Shock initiation of explosives: the idealized condensed-phase model

Many current models of condensed-phase explosives employ reactionrate law models where the form of the rate has a power-law dependenceon pressure (i.e. proportional to pⁿ where n is an adjustableparameter). Here, shock-induced ignition is investigated usinga simple model of this form. In particular, the solutions arecontrasted with those from Arrhenius rate law models as studiedpreviously. A large n asymptotic analysis is first performed,which shows that in this limit the evolution begins with aninduction stage, followed by a sequence of pressure runaways,resulting in a forward propagating, decelerating, shocklesssupersonic reaction wave (a weak detonation). The theory predictssecondary shock and super-detonation formation once the weakdetonation reaches the Chapman–Jouguet speed. However,it is found that secondary shock formation does not occur untilthe weak detonation has reached a point close to the initiatingshock, whereas for Arrhenius rate laws the shock forms closerto the piston. Numerical simulations are then conducted forO(1) values of n, and it is shown that the idealized condensed-phasemodel can qualitatively describe a wide range of experimentallyobserved behaviours, from growth mainly at the shock, to smoothgrowth of a pressure pulse behind the shock, to cases wherea secondary shock and possibly a super-detonation form. Thenumerics are used to reveal the different evolutionary mechanismsfor each of these cases. However, the evolution is found tobe sensitive to n, with the whole range of behaviours coveredby varying n from about 3 to 5. The simulations also confirmthe predictions of the theory that pressure-dependent rate lawsare unable to describe homogeneous explosive scenarios wherea super-detonation forms very close to the point of initialrunaway.