Shock-wave initiation of heterogeneous reactive solids

Abstract

Shock-wave initiation of solid explosives depends on localized regions of high temperature (hot spots) created by heterogeneous deformation in the vicinity of various defects. Current mathematical models of shock initiation tend to fall into two broad categories: (1) thermodynamic-state-dependent reaction-rate models, and (2) the continuum theory of multiphase mixtures. The level of generality possessed by (1) appears to be insufficient for representation of observed initiation phenomena, while that of (2) may exceed necessary requirements based on present measurement capabilities. As a means of bridging the gap between these two models, we present an internal-state-variable theory based on elementary physical principles, relying on specific limiting cases for the determination of functional forms. The appropriate minimum set of internal-state variables are the mass fraction of hot spots μ, their degree of reaction f, and their average creation temperature θ. The overall reaction rate λ̇, then depends on μ, f, and θ in addition to the usual macroscopic thermodynamic variables (current state as well as their history). Two specific forms of this set of equations are applied to time-resolved shock-initiation data on PBX-9404. Numerical solution is achieved by the method of characteristics for rate-dependent chemical reaction. Additional questions such as the effect of thermal equilibrium between phases (solid reactants and gaseous products) on the theoretical results are discussed quantitatively.