Elliptic boundary layer in shells of revolution under surface shock loading of normal type

This article presents a method for solving the boundary-value problem for an elliptic boundary layer occurring in thin-walled shells of revolution under impact loads of normal type applied to the face surfaces. The elliptic boundary layer is formed in the vicinity of the conditional front of Rayleigh surface waves and is described by elliptic equations with boundary conditions determined by hyperbolic equations. In the general case of shells of revolution, methods for solving elliptic boundary layer equations developed for shells of revolution with zero Gaussian curvature cannot be applied. The previously considered approach using Laplace and Fourier integral transforms fails because the governing equations become equations with variable coefficients. The method proposed in this article for solving the equations of the elliptic boundary layer is based on the use of asymptotic representations of the Laplace-transformed solutions (in time) in exponential form. Numerical calculations of normal stresses based on the obtained analytical solutions are provided for the case of a spherical shell.