Singularities at the vertex of connected angular inhomogeneities under thermal and elastic loading

TL;DR

This work extends Eshelby’s equivalent inclusion method to analyze singularities at the vertex of multiply connected angular inhomogeneities under steady-state heat conduction and plane-strain elasticity. By representing each inhomogeneity with a distributed eigen-field (eigen-temperature-gradient for thermal loading and eigenstrain for elastic loading) and solving a Fredholm integral equation of the second kind, the authors obtain the singularity orders from an eigenvalue problem and derive closed-form expressions for diagonal and off-diagonal contributions, including interactions between two inclusions. The formulation is validated against classic wedge and bimaterial solutions (Williams, Chen–Huang, Dempsey–Sinclair) and is used to explore bimaterial-interface effects for parallel and perpendicular orientations, revealing how opening angles and material mismatches govern singularity strength and oscillatory behavior. The framework provides a versatile, analytic tool for predicting extreme local fields near angular inhomogeneities and offers a pathway for high-fidelity numerical enrichment in composite materials.

Abstract

This paper investigates the singularities at the vertex of multiply connected angular inhomogeneities for heat conduction and elastic deformation. With the aid of Eshelby's equivalent inclusion method (EIM), each inhomogeneity is simulated as an equivalent inclusion, exhibiting the same material properties as the matrix but containing a continuously distributed eigen-field with potential singularities at the vertices and edge lines. Specifically, the eigen-temperature-gradient (ETG) and eigenstrain are utilized to simulate material mismatch of thermal conductivity and stiffness, respectively. Using the separation of variables, the eigen-fields can be formulated in terms of distance to vertices and opening angles, and disturbed thermal/elastic fields are evaluated by domain integrals of Green's function multiplied by eigen-fields, which form Fredholm's integral equation of the second kind. The boundary value problem is reduced to solve for eigenvalues, which are used to determine the order of singularity. The present solution is versatile - by placing two identical inhomogeneities together, it recovers the classic solutions for a single wedge in a bimaterial media or infinite domain. The general and analytical formulae take full consideration of interactions of multiple inhomogeneities and reveal the effects of opening angles and material properties on the thermal and elastic singularities.

Figures (11)

• Figure 1: Schematic illustration of an infinite domain $\mathcal{D}$ embedded with two isosceles triangular inhomogeneities ($\Omega^1, \Omega^2)$ with opening angles $2 \beta^1, 2\beta^2$ and height $h^1, h^2$, respectively. The angle between symmetric lines of two subdomains is $\gamma$. The matrix exhibits thermal conductivity $K^0$ and $C_{ijkl}^0$, while subdomains exhibits dissimilar properties $K^I, C^I_{ijkl}$.
• Figure 2: Comparison of the dominant singularity $m$ of a triangular void versus the opening angle $\beta^1$ by Eshelby's EIM with the classic wedge solution of the re-entrant corner.
• Figure 3: Comparison and verification of heat flux singularity $m$ between Eshelby's EIM at the tip of a triangular inhomogeneity and the classic solutions. (a) When the triangular void (opening angle $\beta^1 \in [0, \frac{1}{2}] \pi$) is parallel with the bimaterial interface, and the thermal conductivity ratio $K^0 / K^2 = 2, 5$ and $10$; (b) When the triangular void (slit-like crack $\beta^1 \to 0^+$) is inclinedly intersected with the bimaterial interface, and the thermal conductivity ratio $K^1 / K^0 = 0.2, 0.4, 0.6, 2, 5$ and $10$. The straight lines denote EIM's predictions and colored circles represent Chen and Huang's solution Chen1992
• Figure 4: Comparison and verification of dominant complex stress singularity $\lambda$ between the EIM of singularity at the tip of a triangular inhomogeneity in infinite space and the classic Williams's composite wedges problems, $\lambda_F$ and $\lambda_s$ represents symmetric/anti-symmetric real stress singularity, respectively, and $\xi$ denotes virtual stress singularity, opening angle $\beta^1 \in (0, \frac{\pi}{2})$ and $\beta^2 = \frac{\pi}{2}$ with shear modulus $\mu^1 = 0$, and Poisson's ratio $\nu^0 = \nu^2 = 0.3.$ (a) ratio of shear modulus $\mu^0 / \mu^2 = 1$; (b) ratio of shear modulus $\mu^0 / \mu^2 = 2$; (c)ratio of shear modulus $\mu^0 / \mu^2 = 5$; (d) ratio of shear modulus $\mu^0 / \mu^2 = 10$;
• Figure 5: Comparison and verification of symmetric dominant stress singularity parameter $\lambda_F$ between the EIM of singularity at the tip of a single triangular inhomogeneity embedded in the infinite space. The straight lines denoted EIM's predictions is evaluated by placing two same triangular inhomogeneities bonded together, with opening angles $\beta^1 = \beta^2$, shear moduli $\mu^1 = \mu^2$, and Poisson's ratio $\nu^1 = \nu^2 = 0.3$. The colored symbols is evaluated by the EIM formulation Eq. (54) and Eq. (58) in Wu2024 of a single triangular inhomogeneity, with the opening angle $\beta^a = \beta^1 + \beta^2$, shear modulus $\mu^a = \mu^1$, and Poisson's ratio $\nu^a = \nu^1$. Four ratios of shear moduli are considered: $\mu^0 / \mu^2 = 0.2, 0.5, 2, 5$.