An asymptotic method for solving non-stationary dynamic contact problems for an acoustical layer

Abstract

An asymptotic method is proposed for solving non-stationary dynamic contact problems in elasticity theory and acoustics for the case when the half-thickness of the punch exceeds the layer thickness. The method is demonstrated by solving anti-plane non-stationary dynamic contact problems concerning the displacement by a rigid punch of an elastic layer, such problems are essentially the acoustic case of problems in elasticity theory. The problems are reduced to solving an integral equation of the first kind for the Laplace transforms of the unknown contact stresses. The zero term of the asymptotic solution of the integral equation is constructed as the superposition of solutions of the two corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation over the entire axis [1]. The symbol of the kernel of the integral equation is represented in a special form which enables the solution of the Wiener-Hopf integral equation to be reduced to the solution of an integral equation of the second kind for the Laplace-Fourier transform of the unknown contact stresses. The solution of integral equations of the second kind is constructed by successive approximations. After Laplace inversion of the zero term of the asymptotic solution of the integral equation, the asymptotic solution of the problems under consideration is determined. Formulae are presented that relate the force acting on the punch to the displacement of the punch. A law of motion is obtained for a massive punch on an elastic layer for the case when an initial velocity was communicated to the punch at the initial time.