Explicit analytical solutions for the complete elastic field produced by an ellipsoidal thermal inclusion in a semi-infinite space

Thermal inclusion in an elastic half-space is a classical micromechanical model for describing localized heating near a surface. This paper presents explicit analytical solutions for the complete elastic fields, including displacements, strains, and stresses, produced by an ellipsoidal thermal inclusion in a three-dimensional semi-infinite space. Unlike the famous Eshelby solution corresponding to the infinite space case, the present work demonstrates that the interior strain and stress components are no longer uniform and appear to be much more complex. Nevertheless, the results can be represented in a more compact and geometrically meaningful form by constructing auxiliary confocal ellipsoids. The derived explicit solution indicates that the shear components of the stress and strain may be represented in closed-form. The jump conditions are examined and proven to be exactly identical to the infinite space case. A purposely selected benchmark example is studied to illustrate the free boundary surface effects. The degenerate case of a spherical thermal inclusion may be derived in a closed form, and is verified by the well-known Mindlin solution.