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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.

A method that introduces a new stress vector function ( the hexagonal stress vector ) is applied to obtain, in closed form, the elastic fields due to an inclusion in transversely isotropic solids. The solution is an extension of Eshelby’s solution for an ellipsoidal inclusion in isotropic solids. The Green’s functions for double forces and double forces with moment are derived and these are then used to solve the inclusion problem. The elastic field inside the inclusion is expressed in terms of the newtonian and biharmonic potential functions, which are similar to those needed for the solution in isotropic solids. Two more harmonic potential functions are introduced to express the solution outside the inclusion. The constrained strain inside the inclusion is uniform and the relation between the constrained strain and the misfit strain has the same characteristics as those of the Eshelby tensor for isotropic solids, namely, the coefficients coupling an extension to a shear or one shear to another are zero. The explicit closed form expression of this tensor is given. The inhomogeneity problem is then solved by using Eshelby’s equivalent inclusion method. The solution for the thermoelastic displacements due to thermal inhomogeneities is also presented.

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.

A general method is presented to obtain the elastic field in two joined semi-infinite isotropic solids due to an inclusion of any shape which undergoes a spontaneous change of shape and is located anywhere within one of the semi-infinite solids. The inclusions change of shape is such that, were the surrounding material absent, it would result from some prescribed stress free transformation strain (eigenstrain). The two semi-infinite solids can be either perfectly bonded or in frictionless contact at the planar interface. Examples are given for the inclusion with eigenstrains of practical interest. For the case where the inclusion is an ellipsoid and the given eigenstrain is uniform, the solution can be expressed in terms of well-known elliptic integrals. It is shown that existing solutions for the ellipsoidal inclusion in a semi-infinite solid are special cases of the present general solution.

Thermal residual stresses due to the thermal expansion mismatch in a ceramic-interlayer-metal joint are calculated analytically, A qualitative estimate of the stress distribution is obtained by assuming the joint consists of one or two elastic slabs sandwiched between two semi-infinite isotropic elastic solids. These elastic solids and the interlayer(s) are perfectly bonded to each other at the planar interfaces. The region where temperature change takes place is assumed to be a cylinder with its axis normal to the interface. A simple equation is obtained for the stress value at the center of the ceramic-interlayer interface as a function of the thermal expansion coefficients, elastic constants of the constituent materials, and the thickness of the interlayer(s). The results obtained by this simple model agree well, qualitatively, with those obtained experimentally and numerically (finite element calculations). An appropriate interlayer material to reduce the residual stress in ceramic-metal joints is suggested.