A double-coated epitrochoidal elastic inhomogeneity neutral to a uniform hydrostatic stress field

Abstract

We study the design of a double-coated neutral epitrochoidal elastic inhomogeneity that does not disturb the prescribed uniform hydrostatic stress field in the surrounding elastic matrix. The two coatings have a common shear modulus but distinct Poisson’s ratios. In order to achieve the neutrality condition in the matrix, the plane-strain bulk modulus of the matrix and the Poisson’s ratio of the inner coating can be uniquely determined for given elastic properties of the outer coating and the inhomogeneity and given geometry of the composite by iteratively solving two coupled non-linear equations.

1. Introduction

A neutral elastic inhomogeneity when inserted into a uniformly stressed elastic matrix does not disturb the prescribed field in the matrix [1,2]. The study of neutral inhomogeneities (with neutral holes as a special case) can be dated back to Mansfield [3]. Typical and simple examples of neutral elastic inhomogeneities are a coated sphere and a coated cylinder [4,5]. A coated sphere (or a coated cylinder) does not disturb the uniform hydrostatic stress field (or the uniform in-plane hydrostatic stress field) in an isotropic elastic matrix when the bulk modulus (or the plane-strain bulk modulus) of the matrix is chosen appropriately [4–6]. In Hashin and Rosen [5], the two-dimensional neutral inhomogeneity is of circular shape. It is then quite natural to ask if a non-circular inhomogeneity can be made neutral to a uniform hydrostatic stress field. This study endeavors to answer this question.

In this paper, we solve the inverse problem associated with a double-coated epitrochoidal elastic inhomogeneity neutral to a prescribed uniform hydrostatic stress field in the elastic matrix. The two coatings have the same shear modulus and distinct Poisson’s ratios. Using a two-term mapping function [7] for the simply connected domain occupied by the double-coated epitrochoidal inhomogeneity, the three interfaces of the composite are mapped onto three concentric circles in the image plane. Through satisfaction of the neutrality condition in the matrix, the continuity conditions of displacements and tractions across the three perfect interfaces of the composite, and the boundedness condition of stresses and displacements everywhere within the inhomogeneity, a set of two coupled non-linear equations is finally derived. By iteratively solving this set of non-linear equations, the plane-strain bulk modulus of the matrix and the Poisson’s ratio of the inner coating can be uniquely determined for given elastic properties of the outer coating and the inhomogeneity and given geometry of the composite. In addition, the mean stress within each coating is uniform and the hoop stress is uniformly distributed along the entire coating-matrix interface on the side of the outer coating. The numerical result validates the proposed theory.

2. Muskhelishvili’s complex variable formulation

We first establish a fixed 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 [7]

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

(1)

and

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

(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 [8]

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

(3)

3. A double-coated neutral epitrochoidal inhomogeneity

As shown in Figure 1, we consider an epitrochoidal elastic inhomogeneity bonded to an elastic matrix (finite or infinite in extent) through two coatings. Let $S_1,S_2,S_3$ , and $S_4$ denote the matrix, the outer coating, the inner coating, and the epitrochoidal inhomogeneity, respectively, which are perfectly bonded across the three interfaces $L_1,L_2$ , and $L_3$ . In what follows, the subscripts 1, 2, 3, and 4 are used to denote the quantities in $S_1,S_2,S_3$ , and $S_4$ , respectively. The inner and outer coatings have a common shear modulus ( ${\mu }_2={\mu }_3$ ) and distinct Poisson’s ratios ( ${\nu }_2\ne {\nu }_3$ ). The prescribed stress field in the matrix is uniform and hydrostatic and is given by

$${\sigma }_{11}={\sigma }_{22}={\sigma }^{\infty },{\sigma }_{12}=0,z\in S_1,$$

(4)

where ${\sigma }^{\infty }$ is a given real number.

Figure 1.

A double-coated epitrochoidal elastic inhomogeneity neutral to a prescribed uniform hydrostatic stress field in the matrix.

