A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion

Abstract

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.

1. Introduction

Analytical solutions to interface crack problems in laminate and fibrous composites have been well recorded in the literature with one of the main objectives being the study of the effect of interface fracture on the mechanical behavior of composites (see, e.g., [1–14]). In a recent study, Wang and Schiavone [15] analyzed the plane strain problem of a partially debonded circular elastic inhomogeneity and observed that when the debonded portion is filled with an incompressible liquid slit inclusion, interface crack growth can be completely suppressed as if the debonded portion of the circular interface doesn’t even exist when the elastic matrix is subjected to a hydrostatic far-field load. This fact suggests that liquid inclusions could be used quite effectively to repair interface cracks in composites. Although the micromechanics of liquid inclusions has been studied extensively in the last decade (see, e.g., [16–26]), the mechanism by which liquid inclusions play a role in repairing cracks has not been thoroughly investigated.

In this paper, we continue our recent study and analyze the plane strain problem of a partially debonded rigid elliptical inclusion with the debonded portion being occupied by a liquid slit inclusion when the surrounding infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. Since the initial area of the liquid slit inclusion is zero, the inclusion always behaves as if it is incompressible. The liquid slit inclusion admits an internal uniform hydrostatic stress field and remains perfectly bonded to the surrounding media. The original boundary value problem is reduced to a standard Riemann–Hilbert problem with discontinuous coefficients. An analytical solution to this Riemann–Hilbert problem can then be derived quite conveniently. However, this analytical solution contains the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. By imposing the incompressibility condition of the liquid slit inclusion and the balance of moment on a circular disk of infinite radius, the two unknowns can be uniquely determined by solving a set of two coupled linear algebraic equations. In particular, when the two tips of the debonded portion (or those of the liquid slit inclusion) are symmetric with respect to the major or minor axis of the elliptical interface, the two unknowns can be determined quite simply from two separate linear algebraic equations. Our detailed numerical results demonstrate that the shape of the elliptical interface, the positions of the two tips of the debonded portion, and the Poisson’s ratio of the matrix all exert a significant influence on the internal uniform hydrostatic tension characterizing the liquid inclusion and rigid body rotation characterizing the rigid inclusion. In particular, when the remote loading satisfies the harmonic shape condition for a (perfectly bonded) rigid elliptical inclusion [27,28], the rigid elliptical inclusion is still perfectly bonded to the surrounding matrix as if the debonded portion doesn’t even exist. This fact implies that the insertion of the liquid slit inclusion into the debonded portion will also suppress crack growth along the elliptical interface.

2. Complex variable formulation

We first establish a Cartesian coordinate system $\left\{x_i\right\}(i=1,2,3)$ . For plane strain deformation of an isotropic elastic material, the three in-plane stresses $({\sigma }_{11},{\sigma }_{22},{\sigma }_{12})$ , two in-plane displacements $(u_1,u_2)$ , and two stress functions $({\phi }_1,{\phi }_2)$ are given in terms of two analytic functions $\varphi (z)$ and $\psi (z)$ of the complex variable $z=x_1+i{x}_2$ as [29]

$$\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{11}+{\sigma }_{22}=2[\varphi \prime (z)+\overline{\varphi \prime (z)}],}\\ \multicolumn{1}{c}{{\sigma }_{22}-{\sigma }_{11}+2i{\sigma }_{12}=2[\overline{z}\varphi ″(z)+\psi \prime (z)],}\end{array}$$

(1)

and

$$\begin{array}{c}\multicolumn{1}{c}{2\mu (u_1+i{u}_2)=\kappa \varphi (z)-z\overline{\varphi \prime (z)}-\overline{\psi (z)},}\\ \multicolumn{1}{c}{{\phi }_1+i{\phi }_2=i[\varphi (z)+z\overline{\varphi \prime (z)}+\overline{\psi (z)}],}\end{array}$$

(2)

where $\kappa =3-4\nu$ , and µ and $\nu (0\le \nu \le 1/2)$ are the shear modulus and Poisson’s ratio, respectively. In addition, the stresses are related to the two stress functions through [9]

$$\begin{array}{c}\multicolumn{1}{c}{{\sigma }_{11}=-{\phi }_{1,2},{\sigma }_{12}={\phi }_{1,1},}\\ \multicolumn{1}{c}{{\sigma }_{21}=-{\phi }_{2,2},{\sigma }_{22}={\phi }_{2,1}.}\end{array}$$

