Singularities of the interaction of a vibrating stamp with an inhomogeneous heavy base

Abstract

A method is developed for studying the fundamental characteristics of the wave process on the surface of an initally isotropic prestressed elastic half-space caused by an oscillating rigid stamp. The following is taken as the model of the inhomogeneous medium: an elastic layer 0 ⩽ x 3 ⩽ h, x 1, x 2, < ∞ whose mechanical characteristics as well as the initial stresses are arbitrary, fairly smooth functions of the coordinate x 3 in the general case, lies on the surface of a homogeneous half-space x 3, ⩾ h, x 1, x 2, < ∞ ( x 1, x 2, x 3, are a rectangular Cartesian coordinate system). The linearized boundary value problem of the dynamic theory of elasticity of vibrations with frequency ω for a rigid stamp on the surface of an inhomogeneous medium reduces to an integral equation or to a system of integral equations of the first kind whose integral operator kernel is constructed numerically. Approximation of the kernel of the integral operator by a special kind of function enables an approximate solution of the integral equation to be constructed by the factorization method /1, 2/. On the basis of the latter, an effective investigation of the influence of the parameters characterising the inhomogeneity of the medium and the initial state of stress on the wave process both under (stress wave) and outside the stamp is possible. The construction of a general linearized theory and the regularities of elastic wave propagation in bodies with homogeneous initial stresses are considered in /3/, where a fairly complete survey is also given of the literature on this question. A systematic exposition of the theory of wave propagation in elastic media with an inhomogeneous initial state was first given in /4/. The contact problem of the vibrations of an inhomogeneous half-space subjected to a rigid stamp oscillating on its surface was examined in /5, 6/ without taking the initial stresses into account. A method of investigating the regularities of electric-wave excitation in semibounded bodies (layers and cylinders) with varying properties and magnitude of the initial stresses was proposed in /7/ on the basis of the solution of the contact problem. A method of investigating the singularities of elàstic wave propagation in an inhomogeneous initially strained half-space caused by an oscillating load distributed in a certain domain on the surface of the medium has been developed (∗Kalinchuk V.V., Lysenko I.V. and Polyakova I.B., Singularities of elastic wave excitation and propagation in an inhomogeneous heavy half-space. Deposited Manuscript, December 10, 1986, 8877-B86, VINITI, Rostov, 1986).