Geometrically nonlinear plate bending under the action of moving load

Abstract

Considerable scientific interest is the development of mathematical models that describe the behavior of materials that are sensitive to deformation rate and can improve the accuracy of analytical calculations of their deformation in the region of noticeable changes of loading rates. Nonetheless, in most works, the problems were solved under the assumption of small displacements (geometrically linear statement of the problem). Meanwhile, in practice, this is not always true and bending of cover can be commensurable with its thickness, this article approximately solves the problem of geometrically nonlinear deformation of a thin elastic plate in quasistatic setting under the action of an infinite normal uniformly distributed load moving along its surface at a constant speed. In the article, the methods of mathematical modeling, the analytical method, as well as the methods of spatial characteristics and bicharacteristics are used. The problem is solved in quasistatic formulation and is reduced to a system of two nonlinear differential equations for deflections of the plate and the stress function, which include the speed of the load as a parameter. The results of methodological calculations are presented; on the basis of these solutions of linear and nonlinear problems, they were compared, and the influence of finiteness of displacements on the critical speeds of the forces was determined. Materials of the article can be useful in study of wave dynamics, aircraft, mechanics and engineering.