Effective balance equations for poroelastic composites

We derive the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. We assume that the size of the pores (the microscale) is comparable with the distance between adjacent subphases and is much smaller than the size of the whole domain (the macroscale). We then decouple spatial scales embracing the asymptotic (periodic) homogenization technique to derive the new macroscale model by upscaling the fluid-structure interaction problem between the elastic constituents and the fluid phase. The resulting system of partial differential equations is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model, which are to be computed by solving appropriate cell problems which reflect the complexity of the given microstructure. The model reduces to the limit case of simple composites when there are no pores, and standard Biot's poroelasticity whenever only the matrix-fluid interaction is considered. We further prove rigorous properties of the coefficients, namely (a) major and minor symmetries of the effective elasticity tensor, (b) positive definiteness of the resulting Biot's modulus, and (c) analytical identities which allow us to define an effective Biot's coefficient. This model is applicable when the interactions between multiple solid phases occur at the porescale, as in the case of various systems such as biological aggregates, constructs, bone, tendons, as well as rocks and soil.