Consider a thin, semi-infinite shell of weak' material, initially coincident with the x,y-plane. Let the surface density, pM , of this shell increase linearly, in the positive x-direction, from the value zero at x equal to zero. This uniform surface density gradient, denoted by 6, may be due, for example, to a non-uniform thickness which increases linearly in the x-direction. We suppose that the shell is in contact with an explosive material, for simplicity taken initially as semi-infinite, extending from the x,y-plane ( z = 0) to infinity in the positive z-direction. We shall suppose further that the shell (or liner) is deformed by the pressures exerted on it by the burnt explosive material behind a detonation front which is sweeping through the unburnt explosive. Let us assume that the detonation front is plane and normal to the initial plane of the liner. The normal to the detonation front is, however, inclined a t an angle Oo , measured positive in the counter-clockwise sense from the x-axis (-n/2 < Bo < n/2).
Our problem is to calculate the shape of the deformed shell.
The frame of reference we choose is one in which the detonation front is stationary and contains the origin, 0, so that all the motions are reducible to 'A weak material is one whose yield stress in tension or shear, say, is small compared to the pressures in a detonation wave.