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Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding
International Journal of Engineering Science 48 (2010) 67–77
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Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding T.C.T. Ting a, P. Schiavone b,* a b
Division of Mechanics and Computation, Stanford University, Durand 262, Stanford, CA 94305, USA Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
a r t i c l e
i n f o
Article history: Received 1 March 2009 Received in revised form 20 May 2009 Accepted 26 June 2009 Available online 4 August 2009 Communicated by K.R. Rajagopal Keywords: Anisotropic Antiplane shear Inclusion Perfect bonding Imperfect bonding Uniform stress
a b s t r a c t We consider an anisotropic elastic inclusion of arbitrary shape embedded inside an infinite dissimilar anisotropic elastic medium (matrix) subjected to a uniform antiplane shear loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the micromechanical analysis of inclusion-matrix systems, the problem concerned with the design of inclusions with uniform interior stress has received much attention in the literature (see, e.g. [1–5] and the references contained therein). The main focus has been on designing the shape of the inclusion and the properties of the material interface between the inclusion and its surrounding matrix to achieve a uniform stress distribution inside the inclusion. The primary motivation for the interest in this class of problems lies in the optimal nature of an interior uniform stress field in that such a field does not give rise to stress peaks within the inclusion. It is well-known that these stress peaks are usually responsible for the mechanical failure of the inclusion-matrix system (for example, in composite mechanics [2]). The assumption of perfect bonding between an inclusion and its surrounding matrix is a convenient idealization used to simplify analyses. Consequently, researchers have developed models to describe the more realistic scenario incorporating the presence of an imperfect interface which is capable of describing interface damage (e.g. debonding, sliding or cracking) and its subsequent effect on the stress field inside the inclusion. One of the most widely-accepted models of an imperfect interface is based on the assumption that traction is continuous but displacement is discontinuous across the interface with the displacement jump assumed to be proportional in terms of a ‘spring-factor type’ interface function to the respective interface traction component [6]. This model was used in [1] to identify shapes of inclusions and corresponding interface functions which lead to a uniform interior stress field in isotropic antiplane elasticity.
* Corresponding author. Tel.: +1 780 492 3638; fax: +1 780 492 2200. E-mail addresses: [email protected] (T.C.T. Ting), [email protected] (P. Schiavone). 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.06.008
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
In this paper, we generalize the results obtained in [1] to the case of anisotropic elasticity and demonstrate significant differences which arise as a result of the anisotropy of the materials making up the inclusion and the matrix. In particular, in Section 6 we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and the matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. This is in sharp contrast to the corresponding result from isotropic elasticity [7] where it is shown that the corresponding interior stress distribution is uniform if and only if the inclusion is elliptic or the materials in the matrix and the inclusion are the same (trivial case). In Section 7, we discuss the conditions on the elastic constants necessary to achieve this result. In the case when the imperfect interface model is adopted, we find (in Section 8) that the stress distribution inside the inclusion cannot be uniform unless the inclusion is an ellipse. Finally, we remark that the stress distribution inside a circular inclusion continues to be uniform in the presence of a homogeneously imperfect interface as it does in the more restrictive case of linear isotropic elasticity [4]. In what follows, we make use of a number of well-established symbols and conventions. Thus, unless otherwise stated, we assume that the subscripts i; j; k take the values 1, 2, 3, we sum over repeated indices, x = (x1, x2) and x = (x1, x2, x3) are generic points referred to orthogonal Cartesian coordinates in R2 and R3, respectively, and we adopt that the convention that (. . .),i denotes differentiation with respect to xi. Other notation will be defined as it occurs in the paper. 2. Formulation of antiplane deformation In a fixed rectangular coordinate system xi, the stress–strain relations for an anisotropic elastic material are given by
rij ¼ C ijks uk;s ;
ð2:1Þ
C ijks ¼ C jiks ¼ C ksij ¼ C ijsk ;
ð2:2Þ
where rij represent the components of the stress tensor, ui the components of the displacement vector and the elastic stiffnesses Cijks possess the full symmetry shown in (2.2) and positive definite. We note that the third equality in (2.2) is redundant since the first two imply the third (see [8, p. 32 ] for full details). The equations of equilibrium are given by
rij;j ¼ 0:
ð2:3Þ
Consider antiplane deformations for which the displacement components ui(x1, x2, x3) are given by
u1 ¼ u2 ¼ 0;
u3 ¼ uðx1 ; x2 Þ:
ð2:4Þ
Eq. (2.1) can be written as
rij ¼ C ij31 u;1 þ C ij32 u;2 ;
ð2:5Þ
showing that the stress depends on x1 and x2 only. The equations of equilibrium (2.3) for i = 1, 2 are trivially satisfied if r11 = r12 = r22 = 0 or, by (2.5),
C 14 ¼ C 15 ¼ C 24 ¼ C 25 ¼ C 46 ¼ C 56 ¼ 0:
ð2:6Þ
In the above Cab (a, b = 1, 2, . . . , 5, 6) is the contracted notation of Cijks (see [8, p. 36]). Monoclinic materials with the symmetry plane at x3 = 0 more than satisfy (2.6) because they also possess the properties C34 = C35 = 0 that are not required in (2.6). We will thus consider special anisotropic elastic materials that satisfy the condition (2.6). The non-zero stresses r31, r32, r33 obtained from (2.5) are given by
r31 ¼ C 55 u;1 þ C 45 u;2 ; r32 ¼ C 45 u;1 þ C 44 u;2 ; r33 ¼ C 35 u;1 þ C 34 u;2 :
ð2:7Þ
Eq. (2.3) for i = 3 is
r31;1 þ r32;2 ¼ 0:
ð2:8Þ
This suggests [9] that there exists a stress function / such that
r31 ¼ /;2 ; r32 ¼ /;1 :
ð2:9Þ
Substitution of (2.9) into the first two equations in (2.7) leads to
C 55
0
C 45
1
u;1 /;1
þ
C 45
1
C 44
0
u;2 /;2
¼ 0:
ð2:10Þ
One solution of (2.10) can be taken as
u ¼ xf ðzÞ;
/ ¼ f ðzÞ;
where x is a constant, f is an arbitrary function of
