On the Elastic Field of a Shpherical Inhomogeneity with an Imperfectly Bonded Interface

Journal of Elasticity 46: 91–113, 1997. 91c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

On the Elastic Field of a Spherical Inhomogeneitywith an Imperfectly Bonded Interface

Z. ZHONG? and S. A. MEGUIDEngineering Mechanics and Design Laboratory, Department of Mechanical Engineering,University of Toronto, Toronto, Ontario, Canada M5S 3G8.e-mail: [email protected]

Received 3 June 1996; in revised form 28 January 1997

Abstract. This study is devoted to the development of a unified and explicit elastic solution to theproblem of a spherical inhomogeneity with an imperfectly bonded interface. Both tangential andnormal displacement discontinuities at the interface are considered and a linear interfacial condi-tion, which assumes that the tangential and the normal displacement jumps are proportional to theassociated tractions, is adopted. The elastic disturbance due to the presence of an imperfectly bond-ed inhomogeneity is decomposed into two parts: the first is formulated in terms of an equivalentnonuniform eigenstrain distributed over a perfectly bonded spherical inclusion, while the second isformulated in terms of an imaginary Somigliana dislocation field which models the interfacial slidingand normal separation. The exact form of the equivalent nonuniform eigenstrain and the imaginarySomigliana dislocation are fully determined in this paper.

Key words: inhomogeneity, spherical, interface, debonding, averaged properties.

1. Introduction

The presence of and the interaction between inhomogeneities play a significantrole in the mechanics and micromechanics of failure of advanced composite mate-rials. A comprehensive body of knowledge exists dealing with the inhomogeneityproblems, see Eshelby [1, 2], Willis [3, 4], Walpole [5, 6], Moschovidis and Mura[7], Mura [8, 9], Gong and Meguid [10, 11], Nemat-Nasser and Hori [12], amongothers. However, these works assumed a perfect bonding condition at the inter-face between the inhomogeneity and the matrix. Significantly much less has beenaccomplished when considering an imperfectly bonded inhomogeneity, which isthe subject of the current paper.

Contributions concerning the imperfectly bonded inhomogeneity include thework of, for example, Ghahremani [13], Mura and Furuhashi [14], Mura et al.[15], Benveniste [16], Jasiuk et al. [17], Achenbach and Zhu [18], Shibata et al.[19], Gosz et al. [20], Hashin [21], Qu [22] and Huang [23], among others. Unlikethe perfectly bonded inhomogeneity problem where the general solution of Eshelby[1] covers many specific situations, the general solution of the imperfectly bonded

? On sabbatical leave from the Department of Engineering Mechanics and Technology, TongjiUniversity, Shanghai 200092, P. R. China.

JEFF. **INTERPRINT**: PIPS Nr.:135034 MATHKAPelas1259.tex; 11/07/1997; 10:13; v.7; p.1

92 Z. ZHONG AND S. A. MEGUID

case is too complex and analytically intractable. This is why most of the availableefforts are based on either specific loading or oversimplified interfacial conditionssuch as free sliding.

In the present paper, we obtain a general solution for the imperfectly bondedspherical inhomogeneity problem under arbitrary remote loading in a unified andconsistent manner. In our formulation, both interfacial sliding and normal sepa-ration are considered. The elastic disturbance caused by the imperfectly bondedinhomogeneity is decomposed into two parts: the first is induced by an equivalentnonuniform eigenstrain distributed over a spherical inclusion with a perfectly bond-ed interface, while the second is produced by an imaginary Somigliana dislocationfield which models the interfacial sliding and normal separation. The equivalentnonuniform eigenstrain consists of constant and quadratic terms which are fullydetermined in this paper. It is shown that the imaginary Somigliana dislocation canbe used to model not only the interfacial sliding (including non-free sliding) butalso the interfacial normal separation. For the case where an interfacial sliding isaccompanied by a normal separation, we fully determine the form of the Burger’svector of the imaginary Somigliana dislocation.

This article is divided into five main sections. Following this brief introduction,we state the problem and provide a description of the basic equations. In Section3, we decompose the problem into three subproblems. In Section 4, we providea solution to the different subproblems. In Section 5, we analyze and discuss theresults. Finally, in Section 6, we conclude the paper.

2. Problem Statement and Basic Equations

Consider a spherical inhomogeneity embedded in an infinitely extended elas-tic matrix D � , subjected to an applied stress �0

ij or an applied strain "0ij at

infinity. The inhomogeneity has elastic moduli, C�

ijkl, different from those of thematrix, Cijkl. The interface between the inhomogeneity and the matrix is assumedto be imperfectly bonded, which can be modelled by a linear interfacial condi-tion. A coordinate system, with its origin located at the center of the sphericalinhomogeneity, is introduced to analyze this problem, as depicted in Figure 1.

