A symmetric liquid lip inclusion in an infinite isotropic elastic matrix

Abstract

We study the plane strain problem of a symmetric compressible liquid lip inclusion with two cusps in an infinite isotropic elastic matrix subjected to uniform remote in-plane normal stresses. The pair of analytic functions characterizing the elastic field in the matrix is derived in closed form. Explicit, elementary, and concise expressions in terms of the two Skempton’s induced pore-pressure coefficients are obtained for the internal uniform hydrostatic tension within the liquid inclusion and the mode I stress intensity factor at the cusp tip. When the two remote normal stresses satisfy a single condition, the external loading will not induce any singular stress field at the cusp tips.

1. Introduction

Liquid inclusions trapped in solids such as crystals, rocks, biological tissues, hydrogels, and elastomers are ubiquitous in nature. Micromechanics analysis of a composite consisting of a solid matrix and liquid inclusions has become a focused research topic in the past decade [1–15]. In the majority of previous studies on this topic, only regular and smooth shapes of liquid inclusions such as three-dimensional ellipsoidal and two-dimensional elliptical have been investigated. However, the shapes of liquid inclusions in solids are in many cases more irregular. In some situations, the boundary of a liquid inclusion may also possess cusps and sharp corners which are known to be sources of stress singularity and high-stress gradients. It is therefore of great interest to investigate how irregular and cusped shapes of liquid inclusions affect the microscopic and macroscopic mechanical responses of a solid matrix filled with liquid inclusions. This research endeavors to examine this challenging problem.

In this paper, we study the plane strain problem associated with a symmetric compressible liquid lip inclusion with two cusps perfectly bonded to an infinite isotropic elastic matrix subjected to uniform remote in-plane normal stresses. A closed-form exact solution to the problem is derived by means of the techniques of conformal mapping [16,17] and analytic continuation [18] and the application of the residue theorem, which encounters some difficulty due to the appearance of two first-order poles (not located at the origin) in the mapping function. Explicit and elementary expressions for the internal uniform hydrostatic tension within the symmetric liquid lip inclusion and the mode I stress intensity factor at the cusp tip are obtained in terms of the two Skempton’s induced pore-pressure coefficients A and B [19] and are graphically illustrated. Due to the introduction of the two Skempton’s pore-pressure coefficients, the expressions for the internal uniform hydrostatic tension within the liquid inclusion and the stress intensity factor at the cusp tip are particularly concise. When the two remote normal stresses satisfy a single condition, the stress intensity factor at the cusp tip becomes zero and this particular external loading will not induce any singular stress field at the two cusp tips. For an incompressible liquid inclusion, the two remote normal stresses can have identical as well as opposite signs in order to remove the stress singularity. However, for an infinitely compressible liquid inclusion or a traction-free cusped cavity, the two remote normal stresses must have the same sign in order to remove the stress singularity. As a consequence of this research, the compressibility of a symmetric lip crack is explicitly determined from the acquired Skempton’s pore-pressure coefficient B and the formula for its determination in the work by Zimmerman [20].

2. Muskhelishvili’s 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 $z=x_1+\mathrm{i}{x}_2$ as [21]

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

(1)

and

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

(2)

where $\kappa =3-4\nu$ , μ and $\nu (0\le \nu \le 1/2)$ are the shear modulus and Poisson’s ratio, respectively. In addition, the three in-plane stress components are related to the two stress functions through [22]

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

(3)

3. A symmetric compressible liquid lip inclusion

As shown in Figure 1, we consider a symmetric compressible liquid lip inclusion with two cusps embedded in an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses $({\sigma }_{11}^{\infty },{\sigma }_{22}^{\infty },{\sigma }_{12}^{\infty })$ . Let $S_1$ and $S_2$ denote the liquid inclusion and the elastic matrix, respectively, which are perfectly bonded across the liquid–solid interface L.

Figure 1.

A symmetric compressible liquid lip inclusion embedded in an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses.

