Multiphase circular inhomogeneities in isotropic laminated plates

Abstract

This work is concerned with a circular inhomogeneity bonded to an infinite matrix though N concentric circular interphase layers within the context of Kirchhoff theory for isotropic laminated plates. An elegant and effective procedure is proposed to obtain the stress resultant fields within the internal inhomogeneity and the surrounding matrix under thermomechanical loadings. The boundary value problem is finally reduced to two coupled linear algebraic equations and four coupled linear algebraic equations that determine the six real coefficients of the stress resultant field in the inhomogeneity. In particular, the average stress resultants within the inhomogeneity can be directly determined from these six real coefficients. The other six unknown real coefficients, which control the stress resultant field in the surrounding matrix, can then be simply obtained. The effect of the N interphase layers on the stress resultant field is demonstrated by their influence on these 12 real coefficients. The obtained solution is further applied to the design of neutral and harmonic circular elastic inhomogeneities with a single interphase layer.

Keywords

Isotropic laminated plate inhomogeneity interphase layer thermomechanical loading complex variable method

1. Introduction

The interphase layer between the internal inhomogeneity and the surrounding matrix can significantly influence the stress fields of the composites. Within the context of isotropic plane elasticity, an effective method was proposed by Ru [1] to analyze a circular inhomogeneity with an N-layered stepwise graded interphase under thermomechanical loadings. Instead of solving the (N+2)-phase composite problem, the original boundary value problem was finally reduced to a single linear algebraic equation and two coupled linear algebraic equations for the determination of the three real coefficients of the stress field in the inhomogeneity. The other three coefficients characterizing the stress field in the surrounding matrix can then be determined easily.

On the other hand, laminated composite plate structures have been widely used in mechanical, civil, marine and aviation industries. The Kirchhoff plate theory is the most celebrated two-dimensional model for modeling deformations of a thin plate [2 –5]. This paper endeavors to study the coupled stretching and bending deformations of an infinite isotropic laminated plate reinforced by a circular elastic inhomogeneity with N interphase layers under thermomechanical loadings within the framework of Kirchhoff theory for isotropic and laminated plates. The present N-layered interphase model can be applied to simulate a functionally graded interface whose material properties are continuously varied in both the radial and thickness directions.

In this investigation, a novel procedure is proposed to obtain the stress resultant fields within the circular inhomogeneity and the surrounding matrix without resolving the (N+2)-phase composite problem. The stress resultants inside the inhomogeneity are determined by six real coefficients, whilst the stress resultant disturbance in the matrix is governed by six other real coefficients. The present method reduces the determination of the stress resultant field within the inhomogeneity to two coupled linear algebraic equations and four coupled linear algebraic equations, whose coefficients can be obtained by calculating the first and third columns of the product of (N+1) 4×4 matrices, and the first, fourth, fifth and eighth columns of the product of (N+1) 8×8 matrices. The remaining six unknown real coefficients associated with the stress resultant field in the matrix can then be determined easily.

2. The complex variable formulation

Let the main plane of a plate of uniform thickness h be located at $x_{3} = 0$ in a Cartesian coordinate system ${x_{i}} (i = 1, 2, 3)$ . The plate is composed of an isotropic, linearly elastic material that can be inhomogeneous and laminated in the thickness direction.

The displacement field in the Kirchhoff plate theory takes the following form:

{\tilde{u}}_{α} (x_{i}) = u_{α} + x_{3} ϑ_{α}, {\tilde{u}}_{3} (x_{i}) = w,

(1)

where the in-plane displacements $u_{α}$ , deflection w and the slopes $ϑ_{α} = - w_{, α}$ on the main plane are all independent of x₃.

The coordinate system is chosen in such a way that the two in-plane displacements and the out-of-plane deflection on the main plane are decoupled in the equilibrium equations [6]. Consequently, the membrane stress resultants $N_{α β}$ , bending moments $M_{α β}$ , transverse shearing forces $ℜ_{β} = M_{α β, α}$ , in-plane displacements, deflection and slopes on the main plane of the plate and the four stress functions $φ_{α}$ and $η_{α}$ can be concisely expressed in terms of four complex potentials $ϕ (z)$ , $ψ (z)$ , $Φ (z)$ and $Ψ (z)$ of the complex variable $z = x_{1} + i x_{2}$ as [7]

\begin{matrix} N_{11} + N_{22} = 4 Re {ϕ' (z) + B Φ' (z)}, \\ N_{22} - N_{11} + 2 i N_{12} = 2 [\bar{z} ϕ ″ (z) + ψ' (z) + B \bar{z} Φ ″ (z) + B Ψ' (z)], \end{matrix}

(2)

\begin{matrix} M_{11} + M_{22} = 4 D (1 + ν^{D}) Re {Φ' (z)} + \frac{B (κ^{A} - 1)}{μ} Re {ϕ' (z)}, \\ M_{22} - M_{11} + 2 i M_{12} = - 2 D (1 - ν^{D}) [\bar{z} Φ ″ (z) + Ψ' (z)] - \frac{B}{μ} [\bar{z} ϕ ″ (z) + ψ' (z)], \\ ℜ_{1} - i ℜ_{2} = 4 D Φ ″ (z) + \frac{B (κ^{A} + 1)}{2 μ} ϕ ″ (z), \end{matrix}

(3)

\begin{matrix} 2 μ (u_{1} + i u_{2}) = κ^{A} ϕ (z) - z \bar{ϕ' (z)} - \bar{ψ (z)}, \\ ϑ_{1} + i ϑ_{2} = Φ (z) + z \bar{Φ' (z)} + \bar{Ψ (z)}, w = - Re {\bar{z} Φ (z) + χ (z)}, \\ φ_{1} + i φ_{2} = i [ϕ (z) + z \bar{ϕ' (z)} + \bar{ψ (z)}] + i B [Φ (z) + z \bar{Φ' (z)} + \bar{Ψ (z)}], \\ η_{1} + i η_{2} = i D (1 - ν^{D}) [κ^{D} Φ (z) - z \bar{Φ' (z)} - \bar{Ψ (z)}] + i \frac{B}{2 μ} [κ^{A} ϕ (z) - z \bar{ϕ' (z)} - \bar{ψ (z)}], \end{matrix}

(4)

where $Ψ (z) = χ' (z)$ , and

\begin{matrix} μ = \frac{1}{2} (A_{11} - A_{12}), B = B_{12}, D = D_{11}, ν^{A} = \frac{A_{12}}{A_{11}}, ν^{D} = \frac{D_{12}}{D_{11}}, \\ κ^{A} = \frac{3 A_{11} - A_{12}}{A_{11} + A_{12}} = \frac{3 - ν^{A}}{1 + ν^{A}}, κ^{D} = \frac{3 D_{11} + D_{12}}{D_{11} - D_{12}} = \frac{3 + ν^{D}}{1 - ν^{D}}, \end{matrix}

(5)

where $A_{ij} = {QC}_{ij}$ , $B_{ij} = {Qx}_{3} C_{ij}$ and $D_{ij} = {Qx}_{3}^{2} C_{ij}$ ( $ij = 11, 12$ ). $Q (\dots) = \int_{- h_{0}}^{h - h_{0}} (\dots) d x_{3}$ is an integral operator with $h_{0}$ being the distance between the main plane and the lower surface of the plate. $C_{11}$ and $C_{12}$ can be expressed in terms of the Young’s modulus $E = E (x_{3})$ and Poisson’s ratio $ν = ν (x_{3})$ of the plate as $C_{11} = E / E (1 - ν^{2}) (1 - ν^{2})$ and $C_{12} = ν E / ν E (1 - ν^{2}) (1 - ν^{2})$ . $h_{0}$ is determined as $h_{0} = \int_{0}^{h} X_{3} C_{11} d X_{3} / \int_{0}^{h} X_{3} C_{11} d X_{3} \int_{0}^{h} C_{11} d X_{3} \int_{0}^{h} C_{11} d X_{3}$ with $X_{3} = x_{3} + h_{0}$ being the vertical coordinate of the given point from the lowest surface of the plate.

Moreover, the membrane stress resultants, bending moments, transverse shearing forces and modified Kirchhoff transverse shearing forces $V_{1} = ℜ_{1} + M_{12, 2}$ and $V_{2} = ℜ_{2} + M_{21, 1}$ , which exclusively apply to free edges, can be expressed in terms of the four stress functions $φ_{α}$ and $η_{α}$ [8]:

\begin{matrix} N_{α β} = - \in_{β ω} φ_{α, ω}, \\ M_{α β} = - \in_{β ω} η_{α, ω} - \frac{1}{2} \in_{α β} η_{ω, ω}, ℜ_{α} = - \frac{1}{2} \in_{α β} η_{ω, ω β}, V_{α} = - \in_{α ω} η_{ω, ω ω}, \end{matrix}

(6)

where $\in_{α β}$ are the components of the two-dimensional permutation tensor.

If we choose a new coordinate system ${{\hat{x}}_{i}} (i = 1, 2, 3)$ in which ${\hat{x}}_{3} = 0$ is on an arbitrary plane parallel to the main plane and ${\hat{x}}_{α} = x_{α}$ , the in-plane displacements ${\hat{u}}_{α}$ and slopes ${\hat{ϑ}}_{α}$ on ${\hat{x}}_{3} = 0$ and the stress functions ${\hat{φ}}_{α}$ and ${\hat{η}}_{α}$ in the new coordinate system ${{\hat{x}}_{i}} (i = 1, 2, 3)$ can be simply given by

\begin{matrix} {\hat{ϑ}}_{1} + i {\hat{ϑ}}_{2} = ϑ_{1} + i ϑ_{2}, {\hat{u}}_{1} + i {\hat{u}}_{2} = u_{1} + i u_{2} - \hat{h} (ϑ_{1} + i ϑ_{2}), \\ {\hat{φ}}_{1} + i {\hat{φ}}_{2} = φ_{1} + i φ_{2}, {\hat{η}}_{1} + i {\hat{η}}_{2} = η_{1} + i η_{2} + \hat{h} (φ_{1} + i φ_{2}), \end{matrix}

(7)

where

\hat{h} = h_{1} - h_{0},

(8)

with $h_{1}$ being the distance between ${\hat{x}}_{3} = 0$ and the lower surface of the plate ( $h_{1}$ is positive if ${\hat{x}}_{3} = 0$ is above the lower surface of the plate; it is negative if ${\hat{x}}_{3} = 0$ is below the lower surface of the plate). In the new coordinate system, the stress resultants ${\hat{N}}_{α β} = \hat{Q} σ_{α β}$ and ${\hat{M}}_{α β} = \hat{Q} {\hat{x}}_{3} σ_{α β}$ with $σ_{α β}$ being the in-plane stress components and $\hat{Q} (\dots) = \int_{- h_{1}}^{h - h_{1}} (\dots) d {\hat{x}}_{3}$ , the transverse shearing forces ${\hat{ℜ}}_{β} = {\hat{M}}_{α β, α}$ , and the modified Kirchhoff transverse shearing forces ${\hat{V}}_{1} = {\hat{ℜ}}_{1} + {\hat{M}}_{12, 2}$ and ${\hat{V}}_{2} = {\hat{ℜ}}_{2} + {\hat{M}}_{21, 1}$ can also be expressed in terms of the introduced stress functions ${\hat{φ}}_{α}$ and ${\hat{η}}_{α}$ as

\begin{matrix} {\hat{N}}_{α β} = - \in_{β ω} {\hat{φ}}_{α, ω}, \\ {\hat{M}}_{α β} = - \in_{β ω} {\hat{η}}_{α, ω} - \frac{1}{2} \in_{α β} {\hat{η}}_{ω, ω}, {\hat{ℜ}}_{α} = - \frac{1}{2} \in_{α β} {\hat{η}}_{ω, ω β}, {\hat{V}}_{α} = - \in_{α ω} {\hat{η}}_{ω, ω ω} . \end{matrix}

(9)

3. Methodology

Consider a circular elastic inhomogeneity bonded to an infinite matrix through N coaxial circular interphase layers, as shown in Figure 1. Each one of the (N+2) phases is made of an isotropic laminated plate. Let $S_{0}$ , $S_{k}, (k = 1, 2, \dots, N)$ and $S_{N + 1}$ denote the internal circular inhomogeneity, the N interphase layers and the matrix, respectively, which are perfectly bonded across the (N+1) concentric circles $| z | = R_{K}, (k = 0, 1, \dots, N)$ . The matrix is subjected to remote uniform membrane stress resultants $N_{11}^{\infty}, N_{22}^{\infty}$ and bending moments $M_{11}^{\infty}, M_{22}^{\infty}$ measured on its main plane. Without losing generality, we have assumed that $N_{12}^{\infty} = M_{12}^{\infty} = 0$ .

