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The Galerkin vector stress functions are obtained for the complete set of 40 physically significant nuclei of strain in two joined elastic half-spaces of different elastic properties as an extension of the solutions for the nuclei of strain in the half-space. Two types of boundary condition at the planar interface are considered: perfect bonding and frictionless contact. Simplified expressions for the Galerkin vectors are introduced which reduce the complexity of the expressions for the displacements and stresses in the half-space and the two-material problems. The solutions are obtained by simply solving a set of linear simultaneous algebraic equations to find the strengths of the image and fictitious nuclei of strain which make the resultant elastic field satisfy the boundary conditions and show the proper singularity.

An alternate method of approach for solving the axisymmetric elastic fields in the half space with an isotropic spheriodal inclusion is proposed. This new approach involves the application of the Hankel transformation method for the solution of prismatic dislocation loops and Eshelby’s solution for ellipsoidal inclusions. Existing solutions by other methods for the inclusion with pure dilatational misfit in a half space are shown to be special cases of the present, more general solution.