The hardening elasto-plastic constitutive behavior of materials is modeded by defining and calibrating (by laboratory triaxial compression tests) a yield function, a plastic potential function and a hardening law (Lade, 1977). The yield function defines a stress limit above which plastic deformation occurs. The plastic potential is to describe the plastic strain evolution under a certain given stress state. The hardening (or softning) law is needed to model the evolution of the yield function and the plastic potential function with respect to the microstructural change caused by the increase of the plastic strain. The stress-strain solutions of the problem can be obtained by multi-step numerical calculations.
Here we provide an example of the classical modified Cam-Clay model that is usually employed to model the compaction of clay and soft shale (Borja et al., 1990; Pouya et al., 1998).
Lade, P. V. (1977). Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. International Journal of Solids and Structures, 13(11), 1019-1035.
Pouya, A., Djéran-Maigre, I., Lamoureux-Var, V., & Grunberger, D. (1998). Mechanical behaviour of fine grained sediments: experimental compaction and three-dimensional constitutive model. Marine and Petroleum Geology, 15(2), 129-143.
Borja, R. I., & Lee, S. R. (1990). Cam-clay plasticity, part 1: implicit integration of elasto-plastic constitutive relations. Computer Methods in Applied Mechanics and Engineering, 78(1), 49-72.
Burland, J. B. (1990). On the compressibility and shear strength of natural clays. Géotechnique, 40(3), 329-378.