Solution of the dynamic problem of elasticity theory in curvilinear coordinates

The elastic theory dynamic problem for a transtropic multilayer sphere

“…then an analytical solution exists in a form of quite complex linear superpositions of spherical running and reflected waves (of the D'Alembert type) [3,4]. Obtaining such solutions for quite large time values, especially in case of a multilayer shell, is very difficult, though realizable in principle.…”

“…Obtaining solutions based on the elastic theory equations is quite critical, insofar as results based on various shell theories can be inaccurate due to such factors as large thickness of shell walls, short-term loading and steep wavefronts. Precise analytical solutions of such problems are quite cumbersome even for isotropic layers and are available only for few particular cases, such as one-layered spheres and spherical cavities [3,4]. This study is aimed at development of calculation techniques for solving dynamical centrally symmetrical boundary problems of the elastic theory for multilayered spherically transtropic spheres.…”

“…Let us use the spherical coordinate system r, y, where r is a radial coordinate and y is angular coordinate orthogonal to r. Due to the central symmetry of the problem, usage of the third coordinate is not required, since all directions along y are identically equivalent. Within this coordinate system motion equations will take the following form [3,4] ¶s ¶…”