ð2:11Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
z ¼ x1 þ px2
69
ð2:12Þ
and p is a constant. There is no loss of generality in choosing the coefficient of f(z) for the / in (2.11) to be unity. Substitution of (2.11) into (2.10) yields
p¼
C 45 þ il ; C 44
x¼
i
ð2:13Þ
l
in which
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 44 C 55 C 245 > 0:
ð2:14Þ
The other solution of (2.10) is the complex conjugate of (2.11). The general solution is obtained by a superposition of the two solutions. Hence, we may take the general solution as
1 1 ½if ðzÞ þ if ðzÞ ¼ Im f ðzÞ; 2l l 1 / ¼ ½f ðzÞ þ f ðzÞ ¼ Re f ðzÞ: 2
u¼
ð2:15Þ
In the above Re and Im denote the real and imaginary parts, respectively, and the over bar stands for the complex conjugate. The two equations in (2.15) can be combined into one equation as ([8, p. 71])
/ þ il u ¼ f ðzÞ:
ð2:16Þ
Before we close this section, we consider the surface traction
t3 ¼ r31 n1 þ r32 n2 ;
ð2:17Þ
where n (with components n1 and n2) is a unit outward normal vector to a boundary curve C. Let s be the arclength measured along C such that, when facing the direction of increasing s, the material lies on the right-hand side. We then have
n1 ¼
dx2 ; ds
n2 ¼
dx1 ds
ð2:18Þ
so that
t3 ¼
d/ : ds
ð2:19Þ
If C is an interface between two dissimilar materials, continuity of the surface traction t3 across C demands that the discontinuity in / across C must be a constant independent of s. In most cases it may be sufficient to assume that / is continuous across C. If C is a boundary of a void, vanishing of the surface traction t3 demands that / is a constant on C. Again, we may take / = 0 in most cases. 3. Mapping of a closed region of arbitrary shape Let C be the boundary of a closed region. We may represent C in parametric form with parameter w as (x1(w), x2(w)). Consider the mapping
z ¼ wðfÞ ¼ c0 f þ
N X
cm fm ;
N P 1;
ð3:1Þ
m¼1
where c0 and cm are constants, which maps the region outside C in the (x1, x2)-plane into a region outside a unit circle in the f-plane. There is no loss of generality in omitting the constant term in (3.1) since this term represents only a simple translation and rotation of the region outside C. The mapping is one-to-one provided that the roots of
d wðfÞ ¼ 0 df
ð3:2Þ
are inside the unit circle. On C, (3.1) is written as
x1 ðwÞ þ px2 ðwÞ ¼ wðeiw Þ ¼ c0 eiw þ
N X
cm eimw :
m¼1
If we multiply both sides of (3.3) by eikw and integrate with respect to w from zero to 2p we have
ð3:3Þ
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
c0 ¼ cm ¼
Z 2p
1 2p
½x1 ðwÞ þ px2 ðwÞeiw dw;
0
1 2p
Z 2p
ð3:4Þ ½x1 ðwÞ þ px2 ðwÞeimw dw;
m P 1:
0
In what follows, we assume that c0 and cm (m P 1) are known. We are interested in the problem of an inclusion-matrix system of dissimilar anisotropic elastic materials subjected to a uniform loading at infinity. Before we tackle this problem, we note the special cases when the inclusion is a void or a rigid body. 4. A void or a rigid inclusion subject to loading at infinity Let an infinite region with a void whose boundary is C be subject to a uniform antiplane shear Consider the solution
1 r1 31 and r32 at infinity.
/ þ il u ¼ q1 f þ q1 f1 ;
ð4:1Þ
where z and f are related by (3.1) and q1 and q1 are to be determined. For (x1, x2) at infinity, z ffi c0f so that
/ þ ilu ffi q1 z=c0 :
ð4:2Þ
In what follows, we adopt the notation that a prime and double prime denote the real and imaginary parts, respectively, of a complex quantity. For example, we write
p ¼ p0 þ ip
00
ð4:3Þ
so that p0 and p00 > 0 are, respectively, the real and imaginary parts of p. Taking the real part on the right hand side of (4.2) and using (2.9), we obtain 0 0 00 00 0 1 1 1 r1 r1 31 ¼ ðq =c 0 Þ p þ ðq =c 0 Þ p ; 32 ¼ ðq =c 0 Þ :
ð4:4Þ
From (4.4) we have
q1 ¼ ic0
1 p00 : r1 32 þ pr31
1
This gives q when Noting that
ð4:5Þ
r and r are prescribed. 1 31
1 32
Re½f ðwÞ ¼ Re½f ðwÞ;
Im½f ðwÞ ¼ Im½f ðwÞ
ð4:6Þ
we obtain from (4.1), on C,
1 þ q1 Þeiw ; /jC ¼ Re½ðq
lujC ¼ Im½ðq1 þ q1 Þeiw :
ð4:7Þ
If the region inside C is a void, /jC = 0. By (4.7),
1 : q1 ¼ q
ð4:8Þ
If the region inside C is a rigid body, ujC = 0. By (4.7),
1 : q1 ¼ q
ð4:9Þ
5. Elliptic inclusion For an anisotropic elastic material, the simplest geometry of an inclusion is an ellipse (a circular inclusion is not the simplest since, for an anisotropic elastic material, the circular region still requires a mapping – see below). In the case of an ellipse, the mapping (3.1) simplifies to
z ¼ c0 f þ c1 f1
ð5:1Þ
with the ellipse specified by
x1 ðwÞ ¼ a cos w;
x2 ðwÞ ¼ b sin w;
ð5:2Þ
where a and b are the corresponding major and minor axes. On C, (5.1) is now written as
a cos w þ pb sin w ¼ c0 eiw þ c1 eiw ¼ ðc0 þ c1 Þ cos w þ iðc0 c1 Þ sin w: Hence,
ð5:3Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
c0 ¼
1 ða ipbÞ; 2
c1 ¼
1 ða þ ipbÞ: 2
71
ð5:4Þ
The mapping is not required if c1 = 0 or p = ia/b. For a circle (a = b), we still require the mapping unless p = i, which corresponds to the case of an isotropic elastic material. The solution in the matrix outside the elliptic inclusion is given by (4.1) in which q1 is given by (4.5) while q1 is to be determined. We adopt the notation that the subscript ‘‘o” refers to the material inside the inclusion. The solution in the inclusion is assumed to be that corresponding to uniform shear stress so that
/o þ ilo uo ¼ qo zo :
ð5:5Þ
Here, qo is an unknown constant to be determined and
zo ¼ x1 þ po x2 :
ð5:6Þ
On the interface C, (4.1) and (5.5) have the expression
/jC þ ilujC ¼ q1 eiw þ q1 eiw ;
ð5:7aÞ
/o jC þ ilo uo jC ¼ qo ða cos w þ po b sin wÞ:
ð5:7bÞ
±iw
We could express e in terms of cosw and sinw or the other way around. It turns out that the algebra is simpler if we replace e±iw in terms of cosw and sinw. We therefore obtain from (5.7a,b)
/jC ¼ Re½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w;
ð5:8aÞ
lujC ¼ Im½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w;
ð5:8bÞ
/o jC ¼ Re½qo ða cos w þ po b sin wÞ;
ð5:8cÞ
lo uo jC ¼ Im½qo ða cos w þ po b sin wÞ:
ð5:8dÞ
The continuity of surface traction and displacement at the interface means that
/jC ¼ /o jC ;
ujC ¼ uo jC :
ð5:9Þ
Using (5.8) in (5.9) and equating the coefficients of cosw and sinw gives
ðq1 Þ0 þ q01 ¼ aq0o ;
ð5:10aÞ
ðq1 Þ00 þ q001 ¼ bðp0o q0o p00o q00o Þ;
ð5:10bÞ
ðq1 Þ00 þ q001 ¼ caq00o ;
ð5:11aÞ
ðq1 Þ0 q01 ¼ cbðp00o q0o þ p0o q00o Þ;
ð5:11bÞ
where
c ¼ l=lo : Elimination of
q01
ð5:12Þ between (5.10a) and (5.11b) gives 00
0
2ðq1 Þ0 ¼ ða þ cbpo Þq0o þ cbpo q00o : Elimination of
q001
ð5:13Þ
between (5.10b) and (5.11a) yields 0
00
2ðq1 Þ00 ¼ bpo q0o þ ðca þ bpo Þq00o : Eqs. (5.13) and (5.14) can be solved for
q0o q00o
¼ ¼
ð5:14Þ q0o
and
q00o .
