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European Journal of Mechanics A/Solids 33 (2012) 75e81

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Neutral coated circular inclusions in finite plane elasticity of harmonic materials

Xu Wang a, Peter Schiavone b,*

a School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, ChinabDepartment of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, Alberta, Canada T6G 2G8

a r t i c l e i n f o

Article history:Received 14 September 2011Accepted 24 November 2011Available online 2 December 2011

Keywords:Neutral inclusionsCoatingHarmonic solidsHydrostatic loading

* Corresponding author.E-mail address: P.Schiavone@ualberta.ca (P. Schiav

0997-7538/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.euromechsol.2011.11.006

a b s t r a c t

We study the neutrality of circular inclusions with single or multiple coatings in the context of finiteplane elasticity. All the phases in the composite belong to a particular class of compressible hyperelasticsolids of harmonic-type. For an N-phase structure, at most 2N different remote uniform hydrostaticloading states can be found for which the stress field in the surrounding matrix is the same as fora matrix containing no inclusions. For a three-phase structure, in which the asymptotic behavior of theharmonic materials obeys a constitutive restriction proposed by Knowles and Sternberg (1975) and thematerial parameter b is constant, we identify a critical value of the coating thickness parameter, abovewhich the coated inclusion is neutral with respect to four different hydrostatic loading states, and at orbelow which the coated inclusion is neutral to only two different hydrostatic loading states.

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

The concept of ‘neutrality’ was first introduced by Mansfield(1953) who found that certain holes with membrane reinforce-ment do not disturb the original uniform stress field in the plate.Mansfield referred to these reinforced holes as ‘neutral holes’. Inthe theory of composites, Hashin (1962) found that certain coatedelastic spheres are similarly ‘neutral’ to a remote uniform hydro-static stress field. Apparently certain coated circular inclusions arealso neutral to an isotropic stress field (see, for example, Mansfield,1953; Ru, 1998; Richards, 2001). The importance of the concept ofa neutral inclusion lies not only in the design of composite struc-tures for specific purposes but perhaps more importantly in itsrelevance to the design of cloaking structures (Milton et al., 2006;Liu, 2010).

So far, themajority of the analysis leading to the identification ofneutral inclusions has been undertaken within the framework oflinearized theories (see for example, Mansfield, 1953; Ru, 1998;Milton and Serkov, 2001). Lipton and Talbot (1999) discussedneutral nonlinear spherical inclusions with highly conducting orweakly conducting imperfect interfaces. For a highly conductinginterface, they found that the sphere is neutral to a critical direct-current (dc) applied field strength. On the other hand for

one).

son SAS. All rights reserved.

aweakly conducting interface, they found that the sphere is neutralto a critical dc applied current density.

In this work we will discuss the feasibility of neutral single ormulti-coated circular inclusions in the context of finite plane elas-ticity. In our discussions the internal circular inclusion, the inter-mediate coating(s) and the external infinite matrix belong to thesame class of compressible hyperelastic harmonic materials (see,for example, Varley and Cumberbatch, 1980; Ru, 2002; Kim andSchiavone, 2009). Our analyses and calculations indicate that foran N-phase structure, in which a circular inclusion is bonded to thesurrounding matrix through (N � 2) co-axial annular coatings, thecoated inclusion is neutral to at most 2N different remote uniformhydrostatic loading states. In particular a neutral circular inclusionwith a single coating is rigorously investigated.

2. Complex variable formulation

Let the complex variable z ¼ x1 þ ix2 ¼ rexpðiqÞ be the initialcoordinates of a material particle in the undeformed configurationand wðzÞ ¼ y1ðzÞ þ iy2ðzÞ the corresponding spatial coordinates inthe deformed configuration. Define the deformation gradienttensor as

Fij ¼vyivxj

: (1)

For a particular class of harmonic materials, the strain energydensityW defined with respect to the undeformed unit area can beexpressed by

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e8176

W ¼ 2m½FðIÞ � J� ; F 0ðIÞ ¼ 14a

�I þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2 � 16ab

q �: (2)

