Asymptotic methods in the axisymmetric dynamic non-stationary contact problem for an elastic half-space

Abstract

Asymptotic methods for solving the axisymmetric dynamic non-stationary contact problem for short and long values of the time of indentation of a rigid punch into an elastic half-space are developed. Using Laplace integral transformations (with respect to time) and Hankel integral transformations (with respect to the coordinate) the contact problem is reduced to solving an integral equation in the unknown Laplace transformant of the contact stresses under the punch. The zeroth term of the asymptotic solution of the integral equation for large values of the Laplace parameter (short times) is constructed using a special approximation in the complex plane of the symbol of the integral-equation kernel. The asymptotic solution of the integral equation for small values of the Laplace parameter (long times) is constructed in powers of this parameter. The solution of the contact problem is obtained using an inverse Laplace transformation, applied to the solutions of the integral equation.