Analysis of shock compression of strong single crystals with logarithmic thermoelastic-plastic theory

A finite strain theory is developed for anisotropic single crystals undergoing shock loading. Inelastic deformation may arise from dislocation slip, twinning, or fracture and crack sliding. Internal energy can generally depend on a logarithmic measure of finite elastic strain, entropy, and an internal variable associated with defect accumulation. A closed form analytical solution is derived for the planar shock response in the thermoelastic regime, at axial stresses up to the Hugoniot elastic limit. In the plastic regime, for highly symmetric orientations and rate independent shear strength, the Rankine–Hugoniot conditions and constitutive relations can be reduced to a set of algebraic equations that can be solved for the material response. The theory is applied towards planar shock loading of single crystals of sapphire, diamond, and quartz. Logarithmic elasticity is demonstrated to be more accurate (i.e., require fewer higher-order elastic constants) than Lagrangian or Eulerian theories for sapphire, diamond, and Z-cut quartz. Results provide new insight into criteria for initiation of twinning, slip, and/or fracture in these materials as well as their strength degradation when shocked at increasingly higher pressures above the Hugoniot elastic limit.