A simple formulation for large-strain cyclic hyperelasto-plasticity using elastic correctors. Theory and algorithmic implementation

Abstract

Proper finite element modelling of elastoplastic behavior under cyclic and multiaxial loading requires the consideration of nonlinear kinematic hardening. Popular models available for nonlinear kinematic hardening are based on multiple additive backstresses, whose evolution include a dynamic recovery term and follow the Armstrong-Frederick proposal; among them, the Ohno-Wang models. Whereas the small strain theory and its numerical implementation are satisfying, large strain extensions are more controversial, specially regarding the mathematical treatment of flow kinematics and the numerical implementation. In this work we present a new approach for modelling nonlinear kinematic hardening at large strains, reproducing the Ohno-Wang model at small strains without explicitly employing the backstress concept. The formulation uses only the classical Kröner-Lee multiplicative decomposition. It avoids the Lion decomposition and it is fully hyperelastic, employing only elastic variables both in the elastic and hardening parts, as well as in the flow equations. The theory has no restriction on the form of stored energies or in the amount of elastic strains so it can be used in soft materials, and it is weak-invariant and volume-preserving by construction. Furthermore, it has the same additive structure of classical small strain algorithms. Geometrical mapping tensors are systematically employed to account for large strain kinematics whereas the iterative algorithmic part is identical to the small strains model, which is recovered bypassing the geometrical mappings. The modelling of visco-hyperelastoplasticity is also straightforward by combining the present theory with finite nonlinear viscoelasticity formulations based on the same framework previously developed by our group.