A compressible liquid inclusion of arbitrary shape in an isotropic elastic matrix

Abstract

We use Muskhelishvili’s complex variable formulation to solve the plane-strain problem of a compressible liquid inclusion of arbitrary shape embedded within an infinite isotropic elastic matrix under uniform remote in-plane stresses. First, the exterior of the domain occupied by the liquid inclusion is mapped onto the exterior of a unit circle in the image plane using a mapping function that contains an arbitrary number of terms. With the aid of a modified form of analytic continuation, a set of linear algebraic equations with relatively simple structure is obtained. Once this set of linear algebraic equations is solved, the internal uniform hydrostatic stress field within the liquid inclusion and the elastic field in the matrix (characterized by a pair of analytic functions) are fully determined. We illustrate our theory by deriving a closed-form solution for a hypotrochoidal liquid inclusion and comparing our results with those available in the existing literature. In addition, numerical results for the internal uniform hydrostatic stress within liquid inclusions with an n-fold axis of symmetry are presented graphically to examine the influence of the number of terms used in the mapping function. Finally, we determine the internal hydrostatic stress for the case of a rectangular liquid inclusion with various aspect ratios.

1. Introduction

The study of porous materials and liquid inclusions embedded in a solid matrix has recently been the focus of intensive research in the literature. This topic has attracted considerable interest due to the presence of such materials in nature (e.g., rocks, crystals, ceramics, hydrogels, elastomers, and in biological tissues, such as bone [1–4]) and the role it plays in the design of advanced composite structures consisting of solids into which are embedded liquid inclusions: these structures demonstrate distinct mechanical properties including enhanced compliance, enhanced overall deformability, and the stiffening phenomenon for “small” liquid inclusions [5,6].

Eshelby’s celebrated theory of the mechanics of inclusions most often forms the basis of so many analyses which seek to explain how an inclusion in an elastic matrix deforms in response to remote stresses [7]. In the progression of research in this area, considerable attention has been devoted to the shape-dependence of a liquid inclusion embedded in a solid elastic matrix. However, these investigations have largely been restricted to simple inclusion geometries. Also, most of these studies focus on incompressible liquid inclusions perhaps as a result of the ease of tractability of the corresponding governing boundary value problems. One particularly elegant and relatively simple method used recently to derive closed-form solutions for the problem of a liquid inclusion embedded in a solid matrix involves the use of a conformal mapping function to map the boundary of the liquid inclusion onto a unit circle in the image plane. This technique can be used for different shapes of liquid inclusions, such as circles and ellipses [8–11], and for the case of liquid inclusions and pores having an n-fold axis of symmetry [12–15].

In this paper, we present a detailed analysis of the problem of an arbitrarily shaped compressible liquid inclusion embedded in an infinite isotropic elastic matrix under uniform remote in-plane stresses. We assume that the inclusion is perfectly bonded to the surrounding matrix. A conformal mapping function with N + 1 terms (where N is a positive integer allowed to be arbitrarily large to accommodate any smooth shape of inclusion) is used to map the exterior of the inclusion onto the exterior of a unit circle in the image plane. By imposing a modified form of analytic continuation across the perfect liquid–solid interface, a set of $2N+1$ linear algebraic equations with relatively simple structure is derived. Solving these equations results in the internal uniform hydrostatic stress in the liquid inclusion. Solutions for the case $N=3$ and for a liquid inclusion in the shape of a hypotrochoid are shown to be consistent with those reported in the existing literature. Furthermore, results for a compressible liquid inclusion with an n-fold axis of symmetry (n ≥ 3) are also obtained specifically to analyze the influence of the number N + 1 of terms in the mapping function on the internal stress of the liquid inclusion. Finally, we investigate the effect of the aspect ratio of a rectangular liquid inclusion on the internal stress inside the liquid inclusion.

2. Muskhelishvili’s complex variable formulation

We first establish a rectangular coordinate system $\left\{x_i\right\}(i=1,2,3)$ . For the 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 $({\phi }_1,{\phi }_2)$ can be expressed in terms of two analytic functions $\varphi (z)$ and $\psi (z)$ of the single complex variable $z=x_1+\mathrm{i}{x}_2$ as [16]

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

(1)

and,

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

(2)

where $\kappa =3-4\nu$ , µ and $\nu (0\le \nu \le 1/122)$ 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 [17]

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

(3)

3. General solution

We consider a compressible liquid inclusion of arbitrary shape 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.

