Elastic Wave Propagation in Homogeneous and Inhomogeneous Media

Abstract

A general method is developed for the solution of the linearized equations of elasticity for both homogeneous and inhomogeneous media. This method yields solutions which describe propagating waves which may be pulses, rapidly changing wave forms, or periodic waves. It is not restricted by the usual considerations which depend upon separation of variables. The solution consists of a series of terms, the first of which describes the wave motion predicted by geometrical optics. Subsequent terms account for certain types of diffraction effects. The series is not necessarily convergent but is presumably asymptotic to the solution.

Fracture Mechanics and Contact Problems in Materials Involving Graded Coatings and Interfacial Zones

“…(5), observing that for the mode I problem under consideration f 2 (x) = 0 and q(x) = 0, replacing the condition (23a) by (5a), and substituting from (10), (11), (17), (19) and (2) into (20)- (23), we obtain the following expressions giving Ci,..., C^B^B^ in terms of /i(x):…”

“…In 1946 Friedlander [18] proposed a solution that consists of a series of terms the first of which describes the wave motion predicted by geometrical optics and the subsequent terms account for certain types of diffraction effects. Karal and Keller [19] extended this method to treat general wave propagation problems in nonhomogeneous elastic media by formulating the problem in terms of displacements and displacement potentials. Pekeris [20] used an asymptotic method to solve the problem for a half-space with a variable speed of sound and reduced the solution to Fourier-Bessel series.…”

“…In this solution the eqauilibrium condition (7) and the compatibility relations (36) are used as additional conditions. Also, the following property of Jacobi polynomials is used to regularize the singular parts of the integral equations (24), that is the terms involving the Cauchy kernels and the free term (19) and (20) for the procedure to be followed) and the stress intensity factors at the crack tip (d, 0) may be evaluated by using the results. The stress intensity factors are defined by and calculated from…”

“…where k y j(y,t), (j= 1,2,3) are known kernels corresponding to the in-plane stress components a yy j(0,y), (j= 1,2,3), (see, for example (19) and (20) for the in-plane stress due to sliding contact). Knowing the stress components a xx , a yy and a xy on the surface, at the critical location (that is, the trailing end y = a of the contact region) the cleavage stress aee(r, 9) may be expressed as <TM{r, 9) = a xx sin 2 (9) + a yy cos' 2 (9) -a xy sm (9)cos (9),…”

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

“…[11,12,34]. In elastodynamics and visco-elastodynamics an analogous procedure involves truncation of wave front expansions such as were investigated by Karal and Keller [24]. Exact solution may thereby be obtained to a variety of initial boundary value problems for inhomogeneous elastic and viscoelastic media (see e.g.…”