Three-dimensional acceleration waves are studied for a large class of materials which includes nonlinear elastic materials, finite linear viscoelastic materials, elastic-plastic materials, hypo-elastic materials, and materials with fading memory. Thermodynamic effects are not included. The material is allowed to be inhomogeneous, anisotropic and undergoing an arbitrary motion ahead of the wave. The purpose of this study is to show how the singular surface theory of continuum mechanics can be used to investigate the effect of material motions ahead of the wave on the growth of the wave amplitude. Results are expressed in terms of the Cauchy stress tensor and the geometry of the wave in the current configuration, and also in terms of the Piola-Kirchoff stress tensor and the geometry of the wave in the reference configuration. The general formulae are applied to plane waves in laminated elastic plates and cylindrical waves in laminated cylindrical shells
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