In order to satisfy the neutrality condition in the matrix in equation (4), the pair of analytic functions ${\varphi }_1(z)$ and ${\psi }_1(z)$ defined in the matrix should take the following particular form:

$${\varphi }_1(z)=\frac{{\sigma }^{\infty }}{2}z,{\psi }_1(z)=0,z\in S_1.$$

(5)

The continuity conditions of tractions and displacements across the three perfect interfaces $L_1,L_2$ and $L_3$ can be expressed in terms of the four pairs of analytic functions ${\varphi }_j(z),{\psi }_j(z)(j=1,2,3,4)$ defined in the four phases of the composite as

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\varphi }_2(z)+z\overline{{\varphi \prime }_2(z)}+\overline{{\psi }_2(z)}={\varphi }_1(z)+z\overline{{\varphi \prime }_1(z)}+\overline{{\psi }_1(z)},\hfill \\ \multicolumn{1}{c}{}& \frac{1}{{\mu }_2}[{\kappa }_2{\varphi }_2(z)-z\overline{{\varphi \prime }_2(z)}-\overline{{\psi }_2(z)}]=\frac{1}{{\mu }_1}[{\kappa }_1{\varphi }_1(z)-z\overline{{\varphi \prime }_1(z)}-\overline{{\psi }_1(z)}],z\in L_1;\hfill \end{array}$$

(6a)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\varphi }_3(z)+z\overline{{\varphi \prime }_3(z)}+\overline{{\psi }_3(z)}={\varphi }_2(z)+z\overline{{\varphi \prime }_2(z)}+\overline{{\psi }_2(z)},\hfill \\ \multicolumn{1}{c}{}& {\kappa }_3{\varphi }_3(z)-z\overline{{\varphi \prime }_3(z)}-\overline{{\psi }_3(z)}={\kappa }_2{\varphi }_2(z)-z\overline{{\varphi \prime }_2(z)}-\overline{{\psi }_2(z)},z\in L_2;\hfill \end{array}$$

(6b)

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\varphi }_4(z)+z\overline{{\varphi \prime }_4(z)}+\overline{{\psi }_4(z)}={\varphi }_3(z)+z\overline{{\varphi \prime }_3(z)}+\overline{{\psi }_3(z)},\hfill \\ \multicolumn{1}{c}{}& \frac{1}{{\mu }_4}[{\kappa }_4{\varphi }_4(z)-z\overline{{\varphi \prime }_4(z)}-\overline{{\psi }_4(z)}]=\frac{1}{{\mu }_3}[{\kappa }_3{\varphi }_3(z)-z\overline{{\varphi \prime }_3(z)}-\overline{{\psi }_3(z)}],z\in L_3.\hfill \end{array}$$

(6c)

We now introduce the following conformal mapping function for the simply connected domain occupied by the double-coated epitrochoidal inhomogeneity [7]

$$z=\omega (\xi )=R(\xi +b{\xi }^n),R>0,0<b\le \frac{1}{n},\left|\xi \right|\le 1,$$

(7)

where n is an integer equal to or greater than 2.

As shown in Figure 2, using the mapping function in equation (7), the outer coating $S_2$ is mapped onto the annulus ${\rho }_1^{-\frac{1}{2}}\le \left|\xi \right|\le 1$ , the inner coating $S_3$ is mapped onto the annulus ${\rho }_2^{-\frac{1}{2}}\le \left|\xi \right|\le {\rho }_1^{-\frac{1}{2}}$ , the epitrochoidal inhomogeneity $S_4$ is mapped onto $\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}}$ ; the coating-matrix interface $L_1$ is mapped onto $\left|\xi \right|=1$ , the coating-coating interface $L_2$ is mapped onto $\left|\xi \right|={\rho }_1^{-\frac{1}{2}}$ , the inhomogeneity-coating interface $L_3$ is mapped onto $\left|\xi \right|={\rho }_2^{-\frac{1}{2}}$ . It is seen that the two parameters ${\rho }_1$ and ${\rho }_2$ should satisfy the restriction that ${\rho }_2>{\rho }_1>1$ . The three interfaces $L_1,L_2$ , and $L_3$ are drawn by specifying the parameters ${\rho }_1=2,{\rho }_2=6,n=3,b=1/n$ in equation (7). Examples of the three interfaces $L_1,L_2$ , and $L_3$ determined by using the mapping function in equation (7) for other values of n are illustrated in Figure 3. $n-1$ sharp corners of the coating-matrix interface $L_1$ can be observed in Figure 3 since the fact that $\omega \prime (\xi )=0$ only when $\left|\xi \right|=1$ . For convenience, we write ${\varphi }_j(\xi )={\varphi }_j(\omega (\xi )),{\psi }_j(\xi )={\psi }_j(\omega (\xi ))(j=2,3,4)$ .

