A mode III crack in an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic medium

Abstract

We apply the techniques of one-to-one mapping and analytic continuation to derive a closed-form solution to the anti-plane elasticity problem of a cracked anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to uniform remote anti-plane shear stresses. The anisotropy of the elliptical inhomogeneity results in the finite crack and the elliptical interface being non-confocal. The mode III stress intensity factor at the crack tip is extracted from the solution, with an emphasis on a vanishingly thin inhomogeneity. When a particular condition on the ratio of the two remote anti-plane stresses is met, the remote loading will not induce any singular stresses at the crack tips and the stress field within the cracked inhomogeneity is uniform. Explicit general solutions for an arbitrary singularity located either in the matrix or the cracked inhomogeneity are derived.

Keywords

anti-plane elasticity anisotropic elasticity cracked elliptical inhomogeneity stress intensity factor remote loading arbitrary singularity general solution

1. Introduction

When a crack is entirely embedded in an elastic inhomogeneity, the difference in elastic constants between the inhomogeneity and the surrounding matrix can result in the stress intensity factor being greater or less than that in a homogeneous elastic plane [1]. The problem of a crack entirely surrounded by an elastic inhomogeneity under various loadings has been investigated by several authors (see, for example, Wu and Chen [1], Erdogan and Gupta [2], Anlas and Santare [3], Zhang and Qian [4], Zhang et al. [5], Amenyah et al. [6] and Wang X and Schiavone [7]). This problem arises when studying crack–damage interaction (the damage is due to microcracking surrounding the main crack) and also in the case of fiber cracking in fibrous composites. In previous studies, however, both the inhomogeneity and the matrix were largely assumed to be isotropic elastic. It is then natural to ask whether an analytical solution exists when both the cracked inhomogeneity and the matrix are anisotropic elastic? Of particular interest is the fact that the anisotropy of the inhomogeneity can incorporate anisotropic damage surrounding the finite crack. In this study, we endeavor to answer this question.

In this paper, we solve the anti-plane elasticity problem of a cracked anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to uniform remote anti-plane shear stresses. Both the inhomogeneity and the matrix are monoclinic with symmetry plane at $x_{3} = 0$ . Due to the anisotropy of the elliptical inhomogeneity, the finite crack and the elliptical interface are not confocal, and the two principal axes of the elliptical interface are not necessarily aligned with the two coordinate axes (the crack lies on the horizontal coordinate axis x₁). The elliptical interface and the crack are confocal when viewed in the $z_{1}$ -plane where $z_{1}$ is the single complex variable for the inhomogeneity appearing in the complex variable formulation. With the aid of the techniques of one-to-one mapping [8] and analytic continuation [9,10], a closed-form solution to the problem is derived. The mode III stress intensity factor at the crack tip is extracted from the solution. The expression for the stress intensity factor in the case of a crack in a vanishingly thin inhomogeneity is nominally identical to the result by Wu and Chen [1] when both the inhomogeneity and the matrix are isotropic elastic. When a particular condition on the ratio of the two remote anti-plane shear stresses is met, the remote loading will not induce any singular stress field at the crack tips and the stress field is uniformly distributed inside the cracked inhomogeneity. The solution method is also employed to derive explicit general solutions for an arbitrary singularity (e.g. a screw dislocation, an anti-plane line force, a dipole) applied either in the matrix or in the cracked inhomogeneity.

2. Complex variable formulation

For the anti-plane shear deformation of a monoclinic material with plane of symmetry at $x_{3} = 0$ , the anti-plane shear stresses $σ_{31}$ and $σ_{32}$ , the out-of-plane displacement $u_{3}$ and the single stress function φ can be expressed concisely in terms of a single analytic function $f (z)$ of the complex variable $z = x_{1} + p x_{2}$ as follows [8]

σ_{31} + p σ_{32} = i μ Im {p} \bar{f' (z)},

(1)

φ + i μ u_{3} = μ f (z),

(2)

where

\begin{matrix} p = \frac{- C_{45} + i \sqrt{C_{44} C_{55} - C_{45}^{2}}}{C_{44}}, Im {p} > 0, \\ μ = \sqrt{C_{44} C_{55} - C_{45}^{2}}, \end{matrix}

(3)

with $C_{55}, C_{45}, C_{44}$ being elastic stiffnesses satisfying the restrictions that $C_{55} > 0, C_{44} > 0, C_{44} C_{55} > C_{45}^{2}$ . Here $μ^{2} = C_{44} C_{55} - C_{45}^{2}$ is a principal minor of the 6×6 stiffness matrix [8]. When $p = i$ for an isotropic elastic material, μ is simply the shear modulus.

