Properties and simplifications of constitutive time‐discretized elastoplastic operators

In the paper, a general constitutive elastoplastic model for associated plasticity is investigated. The model is based on the thermodynamical framework with internal variables and can include basic plastic criteria with a combination of kinematic hardening and non-linear isotropic hardening. The corresponding initial value constitutive elastoplastic problem is discretized by the implicit Euler method. The discretized one-time-step constitutive problem defines the elastoplastic operator, which is formulated by a simple generalization of a projection onto a convex set. Properties of the so-called generalized projection are used for deriving basic properties of the elastoplastic operator like potentiality, monotonicity, Lipschitz continuity and local semismoothness. Further, hardening variables are eliminated from the projective definition of the elastoplastic operators, which yields relations among the models with hardening variables and the perfect plasticity model. Also a simplification of the operator for plastic criteria in eigenvalue forms is introduced. The simplifications are useful for a numerical implementation and can be used for deriving other properties like strong semismoothness of the elastoplastic operators for the classical plastic criteria or strong monotonicity of the stress-strain operator for some models with hardening. The derived properties can be important for convergence analyses of Newton-like methods and other mathematical and numerical analyses.