Extended formulations of evolutive laws and constitutive relations in non-smooth plasticity and viscoplasticity

Plastic and viscoplastic constitutive behavior is of interest in the mechanical modeling of many composite materials and structures. In this paper evolutive laws and constitutive relations in non-smooth plasticity and viscoplasticity are presented by means of a formulation which takes advantage of the proper concepts required to deal with non-smooth problems in plasticity and viscoplasticity. This adopted framework is endowed with considerable advantages in comparison with other formulations of non-smooth problems. In fact, subdifferential calculus shows to be the proper tool to deal with non-smooth functions and corners in plasticity and viscoplasticity. Plastic and viscoplastic constitutive models are revisited and expressed in subdifferential form by adopting the more general context presented herein that includes Koiter's theory as a special case. The evolutive equations in plasticity and viscoplasticity are derived as optimality conditions of a suitably defined Lagrangian in a form usually not considered. Consequently, alternative equivalent expressions of the evolution laws and of the loading/unloading conditions are presented and the equivalence among them is described. Furthermore, the present approach shows to be useful for extensions into other types of elastoplastic materials and for clarifications on the relations existing between different constitutive models in non-smooth viscoplasticity.