(3)

3. Closed-form solution

As shown in Figure 1, a rigid elliptical inclusion is partially bonded to an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses $({\sigma }_{11}^{\infty },{\sigma }_{22}^{\infty },{\sigma }_{12}^{\infty })$ . The rigid inclusion remains perfectly bonded to the matrix along the portion $L_b$ of the elliptical interface L, and is debonded from the matrix along the remaining portion $L_c$ of the elliptical interface L. Thus, $L=L_b\cup L_c$ . In addition, the debonded portion $L_c$ is filled with a liquid slit inclusion, which admits an internal uniform hydrostatic stress field and which is still perfectly bonded to its surrounding media.

Figure 1.

A rigid elliptical inclusion partially bonded to an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses. The debonded portion $L_c$ of the elliptical interface is occupied by a liquid slit inclusion.

According to the above statement and the complex variable formulation in Section 2, the boundary value problem takes the following form in the physical z-plane:

$$\begin{array}{c}\multicolumn{1}{c}{\kappa \varphi (z)-z\overline{\varphi \prime (z)}-\overline{\psi (z)}=2i\mu {\varpi }_{21}z,z\in L_b;}\\ \multicolumn{1}{c}{\varphi (z)+z\overline{\varphi \prime (z)}+\overline{\psi (z)}={\sigma }_{0}z,z\in L_c;}\\ \multicolumn{1}{c}{\varphi (z)\cong \mathrm{Az}+O(1),\psi (z)\cong \mathrm{Bz}+O(1),\left|z\right|\to \infty ,}\end{array}$$

(4)

where ${\varpi }_{21}=\frac{1}{2}(u_{2,1}-u_{1,2})$ is the unknown rigid body rotation of the rigid elliptical inclusion to be determined, ${\sigma }_0$ is the unknown internal uniform hydrostatic tension within the liquid slit inclusion to be determined, and

$$A=\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4},B=\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }+2i{\sigma }_{12}^{\infty }}{2}.$$

(5)

We now introduce the following conformal mapping function for the matrix:

$$z=\omega (\xi )=R(\xi +\frac{m}{\xi }),\xi ={\omega }^{-1}(z)=\frac{z+\sqrt{z^2-4m{R}^2}}{2R},R>0,\left|m\right|\le 1,\left|\xi \right|\ge 1,$$

(6)

where m is a complex number.

As shown in Figure 2, using the mapping function in equation (6), the matrix is mapped onto $\left|\xi \right|\ge 1$ , the portion $L_b$ in the z-plane is mapped onto the arc $l_b$ of the unit circle in the ξ-plane, and the portion $L_c$ in the z-plane is mapped onto the remaining arc $l_c$ of the unit circle in the ξ-plane. Furthermore, the center of the arc $l_b$ lies on $\xi =1$ and the central angle subtended by the arc $l_b$ is $2{\theta }_0$ . The two tips of the arc $l_c$ are located at $\xi =a={\mathrm{e}}^{i{\theta }_0}$ and $\xi =\overline{a}={\mathrm{e}}^{-i{\theta }_0}$ . The choice of a complex number m is to account for the non-symmetry of the two tips of the debonded portion $L_c$ with respect to the principal axes of the ellipse L.

Figure 2.

The image ξ-plane.

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

$$\begin{array}{c}\multicolumn{1}{c}{\kappa \varphi (\xi )-\frac{\omega (\xi )\overline{\varphi \prime (\xi )}}{\overline{\omega \prime (\xi )}}-\overline{\psi (\xi )}=2i\mu {\varpi }_{21}\omega (\xi ),\xi \in l_b;}\\ \multicolumn{1}{c}{\varphi (\xi )+\frac{\omega (\xi )\overline{\varphi \prime (\xi )}}{\overline{\omega \prime (\xi )}}+\overline{\psi (\xi )}={\sigma }_0\omega (\xi ),\xi \in l_c;}\\ \multicolumn{1}{c}{\varphi (\xi )\cong \mathrm{AR}\xi +O(1),\psi (\xi )\cong \mathrm{BR}\xi +O(1),\left|\xi \right|\to \infty ,}\end{array}$$

(7)

where, for convenience, we write $\varphi (\xi )=\varphi (\omega (\xi )),\psi (\xi )=\psi (\omega (\xi ))$ .

