Cam-Clay plasticity. Part V: A mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media

We present a mathematical framework for deformation and strain localization analyses of partially saturated granular media using three-phase continuum mixture theory. First, we develop conservation laws governing a three-phase mixture to identify energy-conjugate expressions for constitutive modeling. Energy conjugate expressions identified relate a certain measure of effective stress to the deformation of the solid matrix, the degree of saturation to the matrix suction, the pressure in each phase to the corresponding intrinsic volume change of this phase, and the seepage forces to the corresponding pressure gradients. From the second of law of thermodynamics we obtain the dissipation inequality; from the principle of maximum plastic dissipation we derive a condition for the convexity of the yield function. Then, we formulate expressions describing conditions for the onset of tabular deformation bands under locally drained and locally undrained conditions. Finally, we cast a specific constitutive model for partially saturated soils within the proposed mathematical framework, and implement it in the context of return mapping algorithm of computational plasticity. The proposed constitutive model degenerates to the classical modified Cam-Clay model of soil mechanics in the limit of full saturation. Numerical examples are presented to demonstrate the performance of the return mapping algorithm as well as illustrate the localization properties of the model as functions of imposed deformation and matrix suction histories.