We first introduce the following conformal mapping function for the matrix [18,19]:

$$z=\omega (\xi )=R(\xi +\sum _{n=1}^N{m}_n{\xi }^{-n}),\left|\xi \right|\ge 1,$$

(4)

where R is a real scaling coefficient, $m_n(n=1,2,\cdots ,N)$ are N given complex constants. Using the mapping function in equation (4), the domain occupied by the matrix is mapped onto $\left|\xi \right|\ge 1$ and the liquid–solid interface L is mapped onto $\left|\xi \right|=1$ . In this discussion, the integer N can be arbitrarily large. Thus, the mapping function in equation (4) with $\left|\xi \right|=1$ can describe an arbitrarily shaped liquid–solid interface L.

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

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

(5)

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

The continuity of tractions across the perfect liquid–solid interface L can then be expressed in terms of $\varphi (\xi )=\varphi (\omega (\xi )),\psi (\xi )=\psi (\omega (\xi ))$ as follows:

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

(6)

The following analytic continuation is then introduced [19]

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

(7)

where

$$\varphi (\xi )\cong \mathrm{RA}\xi +O(1),\psi (\xi )\cong \mathrm{RB}\xi +O(1)\mathrm{as}\left|\xi \right|\to \infty ,$$

(8)

with

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

(9)

In view of equation (7), the interface condition in equation (6) can be written more concisely in the form:

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

(10)

where the superscripts “+” and “−” denote the values obtained when approaching the unit circle from within and from outside, respectively.

It follows from equation (10) that

$$\begin{array}{cc}\multicolumn{1}{c}{}& \varphi (\xi )=R[(A-{\sigma }_0)\xi +\sum _{n=1}^N{a}_n{\xi }^{-n}],\left|\xi \right|\le 1;\hfill \\ \multicolumn{1}{c}{}& \varphi (\xi )=R[A\xi +\sum _{n=1}^N(a_n+m_n{\sigma }_0){\xi }^{-n}],\left|\xi \right|\ge 1,\hfill \end{array}$$

(11)

where $a_n(n=1,2,\cdots ,N)$ are N unknown complex constants to be determined.

The derivation will become extremely tedious and cumbersome if the analytic continuation in equation (7) is used directly. Instead, equation (7) can be written in the following equivalent form:

$$\overline{\omega }\prime \left(\frac{1}{\xi }\right)\varphi (\xi )+\omega (\xi )\overline{\varphi }\prime \left(\frac{1}{\xi }\right)+\overline{\omega }\prime \left(\frac{1}{\xi }\right)\overline{\psi }\left(\frac{1}{\xi }\right)=0,\left|\xi \right|\le 1.$$

(12)

Substituting equations (4), (8), and (11) into equation (12) and equating coefficients for like negative powers of ξ: ${\xi }^{-n}(n=1,2,\cdots ,N)$ , we arrive expediently at the following N linear algebraic equations:

$$\begin{array}{cc}\multicolumn{1}{c}{}& a_n-\sum _{j=n+2}^N(j-n-1){\overline{m}}_{j-n-1}{a}_j-\sum _{j=1}^{N-n-1}j{m}_{j+n+1}{\overline{a}}_j\hfill \\ \multicolumn{1}{c}{}& -\sum _{j=1}^{N-n-1}j{\overline{m}}_j{m}_{j+n+1}{\sigma }_0=-m_{n}A-{\delta }_{n1}\overline{B},n=1,2,\cdots ,N-2;\hfill \\ \multicolumn{1}{c}{}& a_{N-1}=-m_{N-1}A,a_N=-m_{N}A,\hfill \end{array}$$

(13)

where ${\delta }_{n1}$ is the Kronecker delta. The structure of the linear algebraic equations in equation (13) is quite simple even when the integer N is arbitrarily large.

