A Theory of Viscoelastic Analogy for Wave Propagation Normal to the Layering of a Layered Medium

Abstract

Plane wave propagation in the direction normal to the layering of a periodically layered medium is studied. A period consists of two layers of homogeneous, linear elastic or viscoelastic materials. A theory is presented in which the layered medium is replaced by an “equivalent” linear homogeneous viscoelastic medium such that the stress response in the latter and in the layered medium are identical at points which are the centers of the odd layers. A means for determining the relaxation function of this equivalent homogeneous viscoelastic medium is presented and the transient waves in the layered medium are obtained by solving the transient waves in the equivalent homogeneous viscoelastic medium. Stieltjes convolutions with “auxiliary” functions are used to determine the solutions at points other than the centers of the odd layers and at points in the even layers. Numerical examples are presented for an elastic layered medium and comparisions are made with solutions obtained by the ray theory. The results show that the present theory can predict satisfactorily the transient response at any point in the layered medium regardless of whether the point is near or far from the impact end.