Here I and J are the scalar invariants of FFT given by

I ¼ l1 þ l2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFijFij þ 2J

q; J ¼ l1l2 ¼ det

�Fij�; (3)

where l1 and l2 are the principal stretches, m is the shear modulusand 1/2 � a < 1, b > 0 are two material constants. This special class

a242ðzÞ þ ij2ðzÞ þb2z

402ðzÞ

¼ a343ðzÞ þ ij3ðzÞ þb3z

403ðzÞ

;

ða2 � 1Þ42ðzÞ þ ij2ðzÞ þb2z

402ðzÞ

¼ G3ða3 � 1Þ43ðzÞ þ iG3j3ðzÞ þG3b3z

403ðzÞ

;

on jzj ¼ R2 (7)

a141ðzÞ þ ij1ðzÞ þb1z

401ðzÞ

¼ a242ðzÞ þ ij2ðzÞ þb2z

402ðzÞ

;

ða1 � 1Þ41ðzÞ þ ij1ðzÞ þb1z

401ðzÞ

¼ a2 � 1G1

42

z

!þ iG1

j2ðzÞ þb2z

G1402ðzÞ

;

on jzj ¼ R1 (8)

of harmonic materials has attracted considerable attention in theliterature (see for example, Varley and Cumberbatch, 1980; Li andSteigmann, 1993; Ru, 2002; Kim and Schiavone, 2009).

According to the formulation developed recently by Ru (2002),the deformationw can bewritten in terms of two analytic functionsfðzÞ and jðzÞ as

iwðz; zÞ ¼ a4ðzÞ þ ijðzÞ þ bz

40ðzÞ; (4)

and the complex Piola stress function F is given by

Fðz; zÞ ¼ 2im�ða� 1Þ4ðzÞ þ ijðzÞ þ bz

40ðzÞ

�: (5)

42ðzÞ ¼ iz�½a3 þ G3ð1� a3Þ�X þ ð1� G3Þb3

X

�;

j2ðzÞ ¼ R22z

0BBB@

½a3ð1� a2Þ � G3a2ð1� a3Þ�X þ b3½G3a2 þ ð1� a2Þ�X

� b2X½a3 þ G3ð1� a3Þ�X2 þ ð1� G3Þb3

1CCCCA;

ðR1 < jzj < R2Þ (10)

In addition, the Piola stress components can be written in termsof the Piola stress function F as

�s21 þ is11 ¼ F;2; s22 � is12 ¼ F;1: (6)

41ðzÞ ¼ iz

½G1a2 þ ð1� a2Þ�½a3 þ G3ð1� a3Þ� þ r�1ðG1 � 1Þ½a3ð1� a2Þ � G3a2ð1� a3Þ�

G1X

þ b3ð1� G3Þ½G1a2 þ ð1� a2Þ� þ r�1b3ðG1 � 1Þ½G3a2 þ ð1� a2Þ�G1X

þ1� r�1ðG1 � 1Þb2X

G1½a3 þ G3ð1� a3Þ�X2 þ G1ð1� G3Þb3

!; ðjzj< R1Þ

(11)

3. Neutral circular inclusions with a single coating

We first consider a three-phase circular inclusion with two co-axial circular interfaces, as shown in Fig. 1. Let S1, S2 and S3denote the inclusion, the single coating (or the interphase layer)

and the matrix, respectively, all of which are perfectly bondedthrough the inner interfaces L1 of radius R1 and the outer interfaceL2 of radius R2, (R2 > R1). Throughout this paper, the subscript j orthe superscript (j) will be used to identify the associated quantitiesin Sj.

The perfect interface conditions across the two interfacesjzj ¼ R1 and jzj ¼ R2 can be expressed in terms of the analyticfunctions 4jðzÞ and jj(z), (j ¼ 1 � 3) as

where G3 ¼ m3/m2 and G1 ¼ m1/m2.In order tomake the coated circular inclusion neutral to a remote

uniformhydrostatic stressfield, the two analytic functions43ðzÞ andj3(z) defined in the matrix should take the following form

f3ðzÞ ¼ iXz; j3ðzÞ ¼ 0; ðjzj>R2Þ (9)

where X is a real number.By enforcing the perfect interface conditions on the outer

circular interface jzj ¼ R2, we arrive at the following expressions of42ðzÞ and j2(z) defined in the intermediate interphase layer

