A Study of Detonation Diffraction in the Ignition-and-Growth Model

Mathematical modeling of heterogeneous high-energy explosives poses a vexing problem. These materials have a complex microstructure in which crystalline fragments of the energetic material are held together by an inert plastic binder, and voids and pores abound within the granular aggregate. It is observed that when a sample of such an explosive is subjected to a sufficiently strong stimulus, such as impact by a high-velocity projectile, a detonation is initiated. Our knowledge of the thermo-mechanical behavior of the material over the broad ranges of pressure and temperature encountered in a detonation is incomplete, and the same is true of the complex set of reactions that are responsible for the liberation of energy. The morphological complexity, the dearth of information, and the multi-scale nature of material response are insurmountable blocks in the way of any attempt at ab-initio modeling of the detonation phenomena, at least at the present time. On certain aspects of the problem, however, there is broad agreement. It is known that while the crystalline explosive possesses strong ignition thresholds, relatively weaker stimuli are sufficient to initiate the heterogeneous explosive. This propensity is attributed to the creation of a nonuniform temperature distribution when the heterogeneous aggregate is exposed to the initiating shock. Discrete sites, where the local temperature far exceeds the bulk average, are generated as a result of mechanical processes such as friction, pore collapse, shear banding and local plastic deformations. These sites, or hot spots, act as preferred locations of ignition, where burning commences and then spreads to consume the entire bulk. With this picture in mind, efforts have been directed at constructing phenomenological, macro-scale, continuum-type models. These include the Forest-fire model [1, 2], the JTF model [3] and the HVRB model [4], but the most celebrated of the lot is the ignition-and-growth model, originally derived by Lee and Tarver [5] and later refined and utilized by Tarver and colleagues in a large number of publications [6, 7, 8, 9, 10]. Additional references can be found in the recent paper by Tarver [11]. The model treats the explosive as a homogeneous mixture of two distinct constituents, the unreacted explosive and the products of reaction. To each constituent is assigned an equation of state, and a single reaction-rate law is postulated for the conversion of the explosive to products. It is assumed that the two constituents are always in pressure and temperature equilibrium, and that the energy and volume of the mixture is the sum of the corresponding quantities for the individual constituents, weighted by the variable that measures the progress of reaction. The model contains a large number of parameters which are experimentally calibrated to the explosive of interest. The model has had considerable success, though not as a tool which, once calibrated to a certain suite of experiments, has the ability to predict behavior broadly, such as in an entirely different set of experiments.