Multiscale formulation for materials composed by a saturated porous matrix and solid inclusions

Abstract

Despite all the progress achieved in the characterization of heterogeneous materials by using multiscale paradigms based on the Representative Volume Element concept (RVE), there are still many aspects that demand ongoing development. We mention, for instance, in-homogeneous media with internal micro-structure comprising a mixture of components that require a dissimilar number/character of primary fields to describe their physical behavior. Micro-structures constituted by a saturated porous matrix (described through the pair displacement/pore pressure fields) endowed with impermeable non-porous solid inclusions (described only by the displacement field) are a typical example among many other practical applications. This new level of heterogeneity, between the set of primal descriptors for each micro-scale constituent, claims detailed revisions and novel adjustments in multiscale RVE-based theories. In this work, we present a first-order RVE-based formulation to address with the aforementioned phenomenology at a smaller length scale. At the macro-scale, the well-known poromechanics theory is preserved, where the constitutive response is provided by homogenization of the corresponding micro-scale problem. Two important attributes feature the model: (i) it is encompassed within the general framework posited by the ‘‘Method of Multiscale Virtual Power (MMVP)’’, this leads to a variationally consistent methodology; (ii) our formulation preserves the objectivity with respect to the RVE-size (an indispensable but uncommon characteristic in RVE-based multiscale approaches for saturated porous media), due to it is based in a recent contribution of the authors (the so-called Selective Order Expansion technique ‘‘SOE’’). The formulation is implemented following the squared Finite Element (FE2 ) numerical scheme. The potentialities of the model are demonstrated through a detailed series of numerical examples, including rigorous comparisons against the Direct Numerical Simulation (DNS) procedure.