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

Abstract

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.