Numerical simulation of elastic–plastic solid mechanics using an Eulerian stretch tensor approach and HLLD Riemann solver

Abstract

An Eulerian, multi-material numerical method is described for computing dynamic problems involving large deformations in elastic–plastic solids. This approach addresses algorithm failures associated with reconnection and change in topology observed in previously proposed formulations. Among the information contained in the deformation gradients commonly employed for defining constitutive laws suitable for solids, only the symmetric matrix tensor obtained from a polar decomposition of the elastic component of the deformation is required to determine the stress state. The numerical utilization of this symmetric tensor, associated with material stretch, eliminates undesirable, discontinuous deformation states produced by local rigid-body rotations at same-material reconnecting interfaces. Such states appear even where stress states in impacting regions are similar. The temporal evolution of the stretches neither modifies the eigenstructure of the system of equations nor changes its size. We also present a new multi-material approximate Riemann solver based on the HLLD approach, previously applied to other hyperbolic systems, in which waves of distinct velocity exist, for example, as in magnetohydrodynamics. This solver is employed in combination with the modified ghost fluid method (M-GFM) in the description of multi-material interfaces. These composite algorithms enable numerical simulations of the Richtmyer–Meshkov instability (i.e., the instability produced by the interaction of an interface separating materials of different density with a shock wave at an angle) in converging geometries for solid materials that would have previously led to the failure of the method.