We first introduce the following conformal mapping function for the matrix [16,17]:

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

(4)

Using the mapping function in equation (4), the region occupied by the matrix is mapped onto $\left|\xi \right|\ge 1$ , the liquid–solid interface L is mapped onto $\left|\xi \right|=1$ , the two cusp tips at $z=\pm 2(1-m^2)$ are mapped onto the two points at $\xi =\pm 1$ on the unit circle. Furthermore, the initial area $A_1$ of the liquid inclusion enclosed by L can be determined using the following formula [23]:

$$A_1=\frac{1}{2\mathrm{i}}{\int }_{\left|\xi \right|=1}\overline{\omega (\xi )}\omega \prime (\xi )\mathrm{d}\xi .$$

(5)

By substituting the specific mapping function in equation (4) into equation (5) and applying the residue theorem, we finally arrive at

$$A_1=\frac{2\pi m^2(1-m^2)(3+2m^2-m^6)}{{(1+m^2)}^2},$$

(6)

which is illustrated in Figure 2 as a function of m. It is seen from Figure 2 that $A_1$ attains its maximum value of 2.8801 when $m=0.6158$ and becomes zero when $m=0$ and $m=1$ . Due to the appearance of the two first-order poles at $\xi =\pm m$ in the mapping function in equation (4), the area theorem [23] cannot be used directly.

Figure 2.

The initial area $A_1$ of the symmetric liquid lip inclusion as a function of m determined from equation (6).

In order to ensure a uniform hydrostatic stress field within the liquid inclusion, the two stress functions within the inclusion should take the following particular form

$${\phi }_1+\mathrm{i}{\phi }_2=\mathrm{i}{\sigma }_{0}z,z\in S_1,$$

(7)

where ${\sigma }_0$ is the internal uniform hydrostatic tension to be determined.

Thus, the continuity of tractions across the perfect liquid–solid interface L can be expressed in terms of $\varphi (\xi )=\varphi (\omega (\xi ))$ and $\psi (\xi )=\psi (\omega (\xi ))$ for $\left|\xi \right|\ge 1$ as

$$\varphi (\xi )+\frac{\omega (\xi )\overline{\varphi \prime (\xi )}}{\overline{\omega \prime (\xi )}}+\overline{\psi (\xi )}={\sigma }_0\omega (\xi ),\left|\xi \right|=1.$$

(8)

By considering the fact that the uniform remote shear stress ${\sigma }_{12}^{\infty }$ will not induce any internal hydrostatic tension within the liquid inclusion, in the ensuing analysis we focus on the particular case when the matrix is loaded by uniform remote in-plane normal stresses ${\sigma }_{11}^{\infty }$ and ${\sigma }_{22}^{\infty }$ . We now introduce the following analytic continuation [18]:

$$\varphi (\xi )=-\frac{\omega (\xi )\overline{\varphi }\prime \left(\frac{1}{\xi }\right)}{\overline{\omega }\prime \left(\frac{1}{\xi }\right)}-\overline{\psi }\left(\frac{1}{\xi }\right),\left|\xi \right|\le 1.$$

(9)

In view of equations (9), equation (8) can be written more concisely as

$${\varphi }^{-}(\xi )-{\varphi }^{+}(\xi )={\sigma }_0\omega (\xi ),\left|\xi \right|=1,$$

(10)

where the superscripts “+” and “−” indicate the values when approaching the unit circle from inside and outside, respectively. Equation (10) can be equivalently written as

$$\begin{array}{c}{\varphi }^{+}(\xi )-(\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4}-{\sigma }_0)\xi -[\frac{m^2({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}]\frac{1}{\xi }-S(\frac{1}{\xi -m}+\frac{1}{\xi +m})\hfill \\ ={\varphi }^{-}(\xi )-\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4}\xi -[\frac{m^2({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}-{\sigma }_0{m}^2]\frac{1}{\xi }\hfill \\ -[\frac{{\sigma }_0{(1-m^2)}^2}{2}+S](\frac{1}{\xi -m}+\frac{1}{\xi +m}),\left|\xi \right|=1,\hfill \end{array}$$

