Preserving non-negative porosity values in a bi-phase elasto-plastic material under Terzaghi's effective stress principle

Poromechanics is a well -established field of continuum mechanics which seeks to model materials with multiple phases, usually a stiff solid phase and fluid phases of liquids or gases. Applications are widespread particularly in geomechanics where Terzaghi's effective stress is widely used to solve engineering soil mechanics problems. This approach assumes that the solid phase is incompressible, an assumption that leads to many advantages and simplifications without major loss of fidelity to the real world. Under the assumption of finite (as opposed to infinitesimal) strains, the poromechanics of two- or bi-phase materials gains complexity and while the compressible solid phase case has received attention from researchers, the incompressible case has received less. For the finite strain - incompressible solid phase case there is a fundamental issue with standard material models, in that for some loadings solid skeleton mass conservation is violated and negative Eulerian porosities are predicted. While, to the authors' best knowledge, acknowledgement of this essential problem has been disregarded in the literature, an elegant solution is presented here, where the constraint on Eulerian porosity can be incorporated into the free energy function for a material. The formulation is explained in detail, soundly grounded in the laws of thermodynamics and validated on a number of illustrative examples.