Finsler-Geometric Continuum Dynamics and Shock Compression

A continuum mechanics theory of deformable solids is formulated to account for large deformations, nonlinear elasticity, inelastic deformation mechanisms, microstructure changes, and time dependent fields, i.e., dynamics. The theory incorporates notions from Finsler differential geometry, and it provides a diffuse interface description of surfaces associated with microstructure. Mechanisms include phase transitions and inelastic shearing, with phase boundaries and shear planes the associated surfaces. A director or internal state vector of pseudo-Finsler space is viewed as an order parameter. Newly derived in the present work are the governing equations for dynamics, including kinematic relations, balances of momentum and energy, and evolution law(s) for the internal state. Also derived are jump conditions pertinent to shock loading. Metric tensors and volume can vary isotropically with internal state via a conformal transformation. The dynamic theory is applied to describe shock loading of ceramic crystals of boron carbide, accounting for inelastic mechanisms of shear accommodation and densification upon amorphization under high pressure loading. Analytical predictions incorporating the pseudo-Finsler metric demonstrate remarkable agreement with experimental data, without parameter fitting. Additional solutions suggest that dynamic shear strength could be improved significantly in boron-based ceramics by increasing surface energy, decreasing inelastic shear accommodation in softened amorphous bands, and to a lesser extent, by increasing the energy barrier for phase transformation.