Conservation laws for three-phase partially saturated granular media

We review the governing conservation laws for three-phase partially saturated media using mixture theory including finite deformation effects and kinetic energy production. Under the assumption of barotropic flows we derive the mass balance equations in their most general form, including the compressibilities of the constituent phases. We then derive the momentum balance equations including the rates of change of linear momentum in Eulerian and Lagrangian descriptions. Next we write the balance of energy equation and illustrate the conjugate relationship of the partial stress tensor with the rate of deformation of the corresponding constituent phase. Using balance of mass and balance of momentum, we rewrite the balance of energy equation once again in an equivalent form showing the conjugate relationship of an effective constitutive stress with the rate of deformation of the solid matrix. This effective constitutive stress is analogous to Bishop’s effective stress for partially saturated soils, and to Terzaghi’s effective stress for fully saturated soils.