Asymptotic method in problems of elliptic boundary layer in shells of revolution under impacts of normal type

The asymptotic method for studying the behavior of non-stationary waves in thin shells generally involves using the separation method of solutions in the phase plane into components with different indices of variability in coordinates and time. In the case of normal type of impact, one of these components is an elliptical boundary layer occurring in a small neighborhood of the surface Rayleigh wave front. Its equations are derived by the method of asymptotic integration from the three-dimensional equations of elasticity theory. And they are partial differential equations of elliptic type with boundary conditions specified by hyperbolic equations. The article presents a general asymptotic method for solving the equations of the boundary layer under consideration in the case of the arbitrary form shell of revolution as an example. It is based on a preliminary study of basic problems for shells of revolution of zero Gaussian curvature using integral Laplace and Fourier transforms. The equations of this boundary layer for different types of normal loading have a common characteristic property: the asymptotically principal components coincide with the corresponding equations for shells of revolution of zero Gaussian curvature. This property, together with the property of different variability of the components of the stress-strain state and geometric parameters, allows, when using the method of exponential representations in the Laplace transform space, to functionally relate the solutions in the case of the arbitrary form shell of revolution with the solutions for shells of revolution of zero Gaussian curvature. The developed general approach is applied in this article to solving the problem of an elliptical boundary layer in shells of revolution under normal type loading. A numerical calculation of the shear stress for the obtained asymptotic solution in the case of a spherical shell is given.