Consequently by enforcing the perfect interface conditions onthe inner circular interface jzj ¼ R1, we arrive at the followingexpressions of 41ðzÞ and j1(z) defined in the circular inclusion

G1j1ðzÞ ¼�½G1a2ð1� a1Þ � a1ð1� a2Þ�

�½a3 þ G3ð1� a3Þ�X þ ð1� G3Þb3

X

�þ ½a1 þ G1ð1� a1Þ�b2X½a3 þ G3ð1� a3Þ�X2 þ ð1� G3Þb3

�R21z

þr�1½a1 þ G1ð1� a1Þ��½a3ð1� a2Þ � G3a2ð1� a3Þ�X þ b3½G3a2 þ ð1� a2Þ�

X� b2X½a3 þ G3ð1� a3Þ�X2 þ ð1� G3Þb3

�R21z

� G21b1�

½G1a2 þ ð1� a2Þ��½a3 þ G3ð1� a3Þ�X þ ð1� G3Þb3

X

�þ ðG1 � 1Þb2X½a3 þ G3ð1� a3Þ�X2 þ ð1� G3Þb3

þ r�1ðG1 � 1Þ�½a3ð1� a2Þ

�G3a2ð1� a3Þ�X þ b3½G3a2 þ ð1� a2Þ�X

� b2X½a3 þ G3ð1� a3Þ�X2 þ ð1� G3Þb3

��R21z; ðjzj< R1Þ (12)

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e81 77

where r ¼ R21=R22, (0 � r � 1) is a measure of the relative thickness

of the coating.It follows from Eq. (12) that j1(z) h 0 in view of the fact that the

analytic function j1(z) must be regular in its region of definition:jzj< R1. Consequently the following quartic equation in X2 can beobtained:

c4X8 þ c3X

6 þ c2X4 þ c1X

2 þ c0 ¼ 0; (13)

where the five real coefficients c4, c3, c2, c1 and c0 are explicitly givenby

c3 ¼ ½a3 þ G3ð1� a3Þ�½G1a2ð1� a1Þ � a1ð1� a2Þ�½a3 þ G3ð1� a3Þ�

��b2

�1� r�1

�ðG1 � 1Þ þ 2b3ð1� G3Þ½G1a2 þ ð1� a2Þ�½a3 þ G

þr�1b3ðG1 � 1Þh2G2

3a2ð1� a3Þ þ G3ð2a2 � 1Þð2a3 � 1Þ þ 2a3ð

�½G1a2ð1� a1Þ � a1ð1� a2Þ� þ r�1b3½a1 þ G1ð1� a1Þ�h2G2

3a2

þb2

�1� r�1

�½a1 þ G1ð1� a1Þ�

��n½G1a2 þ ð1� a2Þ�½a3 þ G

þG3ð1� a3Þ�2;

c2 ¼ ½a3 þ G3ð1� a3Þ�½G1a2ð1� a1Þ � a1ð1� a2Þ�½a3 þ G3ð1� a3Þ

�b23ð1� G3Þnð1� G3Þ½G1a2 þ ð1� a2Þ� þ r�1ðG1 � 1Þ½G3a2 þ

þ�2b3ð1� G3Þ½a3 þ G3ð1� a3Þ�½G1a2ð1� a1Þ � a1ð1� a2Þ� þ

þG3ð2a2 � 1Þð2a3 � 1Þ þ 2a3ð1� a2Þiþ b2

�1� r�1

�½a1 þ G1

þr�1b3ðG1 � 1Þh2G2

3a2ð1� a3Þ þ G3ð2a2 � 1Þð2a3 � 1Þ þ 2a3

�½G1a2ð1� a1Þ � a1ð1� a2Þ� þ r�1½a1 þ G1ð1� a1Þ�½G3a2 þ ð1þr�1ðG1 � 1Þ½a3ð1� a2Þ � G3a2ð1� a3Þ�

