A note on the implementation of return mapping algorithms and consistent tangent operators in isotropic elastoplasticity

It is now well understood that accurate and stable algorithms for integrating the rate constitutive equations in elastoplasticity are of major importance for carrying out efficient stress computation schemes; furthermore, the paramount role of the consistent tangent has been put forward by several authors. Nevertheless, in many cases the exact consistent linearization may be demanding or computationally expensive to obtain. A first source of difficulty in obtaining consistent tangent operators lies in the evaluation of the gradient of the plastic flow i.e., for standard models, in computing the second derivatives of the yield function. This is however only a preliminary task to accomplish since the complete linearization requires the inversion of the jacobian associated with the local stress computation scheme. This topic has been previously discussed by the authors and an intrinsic representation of the consistent tangent and its explicit expression with no use of matrix operations has been arrived at; these ideas have been further elaborated by extending the treatment to the principal axis formulation of isotropic plasticity. Objective of this work is to present an implementation of the return mapping algorithm and of the consistent tangent that aims to take proper advantage of the isotropic properties of the model. In particular, we provide an entirely intrinsic representation of all the tensor variables that enter the stress computation algorithm and, by properly exploiting the linearized form of the residual equations, we derive a novel intrinsic expression of the consistent tangent which, besides being more compact and effective with respect to other existing ones, is also amenable to a direct specialization to the plane stress case.