Simultaneous Lagrangian and Eulerian formulations of constitutive equations with commutative-symmetrical stretch tensor products

Abstract

Driven by the motivation to properly model isotropic elasticity, a Lagrangian formulation within the additive logarithmic strain space, which was introduced thirty years ago in conjunction with the source code development of the special-purpose finite element program AutoForm for the simulation of deep-drawing processes of thin sheet metal, has recently been generelized to a multiplicative logarithmic strain space formulation, which is essentially based on commutative-symmetrical stretch tensor products. In this context, it turns out that commutative-symmetrical stretch tensor products can model not only isotropic elasticity, but also finite-elastic and finite-plastic orthotropy of the material. This is due to the fact that commutative-symmetrical stretch tensor products are defined simultaneously for both the left/Eulerian and the right/Lagrangian (partial) stretch tensors. The Eulerian tensors of the commutative-symmetrical stretch tensor product are relevant for the modeling of isotropic material elasticity, while the Lagrangian tensors are important for the simultaneous modeling of finite-elastic orthotropy and finite-plastic orthotropy of the material. Since only the material-convective rotation tensor of the deformation gradient’s polar decomposition is employed for commutative-symmetrical stretch tensor products (also for the definitions of the left/Eulerian and right/Lagrangian elastic and plastic stretch tensors), all tensors involved are either of proper-Eulerian or of proper-Lagrangian type, i.e. defined with respect to the current/Eulerian configuration or with respect to the reference/Lagrangian configuration. A formulation with commutative-symmetrical stretch tensor products thus does not require a (zero-stress) intermediate configuration and leads to a new class of constitutive equations.