o� 2G2

1b1b3ð1� G3Þ

c1 ¼ b23ð1� G3Þ ð1� G3Þ½G1a2 þ ð1� a2Þ� þ r�1ðG1 � 1Þ½G3a2 þ ð1

��2b3ð1� G3Þ½a3 þ G3ð1� a3Þ�½G1a2ð1� a1Þ � a1ð1� a2Þ� þ r

þG3ð2a2 � 1Þð2a3 � 1Þ þ 2a3ð1� a2Þiþ b2

�1� r�1

�½a1 þ G1

þb23ð1� G3Þnð1� G3Þ½G1a2ð1� a1Þ � a1ð1� a2Þ� þ r�1½a1 þ G

��2b3ð1� G3Þ½G1a2 þ ð1� a2Þ�½a3 þ G3ð1� a3Þ�

þr�1b3ðG1 � 1Þh2G2

3a2ð1� a3Þ þ G3ð2a2 � 1Þð2a3 � 1Þ þ 2a3ð�G2

1b1b23ð1� G3Þ2;

c4 ¼ ½a3þG3ð1�a3Þ�2 ½G1a2ð1�a1Þ�a1ð1�a2Þ�½a3þG3ð1�a3Þ��1

�

þr ½a1þG1ð1�a1Þ�½a3ð1�a2Þ�G3a2ð1�a3Þ�n

� ½G1a2þð1�a2Þ�½a3þG3ð1�a3Þ�þr�1ðG1�1Þ½a3ð1�a2Þo �G3a2ð1�a3Þ� ; (14a)

þ r�1½a1 þ G1ð1� a1Þ�½a3ð1� a2Þ � G3a2ð1� a3Þ�

3ð1� a3Þ�

1� a2Þi�

þ ½a3 þ G3ð1� a3Þ��2b3ð1� G3Þ½a3 þ G3ð1� a3Þ�

ð1� a3Þ þ G3ð2a2 � 1Þð2a3 � 1Þ þ 2a3ð1� a2Þi

3ð1� a3Þ� þ r�1ðG1 � 1Þ½a3ð1� a2Þ � G3a2ð1� a3Þ�o

� G21b1½a3(14b)

� þ r�1½a1 þ G1ð1� a1Þ�½a3ð1� a2Þ � G3a2ð1� a3Þ�

ð1� a2Þ�o

r�1b3½a1 þ G1ð1� a1Þ�h2G2

3a2ð1� a3Þð1� a1Þ�

���2b3ð1� G3Þ½G1a2 þ ð1� a2Þ�½a3 þ G3ð1� a3Þ�

ð1� a2Þiþ b2

�1� r�1

�ðG1 � 1Þ

�þ b23ð1� G3Þ

nð1� G3Þ

� a2Þ�o

� ½a3 þ G3ð1� a3Þ�n½G1a2 þ ð1� a2Þ�½a3 þ G3ð1� a3Þ�

½a3 þ G3ð1� a3Þ�; (14c)

� a2��

�1b3½a1 þ G1ð1� a1Þ�h2G2

3a2ð1� a3Þð1� a1Þ�

�1ð1� a1Þ�½G3a2 þ ð1� a2Þ�

o

1� a2Þiþ b2

�1� r�1

�ðG1 � 1Þ

�(14d)

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e8178

c0 ¼ b43ð1� G3Þ2 ð1� G3Þ½G1a2ð1� a1Þ � a1ð1� a2Þ� þ r�1½a1

þG1ð1� a1Þ�½G3a2 þ ð1� a2Þ�o�nð1� G3Þ½G1a2

þð1� a2Þ� þ r�1ðG1 � 1Þ½G3a2 þ ð1� a2Þ�o: (14e)

At most eight different real solutions of X can be found whichsatisfy Eq. (13). Physically a single-coated circular inclusion isneutral to at most eight different remote uniform hydrostaticloading states. In addition, the internal stress state within a neutralcoated circular inclusion is uniform and hydrostatic. Once permis-sible real solutions of X are determined from Eq. (13), the remoteuniform hydrostatic loading can be easily determined as

sN11 ¼ sN22 ¼ 2m3

�ð1� a3ÞX � b3

X

�; sN12 ¼ sN21 ¼ 0; (15)

and the induced internal uniform hydrostatic stress statewithin thecircular inclusion is given by