Considering equation (7)₁, we introduce an auxiliary function $Y(\xi )$ defined as follows:

$$Y(\xi )=\{\begin{array}{c}\frac{\omega (\xi )\overline{\varphi }\prime \left(\frac{1}{\xi }\right)}{\overline{\omega }\prime \left(\frac{1}{\xi }\right)}+\overline{\psi }\left(\frac{1}{\xi }\right)+R(2i\mu {\varpi }_{21}-\kappa A)\xi -R(\overline{B}+\mathrm{Am})\frac{1}{\xi },\left|\xi \right|\le 1,\hfill \\ \multicolumn{1}{c}{\kappa \varphi (\xi )-\kappa \mathrm{AR}\xi -R(2i\mu m{\varpi }_{21}+\overline{B}+\mathrm{Am})\frac{1}{\xi },\left|\xi \right|\ge 1.}\end{array}$$

(8)

It is seen from equation (8) that $Y(\xi )$ is analytic and single valued everywhere in $\left|\xi \right|\le 1$ , and is analytic and single valued everywhere in $\left|\xi \right|\ge 1$ including the point at infinity. Furthermore, $Y(\xi )$ is continuous across the arc $l_b$ , i.e.,

$$Y^{+}(\xi )-Y^{-}(\xi )=0,\xi \in l_b,$$

(9)

where the superscripts “+” and “−” indicate the values when approaching the unit circle $\left|\xi \right|=1$ from inside and outside, respectively.

Using the definition of $Y(\xi )$ in equation (8), the interface condition in equation (7)₂ becomes

$$\begin{array}{c}\multicolumn{1}{c}{Y^{+}(\xi )+{\mathrm{e}}^{2\pi \epsilon }{Y}^{-}(\xi )}\\ \multicolumn{1}{c}{=R[{\sigma }_0+2i\mu {\varpi }_{21}-(\kappa +1)A]\xi +\frac{R}{\kappa }[\kappa m{\sigma }_0-2i\mu m{\varpi }_{21}-(\kappa +1)(\overline{B}+\mathrm{Am})]\frac{1}{\xi },\xi \in l_c,}\end{array}$$

(10)

where

$$\epsilon =-\frac{\ln \kappa }{2\pi }.$$

(11)

Equations (9) and (10) constitute a standard Riemann–Hilbert problem with discontinuous coefficients. An analytical solution to the Riemann–Hilbert problem in equations (9) and (10) can be derived as

$$Y(\xi )=\frac{R\kappa [{\sigma }_0+2i\mu {\varpi }_{21}-(\kappa +1)A]}{\kappa +1}[\xi -\chi (\xi )]+\frac{R[\kappa m{\sigma }_0-2i\mu m{\varpi }_{21}-(\kappa +1)(\overline{B}+\mathrm{Am})]}{\kappa +1}[\frac{1}{\xi }-\frac{\chi (\xi )}{\xi \chi (0)}],$$

(12)

where the Plemelj function $\chi (\xi )$ is defined by

$$\chi (\xi )=(\xi -a{)}^{1/2+i\epsilon }(\xi -\overline{a}{)}^{1/2-i\epsilon },$$

(13)

and

$$\chi (0)={\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}.$$

(14)

The brunch cut for the Plemelj function $\chi (\xi )$ is taken along $l_c$ so that $\chi (\xi )\cong \xi$ as $\xi \to \infty$ . The auxiliary function $Y(\xi )$ given by equation (12) contains the two unknowns ${\sigma }_0$ and ${\varpi }_{21}$ . Since the initial area of the liquid slit inclusion is zero, the change in the area of the liquid inclusion due to internal uniform hydrostatic stresses is zero. Thus, the liquid slit inclusion always behaves as if it is incompressible. The incompressibility of the liquid inclusion requires that [16,18,30,31]

$${\int }_{L_c}(u_n^{+}-u_n^{-})\mathrm{d}s=0,$$

(15)

where $u_n$ is the normal component of the displacement along $L_c$ . By considering equation (8), the incompressibility condition in equation (15) can be further written in the form

$$\mathrm{Im}\{\oint \frac{1}{{\xi }^2}\overline{\omega }\prime \left(\frac{1}{\xi }\right)Y(\xi )\mathrm{d}\xi \}=R\mathrm{Im}\{\oint (\frac{1}{{\xi }^2}-\overline{m})Y(\xi )\mathrm{d}\xi \}=0,$$

(16)

where the integral contour is taken around $l_c$ .