In addition, the two anti-plane stress components are related to the single stress function by [8]

σ_{32} = φ_{, 1}, σ_{31} = - φ_{, 2} .

(4)

Let $p^{'}$ and $p^{″} (> 0)$ be, respectively, the real and imaginary parts of p, i.e. $p = p^{'} + i p^{″}$ .

3. Closed-form solution

As shown in Figure 1, we consider an infinite anisotropic elastic matrix containing an anisotropic elastic elliptical inhomogeneity which is weakened by a traction-free crack lying on the segment $[- R, R]$ . Let $S_{1}$ and $S_{2}$ denote the cracked inhomogeneity and the matrix, respectively, which are perfectly bonded across the elliptical interface L. Both the inhomogeneity and the matrix are monoclinic with symmetry plane at $x_{3} = 0$ . Due to the anisotropy of the elliptical inhomogeneity, the crack and the elliptical interface L are not confocal, and in addition, the two principal axes of the ellipse L are not necessarily aligned with the two coordinate axes. The crack and the elliptical interface L are confocal only when the inhomogeneity is isotropic [1]. The matrix is subjected to uniform remote anti-plane shear stresses $σ_{31}^{\infty}$ and $σ_{32}^{\infty}$ . Throughout the paper, the subscripts 1 and 2 are used to identify the respective quantities in $S_{1}$ and $S_{2}$ .

We introduce the following one-to-one mapping functions for the cracked inhomogeneity and the matrix:

\begin{matrix} z_{1} = x_{1} + p_{1} x_{2} = ω_{1} (ξ) = \frac{R}{2} (ξ + \frac{1}{ξ}), 1 \leq | ξ | \leq ρ^{- \frac{1}{2}}, \\ z_{2} = x_{1} + p_{2} x_{2} = ω_{2} (ξ) = \frac{i R}{4 {p^{″}}_{1}} {[{\bar{p}}_{1} - ρ p_{1} - p_{2} (1 - ρ)] ξ + \frac{- p_{1} + ρ {\bar{p}}_{1} + p_{2} (1 - ρ)}{ρ ξ}}, | ξ | \geq ρ^{- \frac{1}{2}}, \end{matrix}

(5)

where $0 < ρ < 1$ .

As shown in Figure 2, using the mapping functions in equation (5), the cracked inhomogeneity is mapped onto $1 \leq | ξ | \leq ρ^{- \frac{1}{2}}$ , the matrix is mapped onto $| ξ | \geq ρ^{- \frac{1}{2}}$ , the finite crack is mapped onto $| ξ | = 1$ and the inhomogeneity–matrix elliptical interface L is mapped onto $| ξ | = ρ^{- \frac{1}{2}}$ . It follows from equation (5) that the ellipse L is described by

\begin{matrix} x_{1} = \frac{R}{2} (ρ^{- \frac{1}{2}} + ρ^{\frac{1}{2}}) \cos ψ - \frac{R {p^{'}}_{1}}{2 {p^{″}}_{1}} (ρ^{- \frac{1}{2}} - ρ^{\frac{1}{2}}) \sin ψ, \\ x_{2} = \frac{R}{2 {p^{″}}_{1}} (ρ^{- \frac{1}{2}} - ρ^{\frac{1}{2}}) \sin ψ, \end{matrix}

(6)

where $ξ = ρ^{- \frac{1}{2}} e^{i ψ}$ . It is seen from equation (6) that (1) the two principal axes of the ellipse L are not aligned with the two coordinate axes when $p_{1}^{'} \neq 0$ , they are along the two coordinate axes only when $p_{1}^{'} = 0$ (the inhomogeneity is orthotropic); (2) the ellipse L and the crack are confocal only when $p_{1} = i$ (the inhomogeneity is isotropic). The elliptical interface and the crack are confocal in the $(x_{1} + p_{1}^{'} x_{2}, p_{1}^{″} x_{2})$ -plane or $z_{1}$ -plane. As $ρ \to 1$ , the ellipse L shrinks to the segment $[- R, R]$ ; as $ρ \to 0$ , the size of the ellipse L becomes very large and is described by

| z_{1} | = \frac{1}{2} R ρ^{- \frac{1}{2}}, as ρ \to 0,

(7)

which is circular in the $(x_{1} + p_{1}^{'} x_{2}, p_{1}^{″} x_{2})$ -plane or $z_{1}$ -plane.