The result is
00 0 2½ð a þ bpo Þðq1 Þ0 bpo ðq1 Þ00 =D; 0 00 2½bpo ðq1 Þ0 þ ða þ bpo Þðq1 Þ00 =D;
c
c
c
ð5:15Þ
where 00
2
o : D ¼ ca2 þ ð1 þ c2 Þabpo þ cb po p
ð5:16Þ
With q0o and q00o given by (5.15), q01 and q001 are computed from (5.10a) and (5.11a). 6. Inclusion of arbitrary shape with perfect bonding It is known that the shear stress inside a non-elliptic inclusion with perfect bonding at the interface cannot be uniform if the linearly elastic materials occupying the inclusion and its surrounding matrix are isotropic [7]. In contrast with this situation is the case when certain anisotropic materials are present in the (non-elliptic) inclusion and the matrix. In fact, in this
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
section, we will show that, under the same assumption of perfect bonding at the interface, the shear stress inside an inclusion of arbitrary shape can indeed be uniform for certain anisotropic elastic materials but only in the presence of a uniform shear stress in the matrix. The mapping corresponding to a non-elliptic inclusion is given by (3.1) in which at least one of the cm (m P 2) is non-zero. The (uniform shear stress) solution in the inclusion is again given by (5.5) in which qo is a constant to be determined. Let the solution in the matrix be given by N X
/ þ ilu ¼ q1 f þ
qm fm
ð6:1Þ
m¼1
in which q1 is given by (4.5) and qm (m P 1) are to be determined. At the interface C, (6.1) gives
/jC þ ilujC ¼ q1 eiw þ q1 eiw þ
N X
qm eimw :
ð6:2Þ
m¼2
Hence, using (4.6),
" 1
/jC ¼ Re ðq þ q1 Þe ujC ¼
iw
þ
l
# imw
qm e
m¼2
"
1
N X
1 þ q1 Þeiw þ Im ðq
N X
ð6:3aÞ
; #
qm eimw :
ð6:3bÞ
m¼2
We write (3.3) as 00
x1 ðwÞ þ px2 ðwÞ ¼ w0 þ iw ; 0
ð6:4Þ iw
00
where w and w are the real and imaginary parts of w(e ), respectively. x1(w) and x2(w) can be obtained by equating the real and imaginary parts of (6.4). The result is
x2 ðwÞ ¼
1 00 w ; p00
x1 ðwÞ ¼ w0
p0 00 w : p00
ð6:5Þ
At the interface C, (5.5) is (using (6.5)),
00 /o jC þ ilo uo jC ¼ q0o þ iqo ½w0 þ ðb þ iaÞw00 ;
ð6:6Þ
where
p0o p0 : p00
ð6:7Þ
/o jC ¼ q0o w0 Gw00 ; 1 00 0 ½qo w þ Fw00 uo jC ¼
ð6:8Þ
a¼
p00o ; p00
b¼
Hence,
lo
in which
F ¼ bq00o þ aq0o ;
G ¼ aq00o bq0o :
ð6:9Þ
Noting that
w0 ¼ Re½wðeiw Þ ¼ Im½iwðeiw Þ;
ð6:10Þ
w00 ¼ Im½wðeiw Þ ¼ Re½iwðeiw Þ; (6.8) can be written as
/o jC ¼ Re ðq0o þ iGÞwðeiw Þ ; 00 1 Im ðiqo þ FÞwðeiw Þ : uo jC ¼
ð6:11Þ
lo
Substitution of w(f) with f = eiw from (3.1) and using (4.6) leads to
(
/o jC ¼ Re
) N X ðq0o iGÞco þ ðq0o þ iGÞc1 eiw þ ðq0o þ iGÞ cm eimw ;
ð6:12aÞ
m¼2
uo jC ¼
1
lo
( ) N X 00 iw 00 00 imw : Im ðiqo FÞco þ ðiqo þ FÞc1 e þ ðiqo þ FÞ cm e m¼2
ð6:12bÞ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
73
Substitution of (6.3) and (6.12) into (5.9) yields
1 þ q1 ¼ ðq0o iGÞco þ ðq0o þ iGÞc1 ; q
ð6:13Þ
qm ¼ ðq0o þ iGÞcm ;
ð6:14Þ
ðm P 2Þ; 00
00
1 þ q1 ¼ cðiqo FÞco þ cðiqo þ FÞc1 ; q 00 ðiqo
qm ¼ c
þ FÞcm ;
ð6:15Þ
ðm P 2Þ:
ð6:16Þ
Eqs. (6.14) and (6.16) are compatible if 00
q0o þ iG ¼ cðF þ iqo Þ:
ð6:17Þ q0o ; G
q00o .
When both the inclusion and the matrix are isotropic, a = 1, b = 0 so that F ¼ ¼ Then (6.17) holds only if c = 1. This means that the materials in the matrix and the inclusion are identical. Eqs. (6.13)–(6.16) give q1 = qoco, qm = qocm (m P 1). The right hand side of (6.1), after using (3.1), simplifies to qoz. This is identical to the right hand side of (5.5) since z = zo when both materials are isotropic. Thus the stress is uniform everywhere, inside and outside the inclusion. This is a trivial case since we have a uniform material with uniform stress everywhere so that an interface of arbitrary shape is irrelevant. This again reiterates the fact that, when the materials are isotropic and perfectly bonded, the stress inside the inclusion cannot be uniform unless the inclusion is an ellipse [7] or the materials in the matrix and the inclusion are the same (trivial case). Equating the real and imaginary parts of (6.17) we have, since G and F are real,
F ¼ q0o =c;
G ¼ c q00o :
ð6:18Þ
Substitution of F and G into (6.13)–(6.16) leads to
1 þ q1 ¼ ðq0o icq00o Þco þ ðq0o þ icq00o Þc1 ; q
ð6:19Þ
qm ¼ ðq0o þ icq00o Þcm ; ðm P 2Þ; 1 þ q1 ¼ ðq0o icq00o Þco þ ðq0o þ icq00o Þc1 : q
ð6:21Þ
ð6:20Þ
Eq. (6.16) leads to (6.20). Addition and subtraction of (6.19) and (6.21) yields, respectively,
q1 ¼ ðq0o þ icq00o Þc1 ; 1 ¼ ðq0o icq00o Þco : q
ð6:22Þ ð6:23Þ
By equating the real and imaginary parts of (6.23), we can solve for q0o and q00o . In fact,
Reðco q1 Þ ; co co
q0o ¼
q00o ¼
Imðco q1 Þ : cco co
ð6:24Þ
Eqs. (6.22) and (6.20) now have the unified expression
qm ¼
cm 1 q ; co
ðm P 1Þ:
ð6:25Þ
Thus qo and qm (m P 1) are given by (6.24) and (6.25). Substitution of (6.25) into (6.1) and making use of the mapping (3.1) we obtain
/ þ il u ¼ q1 z=c0 :
ð6:26Þ
This tells us that, necessarily, the stress in the matrix must be uniform. In summary, we have proved that the shear stress in the matrix cannot be non-uniform when the shear stress inside the inclusion is uniform. However, we remark that, in contrast to the case of isotropic elasticity, the materials in the inclusion and the matrix need not be identical. In fact, if we write the two equations in (6.18) in terms of the ratio q0o =q00o , together with (6.24) we have
q0o cb a c cReðco q1 Þ ¼ ¼ ¼ : Imðco q1 Þ q00o 1 ac b
ð6:27Þ
The second and the third equalities impose the corresponding conditions on the elastic constants a, b and c. Only one of the three elastic constants can be prescribed arbitrarily. Noting that a and c are positive and non-zero, we may solve for a and c in terms of b as
a¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQ 1 þ bÞðQ bÞ;
c¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q b Q 2 ðQ 1 þ bÞ
;
ð6:28Þ
where
Q¼
Reðco q1 Þ : Imðco q1 Þ
ð6:29Þ
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
The expression inside the square root in (6.28) must be positive. It is readily seen that
Q 1 < b < Q ;
if Q > 0;
Q < b < Q 1 ;
if Q < 0:
ð6:30Þ