Using the residue theorem, the incompressibility condition in equation (16) is equivalent to

$$\begin{array}{c}\multicolumn{1}{c}{2\pi \mathrm{Re}\{Y\prime (0)\}-\mathrm{Im}\{{\int }_{\left|\xi \right|\to \infty }(\frac{1}{{\xi }^2}-\overline{m})Y(\xi )\mathrm{d}\xi \}}\\ \multicolumn{1}{c}{=2\pi \mathrm{Re}\{Y\prime (0)\}+\mathrm{Im}\{\overline{m}{\int }_{\left|\xi \right|\to \infty }Y(\xi )\mathrm{d}\xi \}=0,}\end{array}$$

(17)

where the integral is taken in the counterclockwise direction.

It is derived from equation (12) that

$$Y\prime (0)=\frac{R\kappa [{\sigma }_0+2i\mu {\varpi }_{21}-(\kappa +1)A]}{\kappa +1}[1-\chi \prime (0)]-\frac{R[\kappa m{\sigma }_0-2i\mu m{\varpi }_{21}-(\kappa +1)(\overline{B}+\mathrm{Am})]}{\kappa +1}\frac{\chi ″(0)}{2\chi (0)},$$

(18)

and

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Im}\{\overline{m}{\int }_{\left|\xi \right|\to \infty }Y(\xi )\mathrm{d}\xi \}}\\ \multicolumn{1}{c}{=\frac{2\pi R}{\kappa +1}[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{\chi (0)}]\{\kappa {\left|m\right|}^2{\sigma }_0-(\kappa +1)[\mathrm{Re}\left\{\mathrm{mB}\right\}+A{\left|m\right|}^2]\}}\\ \multicolumn{1}{c}{-\frac{\pi R\kappa (1+4{\epsilon }^2){\sin }^2{\theta }_0}{\kappa +1}[m\prime {\sigma }_0+2m″\mu {\varpi }_{21}-m\prime A(\kappa +1)],}\end{array}$$

(19)

where $m\prime$ and $m″$ are, respectively, the real and imaginary parts of m, and

$$\begin{array}{c}\multicolumn{1}{c}{\chi \prime (0)=-{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0),}\\ \multicolumn{1}{c}{\chi ″(0)={\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}{\sin }^2{\theta }_0(1+4{\epsilon }^2).}\end{array}$$

(20)

By substituting the results in equations (18) and (19) into equation (17), we obtain the following linear algebraic equation for the two unknowns ${\sigma }_0$ and ${\varpi }_{21}$ :

$$\begin{array}{c}\multicolumn{1}{c}{2\kappa \{1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m\prime (1+4{\epsilon }^2){\sin }^2{\theta }_0+{\left|m\right|}^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}\frac{{\sigma }_0}{\mu }}\\ \multicolumn{1}{c}{-2m″(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0{\varpi }_{21}}\\ \multicolumn{1}{c}{=(\kappa +1)\{2\kappa [1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)]-m\prime (\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0}\\ \multicolumn{1}{c}{+2{\left|m\right|}^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4\mu }}\\ \multicolumn{1}{c}{+(\kappa +1)\{2m\prime [1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]-(1+4{\epsilon }^2){\sin }^2{\theta }_0\}\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2\mu }-2m″(\kappa +1)[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\frac{{\sigma }_{12}^{\infty }}{\mu }.}\end{array}$$

(21)

In addition, the remote asymptotic behavior of $\psi (\xi )$ is

$$\psi (\xi )\cong \mathrm{BR}\xi +\frac{c+\mathrm{iR}\mathrm{Im}\left\{\mathrm{Bm}\right\}}{\xi }+O\left(\frac{1}{{\xi }^2}\right),\left|\xi \right|\to \infty ,$$

(22)

where c is an arbitrary real number (i.e., $\mathrm{Im}\left\{c\right\}=0$ ) to satisfy the condition that the resultant moment about the origin on the disk $\left|z\right|=\infty$ is zero.