By taking the conjugate of equation (13), we obtain the following N linear algebraic equations:

$$\begin{array}{cc}\multicolumn{1}{c}{}& \sum _{j=1}^{N-n-1}j{\overline{m}}_{j+n+1}{a}_j-{\overline{a}}_n+\sum _{j=n+2}^N(j-n-1)m_{j-n-1}{\overline{a}}_j\hfill \\ \multicolumn{1}{c}{}& +\sum _{j=1}^{N-n-1}j{m}_j{\overline{m}}_{j+n+1}{\sigma }_0={\overline{m}}_{n}A+{\delta }_{n1}B,n=1,2,\cdots ,N-2;\hfill \\ \multicolumn{1}{c}{}& {\overline{a}}_{N-1}=-{\overline{m}}_{N-1}A,{\overline{a}}_N=-{\overline{m}}_{N}A.\hfill \end{array}$$

(14)

The change in area $\Delta A_2$ of the boundary L on the side of the matrix due to remote loading and uniform tension on L can be determined using the following formula, which can be adapted from the result in Wu et al. [9], as

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

(15)

where the integral is taken in the counter-clockwise direction.

By substituting equations (4) and (11) into equation (15), we find that

$$\mathrm{\Delta }{A}_2=\frac{\pi R^2}{\mu }\{A(\kappa +1)-(\kappa +1)\sum _{j=1}^{N}j\mathrm{Re}\left\{{\overline{m}}_j{a}_j\right\}-[1+\kappa \sum _{j=1}^{N}j{\left|m_j\right|}^2]{\sigma }_0\}.$$

(16)

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

$$\mathrm{\Delta }{A}_1=\frac{\pi R^2(1-\sum _{j=1}^{N}j{\left|m_j\right|}^2){\sigma }_0}{K},$$

(17)

where $\pi R^2(1-\sum _{j=1}^{N}j{|m_j|}^2)$ is the initial area $A_1$ of the compressible liquid inclusion and K is its plane-strain bulk modulus. The liquid inclusion becomes 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)

By substituting equations (16) and (17) into equation (18), the following linear algebraic equation is obtained

$$\sum _{j=1}^{N}j{\overline{m}}_j{a}_j+\sum _{j=1}^{N}j{m}_j{\overline{a}}_j+\frac{2}{\kappa +1}[1+\beta +(\kappa -\beta )\sum _{j=1}^{N}j{\left|m_j\right|}^2]{\sigma }_0=2A,$$

(19)

where

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

(20)

Now equations (13), (14), and (19) form a set of coupled $2N+1$ linear algebraic equations for the $2N+1$ unknowns $a_n,{\overline{a}}_n(n=1,2,\cdots ,N)$ and ${\sigma }_0$ . Consequently, $a_n,{\overline{a}}_n(n=1,2,\cdots ,N)$ and ${\sigma }_0$ can be uniquely determined.

When $N=3$ , it is obtained from equations (13), (14), and (19) that

$$\begin{array}{cc}\multicolumn{1}{c}{}& a_1=\frac{\left(\begin{array}{c}-\{m_1^2{\overline{m}}_3+\frac{2}{\kappa +1}[1+\beta +(\kappa -\beta )({\left|m_1\right|}^2+2{\left|m_2\right|}^2+3{\left|m_3\right|}^2)]\}\hfill \\ \multicolumn{1}{c}{\times [A(m_1+2{\overline{m}}_1{m}_3+m_1{\left|m_3\right|}^2)+B{m}_3+\overline{B}]}\\ \multicolumn{1}{c}{+m_3({\overline{m}}_1+m_1{\overline{m}}_3)[A(2+4{\left|m_2\right|}^2+6{\left|m_3\right|}^2+{\left|m_1\right|}^2+m_1^2{\overline{m}}_3)+B{m}_1]}\end{array}\right)}{m_1^2{\overline{m}}_3+{\overline{m}}_1^2{m}_3+2{\left|m_1{m}_3\right|}^2+\frac{2}{\kappa +1}(1-{\left|m_3\right|}^2)[1+\beta +(\kappa -\beta )({\left|m_1\right|}^2+2{\left|m_2\right|}^2+3{\left|m_3\right|}^2)]},\hfill \\ \multicolumn{1}{c}{}& {\sigma }_0=\frac{2A[{|m_1+{\overline{m}}_1{m}_3|}^2+(1-{\left|m_3\right|}^2)(1+2{\left|m_2\right|}^2+3{\left|m_3\right|}^2)]+B(m_1+{\overline{m}}_1{m}_3)+\overline{B}({\overline{m}}_1+m_1{\overline{m}}_3)}{m_1^2{\overline{m}}_3+{\overline{m}}_1^2{m}_3+2{\left|m_1{m}_3\right|}^2+\frac{2}{\kappa +1}(1-{\left|m_3\right|}^2)[1+\beta +(\kappa -\beta )({\left|m_1\right|}^2+2{\left|m_2\right|}^2+3{\left|m_3\right|}^2)]},\hfill \\ \multicolumn{1}{c}{}& a_2=-m_{2}A,a_3=-m_{3}A.\hfill \end{array}$$