Thus b can be arbitrarily prescribed subject to (6.30). The constants a and c are given by (6.28). Finally, we note that the conditions (6.27) on the elastic constants depend on c0 and not on cm (m P 1). This again reinforces the fact that the uniform stress solution for the inclusion and the matrix does not depend on the geometry of the inclusion since c0 simply corresponds to a rotation of the coordinate system. 7. Further discussion of the conditions for uniform stress inside the inclusion and the matrix The conditions (6.27) on the elastic constants are necessary for uniform shear stress inside the arbitrarily-shaped inclusion and in the surrounding matrix. In this section, we use a separate derivation to further illuminate the meaning of these conditions. When the shear stresses inside the inclusion (of arbitrary shape) and surrounding matrix are uniform, the condition of perfect bonding at the interface requires that the corresponding stress and strain fields be identical. The required conditions on the elastic constants arising from just this scenario have been discussed in a more general setting (not limited to antiplane shear) in [8] (see p. 390) and [10]. In the specific case of antiplane shear, the non-zero stresses are r32, r33, r31 and the nonzero strains are e32 and e31. The strain-stress laws in terms of the elastic compliances sab are
e11 ¼ s13 r33 þ s14 r32 þ s15 r31 ¼ 0; e22 ¼ s23 r33 þ s24 r32 þ s25 r31 ¼ 0; e33 ¼ s33 r33 þ s34 r32 þ s35 r31 ¼ 0; 2e32 ¼ s43 r33 þ s44 r32 þ s45 r31 ; 2e31 ¼ s53 r33 þ s54 r32 þ s55 r31 ; 2e12 ¼ s63 r33 þ s64 r32 þ s65 r31 ¼ 0:
ð7:1Þ
Solving for r33 from the third equation and substituting into the remaining equations yields
e11 ¼ s014 r32 þ s015 r31 ¼ 0; e22 ¼ s024 r32 þ s025 r31 ¼ 0; 2e32 ¼ s044 r32 þ s045 r31 ; 2e31 ¼ s054 r32 þ s055 r31 ; 2e12 ¼ s064 r32 þ s065 r31 ¼ 0;
ð7:2Þ
sa3 s3b s33
ð7:3Þ
where
s0ab ¼ sab
are the reduced elastic compliances. The strains e11, e22 and e12 in (7.2) vanish for any r32 and r31 if
s014 ¼ s015 ¼ s024 ¼ s025 ¼ s046 ¼ s056 ¼ 0:
ð7:4Þ
This is the counterpart of (2.6) for antiplane shear. It should be noted that (7.4) can hold without requiring any one of the 21 elastic compliances sab to vanish. Applying (7.2) to the materials inside and outside the inclusion leads to, noting that e32, e31, r32 and r31 are the same in both materials,
0 ¼ ½s044 r32 þ ½s045 r31 ;
0 ¼ ½s045 r32 þ ½s055 r31 ;
ð7:5Þ
where
½s0ab ¼ s0ab ðs0ab Þ0 :
ð7:6Þ
Consider the special case in which the applied shear is r32 = 0, r31 – 0. Eq. (7.5) holds if
½s045 ¼ ½s055 ¼ 0: s045
ð7:7Þ
s055
The fact that and in the two materials are identical does not necessarily mean that s45 and s55 are identical. If we require that the stress in the inclusion and the matrix be the same for any shear loading, we must have
½s044 ¼ ½s045 ¼ ½s055 ¼ 0: In terms of sab, (7.8) is
ð7:8Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
½s44 ðs234 =s33 Þ ¼ ½s45 ðs34 s35 =s33 Þ ¼ ½s55 ðs235 =s33 Þ ¼ 0:
75
ð7:9Þ
Thus s44, s45 and s55 in the two materials need not be the same. In fact (7.9) can be satisfied without requiring any one of the 21 elastic compliances sab to be the same for the two materials (in the inclusion and the matrix). The above result applies equally to the more general case when there are several (more than one) inclusions of dissimilar materials and inclusion geometry provided (7.9) holds between any two corresponding materials. 8. Inclusion of arbitrary shape with imperfect bonding In this final section we consider the case of imperfect bonding between the inclusion and the matrix. Following [1,4], we adopt the imperfect interface model described in Section 1:
/jC ¼ /o jC ;
ð8:1Þ
d t3 jC ¼ /jC ¼ gðwÞðujC uo jC Þ; ds
ð8:2Þ
where g(w) > 0 is a real function of w. Three special cases should be noted. When g(w) = 0, the inclusion is a void (see Section 4). If g(w) is taken to be ‘infinite’, we must have ujC = uojC and the problem reduces to the case of perfect bonding discussed in Section 6. If g(w) is independent of w, the bonding is homogeneously imperfect. Otherwise, we refer to the interface as inhomogeneously imperfect. We assume that g(w) is bounded and non-zero. The solutions in the matrix and the inclusion are assumed to have the expressions given in (6.1) and (5.5) in which qo and qm (m P 1) are to be determined. Imposition of interface condition (8.1) leads to (6.13) and (6.14). It remains to consider the interface condition (8.2). On the interface, using (6.5) we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ ðdx1 Þ2 þ ðdx2 Þ2 ¼ jðwÞdw;
ð8:3Þ
where
jðwÞ ¼
1 p00
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 0 p0 w _ 00 Þ2 þ ðw _ 00 Þ2 : ðp00 w
ð8:4Þ
Here the ‘dot’ denotes differentiation with respect to w. The second equality in (8.2) is
k
d /j ¼ lðuo jC ujC Þ; dw C
ð8:5Þ
where we have taken
gðwÞ ¼
l
ð8:6Þ
kjðwÞ
and k is an arbitrary positive constant. In the special case when the interface is an ellipse and the material in the matrix is isotropic, (8.6) recovers Eq. (6) in [5] except for a proportionality factor. Noting that Re[f(w)] = Im[if(w)], substitution of (6.3a,b) and (6.12b) into (8.5) yields
" 1
kIm iðq þ q1 Þe
iw
þi
M X
# imw
mqm e
( ) N X 00 iw 00 00 imw ¼ cIm ðiqo FÞco þ ðiqo þ FÞc1 e þ ðiqo þ FÞ cm e
m¼2
" 1 þ q1 Þeiw þ Im ðq
M X
# qm eimw :
m¼2
ð8:7Þ
m¼2
Equating coefficients of eiw and eimw we have 00
00
1 þ q1 Þ ¼ c½ðiqo FÞco þ ðiqo þ FÞc1 ðq 1 þ q1 Þ; ikðq ikmqm ¼ c
00 ðiqo
þ FÞcm qm ;
ðm P 2Þ:
ð8:8Þ ð8:9Þ
Eqs. (6.13), (6.14), (8.8) and (8.9) provide the equations for qo and qm (m P 1). Elimination of qm between (6.14) and (8.9) leads to 00
ð1 ikmÞðq0o þ iGÞcm ¼ cðiqo þ FÞcm ;
ðm P 2Þ: q0o
q00o
ð8:10Þ
Since G and F are real, (8.10) holds when (i) ¼ ¼ 0 or (ii) cm = 0 (m P 2). For Case (i) it means qo = 0. By (5.5), /o = uo = 0 in the inclusion, which is not an acceptable solution. For Case (ii) the mapping (3.1) consists of only the first two terms so that the interface is an ellipse. Moreover, cm = 0 (m P 2) means that qm = 0 (m P 2) from (8.9). Thus the solution in the matrix is given by (5.7a).
