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.

Thermal inclusions inside a bounded medium

In the context of thermal conduction taken as a prototype of numerous transport phenomena, a general method is elaborated to study Eshelby's problem of inclusions inside a bounded homogeneous anisotropic medium. This method consists in: (i) recasting by a linear transformation the initial problem into Eshelby's problem of the transformed inclusion inside the transformed finite isotropic medium and (ii) decomposing Eshelby's problem of a thermal inclusion embedded in a finite isotropic medium into the sub-problem of the same inclusion inside the associated infinite medium and the sub-problem of the finite ambient isotropic medium including no inclusion but undergoing appropriate compensating boundary conditions. The general method is applied in the two-dimensional situation and the corresponding temperature field and Eshelby's conduction tensor are explicitly expressed in terms of some curvilinear complex integrals for the Dirichlet and Neumann boundary conditions. Thus, the difficulties owing to the unavailability or non-existence of Green's function are overcome. The general results in the two-dimensional case are finally specified and illustrated by considering a finite circular medium with circular or polygonal inclusions.

Stress Analysis of Thermal Inclusions With Interior Voids and Cracks

Thermal mismatch induced residual stresses are identified as one of the major causes of voiding and failure of some critical components in electronic packaging, such as passivated interconnect lines and isolation trenches. In this paper, a general method is presented for thermal stress analysis of an embedded structural element in the presence of internal or nearby voids and cracks. Here, the elastic mismatch between dissimilar materials is ignored. Hence, the embedded structural element is modeled as a thermal inclusion of arbitrary shape surrounded by an infinite elastic medium of the same elastic constants. Thermal stresses are caused by thermal mismatch between the inclusion and the surrounding material due to a uniform change in temperature. With the present method, the problem is reduced to one of an infinite homogeneous medium containing the same voids and cracks, subjected to a set of remote stresses determined by the geometrical shape of the thermal inclusion. In particular, the remote stresses are uniform when the thermal inclusion is an ellipse. The method gives an elementary expression for the internal stress field of a thermal inclusion with a single interior void or crack. Several examples of practical interest are used to illustrate the method. The results show that an internal void or crack can significantly change stress distribution within the inclusion and gives rise to stress concentration around the void or crack. [S1043-7398(00)00303-0]