(21)

In particular, when $m_1={\overline{m}}_1,m_2=0,m_3={\overline{m}}_3$ and the matrix is subjected to uniform remote hydrostatic stresses (i.e., ${\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty },{\sigma }_{12}^{\infty }=0$ ), we find from equation (21) that

$${\sigma }_0=\frac{{\sigma }_{11}^{\infty }(\kappa +1)[m_1^2(1+m_3)+(1-m_3)(1+3m_3^2)]}{2\{(\kappa +1)m_1^2{m}_3+(1-m_3)[1+\beta +(\kappa -\beta )(m_1^2+3m_3^2)]\}},$$

(22)

which recovers the result in Wang and Schiavone [12].

When $m_1={\overline{m}}_1,m_2=m_3=0$ , from equation (21), we have that

$${\sigma }_0=\frac{(\kappa +1)[{\sigma }_{11}^{\infty }{(1-m_1)}^2+{\sigma }_{22}^{\infty }{(1+m_1)}^2]}{4[1+\beta +m_1^2(\kappa -\beta )]},$$

(23)

which is also in agreement with the result in Wang and Schiavone [20].

Once these $2N+1$ unknowns have been obtained by solving equations (13), (14), and (19), the internal uniform hydrostatic stress field within the liquid inclusion can be identified. The elastic field in the matrix is characterized by the pair of analytic functions $\varphi (\xi )$ for $\left|\xi \right|\ge 1$ given by equation (11)₂ and $\psi (\xi )$ obtained from the analytic continuation in equation (7) as follows

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

(24)

For example, the hoop stress ${\sigma }_{\mathrm{tt}}$ along the liquid–solid interface L on the matrix side is given explicitly by

$${\sigma }_{\mathrm{tt}}=4\mathrm{Re}\left\{\frac{A-\sum _{n=1}^{N}n(a_n+m_n{\sigma }_0){\xi }^{-n-1}}{1-\sum _{n=1}^{N}n{m}_n{\xi }^{-n-1}}\right\}-{\sigma }_0,\xi ={\mathrm{e}}^{\mathrm{i}\theta }.$$

(25)

When the matrix is subjected to uniform remote hydrostatic stresses and the plane-strain bulk modulus of the liquid inclusion is equal to that of the elastic matrix, that is, $\beta =(\kappa -1)/(\kappa -1)22$ , it follows from equations (11), (13), (14), and (19) that ${\sigma }_0={\sigma }_{11}^{\infty }={\sigma }_{22}^{\infty }$ and

$$\varphi (z)=\frac{{\sigma }_{11}^{\infty }}{2}z,\psi (z)=0,z\in S_2.$$

(26)

In this case, the presence of the liquid inclusion of arbitrary shape will not disturb the uniform hydrostatic stress field in the elastic matrix (i.e. the liquid inclusion is, in fact, neutral).

4. Numerical results and discussions

We apply the procedure developed earlier to several cases involving specific geometries describing liquid inclusions of various shapes; for example, a hypotrochoid, shapes with an n-fold axis of symmetry and rectangles with various side ratios (see Figure 1). In each case, the hydrostatic stress within the liquid inclusion is computed. In addition, we study the effect of the number of terms taken in the mapping function on the internal uniform hydrostatic stress.

Figure 1.

Different shapes of liquid inclusions analyzed in the numerical section.

4.1. Hypotrochoidal liquid inclusion

The mapping function for a hypotrochoidal liquid inclusion is given by [16]

$$z=\omega \left(\xi \right)=R(\xi +\frac{m_n}{{\xi }^n}),m_n\in [0,1/n].$$

(27)

With this mapping function, equations (13), (14), and (19) for the case of a remote hydrostatic load ( ${\sigma }_{22}^{\infty }={\sigma }_{11}^{\infty }\ne 0,{\sigma }_{12}^{\infty }=0)$ lead to a system of three equations and three unknowns from which ${\sigma }_0$ is obtained as follows

$${\sigma }_0=\frac{{\sigma }_{11}^{\infty }(\kappa +1)(1+m_n^{2}n)}{2(1+\kappa m_n^{2}n+\beta (1-m_n^{2}n))},$$

(28)

which agrees with the result in Wang and Schiavone [12].

