Non-stationary stress state of an elastic layer at the 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 layer for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on one boundary and normal displacement and shear stress (the second boundary problem) for another 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 layer at an arbitrary moment 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. Distinctive features of wave processes are analyzed.The boundary conditions in problems of elasticity theory belong to one of four types; see e.g. [1]. The principal types are the first and second, wherein the stress and displacement vectors, respectively, are specified. The third and fourth types are called mixed boundary conditions because each employs both displacement and stress, i.e. type three specifies the normal component of the displacement vector and the tangential components of the stress vector while type four species the normal stress and tangential displacement components. The mixed boundary conditions are physically less realistic and sometimes considered approximations of types one and two. The choice of which boundary condition to use could significantly inform the likelihood of obtaining an analytic solution for non-stationary problems.The effectively analytical and numerically analytical methods, grounded mainly in the application of the Laplace transform in the time domain and Fourier (Hankel) transform for the linear coordinate, are developed for a solution of relevant boundary value problems. The problem thus is the build-up of originals which primarily depends on the character of the space-time distribution of the load on the boundary and the type thereof. Interested readers are directed to refs.[2-5] for studies of non-stationary wave processes at an operation of the concentrated or distributed surface actions in an isotropic elastic half-space (half-plane).Refs.[6] and [7] give the solutions of many non-stationary problems. The displacements of points on the surface of the semi-space as well as on the axis of symmetry were studied in refs.[8] and [9]. Thereat, for joint inversion of transforms, the authors used the Cagniard technique with account of transform uniformity with respect to transform parameters [10]. The saddle-point was used in ref. [9] for an approximate search for displacements of points remote from the axis of symmetry. Similarly, with the inverse Laplace transform, using the theory of residues, refs. [11][12][13] solved for the displacements and stress...

Non‐stationary axially symmetric displacement of elastic half‐space in mixed boundary conditions

“…The main advantage of the solution obtained upon mixed boundary conditions is its simple form. This fact is proved by the solution of the non‐stationary plane problem for the elastic half‐space and the layer obtained in the publications [9, 13]. This solution can serve as a source of express information and it can be used as a reference point for numerical methods processing.…”