ASYMPTOTIC THEORY OF WAVE PROCESSES IN SHELLS OF REVOLUTION UNDER SURFACE IMPACT AND NORMAL END ACTIONS

• Published in: Mechanics of solids Vol. 57; no. 2; pp. 232 - 243
• Main Authors: Kirillova, I. V., Kossovich, L. Yu
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2022; Springer Nature B.V
• Subjects: Approximation; Asymptotic methods; Boundary layers; Classical Mechanics; Cylindrical shells; Elliptic functions; Fourier transforms; Integral transforms; Kirchhoff theory; Laplace transforms; Mathematical models; Physics; Physics and Astronomy; Shells of revolution; Surface waves; cylindrical shell; phase plane; asymptotic method; Fourier transform; shell of revolution; Laplace transform; conditional front of Rayleigh surface waves
• ISSN: 0025-6544; 1934-7936
• DOI: 10.3103/S0025654422020078

This article summarizes the authors’ work on mathematical modeling of the non-stationary stress-strain state (SSS) of shells under impact actions on their front surfaces and ends. Mathematical modeling is based on the use of an asymptotic method for constructing approximate theories for the components in different areas of the phase plane of the solution, where its fundamentally different properties take place and different values of the variability and dynamism indicators are observed. The areas where the two-dimensional bending component of the Kirchhoff-Love theory, the quasi-plane problem of the theory of elasticity, the hyperbolic boundary layer in the vicinity of the shear wave, and the elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves are identified. The derivation of asymptotically optimal equations for an elliptic boundary layer, which takes place both in the case of tangential surface impact actions and in the case of normal type end impacts, is given. In this case, in the case of end action, the problem for a semi-infinite shell is reduced by extracting a particular solution to an equivalent problem for an infinite shell. As an example, solutions are constructed for an elliptical boundary layer in cylindrical shells for both loading cases. The integral Laplace transforms in time and Fourier transforms in the longitudinal coordinate are used. The inversion of integral transformations made it possible to present solutions using the elementary function arctg of complex arguments. An analysis of these solutions in a small neighborhood of the quasifront (conditional front of Rayleigh surface waves) made it possible to determine their properties as they move away from the quasifront along the longitudinal coordinate.