Figure 2.

The ξ-plane.

Figure 3.

The three interfaces $L_1,L_2$ , and $L_3$ determined by using the mapping function in equation (7) with ${\rho }_1=2,{\rho }_2=6,b=1/n$ .

By enforcing the continuity conditions in equation (6a–c) with the use of equation (5), we find that

$${\varphi }_2(\xi )=\frac{{\sigma }^{\infty }(\beta +1)}{{\kappa }_2+1}\omega (\xi ),{\psi }_2(\xi )=\frac{{\sigma }^{\infty }({\kappa }_2-1-2\beta )}{{\kappa }_2+1}\overline{\omega }\left(\frac{1}{\xi }\right),{\rho }_1^{-\frac{1}{2}}\le \left|\xi \right|\le 1;$$

(8)

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_3(\xi )=\frac{{\sigma }^{\infty }(\beta +1)}{{\kappa }_3+1}\omega (\xi ),}\\ \multicolumn{1}{c}{{\psi }_3(\xi )=\frac{{\sigma }^{\infty }({\kappa }_2-1-2\beta )}{{\kappa }_2+1}\overline{\omega }\left(\frac{1}{\xi }\right)+2{\sigma }^{\infty }(\beta +1)(\frac{1}{{\kappa }_2+1}-\frac{1}{{\kappa }_3+1})\overline{\omega }\left(\frac{1}{{\rho }_1\xi }\right),{\rho }_2^{-\frac{1}{2}}\le \left|\xi \right|\le {\rho }_1^{-\frac{1}{2}};}\end{array}$$

(9)

$$\begin{array}{cc}\multicolumn{1}{c}{{\varphi }_4(\xi )=}& \multicolumn{1}{c}{\frac{{\sigma }^{\infty }(\beta +1)[\Gamma ({\kappa }_3-1)+2]}{({\kappa }_3+1)({\kappa }_4+1)}\omega (\xi )+\frac{{\sigma }^{\infty }(1-\Gamma )({\kappa }_2-1-2\beta )}{({\kappa }_2+1)({\kappa }_4+1)}\omega ({\rho }_2\xi )}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{+\frac{2{\sigma }^{\infty }(1-\Gamma )(\beta +1)}{{\kappa }_4+1}(\frac{1}{{\kappa }_2+1}-\frac{1}{{\kappa }_3+1})\omega \left(\frac{{\rho }_2}{{\rho }_1}\xi \right),}\\ \multicolumn{1}{c}{{\psi }_4(\xi )=}& \multicolumn{1}{c}{\frac{{\sigma }^{\infty }({\kappa }_2-1-2\beta )(\Gamma +{\kappa }_4)}{({\kappa }_2+1)({\kappa }_4+1)}\overline{\omega }\left(\frac{1}{\xi }\right)+\frac{2{\sigma }^{\infty }(\beta +1)(\Gamma +{\kappa }_4)}{{\kappa }_4+1}(\frac{1}{{\kappa }_2+1}-\frac{1}{{\kappa }_3+1})\overline{\omega }\left(\frac{1}{{\rho }_1\xi }\right)}\\ \multicolumn{1}{c}{}& +\frac{2{\sigma }^{\infty }(\beta +1)[{\kappa }_4-1-\Gamma ({\kappa }_3-1)]}{({\kappa }_3+1)({\kappa }_4+1)}\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)+\frac{{\sigma }^{\infty }(\Gamma -1)({\kappa }_2-1-2\beta )}{({\kappa }_2+1)({\kappa }_4+1)}\frac{{\rho }_2\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)\omega \prime ({\rho }_2\xi )}{\omega \prime (\xi )}\hfill \\ \multicolumn{1}{c}{}& +\frac{2{\sigma }^{\infty }(\Gamma -1)(\beta +1)}{{\kappa }_4+1}(\frac{1}{{\kappa }_2+1}-\frac{1}{{\kappa }_3+1})\frac{\frac{{\rho }_2}{{\rho }_1}\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)\omega \prime \left(\frac{{\rho }_2}{{\rho }_1}\xi \right)}{\omega \prime (\xi )},\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}};\hfill \end{array}$$