76
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
In summary, the shear stress inside the inclusion cannot be uniform in the presence of imperfect bonding unless the inclusion is an ellipse. This result generalizes to this class of anisotropic materials the same result obtained in [1] for isotropic materials (in the absence of an eigenstrain inside the inclusion). We now employ (5.7a,b) or (5.8a–d), for the solution in the matrix and the inclusion. Imposition of (8.1) leads to (5.10a,b). Imposition of (8.5) gives
kRe½ðq1 þ q1 Þ sin w þ iðq1 q1 Þ cos w ¼ cIm½qo ða cos w þ po b sin wÞ Im½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w:
ð8:11Þ
Equating coefficients of cosw and sinw yields
k½ðq1 Þ00 þ q001 ¼ caq00o ðq1 Þ00 q001 ;
ð8:12aÞ
k½ðq1 Þ0 þ q01 ¼ cbðp00o q0o þ p0o q00o Þ ðq1 Þ0 þ q01 :
ð8:12bÞ
Eqs. (5.10a,b) and (8.12a,b) provide four equations for the real and imaginary parts of qo and q1. It should be noted that (8.12a,b) reduce to (5.11a,b) when k = 0. Elimination of q01 between (5.10a) and (8.12b) gives 00
0
2ðq1 Þ0 ¼ ½ð1 þ kÞa þ cbpo q0o þ cbpo q00o : Elimination of
q001
ð8:13Þ
between (5.10b) and (8.12a) yields 0
1 00
00
2ðq Þ ¼ ð1 þ kÞbpo q0o þ ½ca þ ð1 þ kÞbpo q00o :
ð8:14Þ
Eqs. (8.13) and (8.14) can be solved for q0o and q00o . The result is 00 0 b q0o ¼ 2f½ca þ ð1 þ kÞbpo ðq1 Þ0 cbpo ðq1 Þ00 g= D;
b q00o ¼ 2fð1 þ kÞbpo ðq1 Þ0 þ ½ð1 þ kÞa þ cbpo ðq1 Þ00 g= D; 0
00
ð8:15Þ
where
b ¼ ð1 þ kÞca2 þ ½ð1 þ kÞ2 þ c2 abp00 þ ð1 þ kÞcb2 p p D o o: o q0o
q00o
q01
ð8:16Þ
q001
With and given by (8.15) and are computed from (5.10a) and (5.10b). We conclude with an interesting observation. In [4], in the case of isotropic elasticity, the authors proved that, for antiplane shear deformations, under the assumption of uniform remote loading, the stress field inside a circular inclusion is uniform even when the interface is homogeneously imperfect (characterized by a constant interface function). This surprising result is in sharp contrast to the results obtained by Hashin [6] and Gao [11] for the corresponding problems in three-dimensional and plane elasticity, respectively, where, in each case, it is shown that the stress field inside the circular inclusion is intrinsically non-uniform. It is of interest to ask whether the result in [4] continues to hold in the more general case of anisotropic elasticity. In fact, for the elliptic inclusion, using (5.2) in (8.3) leads to
jðwÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b þ ða2 b Þ sin w
ð8:17Þ
so that the interface function (8.6) is given by
l
gðwÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 2 k b þ ða2 b Þ sin w
ð8:18Þ
Clearly, uniform stress inside the elliptic inclusion requires that the interface function be inhomogeneous (depends on w). However, in the case of a circular inclusion, a = b so that g(w) is independent of w and uniformity of stress inside the inclusion is again achieved in the presence of a homogeneously imperfect interface. 9. Conclusions We consider an elastic inclusion of arbitrary shape embedded inside an infinite anisotropic elastic medium (matrix) subjected to linear antiplane deformations under the assumption of uniform loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical. Acknowledgement This work is supported by the Natural Sciences and Engineering Research Council of Canada. The authors are indebted to a reviewer whose constructive comments have greatly improved the paper.
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
77
References [1] Y.A. Antipov, P. Schiavone, On the uniformity of stresses inside an inhomogeneity of arbitrary shape, IMA J. Appl. Math. 68 (2003) 299–311. [2] C.Q. Ru, P. Schiavone, A. Mioduchowski, Uniformity of the stresses within a three-phase elliptical inclusion in anti-plane shear, J. Elast. 52 (1999) 121– 128. [3] C.Q. Ru, Three-phase elliptical inclusions with internal uniform hydrostatic stresses, J. Mech. Phys. Solids 47 (1999) 259–273. [4] C.Q. Ru, P. Schiavone, A circular inclusion with circumferentially inhomogeneous interface in antiplane shear, Proc. Royal Soc. Lond. Ser. A 453 (1997) 2551–2572. [5] X. Wang, E. Pan, L.J. Sudak, Uniform stresses inside an elliptical inhomogeneity with an imperfect interface in plane elasticity, J. Appl. Mech. 75 (2008) 054501-1–054501-5. [6] Z. Hashin, The spherical inhomogeneity with imperfect interface, ASME J. Appl. Mech. 58 (1991) 444–449. [7] C.-Q. Ru, P. Schiavone, On the elliptic inclusion in anti-plane shear, Math. Mech. Solids 1 (1996) 327–333. [8] T.C.T. Ting, Anisotropic Elasticity: Theory and Applications, Oxford University Press, New York, 1996. [9] P. Chadwick, G.D. Smith, Foundations of the theory of surface waves in anisotropic elastic materials, Adv. Appl. Mech. 17 (1977) 303–376. [10] G.J. Dvorak, Y. Benveniste, On transformation strain and uniform fields in multiphase elastic media, Proc. Royal Soc. Lond. Ser. A 437 (1992) 291–310. [11] J. Gao, A circular inhomogeneity with imperfect interface: Eshelby’s tensor and related problems, J. Appl. Mech. 62 (1995) 860–866.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding T.C.T. Ting a, P. Schiavone b,* a b
Division of Mechanics and Computation, Stanford University, Durand 262, Stanford, CA 94305, USA Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
a r t i c l e
i n f o
Article history: Received 1 March 2009 Received in revised form 20 May 2009 Accepted 26 June 2009 Available online 4 August 2009 Communicated by K.R. Rajagopal Keywords: Anisotropic Antiplane shear Inclusion Perfect bonding Imperfect bonding Uniform stress
a b s t r a c t We consider an anisotropic elastic inclusion of arbitrary shape embedded inside an infinite dissimilar anisotropic elastic medium (matrix) subjected to a uniform antiplane shear loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the micromechanical analysis of inclusion-matrix systems, the problem concerned with the design of inclusions with uniform interior stress has received much attention in the literature (see, e.g. [1–5] and the references contained therein). The main focus has been on designing the shape of the inclusion and the properties of the material interface between the inclusion and its surrounding matrix to achieve a uniform stress distribution inside the inclusion. The primary motivation for the interest in this class of problems lies in the optimal nature of an interior uniform stress field in that such a field does not give rise to stress peaks within the inclusion. It is well-known that these stress peaks are usually responsible for the mechanical failure of the inclusion-matrix system (for example, in composite mechanics [2]). The assumption of perfect bonding between an inclusion and its surrounding matrix is a convenient idealization used to simplify analyses. Consequently, researchers have developed models to describe the more realistic scenario incorporating the presence of an imperfect interface which is capable of describing interface damage (e.g. debonding, sliding or cracking) and its subsequent effect on the stress field inside the inclusion. One of the most widely-accepted models of an imperfect interface is based on the assumption that traction is continuous but displacement is discontinuous across the interface with the displacement jump assumed to be proportional in terms of a ‘spring-factor type’ interface function to the respective interface traction component [6]. This model was used in [1] to identify shapes of inclusions and corresponding interface functions which lead to a uniform interior stress field in isotropic antiplane elasticity.
