Reduction of boundary value problems of dynamic elasticity to scalar problems for wave potentials in curvilinear coordinates

Abstract

Owing to significant mathematical difficulties arising when solving dynamic problems of elasticity, ever more attention is paid to the study of types of boundary value problems, boundary shapes, and additional assumptions (for example, such as symmetry) for which, in the statement of the problem in potentials, not only the equations of motion lead to separate scalar wave equations but also the boundary conditions split into separate conditions for each of the potentials. It was shown earlier that the boundary conditions prescribing the normal displacement and tangential stresses (condition (a)) or the normal stress and tangential displacements (condition (b)) on the boundary can be separated for potentials on the plane boundary. In connection with the separation of these boundary conditions on curvilinear boundaries, many claims weremade in the literature, some of which are erroneous. In the present paper, we obtain the most complete result, which clarifies this problem and states that the boundary conditions (a) can be separated on the surface of a circular cylinder and a circular cone in the case of axial symmetry, but the boundary conditions (b) cannot be separated on a curvilinear boundary. Several examples illustrating the obtained results are given.