Non-stationary problems for elastic half-plane with moving point of changing boundary conditions

The paper presents a method of solving plane unsteady problems for an elastic half-space with a mobile boundary related to changing the boundary conditions of a mixed type. The movement of the half-space is described with two wave equations in terms of elastic potentials. The initial conditions are assumed to be zero. It became possible to obtain an explicit solution of the problem in an integral form by using integral relations for a normal displacement of the half-space boundary in the form of two-dimensional convolution of stress with the influence function (arising from the principle of superposition), properties of the convolution in two variables, and of the theory of generalized functions. At the same time, this solution is based on the method of splitting the functions of influence, according to which it is represented as a product of two factors which satisfy the necessary conditions. Thus, in order to obtain final results it is required to carry out the factorization of the influence function which has the desired properties. The analysis of the Fourier and Laplace transformation of the influence function revealed the presence of two simple poles and four branch points. Getting the desired factorization influence function is based on the representation of its transformation as a product of two multipliers; each of them contains only one critical point. In case when critical points are simple, the separation is performed by using a simple factorization, while the branch points are separated with the help of Cauchy-type integrals. The described method allows obtaining the required factorization of the influence function in any typical speed range of the separating point: sub Rayleigh, subsonic, transonic and supersonic ones. As a result, it became possible to obtain the explicit integral formulas which allow solving the problem. They allow defining the unknown displacements and stresses at any speed range of the moving separating point of the boundary conditions. Asymptotic representations of stresses and displacements are found in the neighborhood of boundary conditions change.