Homogenization and numerical modelling of poroelastic materials with self-contact in the microstructure

Abstract

We present a two-scale homogenization-based computational model of porous elastic materials subject to external loads inducing the self-contact interaction at the pore level. Microstructures under consideration are constituted as periodic lattices generated by a representative cell consisting of a solid skeleton and a pore. On its surface, the unilateral frictionless contact appears when the porous material is deformed. We focus on microstructures with rigid inclusions whereby the contact process involves opposing surfaces on the rigid and the compliant skeleton parts. A macroscopic model is derived using the periodic unfolding homogenization and the method of oscillating test functions. An efficient algorithm for the two-scale computational analysis is proposed for the numerical model obtained using the finite element discretization of the homogenized model. For this, a sequential linearization of the two-scale elasticity problem leads to the consistent effective elasticity tensor yielding consistent stiffness matrices of the macroscopic incremental formulation. The micro-level contact problem attains the form of a nonsmooth equation solved using the semi-smooth Newton method without any regularization, or problem relaxation. Numerical examples of two-dimensional deforming structures are presented as a proof of the concept. The proposed modelling approach can be extended to treat self-contact in structures subject to finite deformation.