Quantification of stochastically stable representative volumes for random heterogeneous materials

Abstract

This paper details a procedure to determine lower bounds on the size of representative volume elements (RVEs) by which the size of the RVE can be quantified objectively for random heterogeneous materials. Here, attention is focused on granular materials with various distributions of inclusion size and volume fraction of inclusions. An extensive analysis of the RVE size dependence on the various parameters is performed. Both deterministic and stochastic parameters are analysed. Also, the effects of loading mode and the parameter of interest are studied. As the RVE size is a function of the material, some material properties such as Young's modulus and Poisson's ratio are analysed as factors that influence the RVE size. The lower bound of RVE size is found as a function of the stochastically distributed volume fraction of inclusions; thus the stochastic stability of the obtained results is assessed. To this end a newly defined concept of stochastic stability (DH-stability) is introduced by which stochastic effects can be included in the stability considerations. DH-stability can be seen as an extension of classical Lyapunov stability. As is shown, DH-stability provides an objective tool to establish the lower bound nature of RVEs for fluctuations in stochastic parameters.

Monte Carlo simulations of mesoscale fracture of concrete with random aggregates and pores: a size effect study

“…Karihaloo et al [20] and Duan et al [6] used the fictitious crack model to investigate the size effect on concrete. However, mesoscale numerical studies of the size effect of concrete with statistical analyses are still challenging [21][22][23], mainly due to the complex multi-phase composition, the complicated nonlinear multi-cracking behaviour, and substantial computational cost from repetitive simulations. In this paper, we study the size effect in concrete under tension by Monte Carlo simulations (MCS) of mesoscale specimens of varying sizes.…”

Representative Volumes for Numerical Modeling of Mass Transport in Hydrating Cement Paste

“…n = 5). In order to fulfill the dimesionlessness of the chi-square criterion, we employ the approach of Gitman et al (2006), normalising the R i with respect to R , thus rewriting the Eq. 6 as:…”

“…The RVE concept is widely used today in mechanics, e.g. Gitman et al (2006) proposed a numerical-statistical method to determine the size of the RVE. Zhang et al (2010) presented a numerical-statistical approach to determine the REV for transport modelling in cementitious materials.…”

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

“…Let us denote the representative volume element (RVE) of Y as [16,28,29]; Y ⊂ 2 denotes the section of this composite in the x 3 = 0 plane constant along the x 3 -axis, which is parallel to the fiber direction (see Figure 1).…”