The governing equations for this problem are

�ij;j = 0 inD; (1)

�ij = C�

ijkluk;l in; (2)

�ij = Cijkluk;l inD � ; (3)

where �ij are stresses and ui displacements, and the repeated indices imply sum-mation.

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IMPERFECTLY BONDED INTERFACE 93

Figure 1. A schematic of the spherical inhomogeneity problem.

If we consider the case of elastic isotropy and assume that the tangential andthe normal displacement jumps are proportional to the associated tractions, we canwrite the interfacial condition as

[�ij ]nj = 0; (4)

[ui](�ik � nink) = �Tk; (5)

[ui]nink = �Nk; (6)

where [�] =(out)�(in), ni is the outward unit normal on the interface, Ti =�kjnj(�ik�nink) andNi = �kjnknjni represents the shear and the normal tractionat the interface, respectively. �ik denotes the Kronecker �. � and � denote thecompliance in the tangential and the normal directions of the interface, respectively.According to our definition of [ui], � and � should be positive. Equations (4)–(6)imply that the interfacial traction remains continuous, while both the normal andthe tangential displacements experience a jump across the interface. It can be seenthat � and � characterize the interfacial behavior. For example, the case where� = 0 and � = 0 corresponds to a perfectly bonded interface, while for thecase � = 0 and � 6= 0, only interfacial sliding takes place with normal contactremaining intact. Furthermore, the case where � = 0 and � ! 1 represents thefree sliding interface. This linear interfacial condition, which has been employed bymany researchers [16, 21, 22] corresponds, in essence, to modelling the imperfectlybonded interface by a linear spring-layer of vanishing thickness, see Hashin [21].

The remote boundary condition is given by

ui = "0ijxj ; x !1 (7)

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Figure 2. Decomposition of the problem: (a) a schematic of the general equivalent inclusionmethod for an imperfectly bonded inhomogeneity and (b) three sub-problems examined.

or

�ij = �0ij; x !1; (8)

where xi is the coordinate of the point x. The boundary conditions (7) and (8) canbe treated in a similar manner. Therefore, we only take the boundary condition (7)in our following analysis.

Appendix 1 shows that a unique solution can be found which satisfies equations(1) to (6) subject to the boundary condition (7) or (8).

3. Decomposition of the Problem

As in the case of the perfectly bonded inhomogeneity problem, the stress dis-turbance due to the presence of the imperfectly bonded inhomogeneity can besimulated by that of an eigenstrain "�ij distributed over an inclusion with the sameshape and orientation in the identical infinite matrix, see Figure 2(a). This approachis called the equivalent inclusion method (For further details, see Eshelby [1], Mura[8], Nemat-Nasser and Hori [12], among others).

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Using the equivalent inclusion method, we obtain the following equivalentequation:

�ij = C�

ijkluk;l = Cijkl(uk;l � "�

kl) in; (9)

where "�ij is the equivalent eigenstrain defined only in . In contrast to the perfectlybonded inhomogeneity problem, neither the equivalent eigenstrain nor the stressis uniform inside the imperfectly bonded inhomogeneity. Therefore, Eshelby’ssolution is not valid for the imperfectly bonded inhomogeneity problem.

The imperfectly bonded inhomogeneity problem (we call it the original problemin the sequel) is therefore decomposed into three sub-problems (see Figure 2(b)).These sub-problems are solved separately and then superimposed to provide thegeneral solution of the original problem.

In sub-problem I, an infinite and homogeneous medium D is subjected to theremote boundary condition (7). Thus, a linear displacement field uIi and a uniformstress field �Iij are distributed overD, i.e.,

uIi = "

0ijxj ; (10)

�Iij = Cijkl"

0kl: (11)

In sub-problem II, a nonuniform eigenstrain "�ij is given inside a perfectlybonded inclusion in an infinite and homogeneous medium D with a vanishingdisplacement boundary condition at infinity. In this case, the displacement uIIi andthe stress �IIij can be written as (see Mura [8])

uIIi (x) = �

Z

Cmnkl"�

kl(x0)Gim;n(x� x0) dV 0

; (12)

�IIij (x) = �Cijmn

�Z

Cpqkl"�

kl(x0)Gmp;qn(x� x0)dV 0 + "

�

mn(x)�; (13)

where dV 0 = dx0

1dx0

2dx0

3, and Gij(x) is the Green’s function of elasticity forinfinite medium.

