Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

Abstract

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity i/, depends on (i) an unknown quantity ~-behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vector b along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However, b assumes different directions depending on the choice of discontinuity ~k with which one is concerned and the unknown quantity 4)-behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.

Some problems on jump conditions of shock waves in 3-dimensional solids

“…Because of its complexity, people have carried out systemic research on shock waves in fluids, and have developed a set of relatively perfect theory and method in shock dynamics, such as G.B.Whitham's [1] and Han Zhao-yuan's [2] monographies. Though people have spent lots of time and obtained respectable achievements on the shock waves in solids, such as E.H.Lee [3] and Jing Fu-qian [4] , yet most of these research works focused attention on the one-dimensional solids, while the basic research on three-dimensional solids were hardly carried out, and a few works include the T.C.T.Ting, Li Yong-chi and Tang Zhi-jing's works [5][6][7][8][9][10] on the transfer equations and reflection of shock waves. The foundations of the shock wave theory are the jump conditions across the shock wave front and the related Hugoniot relations, and this paper will present a systemic expatiation of some essential problems on jump conditions of shock waves, such as the description and derivation of the jump conditions in different coordinates and configurations, the meaning and relations between these jump conditions, the shock response curves of shock waves in three-dimensional solids, so that it can call the interests and discussion of other researchers and promote the research development of the shock waves in solids.…”