It is obtained from equation (8) that the original pair of analytic functions $\varphi (\xi )$ and $\psi (\xi )$ can be expressed in terms of the auxiliary function $Y(\xi )$ as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\varphi (\xi )=\frac{1}{\kappa }Y(\xi )+\mathrm{AR}\xi +\frac{R}{\kappa }(2i\mu m{\varpi }_{21}+\overline{B}+\mathrm{Am})\frac{1}{\xi },}\\ \multicolumn{1}{c}{\psi (\xi )=\overline{Y}\left(\frac{1}{\xi }\right)+\frac{\xi (1+\overline{m}{\xi }^2)}{m-{\xi }^2}\varphi \prime (\xi )+R(2i\mu {\varpi }_{21}+\kappa A)\frac{1}{\xi }+R(B+A\overline{m})\xi .}\end{array}$$

(23)

Using equation (23) to impose the condition in equation (22), we obtain another linear algebraic equation for the two unknowns ${\sigma }_0$ and ${\varpi }_{21}$ as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{2}m″(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0\frac{{\sigma }_0}{\mu }}\\ \multicolumn{1}{c}{+2\{1-{\mathrm{e}}^{-2\epsilon {\theta }_0}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m\prime (1+4{\epsilon }^2){\sin }^2{\theta }_0+{\left|m\right|}^2[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]\}{\varpi }_{21}}\\ \multicolumn{1}{c}{=m″(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4\mu }+m″(\kappa +1)[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2\mu }}\\ \multicolumn{1}{c}{+(\kappa +1)\{m\prime [1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]-\frac{1}{2}(1+4{\epsilon }^2){\sin }^2{\theta }_0\}\frac{{\sigma }_{12}^{\infty }}{\mu }.}\end{array}$$

(24)

As a result, ${\sigma }_0$ and ${\varpi }_{21}$ can be uniquely determined by solving the two coupled linear algebraic equations in equations (21) and (24) as follows:

$$\frac{{\sigma }_0}{\mu }=\frac{c_{22}{\delta }_1-c_{12}{\delta }_2}{c_{11}{c}_{22}-c_{12}{c}_{21}},{\varpi }_{21}=\frac{-c_{21}{\delta }_1+c_{11}{\delta }_2}{c_{11}{c}_{22}-c_{12}{c}_{21}},$$

(25)

where the six coefficients $c_{11},c_{12},c_{21},c_{22},{\delta }_1$ , and ${\delta }_2$ are defined by

$$\begin{array}{c}\multicolumn{1}{c}{c_{11}=2\kappa \{1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m\prime (1+4{\epsilon }^2){\sin }^2{\theta }_0+{\left|m\right|}^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\},}\\ \multicolumn{1}{c}{c_{12}=-4c_{21}=-2m″(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0,}\\ \multicolumn{1}{c}{c_{22}=2\{1-{\mathrm{e}}^{-2\epsilon {\theta }_0}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m\prime (1+4{\epsilon }^2){\sin }^2{\theta }_0+{\left|m\right|}^2[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]\},}\end{array}$$

(26a)

$$\begin{array}{c}\multicolumn{1}{c}{{\delta }_1=(\kappa +1)\{2\kappa [1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)]-m\prime (\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0}\\ \multicolumn{1}{c}{+2{\left|m\right|}^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4\mu }}\\ \multicolumn{1}{c}{+(\kappa +1)\{2m\prime [1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]-(1+4{\epsilon }^2){\sin }^2{\theta }_0\}\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2\mu }}\\ \multicolumn{1}{c}{-2m″(\kappa +1)[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\frac{{\sigma }_{12}^{\infty }}{\mu },}\\ \multicolumn{1}{c}{{\delta }_2=m″(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4\mu }+m″(\kappa +1)[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2\mu }}\\ \multicolumn{1}{c}{+(\kappa +1)\{m\prime [1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]-\frac{1}{2}(1+4{\epsilon }^2){\sin }^2{\theta }_0\}\frac{{\sigma }_{12}^{\infty }}{\mu }.}\end{array}$$

(26b)