In sub-problem III, an equivalent Somigliana dislocation is introduced to modelthe interfacial sliding and normal separation at the interface, whose Burger’s vectoris defined as

bi = �[ui]: (14)

This dislocation field is given by Volterra’s solution, i.e.,

uIIIi (x) =

Z@Cmnklbl(x

0)nk(x0)Gim;n(x� x0) dS0

: (15)

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Following Asaro [24], we extend bn to the interior of , and as a result (15) can beexpressed in another form by using Gauss’ theorem:

uIIIi (x) = �

Z

Cmnkl"��

kl (x0)Gim;n(x� x0) dV 0 + bi(x); (16)

with

"��

kl = �(bk;l + bl;k)=2: (17)

The corresponding stress is

�IIIij = �Cijmn

�Z

Cpqkl"��

kl (x0)Gmp;qn(x� x0) dV 0 + "

��

mn(x)�: (18)

Therefore, the solution of the original problem can be expressed as the sum ofthe three sub-problems I, II and III

ui = uIi + u

IIi + u

IIIi ; (19)

�ij = �Iij + �

IIij + �

IIIij : (20)

The interfacial traction continuity condition (4) is satisfied automatically, whereasthe nonuniform eigenstrain "�ij for sub-problem II and the Burger’s vector bi forsub-problem III need to be determined by the equivalent equation (9) and theinterfacial conditions (5) and (6).

4. Solution of Sub-problems

4.1. SOLUTION OF SUB-PROBLEM II

In sub-problem II, a nonuniform eigenstrain "�ij is given inside a perfectly bondedspherical inclusion in an infinite and homogeneous medium D with a vanishingdisplacement boundary condition at infinity. Asaro and Barnett [25] have studiedthis problem, but they only discussed the general nature of the solution and didnot present the explicit expressions of the displacement and stress fields. In thissection, the elastic solution for this sub-problem will be explicitly obtained usingthe following assumed form of "�ij :

"�

ij = a2�1�ij + a

2Pij +Qilxlxj +Qjlxlxi +Mklxkxl�ij +Nijjxj2; (21)

where jxj = (xixi)1=2; a is the radius of the spherical inhomogeneity, �1 is a scalar

and Pij , Qij , Mij and Nij are symmetric deviatoric tensors with the requirementthat Pll = Qll =Mll = Nll = 0.

For isotropic materials, the elastic modulus tensor Cijkl can be written as

Cijkl =2��

1� 2��ij�kl + �(�ik�jl + �il�jk); (22)

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where � is the shear modulus and � Poisson’s ratio. Green’s function for infiniteand isotropic medium is given by

Gij(x� x0) =�ij

4��jx� x0j�

116��(1� �)

@2

@xi@xjjx� x0j: (23)

Substituting (22) and (23) into (12), we have

uIIi =

18�(1� �)

[ IIkl;kli � 2��IIll;i � 4(1� �)�IIil;l]; (24)

�IIij =

Z

"�ij(x0) dV 0

jx� x0j; (25)

IIij =

Z

"�

ij(x0)jx� x0jdV 0

; (26)

where "�ij is given by (21).Introducing the integrals

I1 =

Z

dV 0

jx� x0j; (27)

J1 =

Z

jx� x0jdV 0

; (28)

Iij =

Z

x0

ix0

jdV0

jx� x0j; (29)

Jij =

Z

x0

ix0

jjx� x0jdV 0

; (30)

we obtain the following expressions:

�IIij = a

2(�1�ij + Pij)I1 +QilIlj +QjlIli +MklIkl�ij +NijIll; (31)

IIij = a

2(�1�ij + Pij)J1 +QilJlj +QjlJli +MklJkl�ij +NijJll: (32)

Since is a spherical domain, then

I1 = 2�

a

2�jxj2

3

!; (33)

J1 =2�3

3a4

2+ a

2jxj2 �

jxj4

10

!; (34)

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Iij = �

jxj4

35�

2a2jxj2

15+a4

3

!�ij � 2�

jxj2

7�a2

5

!xixj; (35)

Jij =2�9

jxj6

105�

3a2jxj4

35+

3a4jxj2

5+ a

6

!�ij

��

jxj4

63�

2a2jxj2

35+a4

15

!xixj ;

(36)

for point x inside , and

I1 =4�a3

3jxj; (37)

J1 =4�a3

3

jxj+

a2

5jxj

!; (38)

Iij =4�a5

15

1jxj�

a2

7jxj3

!�ij +

4�a7

35jxj5xixj; (39)

Jij =4�a5

105

7jxj+

2a2

jxj�

a4

9jxj3

!�ij

�4�a7

105

1jxj3

�a2

3jxj5

!xixj;