Figure 1.

A circular inhomogeneity with N interphase layers. Each phase is made of an isotropic laminated plate.

The continuity conditions of displacements and stress resultants across the (N+1) circular interfaces $| z | = R_{k}, k = 0, 1, \dots, N$ are

\begin{matrix} {\hat{u}}_{1}^{(k + 1)} + δ_{k + 1} x_{1} = {\hat{u}}_{1}^{(k)} + δ_{k} x_{1}, {\hat{u}}_{2}^{(k + 1)} + δ_{k + 1} x_{2} = {\hat{u}}_{2}^{(k)} + δ_{k} x_{2}, \\ {\hat{ϑ}}_{1}^{(k + 1)} + γ_{k + 1} x_{1} = {\hat{ϑ}}_{1}^{(k)} + γ_{k} x_{1}, {\hat{ϑ}}_{2}^{(k + 1)} + γ_{k + 1} x_{2} = {\hat{ϑ}}_{2}^{(k)} + γ_{k} x_{2}, \\ {\hat{φ}}_{1}^{(k + 1)} = {\hat{φ}}_{1}^{(k)}, {\hat{φ}}_{2}^{(k + 1)} = {\hat{φ}}_{2}^{(k)}, {\hat{η}}_{1}^{(k + 1)} = {\hat{η}}_{1}^{(k)}, {\hat{η}}_{2}^{(k + 1)} = {\hat{η}}_{2}^{(k)}, \end{matrix}

(10)

where $δ_{k}$ and $γ_{k}$ $(k = 0, 1, \dots, N)$ are respectively the volumetric eigenstrain and eigencurvature imposed on $S_{k}$ . By using Equations (4) and (7), the interface conditions across $| z | = R_{k}$ in Equation (10) can be equivalently expressed in terms of the analytic functions in $S_{k}$ and $S_{k + 1}$ as

\begin{matrix} A_{k + 1} [\begin{matrix} ϕ_{k + 1} (z) \\ \bar{z} ϕ'_{k + 1} (z) + ψ_{k + 1} (z) \\ Φ_{k + 1} (z) \\ \bar{z} Φ'_{k + 1} (z) + Ψ_{k + 1} (z) \end{matrix}] + {\bar{A}}_{k + 1} [\begin{matrix} \bar{ϕ_{k + 1} (z)} \\ z \bar{ϕ'_{k + 1} (z)} + \bar{ψ_{k + 1} (z)} \\ \bar{Φ_{k + 1} (z)} \\ z \bar{Φ'_{k + 1} (z)} + \bar{Ψ_{k + 1} (z)} \end{matrix}] = A_{k} [\begin{matrix} ϕ_{k} (z) \\ \bar{z} ϕ'_{k} (z) + ψ_{k} (z) \\ Φ_{k} (z) \\ \bar{z} Φ'_{k} (z) + Ψ_{k} (z) \end{matrix}] + {\bar{A}}_{k} [\begin{matrix} \bar{ϕ_{k} (z)} \\ z \bar{ϕ'_{k} (z)} + \bar{ψ_{k} (z)} \\ \bar{Φ_{k} (z)} \\ z \bar{Φ'_{k} (z)} + \bar{Ψ_{k} (z)} \end{matrix}] \\ + C_{k} [\begin{matrix} z \\ \bar{z} \end{matrix}], \\ B_{k + 1} [\begin{matrix} ϕ_{k + 1} (z) \\ \bar{z} ϕ'_{k + 1} (z) + ψ_{k + 1} (z) \\ Φ_{k + 1} (z) \\ \bar{z} Φ'_{k + 1} (z) + Ψ_{k + 1} (z) \end{matrix}] + {\bar{B}}_{k + 1} [\begin{matrix} \bar{ϕ_{k + 1} (z)} \\ z \bar{ϕ'_{k + 1} (z)} + \bar{ψ_{k + 1} (z)} \\ \bar{Φ_{k + 1} (z)} \\ z \bar{Φ'_{k + 1} (z)} + \bar{Ψ_{k + 1} (z)} \end{matrix}] = B_{k} [\begin{matrix} ϕ_{k} (z) \\ \bar{z} ϕ'_{k} (z) + ψ_{k} (z) \\ Φ_{k} (z) \\ \bar{z} Φ'_{k} (z) + Ψ_{k} (z) \end{matrix}] + {\bar{B}}_{k} [\begin{matrix} \bar{ϕ_{k} (z)} \\ z \bar{ϕ'_{k} (z)} + \bar{ψ_{k} (z)} \\ \bar{Φ_{k} (z)} \\ z \bar{Φ'_{k} (z)} + \bar{Ψ_{k} (z)} \end{matrix}], \end{matrix}

| z | = R_{k}, k = 0, 1, \dots, N,

(11)

where

A = [\begin{matrix} I & - \hat{h} I \\ 0 & I \end{matrix}] [\begin{matrix} \frac{κ^{A}}{2 μ} & - \frac{1}{2 μ} & 0 & 0 \\ - \frac{i κ^{A}}{2 μ} & - \frac{i}{2 μ} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & - i & i \end{matrix}],

(12)

B = [\begin{matrix} I & 0 \\ \hat{h} I & I \end{matrix}] [\begin{matrix} i & - i & i B & - i B \\ 1 & 1 & B & B \\ \frac{i B κ^{A}}{2 μ} & \frac{i B}{2 μ} & i D (3 + ν^{D}) & i D (1 - ν^{D}) \\ \frac{B κ^{A}}{2 μ} & - \frac{B}{2 μ} & D (3 + ν^{D}) & - D (1 - ν^{D}) \end{matrix}],

(13)

C_{k} = [\begin{matrix} δ_{k} - δ_{k + 1} & δ_{k} - δ_{k + 1} \\ - i (δ_{k} - δ_{k + 1}) & i (δ_{k} - δ_{k + 1}) \\ γ_{k} - γ_{k + 1} & γ_{k} - γ_{k + 1} \\ - i (γ_{k} - γ_{k + 1}) & i (γ_{k} - γ_{k + 1}) \end{matrix}] .

(14)

Remark. $A_{k}$ and $B_{k}$ are obtained by attaching the subscript k to the material and geometric parameters in Equations (12) and (13).

In addition, the asymptotic behaviors of $ϕ_{N + 1} (z), ψ_{N + 1} (z), Φ_{N + 1} (z)$ and $Ψ_{N + 1} (z)$ at infinity are

ϕ_{N + 1} (z) ≅ Fz + O (1), ψ_{N + 1} (z) ≅ Gz + O (1), Φ_{N + 1} (z) ≅ Iz + O (1), Ψ_{N + 1} (z) ≅ Jz + O (1), | z | \to \infty .

(15)

where the four real numbers F, G, I and J are given in terms of the remote loading $N_{11}^{\infty}, N_{22}^{\infty}, M_{11}^{\infty}$ and $M_{22}^{\infty}$ as

\begin{matrix} F = \frac{μ_{N + 1} D_{N + 1} (1 + ν_{N + 1}^{D}) (N_{11}^{\infty} + N_{22}^{\infty}) - B_{N + 1} μ_{N + 1} (M_{11}^{\infty} + M_{22}^{\infty})}{4 μ_{N + 1} D_{N + 1} (1 + ν_{N + 1}^{D}) - B_{N + 1}^{2} (κ_{N + 1}^{A} - 1)}, \\ G = \frac{μ_{N + 1} D_{N + 1} (1 - ν_{N + 1}^{D}) (N_{22}^{\infty} - N_{11}^{\infty}) + B_{N + 1} μ_{N + 1} (M_{22}^{\infty} - M_{11}^{\infty})}{2 μ_{N + 1} D_{N + 1} (1 - ν_{N + 1}^{D}) - B_{N + 1}^{2}}, \\ I = \frac{4 μ_{N + 1} (M_{11}^{\infty} + M_{22}^{\infty}) - B_{N + 1} (κ_{N + 1}^{A} - 1) (N_{11}^{\infty} + N_{22}^{\infty})}{16 μ_{N + 1} D_{N + 1} (1 + ν_{N + 1}^{D}) - 4 B_{N + 1}^{2} (κ_{N + 1}^{A} - 1)}, \\ J = \frac{2 μ_{N + 1} (M_{11}^{\infty} - M_{22}^{\infty}) + B_{N + 1} (N_{11}^{\infty} - N_{22}^{\infty})}{4 μ_{N + 1} D_{N + 1} (1 - ν_{N + 1}^{D}) - 2 B_{N + 1}^{2}} . \end{matrix}

(16)

Pre-multiplying the two conditions in Equation (11) by $B_{k + 1}^{T}$ and $A_{k + 1}^{T}$ , respectively, adding the resulting conditions and utilizing the following orthogonality relations

\begin{matrix} B_{k + 1}^{T} A_{k + 1} + A_{k + 1}^{T} B_{k + 1} = [\begin{matrix} 0 & - \frac{i (κ_{k + 1}^{A} + 1)}{μ_{k + 1}} & 0 & 0 \\ - \frac{i (κ_{k + 1}^{A} + 1)}{μ_{k + 1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 i D_{k + 1} \\ 0 & 0 & 8 i D_{k + 1} & 0 \end{matrix}], \\ B_{k + 1}^{T} {\bar{A}}_{k + 1} + A_{k + 1}^{T} {\bar{B}}_{k + 1} = 0, \end{matrix}

(17)

we finally arrive at the following expression:

\begin{matrix} [\begin{matrix} 0 & \frac{κ_{k + 1}^{A} + 1}{μ_{k + 1}} & 0 & 0 \\ \frac{κ_{k + 1}^{A} + 1}{μ_{k + 1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & - 8 D_{k + 1} \\ 0 & 0 & - 8 D_{k + 1} & 0 \end{matrix}] [\begin{matrix} ϕ_{k + 1} (z) \\ \bar{z} ϕ'_{k + 1} (z) + ψ_{k + 1} (z) \\ Φ_{k + 1} (z) \\ \bar{z} Φ'_{k + 1} (z) + Ψ_{k + 1} (z) \end{matrix}] \\ = i (B_{k + 1}^{T} A_{k} + A_{k + 1}^{T} B_{k}) [\begin{matrix} ϕ_{k} (z) \\ \bar{z} ϕ'_{k} (z) + ψ_{k} (z) \\ Φ_{k} (z) \\ \bar{z} Φ'_{k} (z) + Ψ_{k} (z) \end{matrix}] + i (B_{k + 1}^{T} {\bar{A}}_{k} + A_{k + 1}^{T} {\bar{B}}_{k}) [\begin{matrix} \bar{ϕ_{k} (z)} \\ z \bar{ϕ'_{k} (z)} + \bar{ψ_{k} (z)} \\ \bar{Φ_{k} (z)} \\ z \bar{Φ'_{k} (z)} + \bar{Ψ_{k} (z)} \end{matrix}] + i B_{k + 1}^{T} C_{k} [\begin{matrix} z \\ \bar{z} \end{matrix}], \end{matrix}

| z | = R_{k}, k = 0, 1, \dots, N,

(18)

where

\begin{matrix} i (B_{k + 1}^{T} A_{k} + A_{k + 1}^{T} B_{k}) \\ = (\begin{matrix} 0 & \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} μ_{k + 1}} \\ \frac{μ_{k} + κ_{k}^{A} μ_{k + 1}}{μ_{k} μ_{k + 1}} & 0 \\ 0 & 2 ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + \frac{B_{k + 1} - B_{k}}{μ_{k}} \\ 2 ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + \frac{κ_{k}^{A} (B_{k + 1} - B_{k})}{μ_{k}} & 0 \end{matrix} \\ \begin{matrix} 0 & 2 ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + \frac{κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{μ_{k + 1}} \\ 2 ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + \frac{B_{k} - B_{k + 1}}{μ_{k + 1}} & 0 \\ 0 & \begin{matrix} 2 (B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) \\ - 2 D_{k + 1} (3 + ν_{k + 1}^{D}) - 2 D_{k} (1 - ν_{k}^{D}) \end{matrix} \\ \begin{matrix} 2 (B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) \\ - 2 D_{k + 1} (1 - ν_{k + 1}^{D}) - 2 D_{k} (3 + ν_{k}^{D}) \end{matrix} & 0 \end{matrix}), \end{matrix}

