An attempt to calculate analytical elastic fields in two joined quarter spaces due to an interior point load

Abstract

The problem of two joined quarter spaces under a body force can be useful in many applications, such as welded materials, where the free surface is in close proximity. Due to the complexity of the domain, no analytical solutions have been presented for calculating elastic fields within the two regions. In this study, we address the problem of a point load acting at the interior of one of the two joined isotropic quarter spaces. Elastic fields for the two regions are presented in terms of Papkovich-Neuber functions. Traction-free boundary conditions and perfect bonding at the interface were both considered. Green's analysis and proposed potential Green's function of two joined quarter spaces were combined with the integral image method to overcome the boundary conditions. Stress distributions between the two regions and displacement on the free surfaces of the two-quarter spaces for different material combinations are illustrated. The continuity of stresses and displacements at the interface were preserved, however, disturbance of stresses on the free surface was observed. This stress disturbance may be due to the complexity of the domain and/or the proposed harmonic functions. The solution leads back to the solution of point load at the interior of two joined semi-infinite half-spaces (Rongved's solutions) and of point loading at the interior of a semi-infinite space (Mindlin's solutions).