(40)

for point x outside .Therefore, the displacement and the stress fields for sub-problem II can be

obtained as

uIIi =

1 + �

3(1� �)�1a

2xi

+

�2

35(1� �)(Qmn �Nmn) +

1 + �

7(1� �)Mmn

�xmxnxi

+

�23� 21�35(1� �)

Qil +1 + �

7(1� �)Mil +

2(6� 7�)35(1� �)

Nil

�jxj2xl

+

�2(4� 5�)15(1� �)

Pil �7� 5�

15(1� �)Qil �

1 + �

5(1� �)Mil

�a

2xl; (41)

�IIij = �

4�(1 + �)

3(1� �)a

2�1�ij

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�2�a2�(7� 5�)

15(1� �)(Pij +Qij) +

(1 + �)

5(1� �)Mij

�

+2��

2(1� 7�)35(1� �)

Qmn �6(1 + �)

7(1� �)Mmn �

2(1� 7�)35(1� �)

Nmn

�xmxn�ij

+2��(23� 21�)35(1� �)

Qij +(1 + �)

7(1� �)Mij �

(23� 21�)35(1� �)

Nij

�jxj2

�2��

2(5� 7�)35(1� �)

(Qjm �Njm)�2(1 + �)

7(1� �)Mjm

�xmxi

�2��

2(5� 7�)35(1� �)

(Qim �Nim)�2(1 + �)

7(1� �)Mim

�xmxj ; (42)


uIIi =

1 + �

3(1� �)�1a

5 xi

jxj3

+a5

3(1� �)

"1� 2�jxj3

+3a2

5jxj5

#Pilxl

+a5

2(1� �)

1jxj5

�a2

jxj7

!Pklxkxlxi

+2a5

5(1� �)

"1� 2�3jxj3

+a2

7jxj5

#Qilxl

+a5

1� �

1

5jxj5�

a2

7jxj7

!Qklxkxlxi

�2(1 + �)a7

35(1� �)jxj5Milxl +

(1 + �)a7

7(1� �)jxj7Mklxkxlxi

+a5

1� �

"1� 2�5jxj3

+a2

7jxj5

#Nilxl

+a5

2(1� �)

3

5jxj5�

5a2

7jxj7

!Nklxkxlxi; (43)

�IIij =

2�(1 + �)

3(1� �)�1a

5��ij

jxj3�

3xixjjxj5

�

elas1259.tex; 11/07/1997; 10:13; v.7; p.9


+2�a5

3(1� �)

"1� 2�jxj3

+3a2

5jxj5

#Pij

+4�a5

5(1� �)

"1� 2�3jxj3

+3a2

7jxj5

#Qij

�4�(1 + �)a7

35(1� �)jxj5Mij +

2�a5

1� �

"1� 2�5jxj3

+3a2

7jxj5

#Nij

+�a5

1� �

"1� 2�jxj5

�a2

jxj7

#Pklxkxl�ij

��a5

1� �

"5jxj7

�7a2

jxj9

#Pklxkxlxixj

+2�a5

1� �

"1� 2�5jxj5

�a2

7jxj7

#Qklxkxl�ij

�2�a5

1� �

"1jxj7

�a2

jxj9

#Qklxkxlxixj

+2�(1 + �)a7

7(1� �)jxj7Mklxkxl�ij �

2�(1 + �)a7

(1� �)jxj9Mklxkxlxixj

+�a5

1� �

"3(1� 2�)

5jxj5�

5a2

7jxj7

#Nklxkxl�ij

��a5

1� �

"3jxj7

�5a2

jxj9

#Nklxkxlxixj

+2�a5

1� �

�

jxj5�

a2

jxj7

!(Pilxlxj + Pjlxlxi)

+4�a5

1� �

�

5jxj5�

a2

7jxj7

!(Qilxlxj +Qjlxlxi)

+4�(1 + �)a7

7(1� �)jxj7(Milxlxj +Mjlxlxi)

+2�a5

1� �

3�

5jxj5�

5a2

7jxj7

!(Nilxlxj +Njlxlxi); (44)

for point x outside .