In the particular case when n = 1, the inclusion is elliptical and we have

$${\sigma }_0=\frac{{\sigma }_{11}^{\infty }(\kappa +1)(1+m_1^2)}{2(1+\kappa m_1^2+\beta (1-m_1^2))}.$$

(29)

Figure 2 illustrates the ratio $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ for hypotrochoids with cusps ( $m_n=\frac{1}{n})$ when $\beta =0.5$ for various values of the parameter κ. Figure 3 illustrates the same result but for hypotrochoids with rounded corners $(m_n=\frac{2}{n(n+1)})$ .

Figure 2.

The behavior of ${\sigma }_0/{\sigma }_{11}^{\infty }$ as a function of n for various κ values, assuming $(m_n=\frac{1}{n})$ in the case of a hypotrochoidal compressible liquid inclusion with cusps.

Figure 3.

The behavior of ${\sigma }_0/{\sigma }_{11}^{\infty }$ as a function of n for various κ values, assuming $(m_n=\frac{2}{n(n+1)})$ in the case of a hypotrochoidal compressible liquid inclusion with rounded corners.

It can be observed from the figures that as $n\to \infty$ , the shape of the hypotrochoid approaches that of a circle, resulting in the convergence of $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ to $\frac{\kappa +1}{2(1+\beta )}$ . Comparing the results for these two figures shows that for liquid inclusions in the shape of hypotrochoids with rounded corners, the ratio $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ converges at relatively small values of n ( $n\approx 15),$ whereas for hypotrochoids with cusps, the ratio converges at larger values of n ( $n\approx 50)$ . We conclude from the figures that for each shape, $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ increases with increasing κ. Figures 2 and 3 demonstrate that for a compressible hypotrochoidal liquid inclusion $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}=1$ when the bulk modulus of the liquid equates to that of the elastic matrix ( $\kappa =2\beta +1)$ . It is also worth noting that these figures also illustrate that, for $\kappa <1+2\beta =2$ , the internal hydrostatic stress within a hypotrochoidal liquid inclusion with cusps is higher than that within a hypotrochoidal liquid inclusion with rounded corners for $n\ge 2$ . Conversely, for $\kappa >1+2\beta =2$ the internal hydrostatic stress within a hypotrochoidal liquid inclusion with cusps is lower than that within a hypotrochoidal liquid inclusion with rounded corners for $n\ge 2$ . Furthermore, for $\kappa <1+2\beta =2$ , $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ decreases before converging, whereas for $\kappa >1+2\beta =2,$ the trend is reversed.

Figure 4 illustrates the variation of $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ as a function of β for fixed values of n and κ in a hypotrochoidal liquid inclusion with rounded corners. It can be seen from the figure that $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ decreases as β increases. Since $\beta =\frac{\mu }{K}$ , where μ is the shear modulus of the matrix and K is the plane-strain bulk modulus of the liquid, β is directly related to the properties of the matrix and the liquid. Specifically, for the same liquid inclusion, β is larger for stiffer matrices, while for the same matrix, an increase in β indicates a more compressible liquid inclusion.

Figure 4.

Dependence of ${\sigma }_0/{\sigma }_{11}^{\infty }$ on β when $\kappa =1.5\mathrm{and}n=100$ for a hypotrochoidal liquid inclusion with rounded corners.

4.2. Liquid inclusions with an n-fold axis of symmetry

In this section, liquid inclusions with an n-fold axis of symmetry are analyzed. Equations (30)–(33) present the mapping functions for the shapes considered in this study, derived using the Schwarz–Christoffel transformation [18]:

$$\omega \left(\xi \right)=R(\xi +\frac{1}{3}{\xi }^{-2}+\frac{1}{45}{\xi }^{-5}+\frac{1}{162}{\xi }^{-8}+\dots ):\mathrm{equilateral}\mathrm{triangle},$$

(30)

$$\omega \left(\xi \right)=R(\xi -\frac{1}{6}{\xi }^{-3}+\frac{1}{56}{\xi }^{-7}-\frac{1}{176}{\xi }^{-11}+\dots ):\mathrm{square},$$

(31)

$$\omega \left(\xi \right)=R(\xi +\frac{1}{10}{\xi }^{-4}+\frac{1}{75}{\xi }^{-9}+\frac{4}{875}{\xi }^{-14}+\dots ):\mathrm{regular}\mathrm{pentagon},$$

