New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.