(19)

\begin{matrix} i (B_{k + 1}^{T} {\bar{A}}_{k} + A_{k + 1}^{T} {\bar{B}}_{k}) \\ = (\begin{matrix} \frac{κ_{k + 1}^{A}}{μ_{k + 1}} - \frac{κ_{k}^{A}}{μ_{k}} & 0 \\ 0 & \frac{1}{μ_{k + 1}} - \frac{1}{μ_{k}} \\ 2 ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + \frac{κ_{k}^{A} (B_{k} - B_{k + 1})}{μ_{k}} & 0 \\ 0 & 2 ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + \frac{B_{k} - B_{k + 1}}{μ_{k}} \end{matrix} \\ \begin{matrix} 2 ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + \frac{κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{μ_{k + 1}} & 0 \\ 0 & 2 ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + \frac{B_{k} - B_{k + 1}}{μ_{k + 1}} \\ \begin{matrix} 2 (B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) \\ + 2 D_{k} (3 + ν_{k}^{D}) - 2 D_{k + 1} (3 + ν_{k + 1}^{D}) \end{matrix} & 0 \\ 0 & \begin{matrix} 2 (B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) \\ + 2 D_{k} (1 - ν_{k}^{D}) - 2 D_{k + 1} (1 - ν_{k + 1}^{D}) \end{matrix} \end{matrix}), \end{matrix}

(20)

\begin{matrix} i B_{k + 1}^{T} C_{k} \\ = (\begin{matrix} 0 \\ 2 (δ_{k} - δ_{k + 1}) + (2 {\hat{h}}_{k + 1} - \frac{B_{k + 1}}{μ_{k + 1}}) (γ_{k} - γ_{k + 1}) \\ 0 \\ 2 B_{k + 1} (δ_{k} - δ_{k + 1}) + 2 [B_{k + 1} {\hat{h}}_{k + 1} - D_{k + 1} (1 - ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1}) \end{matrix} \\ \begin{matrix} - 2 (δ_{k} - δ_{k + 1}) - (2 {\hat{h}}_{k + 1} + \frac{B_{k + 1} κ_{k + 1}^{A}}{μ_{k + 1}}) (γ_{k} - γ_{k + 1}) \\ 0 \\ - 2 B_{k + 1} (δ_{k} - δ_{k + 1}) - 2 [B_{k + 1} {\hat{h}}_{k + 1} + D_{k + 1} (3 + ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1}) \\ 0 \end{matrix}) . \end{matrix}

(21)

We now define analytic function vectors $f_{k} (z), (k = 0, 1, \dots, N + 1)$ as follows:

f_{k} (z) = [\begin{matrix} ϕ_{k} (z) \\ \frac{R_{k - 1}^{2}}{z} ϕ'_{k} (z) + ψ_{k} (z) \\ Φ_{k} (z) \\ \frac{R_{k - 1}^{2}}{z} Φ'_{k} (z) + Ψ_{k} (z) \end{matrix}], k = 0, 1, \dots, N + 1,

(22)

where $R_{- 1} = 0$ . Consequently, the interface conditions in Equation (18) can be concisely written into the following recurrence relation:

f_{k + 1} (z) = L^{k} f_{k} (z) + g_{k}, k = 0, 1, \dots, N,

(23)

where the components of the 4×4 matrix operator $R_{- 1} = 0$ and the four-dimensional vector operator $g_{k}$ are given by

\begin{matrix} L_{11}^{k} = \frac{μ_{k} + κ_{k}^{A} μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} + \frac{μ_{k} - μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}} z [[\frac{R_{k}^{2}}{z}]] \partial, L_{12}^{k} = \frac{μ_{k} - μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} [[\frac{R_{k}^{2}}{z}]], \\ L_{13}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1} + \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}} z [[\frac{R_{k}^{2}}{z}]] \partial, \\ L_{14}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1} [[\frac{R_{k}^{2}}{z}]], \\ L_{21}^{k} = \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} (κ_{k + 1}^{A} + 1)} \frac{R_{k}^{2} - R_{k - 1}^{2}}{z} \partial + \frac{κ_{k + 1}^{A} μ_{k} - κ_{k}^{A} μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} [[\frac{R_{k}^{2}}{z}]], L_{22}^{k} = \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} (κ_{k + 1}^{A} + 1)}, \\ L_{23}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1} \frac{R_{k}^{2} - R_{k - 1}^{2}}{z} \partial + \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1} [[\frac{R_{k}^{2}}{z}]], \\ L_{24}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1}, \end{matrix}

(24a)

\begin{array}{l} L_{31}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k}^{A} (B_{k} - B_{k + 1})}{8 μ_{k} D_{k + 1}} + \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}}{8 μ_{k} D_{k + 1}} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}} z [[\frac{R_{k}^{2}}{z}]] \partial, \\ L_{32}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}}{8 μ_{k} D_{k + 1}} [[\frac{R_{k}^{2}}{z}]], \\ L_{33}^{k} = \frac{(\begin{matrix} (B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) \\ + D_{k + 1} (1 - ν_{k + 1}^{D}) + D_{k} (3 + ν_{k}^{D}) \end{matrix})}{4 D_{k + 1}} + \frac{(\begin{matrix} (B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) \\ - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D}) \end{matrix})}{4 D_{k + 1}} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}} z [[\frac{R_{k}^{2}}{z}]] \partial, \\ L_{34}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})}{4 D_{k + 1}} [[\frac{R_{k}^{2}}{z}]], \\ L_{41}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{8 μ_{k} D_{k + 1}} \frac{R_{k}^{2} - R_{k - 1}^{2}}{z} \partial + \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + κ_{k}^{A} (B_{k + 1} - B_{k})}{8 μ_{k} D_{k + 1}} [[\frac{R_{k}^{2}}{z}]], \end{array}

\begin{array}{l} L_{42}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{8 μ_{k} D_{k + 1}}, \\ L_{43}^{k} = \frac{(\begin{matrix} (B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) \\ + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D}) \end{matrix})}{4 D_{k + 1}} \frac{R_{k}^{2} - R_{k - 1}^{2}}{z} \partial + \frac{(\begin{matrix} (B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) \\ - D_{k} (3 + ν_{k}^{D}) + D_{k + 1} (3 + ν_{k + 1}^{D}) \end{matrix})}{4 D_{k + 1}} [[\frac{R_{k}^{2}}{z}]], \\ L_{44}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})}{4 D_{k + 1}}, \end{array}

(24b)

g_{k} = [\begin{matrix} \frac{2 μ_{k + 1} (δ_{k} - δ_{k + 1}) + (2 μ_{k + 1} {\hat{h}}_{k + 1} - B_{k + 1}) (γ_{k} - γ_{k + 1})}{κ_{k + 1}^{A} + 1} z \\ - \frac{2 μ_{k + 1} (δ_{k} - δ_{k + 1}) + (2 μ_{k + 1} {\hat{h}}_{k + 1} + B_{k + 1} κ_{k + 1}^{A}) (γ_{k} - γ_{k + 1})}{κ_{k + 1}^{A} + 1} \frac{R_{k}^{2}}{z} \\ - \frac{B_{k + 1} (δ_{k} - δ_{k + 1}) + [B_{k + 1} {\hat{h}}_{k + 1} - D_{k + 1} (1 - ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1})}{4 D_{k + 1}} z \\ \frac{B_{k + 1} (δ_{k} - δ_{k + 1}) + [B_{k + 1} {\hat{h}}_{k + 1} + D_{k + 1} (3 + ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1})}{4 D_{k + 1}} \frac{R_{k}^{2}}{z} \end{matrix}],

(25)

where $[[\frac{R_{k}^{2}}{z}]]$ denotes the transformation that replaces the complex variable z by $R_{k}^{2} / R_{k}^{2} z z$ , and ∂ denotes the derivative. Because all coefficients appearing in Equation (23) are real, all the coefficients in the analytic functions $f_{k} (z), (k = 0, 1, \dots, N + 1)$ must then be real numbers.

It follows from Equation (23) that

f_{N + 1} (z) = L^{N} \times L^{N - 1} \times \dots \times L^{0} f_{0} (z) + g_{N} + L^{N} g_{N - 1} + L^{N} L^{N - 1} g_{N - 2} + \dots + L^{N} L^{N - 1} \times \dots \times L^{1} g_{0} .

(26)

It is easily verified that

g_{N} + L^{N} g_{N - 1} + L^{N} L^{N - 1} g_{N - 2} + \dots + L^{N} L^{N - 1} \times \dots \times L^{1} g_{0} = [\begin{matrix} c_{1} z \\ c_{2} \frac{1}{z} \\ c_{3} z \\ c_{4} \frac{1}{z} \end{matrix}],

(27)

where the real constants $c_{k}, (k = 1, 2, 3, 4)$ are determined by the material and geometric parameters of the composite, and by the volumetric eigenstrains and eigencurvatures. In addition,

\begin{matrix} {L^{N} \times L^{N - 1} \times \dots \times L^{0} f_{0} (z) |}_{ϕ_{0} (z) = {az}^{M}, ψ_{0} (z) = {bz}^{M}, Φ_{0} (z) = {cz}^{M}, Ψ_{0} (z) = {dz}^{M}} \\ = [\begin{matrix} d_{11} z^{M + 2} + d_{12} z^{M} + \frac{d_{13}}{z^{M - 2}} + \frac{d_{14}}{z^{M}} \\ d_{21} z^{M} + d_{22} z^{M - 2} + \frac{d_{23}}{z^{M}} + \frac{d_{24}}{z^{M + 2}} \\ d_{31} z^{M + 2} + d_{32} z^{M} + \frac{d_{33}}{z^{M - 2}} + \frac{d_{34}}{z^{M}} \\ d_{41} z^{M} + d_{42} z^{M - 2} + \frac{d_{43}}{z^{M}} + \frac{d_{44}}{z^{M + 2}} \end{matrix}], \end{matrix}

(28)

where the real constants $d_{ij} (i, j = 1, 2, 3, 4)$ are determined by the four real numbers a, b, c and d appearing in Equation (28), and the geometric and elastic parameters of the composite. In particular,

b = d = 0 \Rightarrow d_{11} = d_{14} = d_{21} = d_{24} = d_{31} = d_{34} = d_{41} = d_{44} = 0 .

(29)

We can then conclude the following:

$f_{k} (z), (k = 0, 1, \dots, N + 1)$ only contain the odd-order powers of z;

the highest-order nonzero term of $f_{0} (z)$ is cubic, which occurs only in $ϕ_{0} (z)$ and $Φ_{0} (z)$ but not in $ψ_{0} (z)$ and $Ψ_{0} (z)$ .

