Mesh insensitive formulation for initiation and growth of shear bands using mixed finite elements

Abstract

An Implicit Nonlinearly Consistent (INC) numerical solution of a partial differential equation (PDE) model for shear bands, which includes a thermo-visco-plastic flow rule and finite thermal conductivity, is presented, and is found to be insensitive to mesh size. Insensitivity is achieved through the use of finite thermal conductivity in the PDE model in conjunction with the INC numerical solver. Finite thermal conductivity gives rise to an inherent physical length scale in the PDE model, governed by competition between shear heating and diffusion. This length scale serves as a localization limiter and will regularize the problem in the strain softening regime. This occurs since diffusion removes heat from the shearband more quickly as localization becomes more severe (i.e. as temperature gradients steepen). The INC solver leaves no splitting error at the end of a time step and is accurate even during phases for which the solution is evolving very rapidly. A key point in this paper is the analytical derivation of the system Jacobian by differentiation of the weak form of the PDE model, thus avoiding the use of numerical approximation formulas. In contrast, solution of the same continuous model using an operator split solution scheme is seen to lead to unreasonably slow convergence. One and two dimensional implementations of the algorithm are presented. For two dimensions, a mixed quadrilateral using discontinuous bilinear functions for plastic strain, and the interpolants associated with the Pian-Sumihara element for the stress is implemented.