On the size of representative volume element for Darcy law in random media

Abstract

Most studies of effective properties of random heterogeneous materials are based on the assumption of the existence of a representative volume element (RVE), without quantitatively specifying its size L relative to that of the micro-heterogeneity d . In this paper, we study the finite-size scaling trend to RVE of the Darcy law for Stokesian flow in random porous media, without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. By analogy to the existing methodology in thermomechanics of random materials, we first formulate a Hill–Mandel condition for the Darcy flow velocity and pressure gradient fields. This dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale-dependent hierarchies on effective (RVE level) permeability. To quantitatively assess the scaling trend towards the RVE, these hierarchies are computed for various porosities of random disc systems, where the disc centres are generated by a planar hard-core Poisson point field. Overall, it turns out that the higher is the density of random discs—or, equivalently, the narrower are the micro-channels in the system—the smaller is the size of RVE pertaining to the Darcy law.

Yield of random elastoplastic materials

“…This approach has primarily been used in linear elasticity [Hazanov and Huet 1994;Cluni and Gusella 2004;Kanit et al 2003;Ostoja-Starzewski 1999;2000], and in new inroads in viscoelasticity [Huet 1995;, elastoplasticity [Jiang et al 2001;Ostoja-Starzewski 2005], plasticity with damage [Clayton and McDowell 2004], thermomechanics with internal variables [Ostoja-Starzewski 2002], and finite (thermo)elasticity [Khisaeva and Ostoja-Starzewski 2006]. Related studies in linear thermoelasticity and Stokesian flow in porous media are currently underway [Du and Ostoja-Starzewski 2006a;2006b]. For elastic-perfectly plastic materials, similar results have been obtained by [He 2001] using a mathematically more rigorous analysis involving gauge functions.…”

Electric-field-induced displacement of a charged spherical colloid embedded in an elastic Brinkman medium

“…However, we assume this medium to be statistically isotropic and homogeneous, so that, at the level of a Representative Volume Element (RVE) of the deterministic continuum, the anisotropy vanishes just as the fluctuations in constitutive response tend to zero; see Ostoja-Starzewski & Wang (1989) for a random elastic model. Such a scale-dependent homogenization (i.e., a passage from a random microstructure to the RVE) was recently studied in the context of Stokesian permeability (Du & Ostoja-Starzewski 2006), albeit the departure from anisotropy was not addressed explicitly; see also Ostoja-Starzewski (2007) for related studies in many other material problems.…”

Weakly periodic boundary conditions for the homogenization of flow in porous media

“…In order to allow for the use of RVEs, the microstructure of the pertinent material should be ergodic and statistically spatially homogeneous. For further reading on the size of the RVE for homogenization of Stokes flow, we refer to [2].…”