The method of ray expansions for investigating transient wave processes in thin elastic plates and shells

摘要

In the present paper, starting from equations of three-dimensional theory of elasticity, the system of the recurrent equations of the ray method is obtained allowing one to describe the dynamic behavior of thin solids. It is shown that the implementation of the recurrent equations of the ray method and the ray series derived from these equations is preferred over the hyperbolic equations of the Timoshenko type for finding the solution for the problems resulting in the generation and propagation of surfaces of discontinuity in thin bodies. This is due to the fact that in the theories of the Timoshenko type there occur two shear waves, on which shear disturbances along and transverse to the plate's middle surface propagate with different velocities, but within the framework of the proposed theory only one shear wave propagates, on which shear takes place both along the middle surface and transversely to the middle surface. The inconsistency of shear wave splitting into two waves is most pronounced in the problems wherein disturbances propagate along the rays possessing not only curvature but torsion as well, for example, along the helical lines located on the surface of a cylindrical shell.