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4.2. SOLUTION OF SUB-PROBLEM III

In sub-problem III, an equivalent Somigliana dislocation is introduced to modelthe interfacial sliding and normal separation at the interface, whose Burger’s vectorbi is assumed to take the form

bi = a2�2xi + a

2Aijxj +Bklxkxlxi; (45)

where a is the radius of the spherical inhomogeneity, �2 is a scalar, Aij and Bij

are symmetric deviatoric tensors with the requirement that All = Bll = 0. Earlierwork indicates that the imaginary Somigliana dislocation was used merely to modelfree sliding (a restrictive case of interfacial sliding), see, for example, the work ofMura and Furuhashi [14], Shibata et al. [19] and Huang [23]. In this work, we haveextended the application of the imaginary Somigliana dislocation to account forboth interfacial sliding and normal separation using the modified Burger’s vectordescription given by (45). In this section, we will present the explicit solution tothis sub-problem.

Substituting (22) and (23) into (16), we have

uIIIi =

18�(1� �)

[ IIIkl;kli � 2��IIIll;i � 4(1� �)�IIIil;l ] + bi; (46)

�IIIij =

Z

"��ij (x0) dV 0

jx� x0j; (47)

IIIij =

Z

"��

ij (x0)jx� x0jdV 0

; (48)

where bi is given by (45) for point x inside and bi = 0 for point x outside , and"��ij is given by substituting (45) into (17) as

"��

ij = �a2(�2�ij +Aij)�Bilxlxj �Bjlxlxi �Bklxkxl�ij : (49)

It can be shown that

�IIIij = �a

2(��ij +Aij)I1 �BilIlj �BjlIli �BklIkl�ij ; (50)

IIIij = �a

2(��ij +Aij)J1 �BilJlj �BjlJli �BklJkl�ij ; (51)

where I1, J1, Iij and Jij have been given in (33)–(40).By adopting a similar technique to that used in Section 4.1, we can obtain the

displacement and the stress fields for sub-problem III, as follows:

uIIIi =

2(1� 2�)3(1� �)

a2�2xi +

7� 5�15(1� �)

a2Ailxl +

4(7� 10�)35(1� �)

Bklxkxlxi

�

�4(7� 4�)35(1� �)

jxj2 �2(5� �)

15(1� �)a

2�Bilxl; (52)

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�IIIij =

4�(1 + �)

3(1� �)a

2�2�ij +

2�(7� 5�)15(1� �)

a2Aij +

24��5(1� �)

Bklxkxl�ij

�4�

35(1� �)[2(7� 4�)Bijjxj

2 + 12�Bilxlxj + 12�Bjlxlxi

�2(7� 10�)Bklxkxl�ij �73(5� �)a2

Bij ]; (53)


uIIIi = �

(1 + �)a5

3(1� �)�2xi

1jxj3

+a5

6(1� �)

�3a2

Aklxkxlxi1jxj7

�3Aklxkxlxi1jxj5

�6a2

5Ailxl

1jxj5

� 2(1� 2�)Ailxl1jxj3

#

�a5

30(1� �)

"30�a2

7Bklxkxlxi

1jxj7

+ 6Bklxkxlxi1jxj5

�12�a2

7Bilxl

1jxj5

+ 4(1� 2�)Bilxl1jxj3

#; (54)

�IIIij = �

2�(1 + �)a5

3(1� �)�2

��ij

jxj3�

3xixjjxj5

�

+�a7

1� �

�Akl

xkxl

jxj7�ij + (Ailxlxj +Ajlxlxi)

1jxj7

�7Aklxkxlxixj

jxj9

�

��a5

1� �

�(1� 2�)Akl

xkxl

jxj5�ij + (Ailxlxj +Ajlxlxi)

1jxj5

�5Aklxkxlxixj

jxj7

��

2�a7

5(1� �)

�Aij

1jxj5

�52(Ailxlxj +Ajlxlxi)

1jxj7

�

�2�(1� 2�)a5

3(1� �)

�Aij

1jxj3

�32(Ailxlxj +Ajlxlxi)

1jxj5

�

�2��a7

7(1� �)

�Bkl

xkxl

jxj7�ij + (Bilxlxj +Bjlxlxi)

1jxj7

elas1259.tex; 11/07/1997; 10:13; v.7; p.12


�7Bklxkxlxixj

jxj9

�

�2�a5

5(1� �)

�(1� 2�)Bkl

xkxl

jxj5�ij + (Bilxlxj +Bjlxlxi)

1jxj5

�5Bklxkxlxixj

jxj7

�

+4��a7

35(1� �)

�Bij

1jxj5

�52 (Bilxlxj +Bjlxlxi)

1jxj7

�

�4�(1� 2�)a5

15(1� �)

�Bij

1jxj3

�32(Bilxlxj +Bjlxlxi)

1jxj5

�; (55)

for point x outside .

4.3. SUPERPOSITION OF SUB-PROBLEMS

Although the solutions of sub-problems II and III have been given in Section 4.1 andSection 4.2, the coefficients �1, �2, Pij , Qij , Mij , Nij , Aij and Bij are unknown.In this section, we will determine these coefficients.

