A first-gradient approach to the remodeling and fluid flow in saturated porous media

We rephrase Gurtin and Anand’s formulation of strain-gradient plasticity (“A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations”. Int. J. Plasticity , 2005) to describe the isochoric structural transformations (remodeling) of multicellular aggregates in in silico compression tests. We consider solid–fluid biphasic media, thereby accounting for interactions that cannot arise in classical elasto-plastic materials. To gain insight into the behavior of the fluid, especially in the proximity of the aggregate’s boundary, we introduce a Darcy–Brinkman model, coupled with the deformation of the solid. This results in a constitutive framework of grade 1 in the fluid velocity (Darcy’s law is of grade 0), wherein the stress tensor of the fluid acquires a dissipative contribution. To obtain the equations determining the system’s evolution, we adopt the Principle of Virtual Power, which allows us to handle explicitly the internal constraints of incompressibility of the solid–fluid mixture and of isochoricity and null-spin of the remodeling rate tensor. Furthermore, we enforce the Principle of Maximum Dissipation to justify the generalized dissipative forces of our model. Finally, we discuss some relevant results of a numerical experiment, and we provide a brief computational background for the initial- and boundary-value problem representing our model.