A mathematical model of ductile hole growth under the application of a mean tensile stress is developed and applied to the problem of spallation in solids. The object is to describe dynamic ductile fracture under a wide range of tensile loading conditions. The mathematical model presented here describes both plate-impact spallation (as observed by postshot examination and time-resolved pressure measurements) and explosively produced spallation (as observed by dynamic x-radiographic techniques) in copper. It is found to be inapplicable to ductile fracture of expanding rings, even in the absence of possible adiabatic shear banding and classical necking instabilities, because of the fact that the mean tensile stress (void growth) and the deviatoric stress (homogeneous plastic shear strain) are not independent. A phenomenological model of void growth under uniaxial stress conditions is developed independently and applied to the numerical finite-difference solution of fracture in an expanding ring. The initial porosity in a material element is a random variable following Poisson statistics and the assumption that all the void radii are equal. The necessary theoretical generalizations and supporting experimental measurements to improve our understanding of fracture and fragmentation in expanding rings are discussed.

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A microphysically based material model for the dynamic inelastic response of a brittle material is developed. The progressive loss of strength as well as the post-failure response of a granular material with friction are included. Crack instability conditions (an inelastic surface in stress space) and inelastic strains are obtained by considering the response of individual microcracks to an applied stress field. The assumptions of material isotropy and an exponential distribution for the crack radius are invoked to provide a tractable formulation. The constitutive model requires a minimal number of physical parameters, is compatible with a previously developed ductile fracture model [J. Appl. Phys. 64, 6699 (1988)] that utilizes inelastic surfaces, and can be formulated as an efficient, robust numerical algorithm for use in three-dimensional computer codes. The material model is implemented into a Lagrangian computer formulation for the demonstration of material response to dynamic loading conditions. Comparisons with one-dimensional, uniaxial impact experiments are provided.

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 p, their degree of reaction J, and their average creation temperature e. The overall reaction rate A, then depends on /1-, f, and e 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 timeresolved 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.

Under certain conditions, plastic wave profiles in 6061-T6 aluminum may achieve a constant wave velocity and steady shape within a few millimeters from the impact surface in a plate-impact experiment. The finite rise time of the steady plastic wave is assumed to be controlled by the motion of dislocations within the solid. The theory of steady-propagating waves is presented and theoretically determined wave profiles are compared with those measured experimentally by means of laser interferometry. These studies provide information on dislocation velocity and multiplication under conditions of shock-wave compression. In particular, if the mobile dislocation density is assumed to be a function of plastic strain alone, the dislocation velocity is found to be proportional to (τ−τ0)n, where τ is the applied shear stress, τ0 is a back stress, and n is a constant approximately equal to 2. Thus, it appears that a linear relationship between dislocation velocity and shear stress, which has been found to apply for strain rates less than 10−2 μsec−1 in aluminum, may not be sufficient to describe rate-dependent behavior at strain rates greater than 10−2 μsec−1, which are achieved in shock-wave compression.