Consequences of Terzaghi’s effective stress decomposition in the context of finite strain poro-mechanics

Poro-mechanics is a branch of mechanics considering the hydro-mechanical behaviour of a porous solid medium whose pores are saturated by a fluid. The presence of both these constituents significantly influences the overall macro-response of the material. Regardless of the application, Terzaghi’s effective stress decomposition is a well-established hypothesis regarding how the total stress tensor is decomposed into a part directly related to strain (effective stress) and a part borne by the fluid phase (interstitial pressure). The assumption of an incompressible solid medium physically justifies Terzaghi’s effective decomposition. However, under the further assumption of finite strain mechanics, a more restrictive constraint on the volume change arises, which the vast majority of formulations neglect. This work details how disregarding this constraint on the volume change can lead to the violation of solid mass balance, which can be counted among the fundamental principles of continuum mechanics. Furthermore, to address this issue practically, an ad hoc stress-strain relationship is designed to respect Terzaghi stress decomposition and solid mass balance in the context of finite-strain hyper-elasticity.