In view of Equations (27) and (28), the general expressions of $ϕ_{0} (z), ψ_{0} (z), Φ_{0} (z), Ψ_{0} (z)$ defined in the internal circular inhomogeneity and $ϕ_{N + 1} (z), ψ_{N + 1} (z), Φ_{N + 1} (z), Ψ_{N + 1} (z)$ defined in the surrounding matrix are given by

\begin{matrix} ϕ_{0} (z) = X_{1} z + X_{3} z^{3}, ψ_{0} (z) = Y_{1} z, Φ_{0} (z) = V_{1} z + V_{3} z^{3}, Ψ_{0} (z) = W_{1} z, z \in S_{0}, \\ ϕ_{N + 1} (z) = Fz + \frac{F_{1}}{z}, ψ_{N + 1} (z) = Gz + \frac{G_{1}}{z} + \frac{G_{3}}{z^{3}}, Φ_{N + 1} (z) = Iz + \frac{I_{1}}{z}, Ψ_{N + 1} (z) = Jz + \frac{J_{1}}{z} + \frac{J_{3}}{z^{3}}, z \in S_{N + 1}, \end{matrix}

(30)

where $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}, W_{1}, F_{1}, G_{1}, G_{3}, I_{1}, J_{1}$ and $J_{3}$ are 12 unknown real coefficients.

The internal stress resultant field is governed by the six real numbers $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}$ and $W_{1}$ , which can be determined from the linear algebraic equations by equating the coefficients of the six positive powers of z ( $z^{3}$ and z in the first and third rows and z in the second and fourth rows) in Equation (26). Once $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}$ and $W_{1}$ have been determined, the other six real coefficients $F_{1}, G_{1}, G_{3}, I_{1}, J_{1}$ and $J_{3}$ , which characterize the perturbed part of the stress resultant field in the matrix, can then be obtained from the remaining conditions of Equation (26).

Once $f_{0} (z)$ in the inhomogeneity and $f_{N + 1} (z)$ in the matrix have been determined, the N analytic function vectors $f_{k} (z), (k = 1, 2, \dots, N)$ defined in the interphase layers can be arrived at by using Equation (23) if necessary. The average stress resultants within the inhomogeneity can be obtained directly from the six real coefficients $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}$ and $W_{1}$ as follows:

\begin{matrix} \bar{N_{12}} = \bar{M_{12}} = \bar{ℜ_{1}} = \bar{ℜ_{2}} = 0, \\ \bar{N_{11}} + \bar{N_{22}} = 4 Re (X_{1} + B_{0} V_{1}), \bar{N_{22}} - \bar{N_{11}} = 2 (Y_{1} + B_{0} W_{1}) + 6 R_{0}^{2} (X_{3} + B_{0} V_{3}), \\ \bar{M_{11}} + \bar{M_{22}} = 4 D_{0} (1 + ν_{0}^{D}) V_{1} + \frac{B_{0} (κ_{0}^{A} - 1)}{μ_{0}} X_{1}, \\ \bar{M_{22}} - \bar{M_{11}} = - 2 D_{0} (1 - ν_{0}^{D}) (3 V_{1} R_{0}^{2} + W_{1}) - \frac{B_{0}}{μ_{0}} (3 X_{1} R_{0}^{2} + Y_{1}), \end{matrix}

(31)

where the overbar means the average.

4. Basic equations

In the previous section, we have obtained in Equation (26) the relationship between $f_{0} (z)$ in the inhomogeneity and $f_{N + 1} (z)$ in the matrix. This relationship is expressed in terms of the matrix operators $L^{k} (k = 0, 1, \dots, N)$ . In this section, this relationship will be presented in a more specified form. It is observed from Equations (22), (23) and (27)–(30) that the general form of $f_{k} (z), (k = 0, 1, \dots, N + 1)$ can be written into

f_{k} (z) = [\begin{matrix} e_{1} z^{3} + e_{9} z + \frac{e_{3}}{z} \\ e_{4} z + \frac{e_{10}}{z} + \frac{e_{2}}{z^{3}} \\ e_{5} z^{3} + e_{11} z + \frac{e_{7}}{z} \\ e_{8} z + \frac{e_{12}}{z} + \frac{e_{6}}{z^{3}} \end{matrix}], k = 0, 1, \dots, N + 1,

(32)

where the subscripts of the real coefficients $e_{i}, (i = 1, 2, \dots, 12)$ have been assigned in such a way that the transformation matrix will have a decomposed form. The 12 coefficients $e_{i}, (i = 1, 2, \dots, 12)$ can be considered as the coordinates of $f_{k} (z)$ . Thus the operator $L^{k}$ is a linear transformation of the coordinates $e_{i}, (i = 1, 2, \dots, 12)$ and can be represented by a 12×12 constant matrix. In fact, if

[\begin{matrix} E_{1} z^{3} + E_{9} z + \frac{E_{3}}{z} \\ E_{4} z + \frac{E_{10}}{z} + \frac{E_{2}}{z^{3}} \\ E_{5} z^{3} + E_{11} z + \frac{E_{7}}{z} \\ E_{8} z + \frac{E_{12}}{z} + \frac{E_{6}}{z^{3}} \end{matrix}] = L^{k} [\begin{matrix} e_{1} z^{3} + e_{9} z + \frac{e_{3}}{z} \\ e_{4} z + \frac{e_{10}}{z} + \frac{e_{2}}{z^{3}} \\ e_{5} z^{3} + e_{11} z + \frac{e_{7}}{z} \\ e_{8} z + \frac{e_{12}}{z} + \frac{e_{6}}{z^{3}} \end{matrix}],

(33)

then the coefficients on the left-hand side of Equation (33) can be obtained from those on the right through

\begin{matrix} {[\begin{matrix} E_{1} & E_{2} & E_{3} & E_{4} & E_{5} & E_{6} & E_{7} & E_{8} \end{matrix}]}^{T} = P^{k} {[\begin{matrix} e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} & e_{8} \end{matrix}]}^{T}, \\ {[\begin{matrix} E_{9} & E_{10} & E_{11} & E_{12} \end{matrix}]}^{T} = Q^{k} {[\begin{matrix} e_{9} & e_{10} & e_{11} & e_{12} \end{matrix}]}^{T}, \end{matrix}

(34)

where the components of the 8×8 real matrix $P^{k}$ and the 4×4 real matrix $Q^{k}$ are explicitly given by

\begin{matrix} P_{11}^{k} = \frac{μ_{k} + κ_{k}^{A} μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{12}^{k} = \frac{μ_{k} - μ_{k + 1}}{R_{k}^{6} μ_{k} (κ_{k + 1}^{A} + 1)}, P_{13}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) (μ_{k} - μ_{k + 1})}{R_{k}^{6} μ_{k} (κ_{k + 1}^{A} + 1)}, \\ P_{14}^{k} = 0, P_{15}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1}, P_{16}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{R_{k}^{6} (κ_{k + 1}^{A} + 1)}, \\ P_{17}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}]}{R_{k}^{6} (κ_{k + 1}^{A} + 1)}, P_{18}^{k} = 0, \end{matrix}

(35a)

\begin{matrix} P_{21}^{k} = \frac{R_{k}^{6} (κ_{k + 1}^{A} μ_{k} - κ_{k}^{A} μ_{k + 1})}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{22}^{k} = \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{23}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) (μ_{k + 1} + κ_{k + 1}^{A} μ_{k})}{μ_{k} (κ_{k + 1}^{A} + 1)}, \\ P_{24}^{k} = 0, P_{25}^{k} = \frac{R_{k}^{6} [2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})]}{κ_{k + 1}^{A} + 1}, P_{26}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1}, \\ P_{27}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})]}{κ_{k + 1}^{A} + 1}, P_{28}^{k} = 0, \end{matrix}

(35b)

\begin{matrix} P_{31}^{k} = \frac{3 R_{k}^{2} (R_{k}^{2} - R_{k - 1}^{2}) (μ_{k} - μ_{k + 1})}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{32}^{k} = 0, P_{33}^{k} = \frac{μ_{k} + κ_{k}^{A} μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{34}^{k} = \frac{R_{k}^{2} (μ_{k} - μ_{k + 1})}{μ_{k} (κ_{k + 1}^{A} + 1)}, \\ P_{35}^{k} = \frac{3 R_{k}^{2} (R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}]}{κ_{k + 1}^{A} + 1}, P_{36}^{k} = 0, \\ P_{37}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1}, P_{38}^{k} = \frac{R_{k}^{2} [2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}]}{κ_{k + 1}^{A} + 1}, \end{matrix}

(35c)

\begin{matrix} P_{41}^{k} = \frac{3 (R_{k}^{2} - R_{k - 1}^{2}) (μ_{k + 1} + κ_{k + 1}^{A} μ_{k})}{μ_{k} (κ_{k + 1}^{A} + 1)}, P_{42}^{k} = 0, P_{43}^{k} = \frac{κ_{k + 1}^{A} μ_{k} - κ_{k}^{A} μ_{k + 1}}{R_{k}^{2} μ_{k} (κ_{k + 1}^{A} + 1)}, P_{44}^{k} = \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} (κ_{k + 1}^{A} + 1)}, \\ P_{45}^{k} = \frac{3 (R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})]}{κ_{k + 1}^{A} + 1}, P_{46}^{k} = 0, \\ P_{47}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{R_{k}^{2} (κ_{k + 1}^{A} + 1)}, P_{48}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1}, \end{matrix}

(35d)

\begin{matrix} P_{51}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k}^{A} (B_{k} - B_{k + 1})}{8 μ_{k} D_{k + 1}}, P_{52}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}}{8 R_{k}^{6} μ_{k} D_{k + 1}}, \\ P_{53}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}]}{8 R_{k}^{6} μ_{k} D_{k + 1}}, P_{54}^{k} = 0, \\ P_{55}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + D_{k + 1} (1 - ν_{k + 1}^{D}) + D_{k} (3 + ν_{k}^{D})}{4 D_{k + 1}}, \\ P_{56}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})}{4 R_{k}^{6} D_{k + 1}}, \\ P_{57}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})]}{4 R_{k}^{6} D_{k + 1}}, P_{58}^{k} = 0, \end{matrix}

(35e)

\begin{matrix} P_{61}^{k} = \frac{R_{k}^{6} [2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + κ_{k}^{A} (B_{k + 1} - B_{k})]}{8 μ_{k} D_{k + 1}}, P_{62}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{8 μ_{k} D_{k + 1}}, \\ P_{63}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}]}{8 μ_{k} D_{k + 1}}, P_{64}^{k} = 0, \\ P_{65}^{k} = \frac{R_{k}^{6} [(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) - D_{k} (3 + ν_{k}^{D}) + D_{k + 1} (3 + ν_{k + 1}^{D})]}{4 D_{k + 1}}, \\ P_{66}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})}{4 D_{k + 1}}, \\ P_{67}^{k} = - \frac{(R_{k}^{2} - R_{k - 1}^{2}) [(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})]}{4 D_{k + 1}}, P_{68}^{k} = 0, \end{matrix}

(35f)

\begin{matrix} P_{71}^{k} = \frac{3 R_{k}^{2} (R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}]}{8 μ_{k} D_{k + 1}}, P_{72}^{k} = 0, \\ P_{73}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k}^{A} (B_{k} - B_{k + 1})}{8 μ_{k} D_{k + 1}}, P_{74}^{k} = \frac{R_{k}^{2} [2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}]}{8 μ_{k} D_{k + 1}}, \\ P_{75}^{k} = \frac{3 R_{k}^{2} (R_{k}^{2} - R_{k - 1}^{2}) [(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})]}{4 D_{k + 1}}, \\ P_{76}^{k} = 0, P_{77}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + D_{k + 1} (1 - ν_{k + 1}^{D}) + D_{k} (3 + ν_{k}^{D})}{4 D_{k + 1}}, \\ P_{78}^{k} = \frac{R_{k}^{2} [(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})]}{4 D_{k + 1}}, \end{matrix}