(10)

where

$$\Gamma =\frac{{\mu }_4}{{\mu }_2}=\frac{{\mu }_4}{{\mu }_3},\beta =\frac{{\mu }_2}{K_1}=\frac{{\mu }_2({\kappa }_1-1)}{2{\mu }_1}$$

(11)

in which $K_1=2{\mu }_1/({\kappa }_1-1)$ is the plane-strain bulk modulus of the elastic matrix. The assumption of ${\mu }_2={\mu }_3$ is to ensure that the resulting pair of analytic functions ${\varphi }_4(\xi )$ and ${\psi }_4(\xi )$ for $\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}}$ is properly defined.

It follows from equations (7) and (10) that: (1) ${\varphi }_4(\xi )$ is analytic everywhere in $\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}}$ ; (2) ${\psi }_4(\xi )$ is singular at $\xi =0$ and is analytic elsewhere in $\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}}$ . By observing the following singular behaviors at $\xi =0$

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{\omega }\left(\frac{1}{\xi }\right)=}& \multicolumn{1}{c}{\mathrm{Rb}\frac{1}{{\xi }^n}+R\frac{1}{\xi },\overline{\omega }\left(\frac{1}{{\rho }_1\xi }\right)=\frac{\mathrm{Rb}}{{\rho }_1^n}\frac{1}{{\xi }^n}+\frac{R}{{\rho }_1}\frac{1}{\xi },\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)=\frac{\mathrm{Rb}}{{\rho }_2^n}\frac{1}{{\xi }^n}+\frac{R}{{\rho }_2}\frac{1}{\xi },}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\frac{{\rho }_2\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)\omega \prime ({\rho }_2\xi )}{\omega \prime (\xi )}\cong \frac{\mathrm{Rb}}{{\rho }_2^{n-1}}\frac{1}{{\xi }^n}+R(1+b^{2}n-\frac{b^{2}n}{{\rho }_2^{n-1}})\frac{1}{\xi }+O({\xi }^{n-2}),}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\frac{\frac{{\rho }_2}{{\rho }_1}\overline{\omega }\left(\frac{1}{{\rho }_2\xi }\right)\omega \prime \left(\frac{{\rho }_2}{{\rho }_1}\xi \right)}{\omega \prime (\xi )}\cong \frac{\mathrm{Rb}}{{\rho }_1{\rho }_2^{n-1}}\frac{1}{{\xi }^n}+R(\frac{1}{{\rho }_1}+\frac{b^{2}n}{{\rho }_1^n}-\frac{b^{2}n}{{\rho }_1{\rho }_2^{n-1}})\frac{1}{\xi }+O({\xi }^{n-2})\mathrm{as}\xi \to 0,}\end{array}$$

(12)

and imposing the condition that ${\psi }_4(\xi )$ must be regular at $\xi =0$ (so that the stresses and displacements are bounded at $z=0$ ), we develop the following relationss when $b\ne 0$ :

