Large deformation plasticity without \({\textbf {F}}^e{\textbf {F}}^p\): a basic Riemannian geometric model for metals

Abstract

We propose a continuum viscoplasticity model for metals where the kinematic aspects and those pertaining to microstructural reorganizations are intrinsically described through Riemannian geometry. Towards this, in addition to a Euclidean deformed manifold, we introduce a time-parametrized Riemannian material manifold where a metric tensor characterizes the irreversible configurational changes due to moving defects, e.g. dislocations or grain boundaries causing plastic deformation. Moreover, we also make use of a time-parametrized Euclidean reference manifold which shares the same macroscopic shape/size as the material manifold. The setup dispenses with the need for a multiplicative decomposition of the deformation gradient. Constitutive closure of the unknown fields, appearing in the metric tensor, is organised through two-temperature non-equilibrium thermodynamics. The approach naturally leads to terms containing higher order gradients of variables describing plastic deformation. Use of the virtual power principle yields a macroscopic force balance for mechanical deformation and a microscopic force balance giving the nonlocal flow rule. Evolution equations for the two temperatures are also coupled with plastic deformation. Numerical simulations on homogeneous and inhomogeneous deformation in oxygen-free high conductivity copper are carried out to validate the model. Simulations of an inhomogeneous deformation scenario, the Taylor impact test to wit, are then performed. To further explore the model, we simulate shear band propagation in a doubly notched plate under impact. The study offers interesting insights into the role of Riemann curvature in band formation.