(11)

where the real constant S is related to $\varphi \prime \left(\frac{1}{m}\right)$ via the relation

$$S=-\frac{{(1-m^4)}^2}{2(1+m^4)(1+m^2+m^4)}\varphi \prime \left(\frac{1}{m}\right).$$

(12)

The left-hand side of equation (11) is analytic and single valued everywhere within the unit circle, and the right-hand side of equation (11) is analytic and single valued everywhere outside the unit circle including the point at infinity. Using Liouville’s theorem, we arrive at

$$\begin{array}{c}\varphi (\xi )=(\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4}-{\sigma }_0)\xi +[\frac{m^2({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}]\frac{1}{\xi }+S(\frac{1}{\xi -m}+\frac{1}{\xi +m}),\left|\xi \right|\le 1;\hfill \\ \varphi (\xi )=\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4}\xi +[\frac{m^2({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}-{\sigma }_0{m}^2]\frac{1}{\xi }+[\frac{{\sigma }_0{(1-m^2)}^2}{2}+S](\frac{1}{\xi -m}+\frac{1}{\xi +m}),\left|\xi \right|\ge 1.\hfill \end{array}$$

(13)

Substitution of equation (13)₂ into equation (12) leads to the following expression for the real constant S in terms of ${\sigma }_0$

$$S=-\frac{{(1-m^4)}^2}{2{(1+m^4)}^2}[\frac{(1-m^4)({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}+\frac{m^2({\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty })}{2}+\frac{{\sigma }_0{m}^2(m^2+m^4+m^6-1)}{{(1+m^2)}^2}].$$

(14)

At this stage, the single unknown ${\sigma }_0$ remains to be determined. The change in area $\mathrm{\Delta }{A}_2$ of the interface L on the side of the matrix due to uniform remote normal stresses and uniform tension on L can be determined using the following formula, which is adapted from that by Wu et al. [3],

$$\mathrm{\Delta }{A}_2=\frac{1}{2\mu }\mathrm{Im}\{{\int }_{\left|\xi \right|=1}\left[\overline{\kappa {\varphi }^{-}(\xi )+{\varphi }^{+}(\xi )}\right]\omega \prime (\xi )\mathrm{d}\xi \},$$

(15)

where the integral is in the counter-clockwise direction.

By substituting equation (13) into equation (15) and applying the residue theorem, we finally arrive at

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{A}_2=\frac{\pi }{\mu }\{\frac{(\kappa +1)({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-{\sigma }_0+(3m^2-1-m^4)\{(\kappa +1)[\frac{m^2({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })}{4}-\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}]-{\sigma }_0\kappa m^2\}}\\ \multicolumn{1}{c}{+\frac{m^2+m^4+m^6-1}{{(1+m^2)}^2}[{\sigma }_0\kappa {(1-m^2)}^2+2S(\kappa +1)]\}.}\end{array}$$

(16)

On the other hand, the change in area $\mathrm{\Delta }{A}_1$ of the linearly compressible liquid inclusion due to internal uniform hydrostatic stresses is given by

$$\mathrm{\Delta }{A}_1=\frac{{\sigma }_0{A}_1}{K}=\frac{\pi {\sigma }_0}{K}\frac{2m^2(1-m^2)(3+2m^2-m^6)}{{(1+m^2)}^2},$$

(17)

where K is the plane-strain bulk modulus of the liquid inclusion. The liquid inclusion is incompressible when $K\to \infty$ and becomes infinitely compressible when $K\to 0$ .