$$\begin{array}{c}({\kappa }_2-1-2\beta )(\Gamma +{\kappa }_4)({\kappa }_3+1)+2{\rho }_1^{-n}(\beta +1)(\Gamma +{\kappa }_4)({\kappa }_3-{\kappa }_2)+2{\rho }_2^{-n}(\beta +1)({\kappa }_2+1)[{\kappa }_4-1-\Gamma ({\kappa }_3-1)]\hfill \\ +{\rho }_2^{1-n}(\Gamma -1)({\kappa }_2-1-2\beta )({\kappa }_3+1)+2{\rho }_1^{-1}{\rho }_2^{1-n}(\Gamma -1)(\beta +1)({\kappa }_3-{\kappa }_2)=0,\hfill \\ ({\kappa }_2-1-2\beta )(\Gamma +{\kappa }_4)({\kappa }_3+1)+2{\rho }_1^{-1}(\beta +1)(\Gamma +{\kappa }_4)({\kappa }_3-{\kappa }_2)+2{\rho }_2^{-1}(\beta +1)({\kappa }_2+1)[{\kappa }_4-1-\Gamma ({\kappa }_3-1)]\hfill \\ +(\Gamma -1)({\kappa }_2-1-2\beta )({\kappa }_3+1)[1+b^{2}n(1-{\rho }_2^{1-n})]+2(\Gamma -1)(\beta +1)({\kappa }_3-{\kappa }_2)\hfill \\ \multicolumn{1}{c}{\times [{\rho }_1^{-1}+b^{2}n({\rho }_1^{-n}-{\rho }_1^{-1}{\rho }_2^{1-n})]=0.}\end{array}$$

(13)

As a check, when $b=0$ and ${\nu }_2={\nu }_3$ , equation (13)₂ reduces to

$$K_1=K_2+\frac{{\rho }_2^{-1}}{\frac{1}{K_4-K_2}+\frac{1-{\rho }_2^{-1}}{K_2+{\mu }_2}},$$

(14)

where $K_2=2{\mu }_2/({\kappa }_2-1)$ and $K_4=2{\mu }_4/({\kappa }_4-1)$ are respectively the plane-strain bulk moduli of the outer coating and the inhomogeneity. Equation (14) is simply the classical result of the composite cylinder assemblage (CCA) model [5,6].

Equation (13) can be considered as two coupled non-linear equations in β and ${\kappa }_3$ for given values of $\Gamma ,{\kappa }_2,{\kappa }_4,{\rho }_1,{\rho }_2,b$ and n. Equation (13) can be further written in the form

$$\begin{array}{cc}\multicolumn{1}{c}{}& \beta =\frac{\left(\begin{array}{c}({\kappa }_2-1)(\Gamma +{\kappa }_4)({\kappa }_3+1)+2{\rho }_1^{-n}(\Gamma +{\kappa }_4)({\kappa }_3-{\kappa }_2)+2{\rho }_2^{-n}({\kappa }_2+1)[{\kappa }_4-1-\Gamma ({\kappa }_3-1)]\hfill \\ \multicolumn{1}{c}{+{\rho }_2^{1-n}(\Gamma -1)({\kappa }_3+1)({\kappa }_2-1)+2{\rho }_1^{-1}{\rho }_2^{1-n}(\Gamma -1)({\kappa }_3-{\kappa }_2)}\end{array}\right)}{2\left(\begin{array}{c}(\Gamma +{\kappa }_4)({\kappa }_3+1)+{\rho }_1^{-n}(\Gamma +{\kappa }_4)({\kappa }_2-{\kappa }_3)+{\rho }_2^{-n}({\kappa }_2+1)[\Gamma ({\kappa }_3-1)-({\kappa }_4-1)]\hfill \\ \multicolumn{1}{c}{+{\rho }_2^{1-n}(\Gamma -1)({\kappa }_3+1)+{\rho }_1^{-1}{\rho }_2^{1-n}(\Gamma -1)({\kappa }_2-{\kappa }_3)}\end{array}\right)},\hfill \\ \multicolumn{1}{c}{}& {\kappa }_3=\frac{\left(\begin{array}{c}(1+2\beta -{\kappa }_2)(\Gamma +{\kappa }_4)+2{\kappa }_2{\rho }_1^{-1}(\beta +1)(\Gamma +{\kappa }_4)-2{\rho }_2^{-1}(\beta +1)({\kappa }_2+1)({\kappa }_4-1+\Gamma )\hfill \\ \multicolumn{1}{c}{+(1-\Gamma )({\kappa }_2-1-2\beta )[1+b^{2}n(1-{\rho }_2^{1-n})]+2{\kappa }_2(\Gamma -1)(\beta +1)[{\rho }_1^{-1}+b^{2}n({\rho }_1^{-n}-{\rho }_1^{-1}{\rho }_2^{1-n})]}\end{array}\right)}{\left(\begin{array}{c}({\kappa }_2-1-2\beta )(\Gamma +{\kappa }_4)+2{\rho }_1^{-1}(\beta +1)(\Gamma +{\kappa }_4)-2\Gamma {\rho }_2^{-1}(\beta +1)({\kappa }_2+1)\hfill \\ \multicolumn{1}{c}{+(\Gamma -1)({\kappa }_2-1-2\beta )[1+b^{2}n(1-{\rho }_2^{1-n})]+2(\Gamma -1)(\beta +1)[{\rho }_1^{-1}+b^{2}n({\rho }_1^{-n}-{\rho }_1^{-1}{\rho }_2^{1-n})]}\end{array}\right)},\hfill \end{array}$$