The boundary value problem has the following form in the ξ-plane:

f_{1} (ξ) + \bar{f_{1} (ξ)} = 0, | ξ | = 1;

(8a)

\begin{array}{l} f_{2} (ξ) - \bar{f_{2} (ξ)} = f_{1} (ξ) - \bar{f_{1} (ξ)}, \\ f_{2} (ξ) + \bar{f_{2} (ξ)} = Γ f_{1} (ξ) + Γ \bar{f_{1} (ξ)}, | ξ | = ρ^{- \frac{1}{2}}; \end{array}

(8b)

f_{2} (ξ) ≅ \frac{R (σ_{31}^{\infty} + {\bar{p}}_{2} σ_{32}^{\infty}) [ρ p_{1} - {\bar{p}}_{1} + p_{2} (1 - ρ)]}{4 μ_{2} {p^{″}}_{1} {p^{″}}_{2}} ξ + O (1), as | ξ | \to \infty,

(8c)

where we write $f_{i} (ξ) = f_{i} (ω_{i} (ξ)) (i = 1, 2)$ for convenience, and

Γ = \frac{μ_{1}}{μ_{2}} .

(9)

By enforcing the boundary and interface conditions in equations (8a) and (8b) with the aid of analytic continuation [9,10], we arrive at the following general solution for $f_{1} (ξ)$ and $f_{2} (ξ)$ expressed in terms of a single unknown analytic function $h (ξ)$ :

\begin{matrix} f_{1} (ξ) = h (ξ) - \bar{h} (\frac{1}{ξ}), 1 \leq | ξ | \leq ρ^{- \frac{1}{2}}; \\ f_{2} (ξ) = \frac{1}{1 - K} [h (ξ) - Kh (ρ ξ) - \bar{h} (\frac{1}{ξ}) + K \bar{h} (\frac{1}{ρ ξ})], | ξ | \geq ρ^{- \frac{1}{2}}, \end{matrix}

(10)

where the mismatch parameter K is defined by

K = \frac{Γ - 1}{Γ + 1} .

(11)

The boundary value problem is solved once $h (ξ)$ is determined. It is stressed that the general solution in equation (10) is still valid for an arbitrary singularity applied either in the matrix or in the cracked inhomogeneity (see the next section for more details). The function $h (ξ)$ for the case of uniform anti-plane shear loading at infinity takes the following simple form

h (ξ) = λ ξ,

(12)

where λ is an unknown complex constant to be determined.

Insertion of equation (12) into equation (10) yields

\begin{matrix} f_{1} (ξ) = λ ξ - \frac{\bar{λ}}{ξ}, 1 \leq | ξ | \leq ρ^{- \frac{1}{2}}; \\ f_{2} (ξ) = \frac{1}{1 - K} [λ (1 - ρ K) ξ + \frac{\bar{λ} (K - ρ)}{ρ ξ}], | ξ | \geq ρ^{- \frac{1}{2}} . \end{matrix}

(13)

Using the expression for $f_{2} (ξ)$ in equation (13) to satisfy the remote asymptotic behavior in equation (8c), the complex constant λ can be uniquely determined as follows:

λ = \frac{R (1 - K) [ρ p_{1} - {\bar{p}}_{1} + p_{2} (1 - ρ)] (σ_{31}^{\infty} + {\bar{p}}_{2} σ_{32}^{\infty})}{4 μ_{2} {p^{″}}_{1} {p^{″}}_{2} (1 - ρ K)} .

(14)

Now the two analytic functions $f_{1} (ξ)$ and $f_{2} (ξ)$ in equation (13) characterizing the elastic fields in the cracked inhomogeneity and in the matrix have been completely determined. In particular, the mode III stress intensity factor $K_{III}$ at the right crack tip $(x_{1}, x_{2}) = (R, 0)$ can be extracted from the solution in equation (13) as follows:

K_{III} = \frac{\sqrt{π R} (1 + K) (1 - ρ)}{2 {p^{″}}_{1} {p^{″}}_{2} (1 - ρ K)} [({p^{'}}_{2} - {p^{'}}_{1}) σ_{31}^{\infty} + ({| p_{2} |}^{2} - {p^{'}}_{1} {p^{'}}_{2} + {p^{″}}_{1} {p^{″}}_{2} \frac{1 + ρ}{1 - ρ}) σ_{32}^{\infty}] .

