2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force

Abstract

The problem of a homogeneous isotropic elastic layer resting on a rigid base and loaded by an internal point force is analytically investigated under plane strain conditions. The displacement field is sought as the superposition of a fundamental solution (doublet state) and a homogeneous solution which allows satisfying the boundary conditions at the upper and lower boundaries of the layer. The displacement field is represented through convergent Fourier integral transforms. Once the closed form solution in the transformed domain is found, the displacement and stress fields are assessed by numerical inversion of transforms. Results concerning both the displacements and stresses for different positions of the load application point are reported and compared to FEM solution, finding very good agreement. The solutions obtained for horizontal and vertical point forces define the Green’s functions for the layer, which can be used to describe the mechanical interaction between the layer and an embedded body. As a striking application, the interaction between an elastic layer and an embedded laterally loaded wall or diaphragm is finally addressed, finding the contact pressure and, in turn, the internal forces in the diaphragm.