A simple model of porous media with elastic deformations and erosion or deposition

This paper deals with a model for solids with porous channels filled by an incompressible isotropic fluid. The Darcy–Brinkman–Stokes law is derived, that represents a rate equation for the local mass flux of the fluid, presenting relaxation times in which this flux evolves towards its local-equilibrium value, viscous effects and a permeability tensor referring to a response of the system to an external agent, i.e. the fluid flow produced by a pressure gradient. The erosion/deposition phenomena in an elastic porous matrix are also studied and particular thermal porous metamaterials, that have interesting functionality, like in fluid flow cloaking, are illustrated as application of the obtained results. This derived model is completely in agreement with a theory formulated in the framework of the rational irreversible thermodynamics, where two internal variables are introduced (a symmetric structural porosity tensor and a symmetric second order tensor influencing viscous phenomena, that is interpreted as the symmetric part of the velocity gradient), when the results are considered in a first approximation and some suitable assumptions are done. The constitutive theory is worked out by using Liu’s and Wang’s theorems. The obtained theory has applications in several technological sectors, like physics of soil, pharmaceutics, physiology, etc., and contributes to the study of new porous metamaterials.