REVISED ANALYSIS OF HOMOGENEOUS SOLUTIONS FOR AN ELASTIC LAYER

Abstract

UDC 539.3 We construct the solution of the homogeneous Lame equations for a layer with mixed boundary conditions imposed on its surfaces. The analyzed stressed state is skew-symmetric about the middle plane of the layer ( x 3 = 0 ). It is shown that the homogeneous solutions corresponding to this case do not contain biharmonic components. For a layer in R 3 with load-free faces, homogeneous solutions were constructed for the first time by Lur’e [1]. He proved that these solutions include biharmonic, vortex, and potential components. Later, the method of homogeneous solutions was extensively developed by different authors. Thus, a procedure of reduction of the homogeneous Lame system to a spectral problem was proposed in [2, 3]. At present, this procedure is called the semiinverse method of Vorovych. However, the problem of existence of biharmonic solutions for different types of boundary conditions imposed on the faces of the layer remains open. Thus, some authors believe that the biharmonic solution must necessarily be present as a component of the homogeneous solution. In the present work, we show that this is not always true. On the basis of the proposed procedure for the construction of homogeneous solutions, which significantly simplifies the procedure used by Lur’e, we consider the case of mixed boundary conditions imposed on the faces of the layer and focus our attention on the skew-symmetric stressed state [about the middle plane of the layer ( x3 = 0 ) ]. It is shown that the homogeneous solutions corresponding to this case do not contain the biharmonic component. We consider the differential equations of equilibrium of an elastic body in displacements [4]