystallographic (crystal orientation) and morphological (grain shape) texture, fiber or particle reinforced composites, foams and laminates. The process of calculating the effective material properties from the arrangement and the properties of the constituents on a smaller scale is termed as homogenization.

For specific material properties, efficient homogenization methods are at hand, see for example Klusemann and Svendsen [26] for the elastic properties, or Fritzen et al. [13] for the yield limit of porous materials. However, in many cases the analytical homogenization is limited, e.g. for the prediction of the crystallographic texture evolution [5]. Then, one follows commonly the pragmatic approach of the RVE or SVE method, which consists of considering a representative section of the material, define appropriate boundary conditions, and solve the initial- and boundary-value problem, usually with the help of numerical methods such as the finite element method. Then, one is able to extract the volume average of the variable of interest, or examine the effect of different microstructures on the overall material behavior. For an account on numerical homogenization by the SVE/RVE method see, e.g., Zohdi and Wriggers [49]. In this work, we will not distinguish strictly between RVE and SVE, which capture the microstructure identically (RVE) or in an approximate sense (SVE). Here, the two terms are used like synonyms.

When focusing on macroscale stress–strain relations, in contrast to analytical techniques, one does not arrive at a closed-form, but obtains an approximation for a specific deformation path. For a coupling with a large-scale FE application, one may use the FE2 method [44], [9], [30], [38]. However, this is computationally very expensive, and one is interested in a reduction of the numerical costs. A relatively new approach to this problem is the coupling of the RVE-method with the nonuniform transformation field analysis (NTFA [29], [12]). Roughly speaking, the RVE method is used to build a database for different deformation modes, from which the actual stress–strain-relations needed in the macroscale calculation are estimated. However, the NTFA is restricted to the small strain setting.

A more direct reduction of the numerical costs of the FE2 method is the optimization of the RVE, and to apply it simultaneously to a large-scale constitutive law. Then, the RVE calculations are carried out only when and where it is necessary, e.g., when the straining is large. Also, the RVE itself should be as representative as possible, but still require an acceptable numerical effort. The problem of determining a RVE with a good balance between representativity and numerical expense, arises. It depends on the material under consideration, and has been subject of many studies, e.g., Kanit et al. [23], Xu and Chen [47], Pelissou et al. [39], Salahouelhadj and Haddadi [41]. Another possibility to increase the ratio representativity/numerical effort is to optimize the material section under consideration [42].

Here, we examine the influence of the shape of the RVE. Usually, cubical RVEs are used. We demonstrate that the use of spherical RVEs is advantageous for two reasons. Firstly, the bias due to specific boundary conditions is weaker, since the surface-to-volume ratio is smaller than for cubical RVEs. Secondly, spherical RVEs do not induce a material-independent anisotropy, unlike cubical RVEs with periodic boundary conditions. The reduction of the shape-induced anisotropy has been discussed by Grasset-Bourdel et al. [16], who considered RVEs with shapes that allow for a complete filling of the space. It was found that a hexagonal arrangement is advantageous, compared to a cubic shape. However, the restriction to shapes that allow for a complete filling of the space appears to be unnecessary, since spherical RVE are used routinely for analytical methods [17], [43], [8], while in numerical calculations the cubical RVEs predominate. Only few exceptions can be found, e.g., Kim et al. [25] and the authors referred to in this work used non-periodic RVEs.

The outline of this work is as follows: We firstly reproduce the fundamentals of the RVE method (Section 2), followed by a discussion of the possible boundary conditions (Section 3) that may be applied to spherical and cubical RVEs (Section 4). In Section 5 the setup for the numerical experiments is described. In Section 6 we examine the convergence while increasing the RVE size for the elastic and plastic properties of a matrix–inclusion-material, and compare different shapes and boundary conditions. In Section 7 we assess the shape-induced anisotropy of the spherical and cubical RVE with periodic boundary conditions by applying them to an effectively isotropic material. Finally we focus on the peculiarities of the localization behavior of the spherical and cubical RVE (Section 8).

Throughout the work a direct tensor notation is preferred. If an expression cannot be represented in the direct notation without introducing new conventions, its components are given with respect to orthonormal base vectors ei, using the summation convention. Vectors are symbolized by lowercase bold letters v = viei, second-order tensors by uppercase bold letters T = Tijei  ej or bold Greek letters. The second-order identity tensor is denoted by I. Fourth-order tensors are symbolized like C. The dyadic product is defined as (a  b) · c =  (b · c)a. A dot represents a scalar contraction. If more than one scalar contraction is carried out, the number of dots corresponds to the number of contractions, thus (a  b  c) · · (d  e) = (b · d)(c · e)a, α = A · · B and σ=C··ε. If only one scalar contraction is carried out, the scalar dot is frequently omitted, e.g., v = Fw, A = BC. ∥x∥ denotes the Frobenius norm.

