Yet another elasto-plasticity formulation

Abstract

An elasto-plasticity formulation is presented that requires no intermediate (stress-free) configuration, since all describing tensors are solely of proper-Eulerian or proper-Lagrangean type. This formulation—based on commutative-symmetrical elastic-plastic stretch tensor products with symmetrizing-rotation tensors in the middle—is discussed and compared with the Bilby-Kroner-Lee formulation, which defines an intermediate (stress-free) configuration that is not well-determined—as noted, e.g., by Casey & Naghdi (1980). For an Eulerian continuum description, it turns out that the symmetric elastic part of the presented formulation (with only proper-Eulerian tensors) has similarities with the elastic tensor factor eJ of the Bilby-Kröner-Lee multiplicative elasto-plastic decomposition F=eJ.pJ) of the deformation gradient F. From a Lagrangean point of view, however, the symmetric elasticity tensors of the two models differ considerably: the elastic right stretch and Cauchy-Green deformation tensor of the new formulation are proper-Lagrangean tensors, while the corresponding tensors of the Bilby-Kroner-Lee formulation are not well-determined, since they refer to an intermediate (stress-free) configuration. As finite orthotropy modeling requires a material reference configuration in which (imaginary) fibers are perpendicular to each other, finite elastic orthotropy and finite plastic orthotropy can only be modeled simultaneously based on proper-Lagrangean elastic and plastic tensors provided by commutative-symmetrical deformation tensor products and not by Bilby-Kröner-Lee formulations.