Multipoles in an elastic multilayer structure in bending and their applications (continual approximation)

Abstract

We consider the equilibrium of a multilayer structure in bending. This structure is assumed to be formed by plane homogeneous and isotropic layers. The strain of each of the layers is described by the classical small bending theory of thin plates with regard to the fact that the layer is acted upon by the neighboring layers, which contact it either directly or through sufficiently compliant interlayers whose flexural rigidity can be neglected. The characteristic scales of variation in all the fields under study treated as functions of coordinates are assumed to exceed the layer thicknesses so much as to justify the description of the structure deformation in the continual approximation. We assume that the layer deflections are completely determined by the transverse loads and are not affected by tangential stresses. We consider multipoles in this structure. Attention is mainly paid to the multipole asymptotics and their applications to the study of solutions far from the regions where the action exerted on structure are mainly concentrated. We present the results of field computations with the use of these asymptotics and of the exact solutions of the corresponding problems for the plane and spatial cases. These results are used to discuss the possibilities of applying the multipole asymptotics for solving the problems approximately.