The equivalent equation (9) can be rewritten as

"�

ij = (D�

ijkl �Dijkl)(�Ikl + �

IIkl + �

IIIkl ) in ; (56)

whereD�

ijkl andDijkl are the respective inverses ofC�

ijkl andCijkl, i.e., the elasticcompliance tensors of the inhomogeneity and the matrix materials.

Substituting (11), (22), (42) and (53) into (56), and comparing the correspondingterms of the polynomial, one can deduce that

g11�1 + g12�2 = �

��1 � 3�2

3a2

��1 + �

1� 2�

�"

0ll; (57)

h11Pij + h12Qij + h13Mij + h15Aij + h16Bij =e0ij

a2 ; (58)

h22Qij + h23Mij + h24Nij + h26Bij = 0; (59)



with

�1 =�

�� 1;

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�2 =��

��(1 + ��)�

�

1 + �;

where e0ij is the deviatoric part of "0

ij , �� and � are the respective shear modulus

of the inhomogeneity and the matrix, �� and � are the respective Poisson’s ratioof the inhomogeneity and the matrix, while the coefficients g11, g12 and hij , (i =1; : : : ; 4; j = 1; : : : ; 6) are given in Appendix 2.

The tangential and the normal displacement discontinuities at the interface are

[ui](�ik � nink) = �a2Aikxi +Aijxixjxk; (62)

[ui]nink = �a2�xk � (Aij +Bij)xixjxk: (63)

The shear and the normal traction at the interface can be written as the sum ofthe corresponding components of the three sub-problems,

Tk = TIk + T

IIk + T

IIIk ; (64)

Nk = NIk +N

IIk +N

IIIk ; (65)

which can be calculated using the stress expressions given by (11), (42) and (53).Combining the results with (5), (6), (62) and (63), we can obtain another threeequations:

g21�1 + g22�2 =(1 + �)

3(1� 2�)"0ll

a2 ; (66)

h51Pij + h52Qij + h53Mij + h54Nij + h55Aij + h56Bij =e0ij

a2 ; (67)

h61Pij + h62Qij + h63Mij + h64Nij + h65Aij + h66Bij =e0ij

a2 ; (68)

where the coefficients g21, g22 and hij , (i = 4; 5; j = 1; : : : ; 6) are given inAppendix 2.

Therefore, �1 and �2 can be obtained by solving equations (57) and (66), whilePij , Qij , Mij , Nij , Aij and Bij can be obtained by solving equations (58)–(61),(67) and (68). These coefficients are uniquely determined, since we generally have

�1 =

�� g11 g12

g21 g22

�� 6= 0; (69)

�2 =

��

h11 h12 h13 0 h15 h16

0 h22 h23 h24 0 h26

0 h32 h33 h34 0 h36

0 h42 h43 h44 0 h46

h51 h52 h53 h54 h55 h56

h61 h62 h63 h64 h65 h66

��6= 0: (70)

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Once these coefficients are obtained, the equivalent eigenstrain "�ij and theBurger’s vector of the equivalent Somigliana dislocation bi can be easily establishedthrough (21) and (45). Accordingly, the elastic field of the original problem iscompletely and uniquely determined by superposition of the respective solutionsof sub-problems I, II and III.

Since we have proven the uniqueness of the solution (see Appendix 1) andobtained a solution which satisfies all the equations and boundary conditions(including interfacial conditions and remote loading conditions) by assuming aform of the equivalent eigenstrain and the Burger’s vector in (21) and (45), itis then reasonable to conclude that the assumed form of the eigenstrain and theBurger’s vector is exact.

It is worth noting that we deal with the linear displacement boundary condition(7) in our preceding analysis. However, if we take the uniform stress boundarycondition (8) into consideration, the analysis is similar. The results can be obtainedby simply substituting (1�2�)�0

ll=2�(1+�) for "0ll, and s0

ij=2� (s0ij is the deviatoric

part of �0ij) for e0

ij in the solution corresponding to the linear displacement boundarycondition.

Hashin [21] has obtained the respective solution of the imperfectly bondedspherical inhomogeneity under spherical symmetric loading, axisymmetric tensionand pure shear. His results can be deduced quite readily from the current generaland unified solution.