The position vector of a material point is denoted by x(x0, t), where x0 indicates the position vector of the same material point in the reference placement. At t = 0, x = x0 holds. The partial derivative of a function with respect to t with x0 kept constant is the material time derivative, indicated by a superimposed dot. The index ”0” indicates that a function or derivative is to be evaluated in the reference placement or with respect to x0. Ω denotes the domain of the RVE under consideration. All unweighted volume averages over this domain are evaluated in the reference placement, denoted as ·1V0Ω0·dV. A superimposed bar indicates a macroscale-quantity.

Section snippets

Scale separation

The scale separation requires that lmicro  lmini  lmacro [18], where lmicro refers to the characteristic size of the heterogeneities, lmini to the RVE size and lmacro to the dimensions of the body. lmicro  lmini ensures the representativity of the RVE, while lmini  lmacro is necessary if one wants to consider the RVE as a material point on the macroscale.

Equilibrium equations

The local balances of linear and angular momentum requireσ·=ρ(x¨-b),σ=σT,where b is a mass-specific force density. They must hold globally for an

Boundary conditions on the RVE

The boundary value problem is complete when at each surface point, with respect to a suitable orthogonal basis bi, either ui, ti or a mixture of both is prescribed. More general, the boundary conditions may be given implicitly in form of constraints, as it is the case for the periodic boundary conditions.

The boundary conditions to which an RVE may be subjected have been exhaustively discussed, see e.g. Suquet [45]. There are no natural or self-evident boundary conditions for the RVE, except for

Non-periodic microstructures

In the case of non-periodic microstructures, the homogenized material response must be approximated by considering possibly large RVE. As the size of the RVE increases, the representativity gets better, and the surface-to-volume ratio tends to zero. Presuming that there is no softening or fracture or other localization-inducing material behavior, the influence of the boundary conditions vanishes (see Section 8).

Considering that the boundary influence is artificial, and therefore preferably

Material behavior of the matrix and inclusions

The material under consideration is a matrix–inclusion material. The matrix is an isotropic, linearly elastic, perfect plastic von Mises material without hardening. The inclusions with a total volume fraction of 0.3 are spherical, isotropic, linearly elastic particles of equal diameter, distributed uniformly without preferred alignment or pattern. They are considerably stiffer than the matrix material. The material parameters are collected in Table 1.

Definition of the RVE

The calculations have been carried out using

Simulation setup

For the study of convergence, we carried out uniaxial tension tests, in which the nominal strain ε is increased to 10%. The latter is accomplished by imposingH¯(uax)ij=ε000-000-.Not prescribing H¯(uax)22 and H¯(uax)33 results in zero stress components T¯22 and T¯33. As characteristic quantities for the statistical evaluation, Young’s modulus E=σ¯11/H¯11 at the onset of the deformation and the Cauchy stress σ¯11 at 10% of nominal strain have been extracted. Six combinations of RVE shapes and

Quantification of the shape-induced anisotropy

The shape-induced elastic anisotropy is determined by imposing mutually orthogonal small strains,H¯(1)ij=δ00000000,H¯(2)ij=0000δ0000,H¯(3)ij=00000000δ,H¯(4)ij=0δ0δ00000,H¯(5)ij=00δ000δ00,H¯(6)ij=00000δ0δ0.where δ is small enough to guarantee a purely elastic material response. The components of the stiffness tetrad are then obtained byCij11=σ¯ij/δimposingH(1),Cij22=σ¯ij/δimposingH(2),Cij33=σ¯ij/δimposingH(3),Cij12=Cij21=σ¯ij/δimposingH(4),Cij13=Cij31=σ¯ij/δimposingH(5),Cij23=Cij32=σ¯ij/δimposing

Localization behavior

In case of non-quasi-convex incremental stress potentials, structural failure of the RVE may occur. For periodic microstructures, a framework for relating the structural failure of an RVE to a material instability on the macroscale is at hand [32]. For non-periodic microstructures, this issue is still not clear. The reason therefor is twofold:

  • While linear displacement boundary conditions prevent localizations to reach the boundary, homogeneous traction boundary conditions allow for an arbitrary

Summary

We compared the performance of spherical and cubical RVE with different boundary conditions, applied to a macroscopically isotropic matrix–inclusion material with hard elastic inclusions and a soft elastoplastic matrix. It is argued that the periodic boundary conditions are not restricted to periodically repeatable unit cells, although the denomination “periodic” is misleading when applied to non-periodic shapes. Then one should speak more generally of coupled boundary conditions.

We could

Acknowledgements

We gratefully acknowledge valuable remarks from Jan Kalisch, Felix Fritzen and Thomas Böhlke. Also, we like to express our gratitude to an unknown reviewer for his/her comments.

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