(35g)

\begin{array}{l} P_{81}^{k} = \frac{3 (R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}]}{8 μ_{k} D_{k + 1}}, P_{82}^{k} = 0, \\ P_{83}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + κ_{k}^{A} (B_{k + 1} - B_{k})}{8 R_{k}^{2} μ_{k} D_{k + 1}}, P_{84}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{8 μ_{k} D_{k + 1}}, \end{array}

\begin{array}{l} P_{85}^{k} = \frac{3 (R_{k}^{2} - R_{k - 1}^{2}) [(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})]}{4 D_{k + 1}}, \\ P_{86}^{k} = 0, P_{87}^{k} = \frac{[(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) - D_{k} (3 + ν_{k}^{D}) + D_{k + 1} (3 + ν_{k + 1}^{D})]}{4 R_{k}^{2} D_{k + 1}}, \\ P_{88}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})}{4 D_{k + 1}}, \end{array}

(35h)

\begin{matrix} Q_{11}^{k} = \frac{μ_{k} + κ_{k}^{A} μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} + \frac{μ_{k} - μ_{k + 1}}{μ_{k} (κ_{k + 1}^{A} + 1)} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}}, Q_{12}^{k} = \frac{μ_{k} - μ_{k + 1}}{R_{k}^{2} μ_{k} (κ_{k + 1}^{A} + 1)}, \\ Q_{13}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1} + \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{κ_{k + 1}^{A} + 1} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}}, \\ Q_{14}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{R_{k}^{2} (κ_{k + 1}^{A} + 1)}, \\ Q_{21}^{k} = \frac{(R_{k}^{2} - R_{k - 1}^{2}) (μ_{k + 1} + κ_{k + 1}^{A} μ_{k}) + R_{k}^{2} (κ_{k + 1}^{A} μ_{k} - κ_{k}^{A} μ_{k + 1})}{μ_{k} (κ_{k + 1}^{A} + 1)}, Q_{22}^{k} = \frac{μ_{k + 1} + κ_{k + 1}^{A} μ_{k}}{μ_{k} (κ_{k + 1}^{A} + 1)}, \\ Q_{23}^{k} = \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})] + R_{k}^{2} [2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})]}{κ_{k + 1}^{A} + 1}, \\ Q_{24}^{k} = \frac{2 μ_{k + 1} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k + 1}^{A} (B_{k} - B_{k + 1})}{κ_{k + 1}^{A} + 1}, \end{matrix}

(36a)

\begin{array}{l} Q_{31}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + κ_{k}^{A} (B_{k} - B_{k + 1})}{8 μ_{k} D_{k + 1}} + \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}}{8 μ_{k} D_{k + 1}} \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}}, \\ Q_{32}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + B_{k + 1} - B_{k}}{8 R_{k}^{2} μ_{k} D_{k + 1}}, \\ Q_{33}^{k} = \frac{(\begin{matrix} [(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) + D_{k + 1} (1 - ν_{k + 1}^{D}) + D_{k} (3 + ν_{k}^{D})] \\ + \frac{R_{k}^{2} - R_{k - 1}^{2}}{R_{k}^{2}} [(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})] \end{matrix})}{4 D_{k + 1}}, \\ Q_{34}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k} - {\hat{h}}_{k + 1}) - D_{k} (1 - ν_{k}^{D}) + D_{k + 1} (1 - ν_{k + 1}^{D})}{4 R_{k}^{2} D_{k + 1}}, \\ Q_{41}^{k} = \frac{(R_{k}^{2} - R_{k - 1}^{2}) [2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}] + R_{k}^{2} [2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + κ_{k}^{A} (B_{k + 1} - B_{k})]}{8 μ_{k} D_{k + 1}}, \\ Q_{42}^{k} = \frac{2 μ_{k} ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + B_{k} - B_{k + 1}}{8 μ_{k} D_{k + 1}}, \end{array}

\begin{array}{l} Q_{43}^{k} = \frac{(\begin{matrix} (R_{k}^{2} - R_{k - 1}^{2}) [(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})] \\ + R_{k}^{2} [(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) - D_{k} (3 + ν_{k}^{D}) + D_{k + 1} (3 + ν_{k + 1}^{D})] \end{matrix})}{4 D_{k + 1}}, \\ Q_{44}^{k} = \frac{(B_{k} + B_{k + 1}) ({\hat{h}}_{k + 1} - {\hat{h}}_{k}) + D_{k + 1} (3 + ν_{k + 1}^{D}) + D_{k} (1 - ν_{k}^{D})}{4 D_{k + 1}} . \end{array}

(36b)

It is seen from Equation (34) that the determination of the first eight coefficients is decoupled from that of the last four coefficients. This fact greatly simplifies the analysis involved. It follows from Equation (30) that

f_{0} (z) = [\begin{matrix} X_{1} z + X_{3} z^{3} \\ Y_{1} z \\ V_{1} z + V_{3} z^{3} \\ W_{1} z \end{matrix}], f_{N + 1} (z) = [\begin{matrix} Fz + \frac{F_{1}}{z} \\ Gz + \frac{G_{1} + {FR}_{N}^{2}}{z} + \frac{G_{3} - F_{1} R_{N}^{2}}{z^{3}} \\ Iz + \frac{I_{1}}{z} \\ Jz + \frac{J_{1} + {IR}_{N}^{2}}{z} + \frac{J_{3} - I_{1} R_{N}^{2}}{z^{3}} \end{matrix}] .

(37)

The condition (26) can then be explicitly expressed into

[\begin{matrix} 0 \\ G_{3} - F_{1} R_{N}^{2} \\ F_{1} \\ G \\ 0 \\ J_{3} - I_{1} R_{N}^{2} \\ I_{1} \\ J \end{matrix}] = S [\begin{matrix} X_{3} \\ 0 \\ 0 \\ Y_{1} \\ V_{3} \\ 0 \\ 0 \\ W_{1} \end{matrix}], S = P^{N} \times P^{N - 1} \times \dots \times P^{1} \times P^{0},

(38)

[\begin{matrix} F \\ G_{1} + {FR}_{N}^{2} \\ I \\ J_{1} + {IR}_{N}^{2} \end{matrix}] = U [\begin{matrix} X_{1} \\ 0 \\ V_{1} \\ 0 \end{matrix}] + g_{N}^{*} + Q^{N} g_{N - 1}^{*} + \dots + Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times g_{0}^{*}, U = Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times Q^{0},

(39)

where $g_{k}^{*} (k = 0, 1, \dots, N)$ are the coefficient vectors of $g_{k} (k = 0, 1, \dots, N)$ defined in Equation (25) and are given by

g_{k}^{*} = [\begin{matrix} \frac{2 μ_{k + 1} (δ_{k} - δ_{k + 1}) + (2 μ_{k + 1} {\hat{h}}_{k + 1} - B_{k + 1}) (γ_{k} - γ_{k + 1})}{κ_{k + 1}^{A} + 1} \\ - \frac{R_{k}^{2} [2 μ_{k + 1} (δ_{k} - δ_{k + 1}) + (2 μ_{k + 1} {\hat{h}}_{k + 1} + B_{k + 1} κ_{k + 1}^{A}) (γ_{k} - γ_{k + 1})]}{κ_{k + 1}^{A} + 1} \\ - \frac{B_{k + 1} (δ_{k} - δ_{k + 1}) + [B_{k + 1} {\hat{h}}_{k + 1} - D_{k + 1} (1 - ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1})}{4 D_{k + 1}} \\ \frac{R_{k}^{2} [B_{k + 1} (δ_{k} - δ_{k + 1}) + [B_{k + 1} {\hat{h}}_{k + 1} + D_{k + 1} (3 + ν_{k + 1}^{D})] (γ_{k} - γ_{k + 1})]}{4 D_{k + 1}} \end{matrix}] .

(40)

The 12 unknowns $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}, W_{1}, F_{1}, G_{1}, G_{3}, I_{1}, J_{1}$ and $J_{3}$ can be determined from Equations (38) and (39) in the following uncoupled manner.

$X_{1}$ and $V_{1}$ can be determined from the two coupled linear algebraic equations obtained by equating the first and third rows of Equation (39) as follows:

X_{1} = \frac{U_{33} \tilde{F} - U_{13} \tilde{I}}{U_{11} U_{33} - U_{13} U_{31}}, V_{1} = \frac{U_{11} \tilde{I} - U_{31} \tilde{F}}{U_{11} U_{33} - U_{13} U_{31}},

(41)

where

\begin{matrix} \tilde{F} = F - [g_{N}^{*} + Q^{N} g_{N - 1}^{*} + \dots + Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times g_{0}^{*}]_{1}, \\ \tilde{I} = I - [g_{N}^{*} + Q^{N} g_{N - 1}^{*} + \dots + Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times g_{0}^{*}]_{3} . \end{matrix}

(42)

Once $X_{1}$ and $V_{1}$ have been determined, $G_{1}$ and $J_{1}$ can be obtained from the second and fourth rows of Equation (39) as

\begin{matrix} G_{1} + {FR}_{N}^{2} = U_{21} X_{1} + U_{23} V_{1} + [g_{N}^{*} + Q^{N} g_{N - 1}^{*} + \dots + Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times g_{0}^{*}]_{2}, \\ J_{1} + {IR}_{N}^{2} = U_{41} X_{1} + U_{43} V_{1} + [g_{N}^{*} + Q^{N} g_{N - 1}^{*} + \dots + Q^{N} \times Q^{N - 1} \times \dots \times Q^{1} \times g_{0}^{*}]_{4} . \end{matrix}

(43)

$X_{3}, Y_{1}, V_{3}$ and $W_{1}$ can be determined from the four coupled linear algebraic equations obtained by equating the first, fourth, fifth and eighth rows of Equation (38) as follows:

[\begin{matrix} X_{3} \\ Y_{1} \\ V_{3} \\ W_{1} \end{matrix}] = {[\begin{matrix} S_{11} & S_{14} & S_{15} & S_{18} \\ S_{41} & S_{44} & S_{45} & S_{48} \\ S_{51} & S_{54} & S_{55} & S_{58} \\ S_{81} & S_{84} & S_{85} & S_{88} \end{matrix}]}^{- 1} [\begin{matrix} 0 \\ G \\ 0 \\ J \end{matrix}] .

(44)

Once $X_{3}, Y_{1}, V_{3}$ and $W_{1}$ have been found, $F_{1}, G_{3}, I_{1}$ and $J_{3}$ can be obtained from the second, third, sixth and seventh rows of Equation (38) as

[\begin{matrix} G_{3} - F_{1} R_{N}^{2} \\ F_{1} \\ J_{3} - I_{1} R_{N}^{2} \\ I_{1} \end{matrix}] = [\begin{matrix} S_{21} & S_{24} & S_{25} & S_{28} \\ S_{31} & S_{34} & S_{35} & S_{38} \\ S_{61} & S_{64} & S_{65} & S_{68} \\ S_{71} & S_{74} & S_{75} & S_{78} \end{matrix}] [\begin{matrix} X_{3} \\ Y_{1} \\ V_{3} \\ W_{1} \end{matrix}] .

(45)

Once the 12 unknowns $X_{1}, X_{3}, Y_{1}, V_{1}, V_{3}, W_{1}, F_{1}, G_{1}, G_{3}, I_{1}, J_{1}$ and $J_{3}$ are determined, the stress resultant fields in the internal inhomogeneity and the surrounding matrix are determined by Equations (2), (3) and (30).

By using the present method, the original boundary value problem is reduced to two coupled linear algebraic equations for the unknowns $X_{1}$ and $V_{1}$ , and four coupled linear algebraic equations for the unknowns $X_{3}, Y_{1}, V_{3}$ and $W_{1}$ . The coefficients of these linear equations can be obtained by calculating the first and third columns of the 4×4 real product matrix U, and the first, fourth, fifth and eighth columns of the 8×8 real product matrix S.

From the above analysis the following is also found.

