Acceleration waves in simple elastic materials

Abstract

The classical method of characteristic surfaces and bicharacteristic rays is used to study acceleration and higher order discontinuities in nonlinear simple elastic materials. No special material symmetry is assumed, and the analysis requires neither material homogeneity nor uniform conditions ahead of the wave in general, although many detailed results do require the latter two conditions. This paper draws heavily on known results, but at the same time many other results, believed to be new, have been included. Analysis of the motion of the singular surface precedes a discussion of the amplitude equations. It is shown that the Jacobian determinant of a certain coordinate transformation, induced by the motion of the surface, plays a dominant role in the equations for transport of amplitude. New results on the kinematics of singular surfaces are given for homogeneous and uniform materials, including a formula for the evolution of surface curvature in anisotropic materials and a complete description of the role of the curvatures of the initial singular surface and of the characteristic slowness surface in producing expansion or focusing of the singular surface.

It is shown that the Jacobian determinant, mentioned above, produces a great simplification in the equations for transport of amplitude and bears significantly on the interpretation of solutions. Discontinuities of all higher orders are shown to be induced by a discontinuity of given order, and the implications of this result for transverse acceleration waves in isotropic materials are pointed out. Coupling of the modes of propagation for uniform multiplicity of slowness surfaces is described. Finally, it is shown that in directions of self intersection or conical intersection of slowness surfaces, the amplitude equations are not ordinary differential equations, but rather they are semilinear hyperbolic partial differential equations, and some of their properties are described. Hyperbolic transport equations have previously been shown to hold for some linear hyperbolic systems, but the kinematical description and reduction to a simple and readily interpretable form, as given here, are new.

It seems clear that the methods employed here could be extended to materials with constitutive behavior far more elaborate than simple elastic.