sð1Þ11 ¼ sð1Þ22 ¼ 2m1

�ð1� a1ÞA� b1

A

�; sð1Þ12 ¼ sð1Þ21 ¼ 0; (16)

where the real constant A can be expressed in terms of X as

A ¼ ½G1a2 þ ð1� a2Þ�½a3 þ G3ð1� a3Þ� þ r�1ðG1 � 1Þ½a3ð1� a2Þ � G3a2ð1� a3Þ�G1

X

þb3ð1� G3Þ½G1a2 þ ð1� a2Þ� þ r�1b3ðG1 � 1Þ½G3a2 þ ð1� a2Þ�G1X

þ1� r�1ðG1 � 1Þb2X

G1½a3 þ G3ð1� a3Þ�X2 þ G1ð1� G3Þb3:

(17)

In particular if we choose a1 ¼ a2 ¼ a3 ¼ 1=2 for the situationinwhich F 0ðIÞ=I approaches unity as I tends to infinity (Knowles andSternberg, 1975; Ru, 2002) and assume that b1 ¼ b2 ¼ b3, all theeight possible roots to Eq. (13) can be explicitly derived as

Xffiffiffiffiffiffiffiffi2b1

p ¼ �1;�1;�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

k1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21 þ k0k2

qk2ð1þ G3Þ

vuut;

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

k1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21 þ k0k2

qk2ð1þ G3Þ

vuut; (18)

where

k2 ¼ �ð1� G1ÞðG3 þ 1Þ þ r�1ðG1 þ 1ÞðG3 � 1Þ��ðG1 þ 1ÞðG3 þ 1Þ þ r�1ðG1 � 1Þð1� G3Þ�>0;

k1 ¼ �ðG1 � 1ÞðG3 þ 1Þ þ r�1ðG1 þ 1Þð1� G3Þ�hðG1 þ 1Þ

�G23 þ 4G3 þ 3

�þ 2ðG1 � 1Þ þ r�1ðG1 � 1Þ

�1� G2

3

�iþhðG1 þ 1ÞðG3 þ 1Þ þ r�1ðG1 � 1Þð1� G3Þ

ihðG1 � 1Þ

�1� G2

3

�þ 2ðG1 þ 1Þ � r�1ðG1 þ 1Þ

�1� G2

3

�i� 8G2

1ðG3 þ 1Þ;

k0 ¼hðG1 � 1ÞðG3 þ 1Þ2þr�1ðG1 þ 1Þ

�1� G2

3

�ihðG1 þ 1Þ

�G23 þ 10G3 þ 13

�þ 12ðG1 � 1Þ þ r�1ðG1 � 1Þ

�1� G2

3

�iþh2ðG1 � 1Þ

�1� G2

3

�þ 4ðG1 þ 1Þ � 2r�1ðG1 þ 1Þ

�1� G2

3

�ihðG1 þ 1Þ

�G23 þ 6G3 þ 5

�þ 4ðG1 � 1Þ þ r�1ðG1 � 1Þ

�1� G2

3

�iþhðG1 � 1Þð1� G3Þ2þr�1ðG1 þ 1Þ

�1� G2

3

�ihðG1 þ 1ÞðG3 þ 1Þ2þr�1ðG1 � 1Þ

�1� G2

3

�i� 16G2

1

�G23 þ 6G3 þ 5

�: (19)

The roots X ¼ �ffiffiffiffiffiffiffiffi2b1

plead to the trivial case corresponding to the

loading sN11 ¼ sN22 ¼ 0, thus they can be ignored. Our rigorousanalysis further indicates that the coated circular inclusion can bemade neutral only when G3 > G1 > 1 (or equivalentlym3 > m1 > m2). In addition when G3 > G1 > 1 and r > rcr,

ðwith rcr ¼ ðG1 � 1ÞðG3 þ 1ÞðG1 þ 1ÞðG3 � 1ÞÞ, there exist four real non-trivial

solutions Xj, (j ¼ 1 � 4) given by

X1ffiffiffiffiffiffiffiffi2b1

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

k1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21 þ k0k2

qk2ð1þ G3Þ

vuut>0;

X2ffiffiffiffiffiffiffiffi2b1

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

k1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21 þ k0k2

qk2ð1þ G3Þ

vuut>X1>0;

X3 ¼ �X1 < 0; X4 ¼ �X2 < X3 < 0; (20)

where X1;X3/0 as r/rcr. As r/rcr, themagnitudes of the loadingstates determined by X1 and X3 will become considerably large inview of Eq. (15).

