Finite strain analysis of shock compression of brittle solids applied to titanium diboride

Abstract

A finite strain theory is developed for polycrystalline brittle materials undergoing shock loading. Inelastic deformation arises principally from extension and opening or sliding of microcracks, and depends on pressure as well as deviatoric stress. In the general theory, internal energy depends on a logarithmic measure of finite elastic strain, entropy, and an internal variable associated with fracture. The theory is applied towards planar shock loading of an isotropic sample under possible static pre-stress. An exact analytical solution is derived when inelasticity is idealized as rate independent. The model and solution are applied to describe polycrystalline ceramic titanium diboride. Results provide new insight into experimental shock data, demonstrating importance of elastic nonlinearity and pressure dependent strength. The model describes shock pressure, mean stress, and shear stress in shocked titanium diboride, including the double yield point, with a minimal number of fitting parameters. The analysis predicts an increase in the Hugoniot elastic limit and suppression of inelasticity with increasing compressive pre-stress, in agreement with recent experiments.