In order to ensure the continuity of displacements across the perfect liquid–solid interface L, it is necessary that

$$\mathrm{\Delta }{A}_1=\mathrm{\Delta }{A}_2.$$

(18)

Substituting equation (16) and (17) into equation (18), the internal uniform hydrostatic tension within the liquid inclusion can be uniquely determined as

$${\sigma }_0=B[{\sigma }_{22}^{\infty }+A({\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty })],$$

(19)

where A and B are the two Skempton’s induced pore-pressure coefficients [19] given explicitly by

$$\begin{array}{c}A=\frac{m^4(3+4m^2+m^8)}{2(1-2m^2+3m^4+3m^8-2m^{10}+m^{12})},\hfill \\ B=\frac{(\kappa +1){(1+m^2)}^2(1-2m^2+3m^4+3m^8-2m^{10}+m^{12})}{\left(\begin{array}{c}(1+m^4{)}^2(1+m^2{)}^2+m^2(1-m^2{)}^2(m^2+m^4+m^6-1{)}^2\hfill \\ \multicolumn{1}{c}{+\kappa [m^2(3m^2-1-m^4){(1+m^4)}^2{(1+m^2)}^2+{(1-m^2)}^2(1-m^4-4m^6-m^8+m^{12})]}\\ \multicolumn{1}{c}{+2\beta m^2(1-m^2)(1+m^4{)}^2(3+2m^2-m^6)}\end{array}\right)},\hfill \end{array}$$

(20)

where

$$\beta =\frac{\mu }{K}.$$

(21)

It is seen from equation (20) that (1) the pore-pressure coefficient A is independent of all the material constants of both the liquid inclusion and the elastic matrix, and depends only on the geometric parameter m; (2) the pore-pressure coefficient B is dependent on the three material and geometric parameters $\beta ,\kappa$ , and m.

When $m=0$ (L reduces to a horizontal slit), we have $A=0,B=1$ and consequently ${\sigma }_0={\sigma }_{22}^{\infty }$ . When $m=1$ (L reduces to a vertical slit), we have $A=B=1$ and consequently ${\sigma }_0={\sigma }_{11}^{\infty }$ . When $\beta =(\kappa -1)/2$ and ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ , we have $B=1$ and consequently ${\sigma }_0={\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ .

We illustrate in Figure 3 the variation of the coefficient A as a function of m. It is seen from Figure 3 that $A(0\le A\le 1)$ is an increasing function of m, and $A=0.5$ when $m=0.6325$ . We illustrate in Figures 4–6 the variations of the coefficient B as a function of $\beta ,\kappa$ and m. It is seen from Figures 4–6 that (1) B is always non-negative, and furthermore, $0\le B\le 1.7119$ ; (2) B is a decreasing function of both ν and β; (3) when both β and κ are fixed, B attains its maximum when $\beta <(\kappa -1)/2$ (see Figure 4) or minimum when $\beta >(\kappa -1)/2$ (see Figure 5) at $m=0.6271$ , which is unaffected by β and κ.

Figure 3.

Variation of the pore-pressure coefficient A as a function of m.

Figure 4.

Variations of the pore-pressure coefficient B as a function of m and κ with $\beta =0$ .

Figure 5.

Variations of the pore-pressure coefficient B as a function of m and κ with $\beta =1$ .

Figure 6.

Variations of the pore-pressure coefficient B as a function of κ and β with $m=0.5$ .

The mode I stress intensity factor at the right cusp tip $z=2(1-m^2)$ can be extracted as follows

$$\begin{array}{c}\multicolumn{1}{c}{K_{\mathrm{I}}=2\sqrt{\frac{\pi }{\omega ″(1)}}\varphi \prime (1)}\\ \multicolumn{1}{c}{=\frac{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}{4}\frac{\sqrt{2\pi {(1-m^2)}^3}}{1+m^2}[1+\frac{{(1+m^2)}^4-2B(1+2m^2+4m^4+2m^6+m^8)}{{(1+m^4)}^2}]}\\ \multicolumn{1}{c}{+\frac{{\sigma }_{22}^{\infty }-{\sigma }_{11}^{\infty }}{2}\frac{\sqrt{2\pi (1-m^2)}}{1+m^2}[1+\frac{m^2{(1+m^2)}^3+B(2A-1)(1-m^2)(1+2m^2+4m^4+2m^6+m^8)}{{(1+m^4)}^2}],}\end{array}$$