In particular, when m is real valued ( $-1\le m\le 1$ ), the two tips of $L_c$ are symmetric with respect to the major or minor axis of the ellipse L. In this case, equations (21) and (24) become decoupled, and ${\sigma }_0$ and ${\varpi }_{21}$ can be determined in a decoupled manner as

$${\sigma }_0=\frac{(\kappa +1)\left(\begin{array}{c}\{2\kappa [1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)]-m(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0\hfill \\ \multicolumn{1}{c}{+2m^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}\\ \multicolumn{1}{c}{-2\{(1+4{\epsilon }^2){\sin }^2{\theta }_0-2m[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}({\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty })}\end{array}\right)}{8\kappa \{1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m(1+4{\epsilon }^2){\sin }^2{\theta }_0+m^2[1+\frac{\cos {\theta }_0-2\epsilon \sin {\theta }_0}{{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}}]\}},$$

(27)

$${\varpi }_{21}=\frac{{\epsilon }_{12}^{\infty }(\kappa +1)\{m[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]-\frac{1}{2}(1+4{\epsilon }^2){\sin }^2{\theta }_0\}}{1-{\mathrm{e}}^{-2\epsilon {\theta }_0}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)-m(1+4{\epsilon }^2){\sin }^2{\theta }_0+m^2[1-{\mathrm{e}}^{2\epsilon {\theta }_0}(\cos {\theta }_0-2\epsilon \sin {\theta }_0)]},$$

(28)

where ${\epsilon }_{12}^{\infty }$ is the remote shear strain defined by

$${\epsilon }_{12}^{\infty }=\frac{{\sigma }_{12}^{\infty }}{2\mu }.$$

(29)

We can see from equations (27) and (28) that ${\sigma }_0$ is unaffected by the remote shear stress ${\sigma }_{12}^{\infty }$ and ${\varpi }_{21}$ is unaffected by the remote normal stresses ${\sigma }_{11}^{\infty }$ and ${\sigma }_{22}^{\infty }$ when the two tips of $L_c$ are symmetric with respect to major or minor axis of the ellipse L. We illustrate in Figures 3–8 the variations of ${\sigma }_0$ and ${\varpi }_{21}$ as functions of ${\theta }_0$ for different values of the two parameters m and κ. We can see from these figures that all three parameters ${\theta }_0,m$ , and κ exert a significant influence on ${\sigma }_0$ and ${\varpi }_{21}$ . In addition, it is seen from Figures 3 and 4 that when m is negative (e.g., $m=-0.8$ ), the value of ${\sigma }_0$ is insensitive to ${\theta }_0$ . In the following, we further discuss several typical cases when m is a real number to demonstrate the analytical results in equations (27) and (28).

Figure 3.

Variations of ${\sigma }_0$ as a function of ${\theta }_0$ for different values of m with $\kappa =2,{\sigma }_{22}^{\infty }=0$ .

Figure 4.

Variations of ${\sigma }_0$ as a function of ${\theta }_0$ for different values of m with $\kappa =2,{\sigma }_{11}^{\infty }=0$ .

Figure 5.

Variations of ${\sigma }_0$ as a function of ${\theta }_0$ for different values of κ with $m=0.5,{\sigma }_{22}^{\infty }=0$ .

Figure 6.

Variations of ${\sigma }_0$ as a function of ${\theta }_0$ for different values of κ with $m=0.5,{\sigma }_{11}^{\infty }=0$ .

Figure 7.

Variations of ${\varpi }_{21}$ as a function of ${\theta }_0$ for different values of m with $\kappa =2$ .

Figure 8.

Variations of ${\varpi }_{21}$ as a function of ${\theta }_0$ for different values of κ with $m=0.5$ .

When the rigid inclusion is completely debonded from the matrix with ${\theta }_0=0$ , equation (27) becomes

$${\sigma }_0=\frac{(\kappa +1)[{(1-m)}^2{\sigma }_{11}^{\infty }+{(1+m)}^2{\sigma }_{22}^{\infty }]}{4(1+m^2\kappa )}.$$

(30)

When the rigid inclusion is circular with $m=0$ , equation (27) becomes

$${\sigma }_0=\frac{(\kappa +1)({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}+\frac{(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0({\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty })}{4\kappa [1+{\mathrm{e}}^{2\epsilon (\pi -{\theta }_0)}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)]},$$

(31)

which is consistent with our recent result for a partially debonded circular elastic inclusion in which the debonded part is filled with an incompressible liquid inclusion [15].