(15)

which serves to iteratively solve for β and ${\kappa }_3$ . We present in Table 1 the calculated β and ${\kappa }_3$ . In Table 1, there is no solution when $\Gamma =0.2,n=2$ ; $\Gamma =3,n=2$ . More specifically, it is calculated using equation (15) that: $\beta =0.3125,{\kappa }_3=0.75<1$ when $\Gamma =0.2,n=2$ ; $\beta =0.6154,{\kappa }_3=3.6667>3$ when $\Gamma =3,n=2$ . If the Poisson’s ratio of the inner coating is relaxed to the range $-1\le {\nu }_3\le 0.5$ or $1\le {\kappa }_3\le 7$ [9,10], there always exists a suitable solution for any value of $\Gamma (>1)$ when ${\kappa }_2={\kappa }_4=2,{\rho }_1=2,{\rho }_2=6,n=2,b=0.5$ . For example, $\beta =0.7143,{\kappa }_3=7$ when $\Gamma \to \infty ,{\kappa }_2={\kappa }_4=2,{\rho }_1=2,{\rho }_2=6,n=2,b=0.5$ (the inhomogeneity becomes rigid).

Table 1.

The calculated β and ${\kappa }_3$ for different values of Γ and n with ${\kappa }_2={\kappa }_4=2,{\rho }_1=2,{\rho }_2=6,b=1/n$ .

	n = 2	n = 3	n = 4	n = 5
Γ = 0.2		β = 0.4290κ3 = 1.0909	β = 0.4690κ3 = 1.2238	β = 0.4856κ3 = 1.2816
Γ = 0.5	β = 0.4118κ3 = 1.2857	β = 0.4681κ3 = 1.5205	β = 0.4863κ3 = 1.6026	β = 0.4936κ3 = 1.6367
Γ = 2	β = 0.5789κ3 = 3.0000	β = 0.5265κ3 = 2.5385	β = 0.5112κ3 = 2.4175	β = 0.5051κ3 = 2.3716
Γ = 3		β = 0.5382κ3 = 2.8333	β = 0.5160κ3 = 2.6342	β = 0.5073κ3 = 2.5606

The result in Table 1 indicates that: (1) the plane-strain bulk modulus of the matrix ( $K_1>0$ or $\beta >0$ ) and the Poisson’s ratio of the inner coating ( $0\le {\nu }_3\le 1/2$ or $1\le {\kappa }_3\le 3$ ) can be uniquely determined for given elastic properties of the outer coating and inhomogeneity and given geometry of the composite; (2) the assumption of ${\mu }_2={\mu }_3$ and ${\nu }_2\ne {\nu }_3$ is feasible for our purpose. Our numerical result also indicates that it is impossible to achieve neutrality of a non-circular elastic inhomogeneity through a single coating.

In addition, the mean stress is uniform in each coating and is given by

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\sigma }_{11}+{\sigma }_{22}=\frac{4{\sigma }^{\infty }(\beta +1)}{{\kappa }_2+1},z\in S_2;\hfill \\ \multicolumn{1}{c}{}& {\sigma }_{11}+{\sigma }_{22}=\frac{4{\sigma }^{\infty }(\beta +1)}{{\kappa }_3+1},z\in S_3.\hfill \end{array}$$

(16)

Moreover, the hoop stress is uniformly distributed along the coating-matrix interface $L_1$ on the outer coating side and is given by

$${\sigma }_{\mathrm{tt}}=\frac{{\sigma }^{\infty }(4\beta +3-{\kappa }_2)}{{\kappa }_2+1},z\in L_1\cap S_2.$$

(17)

Equation (17) suggests that the stresses in the outer coating are still bounded at the possible sharp corners of $L_1$ .