(15)

It is deduced from equation (15) that the remote loading will not induce any singular stress field at the two crack tips when the following condition on the ratio of the two remote anti-plane shear stresses $σ_{32}^{\infty}$ and $σ_{31}^{\infty}$ is met

\frac{σ_{32}^{\infty}}{σ_{31}^{\infty}} = \frac{{p^{'}}_{1} - {p^{'}}_{2}}{{| p_{2} |}^{2} - {p^{'}}_{1} {p^{'}}_{2} + {p^{″}}_{1} {p^{″}}_{2} \frac{1 + ρ}{1 - ρ}},

(16)

which is unaffected by the mismatch parameter K and which is illustrated in Figure 3. When $p_{1}^{'} = p_{2}^{'}$ , equation (16) reduces to $σ_{32}^{\infty} = 0$ .

When the condition in equation (16) is satisfied, the stress field within the cracked inhomogeneity is uniform and is given by

\frac{σ_{31}}{σ_{31}^{\infty}} = \frac{(1 + K) (1 - ρ) [{({p^{'}}_{1} - {p^{'}}_{2})}^{2} + {({p^{″}}_{2} + {p^{″}}_{1} \frac{1 + ρ}{1 - ρ})}^{2}]}{2 (1 - ρ K) ({| p_{2} |}^{2} - {p^{'}}_{1} {p^{'}}_{2} + {p^{″}}_{1} {p^{″}}_{2} \frac{1 + ρ}{1 - ρ})}, σ_{32} = 0, x_{1} + i x_{2} \in S_{1} .

(17)

As $ρ \to 1$ (the inhomogeneity is vanishingly thin), equation (15) becomes

\frac{K_{III}}{\sqrt{π R} σ_{32}^{\infty}} = Γ,

(18)

which is unaffected by $σ_{31}^{\infty}$ and is nominally identical to the result by Wu and Chen [1] when both the inhomogeneity and the matrix are isotropic elastic.

As $ρ \to 0$ (the inhomogeneity is very large and becomes circular in the $(x_{1} + p_{1}^{'} x_{2}, p_{1}^{″} x_{2})$ -plane or $z_{1}$ -plane), equation (15) becomes

K_{III} = \frac{\sqrt{π R} (1 + K)}{2 {p^{″}}_{1} {p^{″}}_{2}} [({p^{'}}_{2} - {p^{'}}_{1}) σ_{31}^{\infty} + ({| p_{2} |}^{2} + {p^{″}}_{1} {p^{″}}_{2} - {p^{'}}_{1} {p^{'}}_{2}) σ_{32}^{\infty}] .

(19)

When $p_{1}^{'} = p_{2}^{'} = 0$ (both the inhomogeneity and the matrix are orthotropic), equation (15) becomes

\frac{K_{III}}{\sqrt{π R} σ_{32}^{\infty}} = \frac{(1 + K) [{p^{″}}_{2} (1 - ρ) + {p^{″}}_{1} (1 + ρ)]}{2 {p^{″}}_{1} (1 - ρ K)},

(20)

which is unaffected by the remote stress component $σ_{31}^{\infty}$ . Furthermore, when $p_{1}^{″} = p_{2}^{″}$ , equation (20) reduces to

\frac{K_{III}}{\sqrt{π R} σ_{32}^{\infty}} = \frac{1 + K}{1 - ρ K},

(21)

which is nominally identical to the result by Wu and Chen [1] when both the inhomogeneity and the matrix are isotropic elastic.

4. An arbitrary singularity

In the previous section, we have considered the case of uniform remote loading, which corresponds to a singularity applied at infinity. In this section, we consider an arbitrary singularity (e.g. a screw dislocation, an anti-plane line force, a dipole) applied at any position in the matrix or in the cracked inhomogeneity. When an arbitrary singularity is located in the matrix, the function $h (ξ)$ has the following form

h (ξ) = (1 - K) \sum_{j = 0}^{+ \infty} K^{j} Ω (ρ^{j} ξ),

(22)

where $Ω (ξ)$ is the principal part of $f_{2} (ξ)$ .

When an arbitrary singularity is located inside the cracked inhomogeneity, the function $h (ξ)$ has the following form

h (ξ) = \sum_{j = 0}^{+ \infty} K^{j} Ω (ρ^{j} ξ) - \sum_{j = 0}^{+ \infty} K^{j + 1} \bar{Ω} (\frac{1}{ρ^{j + 1} ξ}),

(23)

where $Ω (ξ)$ is the principal part of $f_{1} (ξ)$ .