When the remote loading satisfies the following harmonic shape condition for a perfectly bonded rigid elliptical inclusion [27,28]:

$$\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{{\sigma }_{22}^{\infty }+{\sigma }_{11}^{\infty }}=\frac{m(\kappa -1)}{2},{\sigma }_{12}^{\infty }=0,$$

(32)

Equation (27) reduces to

$${\sigma }_0=\frac{(\kappa +1)({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}=\frac{(\kappa +1)[{(1-m)}^2{\sigma }_{11}^{\infty }+{(1+m)}^2{\sigma }_{22}^{\infty }]}{4(1+m^2\kappa )},$$

(33)

which means that the rigid elliptical inclusion is still perfectly bonded to the matrix or that the whole elliptical interface L becomes a liquid–solid interface. This fact implies that the insertion of the liquid inclusion into the debonded portion $L_c$ will suppress crack growth.

When the rigid elliptical inclusion is perfectly bonded to the matrix with ${\theta }_0=\pi$ , equation (28) reduces to

$${\varpi }_{21}=\frac{m(\kappa +1){\epsilon }_{12}^{\infty }}{\kappa +m^2},$$

(34)

which is consistent with our previous studies [32,33].

When the rigid inclusion is circular with $m=0$ , equation (28) becomes

$${\varpi }_{21}=-\frac{{\epsilon }_{12}^{\infty }(\kappa +1)(1+4{\epsilon }^2){\sin }^2{\theta }_0}{2[1-{\mathrm{e}}^{-2\epsilon {\theta }_0}(\cos {\theta }_0+2\epsilon \sin {\theta }_0)]},$$

(35)

which also recovers our recent result for a partially debonded circular elastic inclusion in which the debonded portion is filled with an incompressible liquid inclusion [15].

Now, the auxiliary function $Y(\xi )$ has been completely determined. The original pair of analytic functions $\varphi (\xi )$ and $\psi (\xi )$ can then be obtained using equation (23). Consequently, the elastic field in the matrix is known. E.g., the displacement jumps across the debonded portion $L_c$ of the elliptical interface are given by

$$\begin{array}{c}\multicolumn{1}{c}{(u_1+i{u}_2{)}^{-}-(u_1+i{u}_2{)}^{+}}\\ \multicolumn{1}{c}{=\frac{R}{2\mu }\{\kappa [{\sigma }_0+2i\mu {\varpi }_{21}-(\kappa +1)A]+\frac{\kappa m{\sigma }_0-2i\mu m{\varpi }_{21}-(\kappa +1)(\overline{B}+\mathrm{Am})}{\xi \chi (0)}\}{\chi }^{+}(\xi ),\xi \in l_c.}\end{array}$$

(36)

When m is real valued and the remote loading satisfies equation (32), it is verified using equations (28) and (36) that ${\varpi }_{21}=0$ and there is no displacement jump across the debonded portion $L_c$ with the matrix remaining perfectly bonded to the rigid elliptical inclusion.

4. Conclusions

We have derived a closed-form solution to the problem of a rigid elliptical inclusion partially bonded to an infinite elastic matrix subjected to uniform remote in-plane stresses. The debonded portion of the elliptical interface is filled with a liquid slit inclusion. By enforcing the interface conditions along the elliptical interface, a Riemann–Hilbert problem with discontinuous coefficients is derived in equations (9) and (10), and its solution is given by equation (12) which contains two unknowns ${\sigma }_0$ and ${\varpi }_{21}$ . By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, a set of two coupled linear algebraic equations is derived in equations (21) and (24) from which ${\sigma }_0$ and ${\varpi }_{21}$ are uniquely determined in equation (25). When the two tips of $L_c$ are symmetric with respect to the major or minor axis of the ellipse L, ${\sigma }_0$ and ${\varpi }_{21}$ are obtained in an uncoupled manner in equations (27) and (28). Thus, the auxiliary function $Y(\xi )$ and the original pair of analytic functions $\varphi (\xi )$ and $\psi (\xi )$ characterizing the elastic field in the matrix have been completely determined. Once again, we demonstrate that the insertion of a liquid slit inclusion into the debonded portion will suppress crack growth along the elliptical interface.