On existence-uniqueness of the solution in a nonlinear Biot’s model

The response of elastic porous media under applied loads consists of an instantaneous deformation followed by a time dependent consolidation process associated with the drainage of the pore fluid. In the simulation of the swelling of elastic cartilagineous tissues, the permeability tensor of the porous medium depends on the strain, thus resulting in a nonlinear model. In this paper, we present a nonlinear one dimensional Biot’s model and prove the existence and uniqueness of the solution of this model. Appropriate boundary conditions required for the uniqueness of the solution are to be introduced. Then, the Galerkin method is used to prove that the model has a unique weak solution. Finally, two simple numerical examples of 1-D non linear Biot’s model are discussed.