On the other hand when G3 > G1 > 1 and r � rcr, there existonly two real non-trivial solutions X1 and X2 given by

X1ffiffiffiffiffiffiffiffi2b1

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

k1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik21 þ k0k2

qk2ð1þ G3Þ

vuut>0; X2 ¼ �X1 < 0: (21)

The above results imply that for given material parametersdescribing the composite (with the restriction that G3 > G1 > 1,a1 ¼ a2 ¼ a3 ¼ 1=2 and b1 ¼ b2 ¼ b3), there is a critical value ofthe thickness parameter rcr above which the coated inclusion isneutral to four different hydrostatic loading states, and at or belowwhich the coated inclusion is neutral to only two different hydro-static loading states.

Fig. 2 illustrates different combinations of G1 and G3 satisfying

three values of rcr ¼ ðG1 � 1ÞðG3 þ 1ÞðG1 þ 1ÞðG3 � 1Þ ¼ 0:6;0:8;0:9. Clearly, if the

chosenpair [G1,G3] lies above the curve for afixed value of r ¼ rcr asshown in Fig. 2, the coated inclusion is neutral to four different

hydrostatic loading states; if the chosenpair [G1,G3] lies on or belowthe curve for a fixed value of r ¼ rcr as shown in Fig. 2, the coatedinclusion is neutral to only two different hydrostatic loading states.

It is of interest to probe the existence of neutral inclusions in theabsence of the single coating. In this case we have G1 ¼ 1 and

Fig. 1. A circular inclusion with a single coating.

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e81 79

a1 ¼ a2; b1 ¼ b2. Now the inclusion can be of arbitrary shape (seethe discussion in Sec. 3.2 by Ru et al., 2005). Thus in order to makethe inclusion neutral, the real number X should satisfy thefollowing quadratic equation in X2

Fig. 2. Different combinations of G1 and G3 satisfying three values of

rcr ¼ ðG1 � 1ÞðG3 þ 1ÞðG1 þ 1ÞðG3 � 1Þ ¼ 0:6;0:8; 0:9.

½a3 þ G3ð1� a3Þ�½a3ð1� a2Þ � G3a2ð1� a3Þ�X4

þfb3ð1� G3Þ½a3ð1� a2Þ � G3a2ð1� a3Þ� þ b3½G3a22

þð1� a2Þ�½a3 þ G3ð1� a3Þ� � b2gX

þb23ð1� G3Þ½G3a2 þ ð1� a2Þ� ¼ 0: (22)

Below we consider the practical situation of a2 ¼ a3 ¼ 1=2.Now Eq. (22) is reduced to

�1�G2

3

�X4þ4

hb3

�1þG2

3

��2b2

iX2þ4b23

�1�G2

3

�¼ 0: (23)

There are four real non-trivial solutions of X satisfying Eq. (23) ifðG3�1Þðb3�b2Þ>0. On the other hand if ðG3�1Þðb3�b2Þ � 0,there exists no non-trivial solution of X satisfying Eq. (23). Conse-quently an uncoated arbitrary shaped inclusion will be neutral tofour remote uniform hydrostatic loading states if a2 ¼ a3 ¼ 1=2 andðG3�1Þðb3�b2Þ>0, but cannot be neutral if a2 ¼ a3 ¼ 1=2 andðG3�1Þðb3�b2Þ � 0.

4. Neutral circular inclusions with multiple coatings

In this section we consider an N-phase structure (N � 3):a circular inclusion S1 is bonded to the surrounding matrix SNthrough (N � 2) coatings S2, S3,., SN � 1, as illustrated in Fig. 3. TheN phases are numbered sequentially from “1” for the internalinclusion to “N” for the matrix. The resulting (N� 1) perfect circular

Fig. 3. A circular inclusion with (N � 2) coatings.

