This dissertation explores the physics of compressive and tensile shock waves in compressible solids, focusing on thermoelastic (TE) and thermoviscoelastic (TVE) materials with and without rheology. Shock phenomena in compressible solids are inherently complex, involving nonlinear wave interactions, dissipation mechanisms, and material-specific properties. Through the development of mathematical models, numerical methods, and detailed model problem studies, this work provides a comprehensive understanding of these processes. The mathematical framework is constructed using conservation and balance laws (CBL) of classical continuum mechanics (CCM) and constitutive theories derived from the entropy inequality and the representation theorem. The models incorporate finite deformation and finite strain physics, making them essential for studying compressible materials. For TE and TVE solids, dissipation is modeled through strain-rate-dependent mechanisms, while TVE solids with rheology account for the additional influence of long-chain molecular structures, introducing relaxation phenomena and a spectrum of viscosities and relaxation times. The equations governing these shock phenomena are solved using a space-time coupled finite element method. This variationally consistent approach, based on space-time residual functionals, ensures stability, accuracy, and efficient error estimation. The computational framework is applied to one-dimensional rod geometries to study shock physics. A simple equation of state is considered to illustrate various aspects of the shock physics and its effects on the wave propagation and formation of the shocks. Numerical studies for compressive and tensile shock physics are presented for 1D wave propagation in a rod model of TES, TVES, and TVES with memory. These model problem studies reveal distinct behaviors of compressive and tensile shock waves. Compressive shocks lead to sharp increases in density, while tensile shocks produce density reductions, showcasing their inherently different natures. The introduction of rheology in TVE solids further enriches the shock dynamics, demonstrating enhanced dissipation and wave attenuation due to the interplay of molecular relaxation and viscosity. Beyond one-dimensional studies, the dissertation presents an extension to two-dimensional compressive shock physics, offering a more complete mathematical model and computational approach for multidimensional problems. By bridging theoretical modeling, numerical solutions, and detailed shock physics studies, this work contributes to the fundamental understanding of compressible solid mechanics. The results have implications for advanced materials and applications requiring precise control and prediction of wave dynamics in solids.
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