(32)

$$\omega \left(\xi \right)=R(\xi -\frac{1}{15}{\xi }^{-5}+\frac{1}{99}{\xi }^{-11}-\frac{5}{1377}{\xi }^{-17}+\dots ):\mathrm{regular}\mathrm{hexagon}.$$

(33)

Equations (13), (14), and (19) are solved using the given mapping functions for each shape. Figure 5 illustrates the variation of $\frac{{\sigma }_0}{A}$ as a function of N (the number of terms in the mapping function is N + 1) for an equilateral triangle, a square, a regular pentagon, and a regular hexagon. The results indicate that, as the number of terms in the mapping function increases, $\frac{{\sigma }_0}{A}$ converges to its true value for each shape. This occurs as a result of the increase in the number of terms which essentially eliminates the rounded corners in the shapes. The rate of convergence decreases as the number of sides of the shape increases. Consequently, for the same level of error in $\frac{{\sigma }_0}{A},$ a liquid inclusion in the shape of an equilateral triangle requires fewer terms in the mapping function. The repeated numbers in the figure are the results of zero terms in the mapping function. The figure is generated for $\kappa =1.5$ , which is less than $1+2\beta =2$ . For $\kappa >1+2\beta =2$ , the trend would reverse, exhibiting a decrease instead of an increase with increasing N. It can also be observed that, for $\kappa <1+2\beta$ , $\frac{{\sigma }_0}{A}$ decreases as the number of sides in the shape of the liquid inclusion increases. Therefore, $\frac{{\sigma }_0}{A}$ has the highest value in a triangular liquid inclusion. Moreover, the results obtained are valid for any remote loading condition, indicating that the applied remote deviatoric stress does not influence the internal hydrostatic stress of the liquid inclusion with an n-fold axis of symmetry.

Figure 5.

Internal hydrostatic stress for shapes with an n-fold axis of symmetry.

4.3. Liquid inclusions in the shape of rectangles

In this section, ${\sigma }_0$ for a rectangular liquid inclusion is obtained by solving all $2N+1$ equations and by considering the following mapping function for a rectangle [21]

$$z=\omega \left(\xi \right)=R(\xi +\sum \frac{m_k}{{\xi }^k})(k=1,3,5,7,9,11,\dots )$$

(34)

Figure 6 illustrates the dependence of $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ with N under remote hydrostatic loading for various side ratios of the rectangular liquid inclusion. This figure shows that, for $\kappa <1+2\beta$ , $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ increases as the side ratio of the rectangular liquid inclusion increases.

Figure 6.

The dependence of $\frac{{\sigma }_0}{{\sigma }_{11}^{\infty }}$ to N in a rectangular liquid inclusion under remote hydrostatic loading for different side ratios.

5. Conclusion

A simple and efficient method is proposed to study the problem of a compressible liquid inclusion of arbitrary shape in an infinite isotropic elastic matrix subjected to uniform remote in-plane stresses. Using the modified form of the analytic continuation in equation (12), we derive a set of $2N+1$ linear algebraic equations, as presented in equations (13), (14), and (19). These equations, which remain structurally simple even for arbitrarily large N, are used to find the $2N+1$ unknowns: $a_n,{\overline{a}}_n(n=1,2,\cdots ,N)$ and ${\sigma }_0$ . Once these $2N+1$ unknowns are uniquely determined, the pair of analytic functions $\varphi (\xi )$ and $\psi (\xi )$ for $\left|\xi \right|\ge 1$ characterizing the elastic field in the matrix and the internal uniform hydrostatic stress field within the liquid inclusion are determined. The result obtained for a hypotrochoidal liquid inclusion agrees with that in the existing literature. The results demonstrate that for $\kappa <1+2\beta$ , the internal hydrostatic stress within a hypotrochoidal liquid inclusion with cusps exceeds that of an inclusion with rounded corners. However, for $\kappa >1+2\beta$ , the inclusion with cusps exhibits lower internal hydrostatic stress compared to the one with rounded corners. Furthermore, for liquid inclusions with an n-fold axis of symmetry, the internal hydrostatic stress decreases as n increases when $\kappa <1+2\beta$ , with faster convergence observed for shapes with smaller n. Finally, the method is also applied to rectangular liquid inclusions and the results reveal that the internal hydrostatic stress of the liquid inclusion increases by increasing the side ratio of the shape, for $\kappa <1+2\beta$ .