By substituting equation (22) into equation (10), the explicit general solution for an arbitrary singularity in the matrix is given by

\begin{matrix} f_{1} (ξ) = (1 - K) \sum_{j = 0}^{+ \infty} K^{j} [Ω (ρ^{j} ξ) - \bar{Ω} (\frac{ρ^{j}}{ξ})], 1 \leq | ξ | \leq ρ^{- \frac{1}{2}}; \\ f_{2} (ξ) = Ω (ξ) + K \bar{Ω} (\frac{1}{ρ ξ}) - (1 - K^{2}) \sum_{j = 0}^{+ \infty} K^{j} \bar{Ω} (\frac{ρ^{j}}{ξ}), | ξ | \geq ρ^{- \frac{1}{2}} . \end{matrix}

(24)

The mode III stress intensity factors at the two crack tips can be extracted from the solution in equation (24) as

K_{III} (\pm R) = 2 μ_{1} \sqrt{\frac{π}{R}} (1 - K) \sum_{j = 0}^{+ \infty} {(ρ K)}^{j} Re {Ω^{'} (\pm ρ^{j})},

(25)

where the sign in ± is chosen in such a way that the plus sign corresponds to the right crack tip and the minus sign to the left crack tip.

By substituting equation (23) into equation (10), the explicit general solution for an arbitrary singularity inside the cracked inhomogeneity is given by

\begin{matrix} f_{1} (ξ) = \sum_{j = 0}^{+ \infty} K^{j} [Ω (ρ^{j} ξ) - \bar{Ω} (\frac{ρ^{j}}{ξ})] + \sum_{j = 1}^{+ \infty} K^{j} [Ω (\frac{ξ}{ρ^{j}}) - \bar{Ω} (\frac{1}{ρ^{j} ξ})], 1 \leq | ξ | \leq ρ^{- \frac{1}{2}}; \\ f_{2} (ξ) = (1 + K) \sum_{j = 0}^{+ \infty} K^{j} [Ω (\frac{ξ}{ρ^{j}}) - \bar{Ω} (\frac{ρ^{j}}{ξ})], | ξ | \geq ρ^{- \frac{1}{2}} . \end{matrix}

(26)

The mode III stress intensity factors at the two crack tips can be extracted from the solution in equation (26) as

K_{III} (\pm R) = 2 μ_{1} \sqrt{\frac{π}{R}} [\sum_{j = 0}^{+ \infty} {(ρ K)}^{j} Re {Ω^{'} (\pm ρ^{j})} + \sum_{j = 1}^{+ \infty} {(\frac{K}{ρ})}^{j} Re {Ω^{'} (\pm ρ^{- j})}],

(27)

where, again, the sign in ± is chosen in such a way that the plus sign is for the right crack tip and the minus sign for the left crack tip.

For example, when a screw dislocation with Burgers vector b is applied at $(x_{1}, x_{2}) = (x_{1}^{0}, x_{2}^{0})$ in the matrix, the principal part $Ω (ξ)$ has the form

Ω (ξ) = \frac{b}{2 π} \ln (ξ - ξ_{0}),

(28)

where

x_{1}^{0} + p_{2} x_{2}^{0} = ω_{2} (ξ_{0}), | ξ_{0} | > ρ^{- \frac{1}{2}} .

(29)

Substitution of equation (28) into equation (25) yields the mode III stress intensity factors at the two crack tips as

K_{III} (\pm R) = \frac{μ_{1} b}{\sqrt{π R}} (1 - K) \sum_{j = 0}^{+ \infty} {(ρ K)}^{j} Re {\frac{1}{\pm ρ^{j} - ξ_{0}}} .

(30)

When a screw dislocation with Burgers vector b is applied at $(x_{1}, x_{2}) = (x_{1}^{0}, x_{2}^{0})$ in the cracked inhomogeneity and originates elsewhere other than the crack [11], the principal part $Ω (ξ)$ has the form

Ω (ξ) = \frac{b}{2 π} \ln (ξ - ξ_{0}) - \frac{bK}{2 π (1 + K)} \ln ξ,

(31)

where

x_{1}^{0} + p_{1} x_{2}^{0} = ω_{1} (ξ_{0}), 1 < | ξ_{0} | < ρ^{- \frac{1}{2}} .