It is seen from equation (10) that the bounded stresses are non-uniform within the epitrochoidal inhomogeneity. For example, the mean stress is non-uniformly and regularly distributed within the epitrochoidal inhomogeneity as

$$\begin{array}{ll}\multicolumn{1}{c}{{\sigma }_{11}+{\sigma }_{22}=}& \multicolumn{1}{c}{\frac{4{\sigma }^{\infty }(\beta +1)\left[\Gamma ({\kappa }_3-1)+2\right]}{({\kappa }_3+1)({\kappa }_4+1)}+\frac{4{\sigma }^{\infty }}{({\kappa }_2+1)({\kappa }_3+1)({\kappa }_4+1)}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\times \mathrm{Re}\left\{\frac{(1-\Gamma )({\kappa }_3+1)({\kappa }_2-1-2\beta )({\rho }_2+bn{\rho }_2^n{\xi }^{n-1})+2(1-\Gamma )(\beta +1)({\kappa }_3-{\kappa }_2)\left(\frac{{\rho }_2}{{\rho }_1}+\frac{bn{\rho }_2^n}{{\rho }_1^n}{\xi }^{n-1}\right)}{1+bn{\xi }^{n-1}}\right\},}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\left|\xi \right|\le {\rho }_2^{-\frac{1}{2}}.}\end{array}$$

(18)

Recall that the internal stress field within a coated circular inhomogeneity neutral to a prescribed uniform hydrostatic stress field is uniform and hydrostatic. When $n\ge 4$ , the stresses at $\xi =0$ or $z=0$ is hydrostatic and is given by

$${\sigma }_{11}={\sigma }_{22}={\sigma }_0,{\sigma }_{12}=0\mathrm{at}z=0,$$

(19)

where

$${\sigma }_0=\frac{2{\sigma }^{\infty }(\beta +1)[\Gamma ({\kappa }_3-1)+2]}{({\kappa }_3+1)({\kappa }_4+1)}+\frac{2{\sigma }^{\infty }{\rho }_2(1-\Gamma )[({\kappa }_3+1)({\kappa }_2-1-2\beta )+2{\rho }_1^{-1}(\beta +1)({\kappa }_3-{\kappa }_2)]}{({\kappa }_2+1)({\kappa }_3+1)({\kappa }_4+1)}.$$

(20)

We illustrate in Figure 4 the ratio ${\sigma }_0/{\sigma }^{\infty }$ for different values of $n(\ge 4)$ and Γ determined by equation (20). It is seen from Figure 4 that: (1) ${\sigma }_0/{\sigma }^{\infty }$ is an increasing function of n when $\Gamma <1$ , a decreasing function of n when $\Gamma >1$ ; (2) ${\sigma }_0/{\sigma }^{\infty }$ is always an increasing function of Γ (i.e., the hydrostatic stress level at $z=0$ increases as the inhomogeneity becomes harder).

Figure 4.

Variations of ${\sigma }_0/{\sigma }^{\infty }$ for different values of $n(\ge 4)$ and Γ determined by equation (20) with ${\rho }_1=2,{\rho }_2=6,b=1/n,{\kappa }_2={\kappa }_4=2$ .

4. Conclusion

We have achieved the neutrality of a double-coated epitrochoidal elastic inhomogeneity to a prescribed uniform hydrostatic stress field. In order to satisfy the neutrality condition in equation (4) or (5), a set of two coupled non-linear equations in equation (13) or (15) is derived and is solved iteratively to arrive at the two parameters β and ${\kappa }_3$ . The numerical result in Table 1 demonstrates the validity of the proposed theory. For the first time, we demonstrate that a double-coated non-circular elastic inhomogeneity can become neutral to a prescribed uniform hydrostatic stress field.