Residual-based shock capturing in solids

This paper adapts the concept of residual-based shock-capturing viscosity to the setting of solid mechanics. To evaluate the residual of the momentum balance equation, one requires the divergence of the Cauchy stress. Solid constitutive models used in simulations of extreme events involving shocks typically specify the Cauchy stress in terms of a local rate equation rather than an explicit formula involving the current deformation gradient and/or strain rate; unlike in widely-used models for fluid mechanics, there is usually no closed-form expression for spatial derivatives of the solid Cauchy stress. We therefore investigate the evolution of stress gradients given rate-form models involving various objective rates of Cauchy stress. The rate equations we derive for the stress gradient are then integrated numerically to furnish the stress divergence needed to define the shock viscosity. The properties of this shock viscosity are demonstrated in benchmark problems using Lagrangian isogeometric analysis and an immersed isogeometric–meshfree simulation framework.