(32)

Substitution of equation (31) into equation (27) yields the mode III stress intensity factors at the two crack tips as

K_{III} (\pm R) = \frac{μ_{1} b}{\sqrt{π R}} [\pm \frac{- K}{1 - K} + \sum_{j = 0}^{+ \infty} {(ρ K)}^{j} Re {\frac{1}{\pm ρ^{j} - ξ_{0}}} + \sum_{j = 1}^{+ \infty} {(\frac{K}{ρ})}^{j} Re {\frac{1}{\pm ρ^{- j} - ξ_{0}}}] .

(33)

It is verified that when the screw dislocation lies on the elliptical interface L with $ξ_{0} {\bar{ξ}}_{0} = ρ^{- 1}$ , the results (using equations (30) and (33)) are the same. We illustrate in Figures 4 and 5 the normalized stress intensity factors $K_{3} (\pm R) = \sqrt{π R} K_{III} (\pm R) / (μ_{2} b)$ induced by a screw dislocation lying on the elliptical interface with $ξ_{0} = ρ^{- \frac{1}{2}} e^{i ψ_{0}}$ according to equation (30). $K \leq 0$ (i.e. $Γ \leq 1$ ) in Figures 4 and 5 means that the inhomogeneity is not stiffer than the matrix, which is the situation when treating the inhomogeneity as the damaged zone. It is seen from Figures 4 and 5 that (1) the screw dislocation exerts a shielding or anti-shielding effect on the two crack tips depending on the specific value of $ψ_{0}$ ; (2) the magnitudes of the two normalized stress intensity factors in general are reduced as $K (\leq 0)$ decreases (the inhomogeneity becomes softer); (3) when K is fixed, $K_{3} (+ R) = 0$ or $K_{3} (- R) = 0$ at two particular points of the elliptical interface; (4) when K is fixed, the magnitude of $K_{3} (+ R)$ attains its maximum when $ψ_{0} = 0$ and that of $K_{3} (- R)$ attains its maximum when $ψ_{0} = π$ . According to equation (6), $ψ_{0} = 0$ corresponds to $x_{1}^{0} = \frac{1}{2} R (ρ^{- \frac{1}{2}} + ρ^{\frac{1}{2}}), x_{2}^{0} = 0$ and $ψ_{0} = π$ corresponds to $x_{1} = - \frac{1}{2} R (ρ^{- \frac{1}{2}} + ρ^{\frac{1}{2}}), x_{2} = 0,$ both of which are unaffected by $p_{1}$ . In order to see observation (4) in more detail, we illustrate in Figure 6 $max {| K_{3} (+ R) |}$ as a function of K and ρ. It is seen from Figure 6 that $max {| K_{3} (+ R) |}$ is an increasing function of both K and ρ.

When a screw dislocation with Burgers vector b is applied at $(x_{1}, x_{2}) = (x_{1}^{0}, x_{2}^{0})$ in the cracked inhomogeneity and is emitted from the crack [11], the principal part $Ω (ξ)$ has the form

Ω (ξ) = \frac{b}{2 π} \ln (ξ - ξ_{0}) - \frac{b}{4 π} \ln ξ,

(34)

where $ξ_{0}$ is defined in equation (32).

When the singularity is applied at a finite point, only series form solutions in equations (24) and (25) can be derived. The solutions in equations (24) and (25) converge very fast even when the singularity approaches the elliptical interface L (see Figures 4 and 5) in view of the fact that all the image singularities have been fully accounted for.

5. Conclusion

We solve the anti-plane shear problem of a cracked anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to uniform remote anti-plane shear stresses. The elliptical interface and the crack are confocal in the $z_{1}$ -plane, and they are not confocal in the $(x_{1}, x_{2})$ -plane. A closed-form solution is obtained in equation (13). The mode III stress intensity factor at the crack tip is extracted in equation (15). The stress intensity factor for a crack in a vanishingly thin inhomogeneity is given by equation (18). When the condition on the ratio of the two remote anti-plane stresses in equation (16) is met, the remote loading will not induce any singular stress field at the two crack tips and the stress field within the cracked inhomogeneity is uniform and is given by equation (17), which indicates that the stress component $σ_{32}$ is always zero within the cracked inhomogeneity. The explicit general solution is derived in equation (24) for an arbitrary singularity applied in the matrix. The explicit general solution is derived in equation (25) for an arbitrary singularity applied in the cracked inhomogeneity.

Footnotes

ORCID iD

Peter Schiavone

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2023-03227 Schiavo).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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