* Corresponding author. Tel.: +1 780 492 3638; fax: +1 780 492 2200. E-mail addresses: [email protected] (T.C.T. Ting), [email protected] (P. Schiavone). 0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.06.008
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
In this paper, we generalize the results obtained in [1] to the case of anisotropic elasticity and demonstrate significant differences which arise as a result of the anisotropy of the materials making up the inclusion and the matrix. In particular, in Section 6 we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and the matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. This is in sharp contrast to the corresponding result from isotropic elasticity [7] where it is shown that the corresponding interior stress distribution is uniform if and only if the inclusion is elliptic or the materials in the matrix and the inclusion are the same (trivial case). In Section 7, we discuss the conditions on the elastic constants necessary to achieve this result. In the case when the imperfect interface model is adopted, we find (in Section 8) that the stress distribution inside the inclusion cannot be uniform unless the inclusion is an ellipse. Finally, we remark that the stress distribution inside a circular inclusion continues to be uniform in the presence of a homogeneously imperfect interface as it does in the more restrictive case of linear isotropic elasticity [4]. In what follows, we make use of a number of well-established symbols and conventions. Thus, unless otherwise stated, we assume that the subscripts i; j; k take the values 1, 2, 3, we sum over repeated indices, x = (x1, x2) and x = (x1, x2, x3) are generic points referred to orthogonal Cartesian coordinates in R2 and R3, respectively, and we adopt that the convention that (. . .),i denotes differentiation with respect to xi. Other notation will be defined as it occurs in the paper. 2. Formulation of antiplane deformation In a fixed rectangular coordinate system xi, the stress–strain relations for an anisotropic elastic material are given by
rij ¼ C ijks uk;s ;
ð2:1Þ
C ijks ¼ C jiks ¼ C ksij ¼ C ijsk ;
ð2:2Þ
where rij represent the components of the stress tensor, ui the components of the displacement vector and the elastic stiffnesses Cijks possess the full symmetry shown in (2.2) and positive definite. We note that the third equality in (2.2) is redundant since the first two imply the third (see [8, p. 32 ] for full details). The equations of equilibrium are given by
rij;j ¼ 0:
ð2:3Þ
Consider antiplane deformations for which the displacement components ui(x1, x2, x3) are given by
u1 ¼ u2 ¼ 0;
u3 ¼ uðx1 ; x2 Þ:
ð2:4Þ
Eq. (2.1) can be written as
rij ¼ C ij31 u;1 þ C ij32 u;2 ;
ð2:5Þ
showing that the stress depends on x1 and x2 only. The equations of equilibrium (2.3) for i = 1, 2 are trivially satisfied if r11 = r12 = r22 = 0 or, by (2.5),
C 14 ¼ C 15 ¼ C 24 ¼ C 25 ¼ C 46 ¼ C 56 ¼ 0:
ð2:6Þ
In the above Cab (a, b = 1, 2, . . . , 5, 6) is the contracted notation of Cijks (see [8, p. 36]). Monoclinic materials with the symmetry plane at x3 = 0 more than satisfy (2.6) because they also possess the properties C34 = C35 = 0 that are not required in (2.6). We will thus consider special anisotropic elastic materials that satisfy the condition (2.6). The non-zero stresses r31, r32, r33 obtained from (2.5) are given by
r31 ¼ C 55 u;1 þ C 45 u;2 ; r32 ¼ C 45 u;1 þ C 44 u;2 ; r33 ¼ C 35 u;1 þ C 34 u;2 :
ð2:7Þ
Eq. (2.3) for i = 3 is
r31;1 þ r32;2 ¼ 0:
ð2:8Þ
This suggests [9] that there exists a stress function / such that
r31 ¼ /;2 ; r32 ¼ /;1 :
ð2:9Þ
Substitution of (2.9) into the first two equations in (2.7) leads to
C 55
0
C 45
1
u;1 /;1
þ
C 45
1
C 44
0
u;2 /;2
¼ 0:
ð2:10Þ
One solution of (2.10) can be taken as
u ¼ xf ðzÞ;
/ ¼ f ðzÞ;
where x is a constant, f is an arbitrary function of
ð2:11Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
z ¼ x1 þ px2
69
ð2:12Þ
and p is a constant. There is no loss of generality in choosing the coefficient of f(z) for the / in (2.11) to be unity. Substitution of (2.11) into (2.10) yields
p¼
C 45 þ il ; C 44
x¼
i
ð2:13Þ
l
in which
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 44 C 55 C 245 > 0:
ð2:14Þ
The other solution of (2.10) is the complex conjugate of (2.11). The general solution is obtained by a superposition of the two solutions. Hence, we may take the general solution as
1 1 ½if ðzÞ þ if ðzÞ ¼ Im f ðzÞ; 2l l 1 / ¼ ½f ðzÞ þ f ðzÞ ¼ Re f ðzÞ: 2
u¼
ð2:15Þ
In the above Re and Im denote the real and imaginary parts, respectively, and the over bar stands for the complex conjugate. The two equations in (2.15) can be combined into one equation as ([8, p. 71])
/ þ il u ¼ f ðzÞ:
ð2:16Þ
Before we close this section, we consider the surface traction
t3 ¼ r31 n1 þ r32 n2 ;
ð2:17Þ
where n (with components n1 and n2) is a unit outward normal vector to a boundary curve C. Let s be the arclength measured along C such that, when facing the direction of increasing s, the material lies on the right-hand side. We then have
n1 ¼
dx2 ; ds
n2 ¼
dx1 ds
ð2:18Þ
so that
t3 ¼
d/ : ds
ð2:19Þ
If C is an interface between two dissimilar materials, continuity of the surface traction t3 across C demands that the discontinuity in / across C must be a constant independent of s. In most cases it may be sufficient to assume that / is continuous across C. If C is a boundary of a void, vanishing of the surface traction t3 demands that / is a constant on C. Again, we may take / = 0 in most cases. 3. Mapping of a closed region of arbitrary shape Let C be the boundary of a closed region. We may represent C in parametric form with parameter w as (x1(w), x2(w)). Consider the mapping
z ¼ wðfÞ ¼ c0 f þ
N X
cm fm ;
N P 1;
ð3:1Þ
m¼1
where c0 and cm are constants, which maps the region outside C in the (x1, x2)-plane into a region outside a unit circle in the f-plane. There is no loss of generality in omitting the constant term in (3.1) since this term represents only a simple translation and rotation of the region outside C. The mapping is one-to-one provided that the roots of
d wðfÞ ¼ 0 df
ð3:2Þ
are inside the unit circle. On C, (3.1) is written as
x1 ðwÞ þ px2 ðwÞ ¼ wðeiw Þ ¼ c0 eiw þ
N X
cm eimw :
m¼1
If we multiply both sides of (3.3) by eikw and integrate with respect to w from zero to 2p we have
ð3:3Þ
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
c0 ¼ cm ¼
Z 2p
1 2p
½x1 ðwÞ þ px2 ðwÞeiw dw;
0
1 2p
Z 2p
ð3:4Þ ½x1 ðwÞ þ px2 ðwÞeimw dw;
m P 1:
0
In what follows, we assume that c0 and cm (m P 1) are known. We are interested in the problem of an inclusion-matrix system of dissimilar anisotropic elastic materials subjected to a uniform loading at infinity. Before we tackle this problem, we note the special cases when the inclusion is a void or a rigid body. 4. A void or a rigid inclusion subject to loading at infinity Let an infinite region with a void whose boundary is C be subject to a uniform antiplane shear Consider the solution
1 r1 31 and r32 at infinity.
