RVE size for random composites with local volume fraction variation: attainability of a usable finite RVE at high stiffness contrast

In computational homogenization of random two-phase composites, the representative volume element (RVE) is the smallest volume on which effective (homogenized) properties can be computed within a prescribed tolerance. Savvas, Stefanou and Papadrakakis (CMAME 305 (2016) 340–358) determined RVE sizes for random composites accounting for local volume fraction variation, using a moving-window XFEM/Monte Carlo procedure in which the RVE is attained when Dirichlet and Neumann bounds on apparent properties converge. Their study reported that at stiffness ratio $E_{in}/E_m=1000$ the RVE size could not be defined within a reasonable tolerance (20% in the CMAME paper); further reported findings are that different elasticity-tensor components converge at different rates, that the study was restricted to 2D circular monodisperse inclusions, and that thousands of moving windows must be analyzed. The open problem is whether a usable finite RVE can be defined at the high stiffness ratio $E_{in}/E_m = 1000$ in the source study's setting under its Dirichlet/Neumann bound-convergence criterion, or whether the required mesoscale is prohibitive, with property-specific convergence rates, generalization to 3D, non-circular, and polydisperse microstructures, and moving-window sampling cost as motivating context.

Background

Homogenization replaces a heterogeneous material by an equivalent homogeneous continuum whose effective elastic properties are computed by solving boundary value problems on a subvolume. The representative volume element (RVE) is a volume large enough that the computed effective properties no longer depend on the subvolume size or on the applied boundary conditions; for random microstructures it is approached only asymptotically, and finite-size domains are called statistical volume elements (SVEs). The effective properties computed on an SVE are called apparent properties: under kinematic uniform boundary conditions (KUBC — prescribed displacements, an analogue of Dirichlet conditions) one obtains an upper bound on the effective stiffness, while static uniform boundary conditions (SUBC — prescribed tractions, a Neumann analogue) give a lower bound, with both bounds converging toward the effective properties as the domain grows. A standard criterion (Kanit, Forest, Galliet, Mounoury and Jeulin, Int. J. Solids Struct. 40 (2003) 3647–3679; Ostoja-Starzewski, Probab. Eng. Mech. 21 (2006) 112–132) declares the RVE attained when the two bounds, or the ensemble statistics of the apparent properties, agree within a prescribed tolerance; the non-dimensional mesoscale $\delta=L/d$ compares the SVE side length $L$ to a microstructural length such as the inclusion diameter $d$. Ostoja-Starzewski's scale-dependent homogenization program (reviewed in Adv. Appl. Mech. 49 (2016) 111–211) established that the convergence rate of the mesoscale bounds is governed by the microstructure statistics and, crucially, by the phase contrast; Sab and Nedjar (C. R. Mécanique 333 (2005) 187–195) proposed periodization (making each realization periodic) as an alternative RVE definition that converges faster for linear composites at moderate contrast. A key statistical input is the local volume fraction (LVF) — the inclusion fraction within each window — which varies spatially in real random composites; earlier work (Borbély, Kenesei and Biermann, Acta Mater. 54 (2006) 2735–2744) showed that the linearity of window properties in the LVF degrades as the stiffness contrast grows, foreshadowing the contrast difficulty. Savvas, Stefanou and Papadrakakis (CMAME 305 (2016) 340–358) combined these ideas: from computer-simulated images of random composites with randomly dispersed circular inclusions, they extracted SVEs by the moving window technique (overlapping square windows scanned across the image), solved each window problem with the extended finite element method (XFEM — a finite element method in which the inclusion boundaries are captured by enrichment functions, avoiding conforming remeshing), and used Monte Carlo simulation over many windows to track the convergence of KUBC/SUBC bounds on the apparent elasticity tensor components, with the RVE declared attained when the bound discrepancy fell below tolerance. Their findings: the required $\delta$ grows sharply with stiffness contrast ($\delta>2.8$ at $E_{in}/E_m=10$; $\delta>18$ at $E_{in}/E_m=100$; unattainable within the 20% tolerance at $E_{in}/E_m=1000$); compliant inclusions were reported to converge even more slowly; higher volume fractions require larger RVEs; and the shear component $C_{33}$ of the apparent elasticity tensor (Voigt notation, in which $C_{11}$ and $C_{22}$ are axial coefficients and $C_{33}$ the in-plane shear coefficient) converges faster than the average axial stiffness $(C_{11}+C_{22})/2$, so different properties demand different RVE sizes. The central difficulty recorded by the study is that at the high stiffness ratio the RVE size could not be defined within a reasonable tolerance; the component-dependent convergence rates, the restriction to 2D circular monodisperse inclusions, and the cost of analyzing thousands of windows ($n_w=3721$ at $\delta=5.605$) are further reported findings and motivations for the open problem below.

Problem Statement

For random two-phase composites with local volume fraction variation, in the setting of the source study (two-dimensional computer-simulated microstructures with randomly dispersed circular monodisperse inclusions at volume fractions of 20–40%, analyzed by the image-based moving-window XFEM/Monte Carlo procedure of Savvas, Stefanou and Papadrakakis, CMAME 305 (2016) 340–358), determine whether a usable finite RVE can be defined at the high matrix/inclusion stiffness ratio $E_{in}/E_m = 1000$ tested by that study (where $E_{in}$ and $E_m$ denote the Young's moduli of the inclusions and of the matrix), at which the study reported that the RVE size could not be defined within a reasonable tolerance. Attainment is to be judged by the study's own criterion: the discrepancy

$e_\delta = (C_{ij}^D - C_{ij}^N)/C_{ij}^D$

between the KUBC/Dirichlet and SUBC/Neumann bounds on the spatially averaged apparent elasticity tensor components — the average axial stiffness $(C_{11}+C_{22})/2$ and the shear component $C_{33}$ — falling below an explicitly declared prescribed tolerance as the mesoscale $\delta = L/d$ grows. The criterion is evaluated within this bound-convergence framework; the 20% tolerance reported for the CMAME study is the natural reference. Because the KUBC and SUBC bounds of this ergodic model class converge toward the effective properties as the domain grows, a finite $\delta$ satisfying any fixed positive tolerance is in principle assured, so the operative question is not existence but how large the required $\delta$ is at $E_{in}/E_m = 1000$ and whether it is attained within a feasible window range — whether a finite $\delta$ at which the criterion is met is attainable in practice, or whether the required $\delta$ is prohibitive for the stated model class and tolerance. In the latter case an alternative RVE definition or modeling strategy may be proposed, but it must be validated against the same Dirichlet/Neumann bound framework; the question admits a negative resolution, since a prohibitive required $\delta$ answers it.

The property-dependent convergence rates reported by the source (shear $C_{33}$ versus average axial stiffness), the generalization beyond 2D circular monodisperse inclusions to three-dimensional microstructures, non-circular inclusion shapes, and polydisperse size distributions, and the computational cost of the moving-window sampling are background context and motivations for this question, not conjunctive deliverables of the answer.

References

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