A micropolar isotropic plasticity formulation for non-associated flow rule and softening featuring multiple classical yield criteria Part I -- Theory

The Cosserat continuum is used in this paper to regularize the ill-posed governing equations of the Cauchy/Maxwell continuum. Most available constitutive models adopt yield and plastic potential surfaces with a circular deviatoric section. This is a too crude an approximation which hinders the application of the Cosserat continuum into practice, particularly in the geotechnical domain. An elasto-plastic constitutive model for the linear formulation of the Cosserat continuum is here presented, which features non-associated flow and hardening/softening behaviour, whilst linear hyper-elasticity is adopted to reproduce the recoverable response. For the formulation of the yield and plastic potential functions, a definition of the \textit{equivalent von Mises stress} is used which is based on Hencky's interpretation of the von Mises criterion and also on the theory of representations. The dependency on the Lode's angle of both the yield and plastic potential functions is introduced through the adoption of a recently proposed \textit{Generalized classical} criterion, which rigorously defines most of the classical yield and failure criteria.