Shock initiation of explosives: the idealized condensed-phase model

Many current models of condensed-phase explosives employ reaction rate law models where the form of the rate has a power-law dependence on pressure (i.e. proportional to p(n) where n is an adjustable parameter). Here, shock-induced ignition is investigated using a simple model of this form. In particular, the solutions are contrasted with those from Arrhenius rate law models as studied previously. A large n asymptotic analysis is first performed, which shows that in this limit the evolution begins with an induction stage, followed by a sequence of pressure runaways, resulting in a forward propagating, decelerating, shockless supersonic reaction wave (a weak detonation). The theory predicts secondary shock and super-detonation formation once the weak detonation reaches the Chapman-Jouguet speed. However, it is found that secondary shock formation does not occur until the weak detonation has reached a point close to the initiating shock, whereas for Arrhenius rate laws the shock forms closer to the piston. Numerical simulations are then conducted for O(1) values of n, and it is shown that the idealized condensed-phase model can qualitatively describe a wide range of experimentally observed behaviours, from growth mainly at the shock, to smooth growth of a pressure pulse behind the shock, to cases where a secondary shock and possibly a super-detonation form. The numerics are used to reveal the different evolutionary mechanisms for each of these cases. However, the evolution is found to be sensitive to n, with the whole range of behaviours covered by varying n from about 3 to 5. The simulations also confirm the predictions of the theory that pressure-dependent rate laws are unable to describe homogeneous explosive scenarios where a super-detonation forms very close to the point of initial runaway.