5. Analysis of Results and Discussion

For the case of a perfectly bonded interface where � = � = 0, it can be foundthat �2 = Aij = Bij = Qij = Mij = Nij = 0 and our general solutionreduces to Eshelby’s solution for spherical inhomogeneity [8]. For the case of puresliding without normal separation at the interface (� = 0), we have �2 = 0 andAij = �Bij . When the inhomogeneity has the same elastic constants as those ofthe matrix, i.e., �� = � and �� = �, we can show that �1 = Pij = Qij = Mij =Nij = 0, which means that the equivalent eigenstrain is nonexistent ("�ij = 0).

As reported by Ghahremani [13], Mura et al. [15] and Hashin [21], the stressinside the imperfectly bonded inhomogeneity is nonuniform. This is confirmedby our general solution. In fact, our solution indicates that the stress inside theimperfectly bonded inhomogeneity is nonuniform, except for two special casesdescribed below.

The first case is when e0ij = 0 and "0

ll 6= 0. In this case, we can find fromequations (58)–(61), (67) and (68) that Pij = Qij =Mij = Nij = Aij = Bij = 0,and we have

"�

ij = a2�1�ij ; (71)

bi = a2�2xi; (72)

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Figure 3. Variation of normal stress �11=�0 versus x1=a(x2 = x3 = 0) under remote uniaxial

tension, �011 = �0, with ��=� = 20.

which leads to a uniform stress field inside the inhomogeneity. This result alsoindicates that the remote volumetric-type straining only causes normal separation(without sliding) at the inhomogeneity-matrix interface.

The second case is when � = �. In this case, we can show that Qij = Mij =Nij = Bij = 0 and

"�

ij = a2�1�ij + a

2Pij ; (73)

bi = a2�2xi + a

2Aijxj; (74)

which leads to a uniform stress distribution inside the inhomogeneity.Figure 3 demonstrates the variation of the normal stress �11=�

0 along the x1-axis, under the remote uniaxial tension in which �0

11 = �0 and �022 = �0

33 = �012 =

�023 = �0

31 = 0, for different dimensionless interfacial compliances �0 and �0

(�0 = ��=a and �0 = ��=a), with ��=� = 20, �� = 0:2 and � = 0:35. Thisfigure shows that the stress is nonuniform inside the inhomogeneity (0 6 x1=a 6 1)with an imperfectly bonded interface, and that the interfacial compliance �0 and�0 have important influence on the stress distribution inside the inhomogeneity.For a pure sliding (�0 = 0), the stress increases inside the inhomogeneity with theincrease of �0.

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Figure 4. Variation of shear stress �12=�0 versus x1=a(x2 = x3 = 0) under remote pure shear,

�012 = �0, with ��=� = 20.

However, the results are quite different for the case of a sliding accompaniedby a comparable normal separation (�0 6= 0) in which the stress decreases withthe increase of �0. We can deduce that in the case of complete debonding (�0 !

1; �0 ! 1), the stress vanishes inside the inhomogeneity. We can also findfrom this figure that the stress tends to the remote applied stress �0 in the matrixwhen x1=a > 5, which means that the stress disturbance due to the presence of theimperfectly bonded inhomogeneity only influences a small local region (x1=a < 5).

Figure 4 shows the variation of the shear stress �12=�0 along the x1-axis under

the remote pure shear in which �012 = � 0 and �0

11 = �022 = �0

33 = �023 = �0

31 = 0,for different interfacial compliances �0 and �0, with ��=� = 20, �� = 0:2 and� = 0:35. It can be seen that the stress is more nonuniform inside the inhomogeneitythan in Figure 3, which means that the stress nonuniformity is greatly dependenton the remote loading type.

Figure 5 demonstrates the variation of the normal stress �11=�0 along the x1-

axis, under the same conditions as in Figure 3 except that ��=� = 1=20, whichrepresents the case where the inhomogeneity is much softer than the matrix.

In this case, it is interesting to observe that the stress distribution is almost thesame for different �0 and �0. Moreover, the stresses are nearly uniform inside theinhomogeneity.

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Figure 5. Variation of normal stress �11=�0 versus x1=a(x2 = x3 = 0) under remote uniaxial

tension, �011 = �0, with ��=� = 1=20.

Obviously, the effect of the imperfectly bonded interface is more pronouncedfor the hard inhomogeneity than for the soft inhomogeneity, since in the latter casethe inhomogeneity tends to behave like a void. In this limiting case, the influenceof interface disappears, i.e., all considered boundary conditions will yield thesame results. These observations indicate that the ratio of the shear moduli ��=�plays an important role in determining the stress distribution for this class ofproblems. Furthermore, it must be realized that the presence of a jump in thenormal displacement may give rise to interpenetration (overlap) of materials.

