Construction of mathematical models of the stressed-strained state of a material with a porous water-saturated base under dynamic load

Abstract

Materials of beams, plates, slabs, strips have been commonly applied in various fields of industry and agriculture as flat elements in the structures for machinery and construction. They are associated with the design of numerous engineering structures and facilities, such as the foundations of various buildings, airfield and road surfaces, floodgates, including underground structures. This paper reports a study into the interaction of the material (of beams, plates, slabs, strips) with the deformable base as a three-dimensional body and in the exact statement of a three-dimensional problem of mathematical physics under dynamic loads. The tasks of studying the interaction of a material (beams, plates, slabs, strips) with a deformable base have been set. A material lying on a porous water-saturated viscoelastic base is considered as a viscoelastic layer of the same geometry. It is assumed that the lower surface of the layer is flat while the upper surface, in a general case, is not flat and is given by some equation. Classical approximate theories of the interaction of a layer with a deformable base, based on the Kirchhoff hypothesis, have been considered. Using the well-known hypothesis by Timoshenko and others, the general three-dimensional problem is reduced to a two-dimensional one relative to the displacement of points of the median plane of the layer, which imposes restrictions on external efforts. In the examined problem, there is no median plane. Therefore, as the desired values, displacements and deformations of the points in the plane have been considered, which, under certain conditions, pass into the median plane of the layer. It is not possible to find a closed analytical solution for most problems while experimental studies often turn out to be time-consuming and dangerous processes