(22)

which is illustrated in Figures 7–12. It is seen from Figures 7–12 that (1) under a hydrostatic far-field load, $K_{\mathrm{I}}=0$ when $\beta =(\kappa -1)/2$ ; (2) when $m=0.6325$ , we have $A=0.5$ in Figure 3 and consequently $K_{\mathrm{I}}=2.518{\sigma }_{22}^{\infty }$ for any values of β and κ under a deviatoric far-field load (see Figures 8, 10, and 12); (3) under a hydrostatic far-field load, $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }\le 0$ when $0\le \beta <(\kappa -1)/2$ and $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }\ge 0$ when $\beta >(\kappa -1)/2$ (see Figure 7, 9, and 11); (4) under a hydrostatic far-field load, the magnitude of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ for a finite value of β is always lower than that for an infinitely compressible liquid inclusion with $\beta =\infty$ (see Figure 7); (5) under a deviatoric far-field load, the magnitude of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ for a finite value of β is lower than that for $\beta =\infty$ when $m<0.6325$ and is higher than that for $\beta =\infty$ when $m>0.6325$ (see Figure 8); (6) $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }\to 0$ as $m\to 0$ for a finite value of β, whereas $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }=2.5$ as $m\to 0$ for an infinite value of β (see Figures 7 and 8). The explicit expression for the stress intensity factor in equation (22) becomes concise due to the introduction of the two pore-pressure coefficients A and B.

Figure 7.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and β with $\kappa =2$ under a hydrostatic far-field load with ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ .

Figure 8.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and β with $\kappa =2$ under a deviatoric far-field load with ${\sigma }_{11}^{\infty }=-{\sigma }_{22}^{\infty }$ .

Figure 9.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and κ with $\beta =0$ for an incompressible liquid inclusion under a hydrostatic far-field load with ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ .

Figure 10.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and κ with $\beta =0$ for an incompressible liquid inclusion under a deviatoric far-field load with ${\sigma }_{11}^{\infty }=-{\sigma }_{22}^{\infty }$ .

Figure 11.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and κ with $\beta =1$ under a hydrostatic far-field load with ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ .

Figure 12.

Variations of $K_{\mathrm{I}}/{\sigma }_{22}^{\infty }$ as a function of m and κ with $\beta =1$ under a deviatoric far-field load with ${\sigma }_{11}^{\infty }=-{\sigma }_{22}^{\infty }$ .

It is derived from the stress intensity factor in equation (22) that the remote loading will not induce any singular stress field at the cusp tips when the following condition is met:

$$\frac{{\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty }}{{\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty }}=\frac{(1-m^2)[{(1+m^4)}^2+{(1+m^2)}^4-2B(1+2m^2+4m^4+2m^6+m^8)]}{2[{(1+m^4)}^2+m^2{(1+m^2)}^3+B(2A-1)(1-m^2)(1+2m^2+4m^4+2m^6+m^8)]},$$

(23)

which is illustrated in Figure 13. It is seen from Figure 13 that (1) the ratio $({\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty })/({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })$ lies between 0 and 1 when $\beta >0.5$ (see Figure 13(a)); (2) as expected, ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ for any value of m when $\beta =0.5$ (see Figure 13(a)); (3) the ratio $({\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty })/({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })$ lies in the two ranges of $(-\infty ,0]$ and $[1,+\infty )$ when $0\le \beta <0.5$ (see Figure 13(b)). The first observation implies that ${\sigma }_{11}^{\infty }$ and ${\sigma }_{22}^{\infty }$ should have the same sign and in addition $\left|{\sigma }_{11}^{\infty }\right|\ge \left|{\sigma }_{22}^{\infty }\right|$ in order to remove the singular stress field. The third observation implies that in order to remove the singular stress field, ${\sigma }_{11}^{\infty }$ and ${\sigma }_{22}^{\infty }$ have the same sign and in addition $\left|{\sigma }_{22}^{\infty }\right|\ge \left|{\sigma }_{11}^{\infty }\right|$ when the ratio lies between −1 and 0 or have opposite signs when the magnitude of the ratio is greater than unity. In addition, it is observed from Figure 13(b) that ${\sigma }_{11}^{\infty }=-{\sigma }_{22}^{\infty }$ when $m=0.365,\beta =0$ , which is reflected in Figure 8, or when $m=0.177,\beta =0.4$ .

