The explicit elastic field for two perfectly bonded half-spaces with an ellipsoidal thermal inclusion

Abstract

The analytical solution for a full-space or half-space with an ellipsoidal inclusion is available in literature, but the explicit analytical solution for two bonded half-spaces with a three-dimensional inclusion was more complex and less reported. Deriving a complete explicit analytical solution for two bonded half-spaces with an ellipsoidal inclusion involves complicated domain integrations/derivatives. This work first derived the explicit analytic solution of two perfectly bonded half-spaces due to an ellipsoidal thermal inclusion and presented it in a more concise and compact tensorial structure. These explicit expressions became easier to program and geometrically meaningful after introducing the normal unit vector of the imaginary confocal ellipsoid. Further, the current solution can be simplified and matches exactly with the published analytical solutions in the case of either semi-infinite space or full space, by varying Young's modulus ratio of the two half-spaces. The closed-form solutions of two perfectly bonded half-spaces embedded with a spherical or spheroidal inclusion were derived and validated as well. The strain/stress jump conditions across the bonded interface were comprehensively discussed for a mechanistic understanding of the delamination around the bonded interface. Moreover, the agreement between the present analytical solution and the photo-elastic experimental results justifies the engineering practicability of this work.