On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

Abstract

An exact analytical solution has been constructed for the plane problem on action of a non‐stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non‐stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the object at an arbitrary point of time. Some variants of non‐stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.