Multiscale formulation for saturated porous media preserving the representative volume element size objectivity

Abstract

A multiscale model for saturated porous media is proposed, based on the concept of representative volume element (RVE). The physics between macro and micro-scales is linked in terms of virtual power measures given by the general theory of poromechanics. Then, applying the so-called Principle of Multiscale Virtual Power, together with suitable admissible constraints on micro-scale displacement and pore pressure fields, a well-established variational framework is obtained. This setting allows deriving the weak form of micro-scale balance equations as well as the homogenization rules for the macro-scale stress-like variables and body forces. Whenever the micro-scale mechanical constitutive functionals admit, as input arguments, the full-order expansion of the micro-scale pore pressure field, a size effect is inherited on the macro-scale material response. The current literature attributes this issue to the so-called “dynamical” or “second-order” term of the homogenized flux velocity. It has been commonly suggested that the influence of this term is negligible by assuming infinitely small micro-scale dimensions. However, such an idea compromises the fundamental notion of the existence of RVE for highly heterogeneous media. In this work, we show that the micro-scale size dependence can be consistently eliminated by a simple constitutive-like assumption. Accordingly, slight and selective redefinitions in the input arguments of micro-scale material laws are proposed, leading to a constitutive formulation that allows the combination of micro-scale variables with different orders of expansion. Just at this specific (constitutive) level, a reduced-order expansion is selectively adopted for the micro-scale pore pressure field. Thus, the RVE notion is restored while still retaining the major effects of the “dynamical” component of the homogenized flux velocity. The proposed formulation is implemented within a finite element squared (FE (Formula presented.)) environment. Some numerical experiments are presented showing the viability of the methodology, including comparisons against analytical, mono-scale and DNS solutions.