/ þ il u ¼ q1 f þ q1 f1 ;
ð4:1Þ
where z and f are related by (3.1) and q1 and q1 are to be determined. For (x1, x2) at infinity, z ffi c0f so that
/ þ ilu ffi q1 z=c0 :
ð4:2Þ
In what follows, we adopt the notation that a prime and double prime denote the real and imaginary parts, respectively, of a complex quantity. For example, we write
p ¼ p0 þ ip
00
ð4:3Þ
so that p0 and p00 > 0 are, respectively, the real and imaginary parts of p. Taking the real part on the right hand side of (4.2) and using (2.9), we obtain 0 0 00 00 0 1 1 1 r1 r1 31 ¼ ðq =c 0 Þ p þ ðq =c 0 Þ p ; 32 ¼ ðq =c 0 Þ :
ð4:4Þ
From (4.4) we have
q1 ¼ ic0
1 p00 : r1 32 þ pr31
1
This gives q when Noting that
ð4:5Þ
r and r are prescribed. 1 31
1 32
Re½f ðwÞ ¼ Re½f ðwÞ;
Im½f ðwÞ ¼ Im½f ðwÞ
ð4:6Þ
we obtain from (4.1), on C,
1 þ q1 Þeiw ; /jC ¼ Re½ðq
lujC ¼ Im½ðq1 þ q1 Þeiw :
ð4:7Þ
If the region inside C is a void, /jC = 0. By (4.7),
1 : q1 ¼ q
ð4:8Þ
If the region inside C is a rigid body, ujC = 0. By (4.7),
1 : q1 ¼ q
ð4:9Þ
5. Elliptic inclusion For an anisotropic elastic material, the simplest geometry of an inclusion is an ellipse (a circular inclusion is not the simplest since, for an anisotropic elastic material, the circular region still requires a mapping – see below). In the case of an ellipse, the mapping (3.1) simplifies to
z ¼ c0 f þ c1 f1
ð5:1Þ
with the ellipse specified by
x1 ðwÞ ¼ a cos w;
x2 ðwÞ ¼ b sin w;
ð5:2Þ
where a and b are the corresponding major and minor axes. On C, (5.1) is now written as
a cos w þ pb sin w ¼ c0 eiw þ c1 eiw ¼ ðc0 þ c1 Þ cos w þ iðc0 c1 Þ sin w: Hence,
ð5:3Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
c0 ¼
1 ða ipbÞ; 2
c1 ¼
1 ða þ ipbÞ: 2
71
ð5:4Þ
The mapping is not required if c1 = 0 or p = ia/b. For a circle (a = b), we still require the mapping unless p = i, which corresponds to the case of an isotropic elastic material. The solution in the matrix outside the elliptic inclusion is given by (4.1) in which q1 is given by (4.5) while q1 is to be determined. We adopt the notation that the subscript ‘‘o” refers to the material inside the inclusion. The solution in the inclusion is assumed to be that corresponding to uniform shear stress so that
/o þ ilo uo ¼ qo zo :
ð5:5Þ
Here, qo is an unknown constant to be determined and
zo ¼ x1 þ po x2 :
ð5:6Þ
On the interface C, (4.1) and (5.5) have the expression
/jC þ ilujC ¼ q1 eiw þ q1 eiw ;
ð5:7aÞ
/o jC þ ilo uo jC ¼ qo ða cos w þ po b sin wÞ:
ð5:7bÞ
±iw
We could express e in terms of cosw and sinw or the other way around. It turns out that the algebra is simpler if we replace e±iw in terms of cosw and sinw. We therefore obtain from (5.7a,b)
/jC ¼ Re½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w;
ð5:8aÞ
lujC ¼ Im½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w;
ð5:8bÞ
/o jC ¼ Re½qo ða cos w þ po b sin wÞ;
ð5:8cÞ
lo uo jC ¼ Im½qo ða cos w þ po b sin wÞ:
ð5:8dÞ
The continuity of surface traction and displacement at the interface means that
/jC ¼ /o jC ;
ujC ¼ uo jC :
ð5:9Þ
Using (5.8) in (5.9) and equating the coefficients of cosw and sinw gives
ðq1 Þ0 þ q01 ¼ aq0o ;
ð5:10aÞ
ðq1 Þ00 þ q001 ¼ bðp0o q0o p00o q00o Þ;
ð5:10bÞ
ðq1 Þ00 þ q001 ¼ caq00o ;
ð5:11aÞ
ðq1 Þ0 q01 ¼ cbðp00o q0o þ p0o q00o Þ;
ð5:11bÞ
where
c ¼ l=lo : Elimination of
q01
ð5:12Þ between (5.10a) and (5.11b) gives 00
0
2ðq1 Þ0 ¼ ða þ cbpo Þq0o þ cbpo q00o : Elimination of
q001
ð5:13Þ
between (5.10b) and (5.11a) yields 0
00
2ðq1 Þ00 ¼ bpo q0o þ ðca þ bpo Þq00o : Eqs. (5.13) and (5.14) can be solved for
q0o q00o
¼ ¼
ð5:14Þ q0o
and
q00o .