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e8180

interfaces L1, L2,., LN � 1 are concentric with respective radii R1, R2,., RN � 1, (R1 < R2 < .< RN � 1).

In order to make the circular inclusion with multiple coatingsneutral to remote uniform hydrostatic loading, the two analyticfunctions 4jðzÞ and jj(z) defined in phase j must take the followingforms

4jðzÞ ¼ iXjz;jjðzÞ ¼ YjR2N�1z

�1; (24)

where Xj and Yj are unknown real numbers. Consequently byenforcing the perfect interface conditions across the interface jzj ¼Rj�1 separating the inner phase (j� 1) and the outer phase j, we caneasily arrive at the following recursive relationship

Xj�1 ¼ �ajþGj1�aj

�Xjþ

bj1�Gj

Xj

þrðN�1Þðj�1Þ1�Gj

Yj;

rðN�1Þðj�1ÞYj�1 ¼ ajXjþbjXj

þrðN�1Þðj�1ÞYj�aj�1Xj�1�bj�1

Xj�1;

ðj ¼ 2;3;/;NÞ (25)

where

rðN�1Þðj�1Þ ¼ R2N�1

R2j�1

� 1;Gj ¼mj

mj�1: (26)

In addition, by enforcing XN ¼ X;YN ¼ 0 (the prescribedhydrostatic stress field in the matrix is undisturbed) and Y1 ¼ 0(the analytic function j1ðzÞ should be regular in its definitionregion jzj< R1), we can finally obtain an algebraic equation of order2N�1 in X2 for the N-phase structure

X2N�1

n¼0

cnX2n ¼ 0; (27)

where cn, (n ¼ 0,1,., 2N � 1) are (2N � 1 þ 1) real coefficients. Theabove equation suggests that a circular inclusion with (N � 2)coatings can be neutral to at most 2N different remote uniformhydrostatic loading states.

For example, we consider a circular inclusion with two coatings(N ¼ 4). We take the following material parameters:

a1 ¼ a2 ¼ a3 ¼ a4 ¼ 1=2; b1 ¼ b3 ¼ b4 ¼ 0:6;b2¼ 0:4;G2 ¼ 0:2;G3 ¼ G4 ¼ 5;

X. Wang, P. Schiavone / European Journal of Mechanics A/Solids 33 (2012) 75e81 81

and take the geometric parameters: r32 ¼ R23=R22 ¼ 1:345;

r31 ¼ R23=R21 ¼ 1:69. Our calculations indicate that the circular

inclusion with two coatings will be neutral to the following 24 ¼ 16remote hydrostatic loading states:

sN11m4

¼ sN22m4

¼ �ð0:0241;0:0608;0:0832;0:3664;

0:4032;0:4059;0:422;1:6882Þ: (28)

Finally we consider a circular inclusion with a graded coating inwhich the material property is continuously varied along the radialdirection. Because a graded coating can be approximated bya sufficiently large number of bonded thin coatings with stepwisematerial constants, the permissible values of the hydrostaticloading, for which the prescribed uniform stress field is undis-turbed in the matrix, are expected to be continuously distributed.

5. Conclusions

We have shown rigorously that single or multi-coated circularinclusions made of compressible hyperelastic harmonic solids canbe neutral with respect to certain applied uniform hydrostaticstresses. The analysis in Section 3 suggests that a circular inclusionwith a single coating under the restriction that a1 ¼ a2 ¼ a3 ¼1=2 and b1 ¼ b2 ¼ b3 can be neutral only when m3 > m1 > m2(when thematrix is stiffer than the inclusion, which is in turn stifferthan the coating). The observation of m3 > m1 > m2 is unexpected ifwe discuss the problem within the framework of linear elasticity.Our result clearly indicates the very interesting observation that inthe context of nonlinear elasticity, the neutrality of inclusions isdependent on the specific values of the applied loading. This fact isin qualitative agreement with that obtained by Lipton and Talbot(1999).

Acknowledgments

The authors would like to thank an anonymous reviewer forvaluable comments. X.W. was supported by Innovation Program ofShanghai Municipal Education Commission (No. 12ZZ058). P.S.acknowledges the support of the Natural Sciences and EngineeringResearch Council of Canada.

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