Elastic Solution of a Polyhedral Particle With a Polynomial Eigenstrain and Particle Discretization

Abstract

The paper extends the recent work (Wu, C., and Yin, H., 2021, “Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain,” ASME J. Appl. Mech., 88(6), p. 061002) of Eshelby’s tensors for polynomial eigenstrains from a two-dimensional (2D) to three-dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear, and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity using Eshelby’s equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby’s tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green’s function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the C0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of Eshelby’s tensor, the elastic analysis is robust, stable, and efficient.