The internal stress resultant field is uniform and hydrostatic if the composite is subjected to thermal loading or the remote mechanical loading characterized by G = J = 0, which implies that $N_{11}^{\infty} = N_{22}^{\infty}$ and $M_{11}^{\infty} = M_{22}^{\infty}$ in view of Equation (16) (i.e., the remote membrane stress resultants and bending moments are hydrostatic). The internal uniform and hydrostatic stress resultant field is specifically given by

\begin{matrix} N_{12} = M_{12} = ℜ_{1} = ℜ_{2} = 0, \\ N_{11} = N_{22} = 2 (X_{1} + B_{0} V_{1}), M_{11} = M_{22} = 2 D_{0} (1 + ν_{0}^{D}) V_{1} + \frac{B_{0} (κ_{0}^{A} - 1)}{2 μ_{0}} X_{1}, z \in S_{0} . \end{matrix}

(46)

The internal stress resultants are quadratic functions of the two in-plane coordinates $x_{1}$ and $x_{2}$ if the composite is subjected to the remote mechanical loading characterized by F = I = 0. The internal stress resultant field is specifically given by

\begin{matrix} N_{12} = M_{12} = 0, \\ N_{11} = - (Y_{1} + B_{0} W_{1}) - 12 (X_{3} + B_{0} V_{3}) x_{2}^{2}, N_{22} = Y_{1} + B_{0} W_{1} + 12 (X_{3} + B_{0} V_{3}) x_{1}^{2}, \\ M_{11} = D_{0} (1 - ν_{0}^{D}) W_{1} + \frac{B_{0} Y_{1}}{2 μ_{0}} + [12 D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} + 1) X_{3}}{2 μ_{0}}] x_{1}^{2} - [12 ν_{0}^{D} D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} - 3) X_{3}}{2 μ_{0}}] x_{2}^{2}, \\ M_{22} = - D_{0} (1 - ν_{0}^{D}) W_{1} - \frac{B_{0} Y_{1}}{2 μ_{0}} + [12 ν_{0}^{D} D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} - 3) X_{3}}{2 μ_{0}}] x_{1}^{2} - [12 D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} + 1) X_{3}}{2 μ_{0}}] x_{2}^{2}, \\ ℜ_{1} = [24 D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} + 1) X_{3}}{μ_{0}}] x_{1}, ℜ_{2} = - [24 D_{0} V_{3} + \frac{3 B_{0} (κ_{0}^{A} + 1) X_{3}}{μ_{0}}] x_{2}, z \in S_{0} . \end{matrix}

(47)

5. Application

We consider a circular inhomogeneity with a single interphase layer (N = 1) under remote mechanical loading. Firstly, the mechanical loading is characterized by G = J = 0. In this special loading case, it is enough to calculate the first and third columns of the 4×4 product matrix U.

The first column of U is

\begin{matrix} U_{11} = \frac{[2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [2 μ_{1} + (κ_{1}^{A} - 1) μ_{2}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} [\frac{2 (μ_{2} - μ_{1}) [(κ_{0}^{A} - 1) μ_{1} - (κ_{1}^{A} - 1) μ_{0}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)}], \end{matrix}

(48a)

\begin{matrix} U_{21} = \frac{R_{1}^{2} [2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [2 κ_{2}^{A} μ_{1} - (κ_{1}^{A} - 1) μ_{2}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{1}^{2} [4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{2 R_{0}^{2} (μ_{2} + κ_{2}^{A} μ_{1}) [(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2} [4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(48b)

\begin{array}{l} U_{31} = \frac{[2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + (κ_{1}^{A} - 1) (B_{1} - B_{2})]}{8 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)}, \\ + \frac{[4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) + D_{1} (1 + ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})]}{16 μ_{0} D_{1} D_{2}} \end{array}

(48c)

\begin{array}{l} + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + B_{2} - B_{1}]}{4 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) - D_{1} (1 - ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})]}{16 μ_{0} D_{1} D_{2}}, \end{array}

\begin{matrix} U_{41} = \frac{R_{1}^{2} [2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [4 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{2} - B_{1})]}{8 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{1}^{2} [4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) - D_{1} (1 + ν_{1}^{D}) + D_{2} (3 + ν_{2}^{D})]}{16 μ_{0} D_{1} D_{2}} \\ + \frac{R_{0}^{2} [(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}] [2 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{4 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{0}^{2} [4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) + D_{2} (3 + ν_{2}^{D}) + D_{1} (1 - ν_{1}^{D})]}{16 μ_{0} D_{1} D_{2}} . \end{matrix}

(48d)

The third column of U is

\begin{matrix} U_{13} = \frac{2 [2 μ_{1} + (κ_{1}^{A} - 1) μ_{2}] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{2 (μ_{1} - μ_{2}) [4 μ_{1} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{0} - B_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(49a)

\begin{matrix} U_{23} = \frac{2 R_{1}^{2} [2 κ_{2}^{A} μ_{1} - (κ_{1}^{A} - 1) μ_{2}] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{1}^{2} [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)} \\ + \frac{2 R_{0}^{2} (μ_{2} + κ_{2}^{A} μ_{1}) [4 μ_{1} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{0} - B_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2} [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(49b)

\begin{matrix} U_{33} = \frac{[2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (B_{0} - B_{1})] [4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + (κ_{1}^{A} - 1) (B_{1} - B_{2})]}{4 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{[(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) + D_{1} (1 + ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})]}{4 D_{1} D_{2}} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[4 μ_{1} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{0} - B_{1})] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + B_{2} - B_{1}]}{4 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) - D_{1} (1 - ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})]}{4 D_{1} D_{2}}, \end{matrix}

(49c)

\begin{matrix} U_{43} = \frac{R_{1}^{2} [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}] [4 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{2} - B_{1})]}{4 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{1}^{2} [(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) - D_{1} (1 + ν_{1}^{D}) + D_{2} (3 + ν_{2}^{D})]}{4 D_{1} D_{2}} \\ + \frac{R_{0}^{2} [4 μ_{1} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{0} - B_{1})] [2 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{4 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{0}^{2} [(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) + D_{2} (3 + ν_{2}^{D}) + D_{1} (1 - ν_{1}^{D})]}{4 D_{1} D_{2}} . \end{matrix}

(49d)

$X_{1}$ and $V_{1}$ characterizing the hydrostatic stress resultant field in the inhomogeneity can then be uniquely determined by using Equation (41) with $\tilde{F} = F$ and $\tilde{I} = I$ (in the absence of thermal loading); $G_{1}$ and $J_{1}$ characterizing the disturbance field in the matrix can be obtained by using Equation (43). In general, both $G_{1}$ and $J_{1}$ are nonzero. This fact implies that the presence of the inhomogeneity will alter the original stress resultant field in the infinite matrix. It has been observed that certain reinforced “neutral” holes may be designed for laminated plates and shells that do not alter the original stress distribution in the cut sheet [9 –12]. Here we will address a similar issue by virtue of a circular inhomogeneity with a single interphase layer. If there exists no stress resultant disturbance in the surrounding matrix (the inhomogeneity is termed “neutral”), we have $G_{1} = J_{1} = 0$ . It is then deduced from Equation (39) that

[\begin{matrix} R_{1}^{2} U_{11} - U_{21} & R_{1}^{2} U_{13} - U_{23} \\ R_{1}^{2} U_{31} - U_{41} & R_{1}^{2} U_{33} - U_{43} \end{matrix}] [\begin{matrix} X_{1} \\ V_{1} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] .

(50)

In order to have a non-trivial solution of ${[\begin{matrix} X_{1} & V_{1} \end{matrix}]}^{T}$ , we arrive at the following condition:

(R_{1}^{2} U_{11} - U_{21}) (R_{1}^{2} U_{33} - U_{43}) - (R_{1}^{2} U_{13} - U_{23}) (R_{1}^{2} U_{31} - U_{41}) = 0 .

(51)

By using Equations (48) and (49), the above condition can be explicitly expressed into

\frac{R_{0}^{4}}{R_{1}^{4}} (q_{1} q_{2} - q_{3} q_{4}) + \frac{R_{0}^{2}}{R_{1}^{2}} (p_{1} q_{2} + p_{2} q_{1} - p_{3} q_{4} - p_{4} q_{3}) + p_{1} p_{2} - p_{3} p_{4} = 0,

(52)

where the eight real coefficients $p_{k}, q_{k}, (k = 1, 2, 3, 4)$ are completely determined by the material parameters of the three-phase composite and are given by

\begin{matrix} p_{1} = \frac{2 [2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [(κ_{1}^{A} - 1) μ_{2} - (κ_{2}^{A} - 1) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [4 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) - (κ_{2}^{A} - 1) (B_{1} - B_{2})]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \\ q_{1} = \frac{2 [2 μ_{2} + (κ_{2}^{A} - 1) μ_{1}] [(κ_{0}^{A} - 1) μ_{1} - (κ_{1}^{A} - 1) μ_{0}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [4 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) - (κ_{2}^{A} - 1) (B_{1} - B_{2})]}{4 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \\ p_{2} = \frac{[2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (B_{0} - B_{1})] [4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + (κ_{1}^{A} - 1) (B_{1} - B_{2})]}{2 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{[(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) + D_{1} (1 + ν_{1}^{D}) - D_{2} (1 + ν_{2}^{D})]}{2 D_{1} D_{2}}, \\ q_{2} = \frac{[4 μ_{1} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{0} - B_{1})] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + B_{2} - B_{1}]}{2 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{[(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) - D_{1} (1 - ν_{1}^{D}) - D_{2} (1 + ν_{2}^{D})]}{2 D_{1} D_{2}}, \end{matrix}

(53)

\begin{matrix} p_{3} = \frac{4 [(κ_{1}^{A} - 1) μ_{2} - (κ_{2}^{A} - 1) μ_{1}] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (B_{0} - B_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[4 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + (κ_{2}^{A} - 1) (B_{2} - B_{1})] [(B_{0} + B_{1}) ({\hat{h}}_{0} - {\hat{h}}_{1}) + D_{0} (1 + ν_{0}^{D}) + D_{1} (1 - ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)}, \\ q_{3} = \frac{2 [2 μ_{2} + (κ_{2}^{A} - 1) μ_{1}] [4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{1}^{A} - 1) (B_{1} - B_{0})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[4 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + (κ_{2}^{A} - 1) (B_{2} - B_{1})] [(B_{0} + B_{1}) ({\hat{h}}_{1} - {\hat{h}}_{0}) - D_{0} (1 + ν_{0}^{D}) + D_{1} (1 + ν_{1}^{D})]}{D_{1} (κ_{2}^{A} + 1)}, \\ p_{4} = \frac{[2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + (κ_{1}^{A} - 1) (B_{1} - B_{2})]}{4 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{[4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} - 1) (B_{0} - B_{1})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) + D_{1} (1 + ν_{1}^{D}) - D_{2} (1 + ν_{2}^{D})]}{8 μ_{0} D_{1} D_{2}}, \\ q_{4} = \frac{[(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}] [2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + B_{2} - B_{1}]}{2 μ_{0} μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{[4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + (κ_{0}^{A} - 1) (B_{1} - B_{0})] [(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) - D_{1} (1 - ν_{1}^{D}) - D_{2} (1 + ν_{2}^{D})]}{8 μ_{0} D_{1} D_{2}} . \end{matrix}

(54)

For given material parameters of the composite, $R_{0}^{2} / R_{1}^{2}$ can be solved from Equation (52) as

\frac{R_{0}^{2}}{R_{1}^{2}} = \frac{p_{3} q_{4} + p_{4} q_{3} - p_{1} q_{2} - p_{2} q_{1} \pm \sqrt{{(p_{1} q_{2} + p_{2} q_{1} - p_{3} q_{4} - p_{4} q_{3})}^{2} - 4 (p_{1} p_{2} - p_{3} p_{4}) (q_{1} q_{2} - q_{3} q_{4})}}{2 (q_{1} q_{2} - q_{3} q_{4})},

0 < \frac{R_{0}^{2}}{R_{1}^{2}} < 1 .

