A strain space-based covariant formulation of nonlinear kinematic hardening finite deformation elasto-plasticity

This article proposes a strain space-based covariant formulation of nonlinear kinematic hardening finite deformation elasto-plasticity using the metric tensor in the spatial configuration. A nonlinear kinematic hardening evolution equation is formulated in the current configuration using the Lie derivative of the kinematic hardening tensor, allowing a consistent pull-back to the reference configuration within the proposed covariant framework. The two conditions to ensure isochoric plastic deformation are derived consistently as traces of Lie derivatives of elastic left Cauchy Green and kinematic hardening tensors equal to zero. The constraint on the deviatoric parts of elastic left Cauchy Green and kinematic hardening tensors due to these conditions leads to the accumulation of error in the deviatoric parts, resulting in an unstabilized response with loading–unloading cycles. A method is proposed to correctly calculate the deviatoric and trace parts of the elastic left Cauchy Green tensor and the deviatoric part of the kinematic hardening tensor. Numerical simulations depict that the formulation based on these modifications can predict both stabilized and unstabilized responses as per the expected behavior. Further, from a numerical implementation perspective, the covariant structure of constitutive equations facilitates the straightforward integration of the constitutive equations using the covariant backward Euler integration scheme and the derivation of a closed-form expression for the tangent constitutive tensor, leading to quadratic global convergence. The developed model uses a neural network optimization and finite element model evaluation in MATLAB to quantify different material parameters from available experimental results in the literature.