Deviatoric plastic strain increment

Norm of deviatoric plastic strain increment Hardening parameter... 

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor... 

The result of a simple tension experiment for a metal is schematically shown in Fig. 2.17 with axes of axial stress ai and axial sdain 1 or deviatoric stress s and deviatoric strain e. In metals the volumetric plastic strain can generally be ignored (sP = 0) therefore we can treat the behavior as a uniaxial response. On the other hand, the shearing behavior of geomaterials is inevitably accompanied by volume changes that are plastic, therefore we have to modify the original flow theory developed for metallic materials (Kachanov 2005 Lubliner 1990). Note that in small strain plasticity we assume that the plastic increment de can be decomposed into incremental elastic and plastic components ... 

The incremental theory of plasticity assumes that the strains depend on the entire history of loading. The total increments of deviatoric strain Asy is composed of a viscoplastic component... 

---