(55)

It is observed from Equation (52) that if

(p_{1} p_{2} - p_{3} p_{4}) (p_{1} p_{2} - p_{3} p_{4} + p_{1} q_{2} + p_{2} q_{1} - p_{3} q_{4} - p_{4} q_{3} + q_{1} q_{2} - q_{3} q_{4}) < 0,

(56)

there is a single root of $R_{0}^{2} / R_{0}^{2} R_{1}^{2} R_{1}^{2}$ $(0 < R_{0}^{2} / R_{0}^{2} R_{1}^{2} R_{1}^{2} < 1)$ . It is deduced from Equations (16), (39) and (50) that

\frac{4 μ_{2} D_{2} (1 + ν_{2}^{D}) N_{11}^{\infty} - 4 B_{2} μ_{2} M_{11}^{\infty}}{4 μ_{2} M_{11}^{\infty} - B_{2} (κ_{2}^{A} - 1) N_{11}^{\infty}} = \frac{U_{13} U_{21} - U_{11} U_{23}}{U_{33} U_{21} - U_{23} U_{31} + R_{1}^{2} (U_{13} U_{31} - U_{11} U_{33})},

(57)

which implies that the remote hydrostatic bending moments and membrane stress resultants cannot be imposed arbitrarily and should satisfy the above restriction in order to ensure that there is no stress resultant disturbance in the surrounding matrix.

As a check, when all the three phases are homogeneous in the thickness direction and the mid-planes of the three phases are on the same plane, we have $B_{0} = B_{1} = B_{2} = 0$ and ${\hat{h}}_{1} = {\hat{h}}_{2} = {\hat{h}}_{3}$ . Consequently, Equation (52) reduces to

\begin{matrix} (\frac{R_{0}^{2}}{R_{1}^{2}} - \frac{[2 μ_{0} + (κ_{0}^{A} - 1) μ_{1}] [(κ_{2}^{A} - 1) μ_{1} - (κ_{1}^{A} - 1) μ_{2}]}{[2 μ_{2} + (κ_{2}^{A} - 1) μ_{1}] [(κ_{0}^{A} - 1) μ_{1} - (κ_{1}^{A} - 1) μ_{0}]}) \\ \times (\frac{R_{0}^{2}}{R_{1}^{2}} - \frac{[(1 + ν_{0}^{D}) D_{0} + (1 - ν_{1}^{D}) D_{1}] [(1 + ν_{1}^{D}) D_{1} - (1 + ν_{2}^{D}) D_{2}]}{[(1 - ν_{1}^{D}) D_{1} + (1 + ν_{2}^{D}) D_{2}] [(1 + ν_{1}^{D}) D_{1} - (1 + ν_{0}^{D}) D_{0}]}) = 0, \end{matrix}

(58)

which permits a root of $R_{0}^{2} / R_{0}^{2} R_{1}^{2} R_{1}^{2}$ if

\begin{matrix} [(κ_{2}^{A} - 1) μ_{1} - (κ_{1}^{A} - 1) μ_{2}] [(κ_{0}^{A} - 1) μ_{2} - (κ_{2}^{A} - 1) μ_{0}] \\ \times [(1 + ν_{1}^{D}) D_{1} - (1 + ν_{2}^{D}) D_{2}] [(1 + ν_{2}^{D}) D_{2} - (1 + ν_{0}^{D}) D_{0}] < 0 . \end{matrix}

(59)

If the three-phase composite is subjected to mechanical loading characterized by F = I = 0, we have to calculate the first, fourth, fifth and eighth columns of the 8×8 real product matrix S. For example:

\begin{matrix} S_{11} = \frac{(μ_{0} + κ_{0}^{A} μ_{1}) (μ_{1} + κ_{1}^{A} μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + κ_{0}^{A} (B_{0} - B_{1})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ - \frac{3 R_{0}^{4}}{R_{1}^{4}} [\frac{(μ_{0} - μ_{1}) (μ_{1} - μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + B_{1} - B_{0}] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}] \\ + \frac{R_{0}^{6}}{R_{1}^{6}} [\frac{(μ_{1} - μ_{2}) [(κ_{1}^{A} + 3) μ_{0} - (κ_{0}^{A} + 3) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} + 3) (B_{1} - B_{0})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}], \end{matrix}

(60a)

\begin{matrix} S_{21} = \frac{R_{1}^{6} (μ_{0} + κ_{0}^{A} μ_{1}) (κ_{2}^{A} μ_{1} - κ_{1}^{A} μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{R_{1}^{6} [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + κ_{0}^{A} (B_{0} - B_{1})] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ - \frac{3 R_{1}^{2} R_{0}^{4} (μ_{0} - μ_{1}) (μ_{2} + κ_{2}^{A} μ_{1})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} - \frac{3 R_{1}^{2} R_{0}^{4} [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + B_{1} - B_{0}] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{6} (μ_{2} + κ_{2}^{A} μ_{1}) [(κ_{1}^{A} + 3) μ_{0} - (κ_{0}^{A} + 3) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{6} [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [4 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + (κ_{0}^{A} + 3) (B_{1} - B_{0})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(60b)

\begin{matrix} S_{31} = \frac{3 R_{1}^{4} (μ_{0} + κ_{0}^{A} μ_{1}) (μ_{1} - μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{3 R_{1}^{4} [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + κ_{0}^{A} (B_{0} - B_{1})] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{3 R_{1}^{2} R_{0}^{2} (μ_{1} - μ_{2}) [(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{3 R_{1}^{2} R_{0}^{2} [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) - (κ_{0}^{A} - 1) (B_{0} - B_{1})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{3 R_{0}^{4} (μ_{0} - μ_{1}) (μ_{1} + κ_{1}^{A} μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{3 R_{0}^{4} [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + B_{1} - B_{0}] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(60c)

\begin{matrix} S_{41} = \frac{3 R_{1}^{2} (μ_{0} + κ_{0}^{A} μ_{1}) (μ_{2} + κ_{2}^{A} μ_{1})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{3 R_{1}^{2} [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + κ_{0}^{A} (B_{0} - B_{1})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{3 R_{0}^{2} (μ_{2} + κ_{2}^{A} μ_{1}) [(κ_{1}^{A} - 1) μ_{0} - (κ_{0}^{A} - 1) μ_{1}]}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{3 R_{0}^{2} [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [4 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) - (κ_{0}^{A} - 1) (B_{0} - B_{1})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} \\ + \frac{3 R_{0}^{4} (μ_{0} - μ_{1}) (κ_{2}^{A} μ_{1} - κ_{1}^{A} μ_{2})}{R_{1}^{2} μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{3 R_{0}^{4} [2 μ_{0} ({\hat{h}}_{0} - {\hat{h}}_{1}) + B_{1} - B_{0}] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{8 R_{1}^{2} μ_{0} D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(60d)

S_{14} = \frac{R_{0}^{2} (R_{1}^{2} - R_{0}^{2})}{R_{1}^{6}} [\frac{(μ_{1} - μ_{0}) (μ_{1} - μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[2 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}],

(61a)

S_{24} = R_{0}^{2} (R_{1}^{2} - R_{0}^{2}) [\frac{(μ_{1} - μ_{0}) (μ_{2} + κ_{2}^{A} μ_{1})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{[2 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}],

(61b)

\begin{matrix} S_{34} = \frac{R_{0}^{2} (μ_{0} - μ_{1}) (μ_{1} + κ_{1}^{A} μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{R_{1}^{2} (μ_{1} - μ_{2}) (μ_{1} + κ_{1}^{A} μ_{0})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{(R_{1}^{2} - R_{0}^{2}) [2 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}] [2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(61c)

\begin{matrix} S_{44} = \frac{(μ_{1} + κ_{1}^{A} μ_{0}) (μ_{2} + κ_{2}^{A} μ_{1})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{(μ_{0} - μ_{1}) (κ_{2}^{A} μ_{1} - κ_{1}^{A} μ_{2})}{μ_{0} μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{1}^{2} - R_{0}^{2}}{R_{1}^{2}} \frac{[2 μ_{0} ({\hat{h}}_{1} - {\hat{h}}_{0}) + B_{0} - B_{1}] [2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})]}{8 μ_{0} D_{1} (κ_{2}^{A} + 1)} . \end{matrix}

(61d)

The other expressions of S_ij are suppressed here. When $B_{0} = B_{1} = B_{2}$ and ${\hat{h}}_{1} = {\hat{h}}_{2} = {\hat{h}}_{3}$ , Equation (48a,b) reduces to Equation (4.1) of Ru [1] and Equations (60) and (61) reduce to Equations (4.3) and (4.4) of Ru [1].

In the context of isotropic plane elasticity, Ru [1] designed a three-phase “harmonic elastic inclusion” that does not cause any disturbance to the mean stress $(σ_{11} + σ_{22})$ in the surrounding matrix. Here an extension of this concept in the context of isotropic laminated plates is that: a harmonic elastic inclusion does not disturb $(N_{11} + N_{22})$ and $(M_{11} + M_{22})$ when inserted into a uniformly loaded matrix. In order to keep the two sums $(N_{11} + N_{22})$ and $(M_{11} + M_{22})$ undisturbed in the matrix, we must have $F_{1} = I_{1} = 0$ . Consequently, it follows from Equations (44) and (45) that

[\begin{matrix} S_{31} {\overset{⌢}{S}}_{41} + S_{34} {\overset{⌢}{S}}_{44} + S_{35} {\overset{⌢}{S}}_{45} + S_{38} {\overset{⌢}{S}}_{48} & S_{31} {\overset{⌢}{S}}_{81} + S_{34} {\overset{⌢}{S}}_{84} + S_{35} {\overset{⌢}{S}}_{85} + S_{38} {\overset{⌢}{S}}_{88} \\ S_{71} {\overset{⌢}{S}}_{41} + S_{74} {\overset{⌢}{S}}_{44} + S_{75} {\overset{⌢}{S}}_{45} + S_{78} {\overset{⌢}{S}}_{48} & S_{71} {\overset{⌢}{S}}_{81} + S_{74} {\overset{⌢}{S}}_{84} + S_{75} {\overset{⌢}{S}}_{85} + S_{78} {\overset{⌢}{S}}_{88} \end{matrix}] [\begin{matrix} G \\ J \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}],

(62)

where ${\overset{⌢}{S}}_{i j} (i = 4, 8, j = 1, 4, 5, 8)$ are defined as

\begin{array}{l} {\overset{⌢}{S}}_{41} = S_{14} (S_{58} S_{85} - S_{55} S_{88}) + S_{54} (S_{15} S_{88} - S_{18} S_{85}) + S_{84} (S_{18} S_{55} - S_{15} S_{58}), \\ {\overset{⌢}{S}}_{44} = S_{11} (S_{55} S_{88} - S_{58} S_{85}) + S_{51} (S_{18} S_{85} - S_{15} S_{88}) + S_{81} (S_{15} S_{58} - S_{18} S_{55}), \\ {\overset{⌢}{S}}_{45} = S_{11} (S_{58} S_{84} - S_{54} S_{88}) + S_{51} (S_{14} S_{88} - S_{18} S_{84}) + S_{81} (S_{18} S_{54} - S_{14} S_{58}), \\ {\overset{⌢}{S}}_{48} = S_{11} (S_{54} S_{85} - S_{55} S_{84}) + S_{51} (S_{15} S_{84} - S_{14} S_{85}) + S_{81} (S_{14} S_{55} - S_{15} S_{54}), \\ {\overset{⌢}{S}}_{81} = S_{14} (S_{48} S_{55} - S_{45} S_{58}) + S_{44} (S_{15} S_{58} - S_{18} S_{55}) + S_{54} (S_{18} S_{45} - S_{15} S_{48}), \\ {\overset{⌢}{S}}_{84} = S_{11} (S_{45} S_{58} - S_{48} S_{55}) + S_{41} (S_{18} S_{55} - S_{15} S_{58}) + S_{51} (S_{15} S_{48} - S_{18} S_{45}), \\ {\overset{⌢}{S}}_{85} = S_{11} (S_{48} S_{54} - S_{44} S_{58}) + S_{41} (S_{14} S_{58} - S_{18} S_{54}) + S_{51} (S_{18} S_{44} - S_{14} S_{48}), \\ {\overset{⌢}{S}}_{88} = S_{11} (S_{44} S_{55} - S_{45} S_{54}) + S_{41} (S_{15} S_{54} - S_{14} S_{55}) + S_{51} (S_{14} S_{45} - S_{15} S_{44}) . \end{array}