6. Concluding Remarks

A unified solution is obtained for the elastic field of a spherical inhomogeneity withan imperfectly bonded interface under remote uniform loading. The imperfectlybonded interface is modelled by a linear spring-layer of vanishing thickness withboth the tangential and the normal displacement discontinuities being considered.The elastic disturbance due to the presence of the inhomogeneity is decomposedinto two parts: the first is described using an equivalent nonuniform eigenstraindistributed over a spherical inclusion with a perfectly bonded interface, while thesecond is described by an imaginary Somigliana dislocation field, modelling the

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interfacial sliding and normal separation. The exact form of the equivalent nonuni-form eigenstrain and the imaginary Somigliana dislocation are fully determined inthis paper. The results show that, unlike the case of a perfectly bonded interface, thestresses are not uniform inside the spherical inhomogeneity except for two specialcases. The numerical results reveal that the local stress field is strongly influencedby the elastic mismatch between the inhomogeneity and the matrix, the interfacialcompliance and the remote loading.

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Coun-cil of Canada (NSERC), Ontario Centre for Materials Research (OCMR) andALCOA Foundation of USA. Partial support of Dr. Z. Zhong has also been pro-vided by the National Natural Science Foundation of China.

Appendix 1

THE UNIQUENESS OF THE SOLUTION

Assume that there are two solutions for displacements, u1i and u2

i , with �1ij and

�2ij being the corresponding stresses. Assuming that �ui = u1

i � u2i and ��ij =

�1ij � �2

ij , then we have

��ij;j = 0 in D; (A1.1)

��ij = C�

ijkl�uk;l in ; (A1.2)

��ij = Cijkl�uk;l in D � ; (A1.3)

[��ij]nj = 0 on @; (A1.4)

[�ui](�ik � nink) = �(�Tk) on @; (A1.5)

[�ui]nink = �(�Nk) on @; (A1.6)

and �ui = 0 at infinity for the boundary condition (7), or ��ij = 0 at infinity forthe boundary condition (8). Let us introduce a positive-definite quantity such that

I =

Z

��ij�ui;jdV +

ZD�

��ij�ui;jdV > 0 (A1.7)

If I is transformed using Gauss’ theorem to surface integrals, we can show that

I = �

Z@[�(�Ti)(�Ti) + �(�Ni)(�Ni)] dS 6 0 (A1.8)

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since � > 0 and � > 0. Combining (A1.7) and (A1.8), we have I = 0, andconclude that �1

ij = �2ij . Moreover, if the impotent terms related to the rigid-body

translation and rotation are excluded, we also have u1i = u2

i .

Appendix 2

COEFFICIENTS OF EQUATIONS (57)–(61)

g11 = �2(1 + �)

3(1� �)(�1 � 3�2)� 1

g12 =2(1 + �)

3(1� �)(�1 � 3�2)

h11 =7� 5�

15(1� �)+

1�1

h12 = �h15 =7� 5�

15(1� �)

h13 =1 + �

5(1� �)

h16 = �2(5� �)

15(1� �)

h22 =2(5� 7�)35(1� �)

+1�1

h23 = �2(1 + �)

7(1� �)

h24 = �2(5� 7�)35(1� �)

h26 =24�

35(1� �)

h32 = �h34 =2(1� 7�)35(1� �)

�1 +2(1 + �)

5(1� �)�2

h33 = �6(1 + �)

7(1� �)�1 +

2(1 + �)

1� ��2 � 1

h36 =4(7 + 11�)35(1� �)

�1 �12(1 + �)

5(1� �)�2

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h42 =23� 21�35(1� �)

h43 =1 + �

7(1� �)

h44 = �23� 21�35(1� �)

�1�1

h46 = �4(7� 4�)35(1� �)

�1 =�

�� 1

�2 =��

��(1 + ��)�

�

1 + �

COEFFICIENTS OF EQUATIONS (66)–(68)

g21 =2(1 + �)

3(1� �)

g22 = �2(1 + �)

3(1� �)�

12�0

h51 = h61 =7� 5�

15(1� �)

h52 =2(17� 7�)105(1� �)

h53 =12(1 + �)

35(1� �)

h54 =5� 7�

35(1� �)

h55 = �7� 5�

15(1� �)�

12�0

h56 = �2(35 + 11�)105(1� �)

�1

2�0

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h62 =2(5� 7�)

105(1� �)

h63 = �8(1 + �)

35(1� �)

h64 =13� 7�

35(1� �)

h65 = �7� 5�

15(1� �)�

12�0

h66 =2(7 + 19�)105(1� �)

�0 =��

a

�0 =��

a

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Documents

On the Elastic Field of a Shpherical Inhomogeneity with an Imperfectly Bonded Interface