Figure 13.

The ratio $({\sigma }_{11}^{\infty }-{\sigma }_{22}^{\infty })/({\sigma }_{11}^{\infty }+{\sigma }_{22}^{\infty })$ as a function of m for different values of β with $\kappa =2$ determined from equation (23): (a) $\beta \ge (\kappa -1)/2=0.5$ and (b) $0\le \beta <(\kappa -1)/2=0.5$ .

As a consequence, the parameter $C_{\mathrm{pc}}$ [24,25], the fractional decrease in the area of a symmetric traction-free lip crack due to a remote hydrostatic pressure of unit magnitude, can be determined from the expression for the Skempton’s induced pore-pressure coefficient B in equation (20)₂ and the formula for the determination of the pore-pressure coefficient B in Zimmerman [20] as follows:

$$\frac{C_{\mathrm{pc}}\mu }{1-\nu }=\frac{2{(1+m^2)}^2(1-2m^2+3m^4+3m^8-2m^{10}+m^{12})}{m^2(1-m^2)(3+2m^2+6m^4+3m^6+3m^8-m^{14})},$$

(24)

which is illustrated in Figure 14. It is seen from Figure 14 that $C_{\mathrm{pc}}\mu /(1-\nu )$ attains its minimum of 2.4046 at $m=0.6271$ , precisely the value of m to attain $max(B)$ in Figure 4 and $min(B)$ in Figure 5. The compressibility $C_{\mathrm{pc}}$ in equation (24) is proportional to $(1-\nu )/\mu$ , a result in agreement with that for a pore of any shape by Ekneligoda and Zimmerman [25].

Figure 14.

The inverse of the compressibility $C_{\mathrm{pc}}$ of a symmetric lip crack as a function of m determined from equation (24).

4. Conclusion

A rigorous analysis is performed to investigate the plane strain problem of a symmetric compressible liquid lip inclusion embedded in an infinite isotropic elastic matrix subjected to uniform remote in-plane normal stresses. The analytic function $\varphi (\xi )$ for $\left|\xi \right|\ge 1$ and its analytic continuation within the unit circle are obtained in equation (13). The internal uniform hydrostatic tension within the liquid inclusion is obtained in equation (19) in terms of the two Skempton’s induced pore-pressure coefficients A and B. The coefficient B is dependent on the three parameters $\beta ,\kappa$ , and m, while the other coefficient A depends only on m. The mode I stress intensity factor at the cusp tip is obtained in equation (22) also in terms of A and B. The remote loading condition leading to the disappearance of the singular stress field at the cusp tips is obtained in equation (23). As a consequence of our analysis, we also present in equation (24) the compressibility $C_{\mathrm{pc}}$ of a symmetric lip crack. Due to the adoption of the non-standard mapping function in equation (4) which contains the two first-order poles at $\xi =\pm m$ , the analytical determination of $A_1$ and $\mathrm{\Delta }{A}_2$ in equations (6) and (16) poses considerable challenge when applying the residue theorem. In general, a closed-form solution to a linearly compressible liquid inclusion of any irregular and cusped shape embedded in an infinite isotropic elastic matrix can be derived as long as the given mapping function contains a finite number of first-order or higher-order poles within the unit circle, irrespective of the complexity of the mapping function.