A TRIANGULAR ELEMENT DISCRETIZATION FOR COMPUTING DISPALCEMENT OF AN ARBITRARILY SHAPED THERMAL INCLUSION

Inclusion models have been widely used to explore the micromechanical properties of fiber-reinforced composite materials. The composite materials usually contain irregularly shaped inclusions, which can severely affect the mechanical properties of the materials. Abundant research has demonstrated that the stress localizations as well as the sites of crack initiation are predominantly detected in the neighborhood of nonmetallic inclusions. Previous studies on polygonal inclusion mainly focused on the stress/strain solutions under uniform eigenstrains, while the analyses on displacement are limited. Based on the method of Green's function and contour integral, this work presents a closed-form solution for a line element along the boundary of a two-dimensional thermal inclusion. The proposed method of solution is effective for determining the displacement of an arbitrarily shaped inclusion subjected to any distributed dilatational eigenstrain. In the case of uniform eigenstrain, only the boundary of the inclusion needs to be discretized into line elements; therefore, the proposed method analytically yields the closed-form solution for the displacement of an arbitrary polygonal inclusion subjected to uniform thermal eigenstrain. When the eigenstrain is non-uniformly distributed in the inclusion, the resulting displacements may be evaluated by discretizing the thermal inclusion into a system of triangular elements. It is known that the stress and strain fields exhibit singularities at the vertices of a polygonal inclusion. Such singularity issue can be intractable in numerical evaluations of the stresses/strains in the vicinity of the vertices, leading to a commonly seen yet tricky phenomenon of numerical instability. In contrast, the present work shows that the displacement is continuous and bounded at the corners of the polygon. Other than the merit of numerical discretization, the derived closed-form solutions may be conveniently programmed on a personal computer, while the corresponding algorithm seems to be straightforward, facilitating a high accurate and expeditious evaluation of the displacements. Benchmark examples demonstrate the computational efficiency and numerical robustness of the proposed method.