(63)

In order to have a non-trivial solution of ${[\begin{matrix} G & J \end{matrix}]}^{T}$ in Equation (62), we arrive at the following condition:

\begin{array}{l} (S_{31} {\overset{⌢}{S}}_{41} + S_{34} {\overset{⌢}{S}}_{44} + S_{35} {\overset{⌢}{S}}_{45} + S_{38} {\overset{⌢}{S}}_{4}) (S_{71} {\overset{⌢}{S}}_{81} + S_{74} {\overset{⌢}{S}}_{84} + S_{75} {\overset{⌢}{S}}_{85} + S_{78} {\overset{⌢}{S}}_{88}) \\ - (S_{31} {\overset{⌢}{S}}_{81} + S_{34} {\overset{⌢}{S}}_{84} + S_{35} {\overset{⌢}{S}}_{85} + S_{38} {\overset{⌢}{S}}_{88}) (S_{71} {\overset{⌢}{S}}_{41} + S_{74} {\overset{⌢}{S}}_{44} + S_{75} {\overset{⌢}{S}}_{45} + S_{78} {\overset{⌢}{S}}_{48}) = 0 . \end{array}

(64)

The condition (64) leads to an octet equation in $R_{0}^{2} / R_{0}^{2} R_{1}^{2} R_{1}^{2}$ . In addition, it is deduced from Equations (16) and (62) that

\frac{G}{J} = \frac{2 μ_{2} D_{2} (1 - ν_{2}^{D}) (N_{22}^{\infty} - N_{11}^{\infty}) + 2 B_{2} μ_{2} (M_{22}^{\infty} - M_{11}^{\infty})}{2 μ_{2} (M_{11}^{\infty} - M_{22}^{\infty}) + B_{2} (N_{11}^{\infty} - N_{22}^{\infty})} = - \frac{S_{31} {\overset{⌢}{S}}_{81} + S_{34} {\overset{⌢}{S}}_{84} + S_{35} {\overset{⌢}{S}}_{85} + S_{38} {\overset{⌢}{S}}_{88}}{S_{31} {\overset{⌢}{S}}_{41} + S_{34} {\overset{⌢}{S}}_{44} + S_{35} {\overset{⌢}{S}}_{45} + S_{38} {\overset{⌢}{S}}_{48}},

(65)

which implies that $(N_{22}^{\infty} - N_{11}^{\infty})$ and $(M_{22}^{\infty} - M_{11}^{\infty})$ are not arbitrary and should satisfy condition (65) in order to ensure that $(N_{11} + N_{22})$ and $(M_{11} + M_{22})$ are not disturbed in the surrounding matrix. It is stressed that the above result of a harmonic elastic inclusion is still valid for the general loading case when F or I is nonzero.

Finally, if the three-phase composite is only subjected to thermal loading, we have to calculate the four-dimensional vector $(g_{1}^{*} + Q^{1} g_{0}^{*})$ . This vector can be finally determined as

\begin{matrix} [g_{1}^{*} + Q^{1} g_{0}^{*}]_{1} \\ = \frac{2 μ_{2} δ_{1} + (2 μ_{2} {\hat{h}}_{2} - B_{2}) γ_{1}}{κ_{2}^{A} + 1} + \frac{[2 μ_{1} + (κ_{1}^{A} - 1) μ_{2}] [2 μ_{1} (δ_{0} - δ_{1}) + (2 μ_{1} {\hat{h}}_{1} - B_{1}) (γ_{0} - γ_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [[D_{1} (1 - ν_{1}^{D}) - B_{1} {\hat{h}}_{1}] (γ_{0} - γ_{1}) - B_{1} (δ_{0} - δ_{1})]}{2 D_{1} (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{(μ_{2} - μ_{1}) [4 μ_{1} (δ_{0} - δ_{1}) + [4 μ_{1} {\hat{h}}_{1} + B_{1} (κ_{1}^{A} - 1)] (γ_{0} - γ_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[2 μ_{2} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} γ_{0} + D_{1} (1 + ν_{1}^{D})] (γ_{0} - γ_{1})]}{2 D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(66a)

\begin{matrix} [g_{1}^{*} + Q^{1} g_{0}^{*}]_{2} \\ = - R_{1}^{2} \frac{[2 μ_{2} δ_{1} + (2 μ_{2} {\hat{h}}_{2} + B_{2} κ_{2}^{A}) γ_{1}]}{κ_{2}^{A} + 1} + R_{1}^{2} \frac{[2 κ_{2}^{A} μ_{1} - (κ_{1}^{A} - 1) μ_{2}] [2 μ_{1} (δ_{0} - δ_{1}) + (2 μ_{1} {\hat{h}}_{1} - B_{1}) (γ_{0} - γ_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ - R_{1}^{2} \frac{[2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} - D_{1} (1 - ν_{1}^{D})] (γ_{0} - γ_{1})]}{2 D_{1} (κ_{2}^{A} + 1)} \\ - R_{0}^{2} \frac{(μ_{2} + κ_{2}^{A} μ_{1}) [4 μ_{1} (δ_{0} - δ_{1}) + [4 μ_{1} {\hat{h}}_{1} + B_{1} (κ_{1}^{A} - 1)] (γ_{0} - γ_{1})]}{μ_{1} (κ_{1}^{A} + 1) (κ_{2}^{A} + 1)} \\ + R_{0}^{2} \frac{[2 μ_{2} ({\hat{h}}_{1} - {\hat{h}}_{2}) + κ_{2}^{A} (B_{1} - B_{2})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} + D_{1} (1 + ν_{1}^{D})] (γ_{0} - γ_{1})]}{2 D_{1} (κ_{2}^{A} + 1)}, \end{matrix}

(66b)

\begin{array}{l} [g_{1}^{*} + Q^{1} g_{0}^{*}]_{3} \\ = \frac{[D_{2} (1 - ν_{2}^{D}) - B_{2} {\hat{h}}_{2}] γ_{1} - B_{2} δ_{1}}{4 D_{2}} + \frac{[4 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + (κ_{1}^{A} - 1) (B_{1} - B_{2})] [2 μ_{1} (δ_{0} - δ_{1}) + (2 μ_{1} {\hat{h}}_{1} - B_{1}) (γ_{0} - γ_{1})]}{8 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ - \frac{[(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) + D_{1} (1 + ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} - D_{1} (1 - ν_{1}^{D})] (γ_{0} - γ_{1})]}{8 D_{1} D_{2}} \end{array}

(66c)

\begin{array}{l} - \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[2 μ_{1} ({\hat{h}}_{1} - {\hat{h}}_{2}) + B_{2} - B_{1}] [4 μ_{1} (δ_{0} - δ_{1}) + [4 μ_{1} {\hat{h}}_{1} + B_{1} (κ_{1}^{A} - 1)] (γ_{0} - γ_{1})]}{8 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + \frac{R_{0}^{2}}{R_{1}^{2}} \frac{[(B_{1} + B_{2}) ({\hat{h}}_{1} - {\hat{h}}_{2}) - D_{1} (1 - ν_{1}^{D}) + D_{2} (1 - ν_{2}^{D})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} + D_{1} (1 + ν_{1}^{D})] (γ_{0} - γ_{1})]}{8 D_{1} D_{2}}, \end{array}

\begin{matrix} [g_{1}^{*} + Q^{1} g_{0}^{*}]_{4} \\ = R_{1}^{2} \frac{B_{2} δ_{1} + [B_{2} {\hat{h}}_{2} + D_{2} (3 + ν_{2}^{D})] γ_{1}}{4 D_{2}} \\ + \frac{R_{1}^{2} [4 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + (κ_{1}^{A} - 1) (B_{2} - B_{1})] [2 μ_{1} (δ_{0} - δ_{1}) + (2 μ_{1} {\hat{h}}_{1} - B_{1}) (γ_{0} - γ_{1})]}{8 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ - R_{1}^{2} \frac{[(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) - D_{1} (1 + ν_{1}^{D}) + D_{2} (3 + ν_{2}^{D})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} - D_{1} (1 - ν_{1}^{D})] (γ_{0} - γ_{1})]}{8 D_{1} D_{2}} \\ - \frac{R_{0}^{2} [2 μ_{1} ({\hat{h}}_{2} - {\hat{h}}_{1}) + B_{1} - B_{2}] [4 μ_{1} (δ_{0} - δ_{1}) + [4 μ_{1} {\hat{h}}_{1} + B_{1} (κ_{1}^{A} - 1)] (γ_{0} - γ_{1})]}{8 μ_{1} D_{2} (κ_{1}^{A} + 1)} \\ + R_{0}^{2} \frac{[(B_{1} + B_{2}) ({\hat{h}}_{2} - {\hat{h}}_{1}) + D_{2} (3 + ν_{2}^{D}) + D_{1} (1 - ν_{1}^{D})] [B_{1} (δ_{0} - δ_{1}) + [B_{1} {\hat{h}}_{1} + D_{1} (1 + ν_{1}^{D})] (γ_{0} - γ_{1})]}{8 D_{1} D_{2}} . \end{matrix}

(66d)

$X_{1}$ and $V_{1}$ characterizing the hydrostatic stress resultant field in the inhomogeneity can then be uniquely determined by using Equation (41) with $\tilde{F} = - {[g_{1}^{*} + Q^{1} g_{0}^{*}]}_{1}$ and $\tilde{I} = - {[g_{1}^{*} + Q^{1} g_{0}^{*}]}_{3}$ (in the absence of mechanical loading); $G_{1}$ and $J_{1}$ characterizing the disturbance field in the matrix can be obtained by using Equation (43) as

G_{1} = U_{21} X_{1} + U_{23} V_{1} + [g_{1}^{*} + Q^{1} g_{0}^{*}]_{2}, J_{1} = U_{41} X_{1} + U_{43} V_{1} + [g_{1}^{*} + Q^{1} g_{0}^{*}]_{4} .

(67)

When $B_{0} = B_{1} = B_{2} = 0$ and ${\hat{h}}_{1} = {\hat{h}}_{2} = {\hat{h}}_{3}$ , Equation (66a) reduces to the rest of the left-hand side of Equation (4.16) of Ru [1], excluding the first term $[Q_{1} Q_{0}]_{11} X_{1}$ ; meanwhile, Equation (66b) reduces to the rest of the right-hand side of Equation (4.17) of Ru [1], excluding the first term $[Q_{1} Q_{0}]_{21} X_{1}$ .

6. Conclusions

The stress resultant fields induced by a circular inhomogeneity with N concentric circular interphase layers in an infinite isotropic laminated plate are investigated. A short-cut method is presented to obtain the stress resultant fields in the internal inhomogeneity and the surrounding matrix without resolving the coupled (N+2)-phase composite problem. With the present method, the boundary value problem is finally reduced to two coupled linear algebraic equations and four coupled linear algebraic equations, whose coefficients can be obtained by calculating the product of 4×4 transfer matrices $Q^{k} (k = 0, 1, \dots, N)$ and that of 8×8 transfer matrices $P^{k} (k = 0, 1, \dots, N)$ . Furthermore, all calculations involve only real coefficients whilst the complex variable method is employed to make the analysis elegant. A circular inhomogeneity with a single interphase layer under mechanical loading is presented to demonstrate the effectiveness of the proposed method. In particular, neutral and harmonic circular elastic inhomogeneities with a single interphase layer have been successfully designed.

Footnotes

Acknowledgements

We are grateful to Prof. Peter Schiavone for helpful discussions.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No: 11272121) and the Innovation Program of Shanghai Municipal Education Commission, China (Grant No: 12ZZ058).

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