The result is
00 0 2½ð a þ bpo Þðq1 Þ0 bpo ðq1 Þ00 =D; 0 00 2½bpo ðq1 Þ0 þ ða þ bpo Þðq1 Þ00 =D;
c
c
c
ð5:15Þ
where 00
2
o : D ¼ ca2 þ ð1 þ c2 Þabpo þ cb po p
ð5:16Þ
With q0o and q00o given by (5.15), q01 and q001 are computed from (5.10a) and (5.11a). 6. Inclusion of arbitrary shape with perfect bonding It is known that the shear stress inside a non-elliptic inclusion with perfect bonding at the interface cannot be uniform if the linearly elastic materials occupying the inclusion and its surrounding matrix are isotropic [7]. In contrast with this situation is the case when certain anisotropic materials are present in the (non-elliptic) inclusion and the matrix. In fact, in this
72
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
section, we will show that, under the same assumption of perfect bonding at the interface, the shear stress inside an inclusion of arbitrary shape can indeed be uniform for certain anisotropic elastic materials but only in the presence of a uniform shear stress in the matrix. The mapping corresponding to a non-elliptic inclusion is given by (3.1) in which at least one of the cm (m P 2) is non-zero. The (uniform shear stress) solution in the inclusion is again given by (5.5) in which qo is a constant to be determined. Let the solution in the matrix be given by N X
/ þ ilu ¼ q1 f þ
qm fm
ð6:1Þ
m¼1
in which q1 is given by (4.5) and qm (m P 1) are to be determined. At the interface C, (6.1) gives
/jC þ ilujC ¼ q1 eiw þ q1 eiw þ
N X
qm eimw :
ð6:2Þ
m¼2
Hence, using (4.6),
" 1
/jC ¼ Re ðq þ q1 Þe ujC ¼
iw
þ
l
# imw
qm e
m¼2
"
1
N X
1 þ q1 Þeiw þ Im ðq
N X
ð6:3aÞ
; #
qm eimw :
ð6:3bÞ
m¼2
We write (3.3) as 00
x1 ðwÞ þ px2 ðwÞ ¼ w0 þ iw ; 0
ð6:4Þ iw
00
where w and w are the real and imaginary parts of w(e ), respectively. x1(w) and x2(w) can be obtained by equating the real and imaginary parts of (6.4). The result is
x2 ðwÞ ¼
1 00 w ; p00
x1 ðwÞ ¼ w0
p0 00 w : p00
ð6:5Þ
At the interface C, (5.5) is (using (6.5)),
00 /o jC þ ilo uo jC ¼ q0o þ iqo ½w0 þ ðb þ iaÞw00 ;
ð6:6Þ
where
p0o p0 : p00
ð6:7Þ
/o jC ¼ q0o w0 Gw00 ; 1 00 0 ½qo w þ Fw00 uo jC ¼
ð6:8Þ
a¼
p00o ; p00
b¼
Hence,
lo
in which
F ¼ bq00o þ aq0o ;
G ¼ aq00o bq0o :
ð6:9Þ
Noting that
w0 ¼ Re½wðeiw Þ ¼ Im½iwðeiw Þ;
ð6:10Þ
w00 ¼ Im½wðeiw Þ ¼ Re½iwðeiw Þ; (6.8) can be written as
/o jC ¼ Re ðq0o þ iGÞwðeiw Þ ; 00 1 Im ðiqo þ FÞwðeiw Þ : uo jC ¼
ð6:11Þ
lo
Substitution of w(f) with f = eiw from (3.1) and using (4.6) leads to
(
/o jC ¼ Re
) N X ðq0o iGÞco þ ðq0o þ iGÞc1 eiw þ ðq0o þ iGÞ cm eimw ;
ð6:12aÞ
m¼2
uo jC ¼
1
lo
( ) N X 00 iw 00 00 imw : Im ðiqo FÞco þ ðiqo þ FÞc1 e þ ðiqo þ FÞ cm e m¼2
ð6:12bÞ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
73
Substitution of (6.3) and (6.12) into (5.9) yields
1 þ q1 ¼ ðq0o iGÞco þ ðq0o þ iGÞc1 ; q
ð6:13Þ
qm ¼ ðq0o þ iGÞcm ;
ð6:14Þ
ðm P 2Þ; 00
00
1 þ q1 ¼ cðiqo FÞco þ cðiqo þ FÞc1 ; q 00 ðiqo
qm ¼ c
þ FÞcm ;
ð6:15Þ
ðm P 2Þ:
ð6:16Þ
Eqs. (6.14) and (6.16) are compatible if 00
q0o þ iG ¼ cðF þ iqo Þ:
ð6:17Þ q0o ; G
q00o .
When both the inclusion and the matrix are isotropic, a = 1, b = 0 so that F ¼ ¼ Then (6.17) holds only if c = 1. This means that the materials in the matrix and the inclusion are identical. Eqs. (6.13)–(6.16) give q1 = qoco, qm = qocm (m P 1). The right hand side of (6.1), after using (3.1), simplifies to qoz. This is identical to the right hand side of (5.5) since z = zo when both materials are isotropic. Thus the stress is uniform everywhere, inside and outside the inclusion. This is a trivial case since we have a uniform material with uniform stress everywhere so that an interface of arbitrary shape is irrelevant. This again reiterates the fact that, when the materials are isotropic and perfectly bonded, the stress inside the inclusion cannot be uniform unless the inclusion is an ellipse [7] or the materials in the matrix and the inclusion are the same (trivial case). Equating the real and imaginary parts of (6.17) we have, since G and F are real,
F ¼ q0o =c;
G ¼ c q00o :
ð6:18Þ
Substitution of F and G into (6.13)–(6.16) leads to
1 þ q1 ¼ ðq0o icq00o Þco þ ðq0o þ icq00o Þc1 ; q
ð6:19Þ
qm ¼ ðq0o þ icq00o Þcm ; ðm P 2Þ; 1 þ q1 ¼ ðq0o icq00o Þco þ ðq0o þ icq00o Þc1 : q
ð6:21Þ
ð6:20Þ
Eq. (6.16) leads to (6.20). Addition and subtraction of (6.19) and (6.21) yields, respectively,
q1 ¼ ðq0o þ icq00o Þc1 ; 1 ¼ ðq0o icq00o Þco : q
ð6:22Þ ð6:23Þ
By equating the real and imaginary parts of (6.23), we can solve for q0o and q00o . In fact,
Reðco q1 Þ ; co co
q0o ¼
q00o ¼
Imðco q1 Þ : cco co
ð6:24Þ
Eqs. (6.22) and (6.20) now have the unified expression
qm ¼
cm 1 q ; co
ðm P 1Þ:
ð6:25Þ
Thus qo and qm (m P 1) are given by (6.24) and (6.25). Substitution of (6.25) into (6.1) and making use of the mapping (3.1) we obtain
/ þ il u ¼ q1 z=c0 :
ð6:26Þ
This tells us that, necessarily, the stress in the matrix must be uniform. In summary, we have proved that the shear stress in the matrix cannot be non-uniform when the shear stress inside the inclusion is uniform. However, we remark that, in contrast to the case of isotropic elasticity, the materials in the inclusion and the matrix need not be identical. In fact, if we write the two equations in (6.18) in terms of the ratio q0o =q00o , together with (6.24) we have
q0o cb a c cReðco q1 Þ ¼ ¼ ¼ : Imðco q1 Þ q00o 1 ac b
ð6:27Þ
The second and the third equalities impose the corresponding conditions on the elastic constants a, b and c. Only one of the three elastic constants can be prescribed arbitrarily. Noting that a and c are positive and non-zero, we may solve for a and c in terms of b as
a¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQ 1 þ bÞðQ bÞ;
c¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q b Q 2 ðQ 1 þ bÞ
;
ð6:28Þ
where
Q¼
Reðco q1 Þ : Imðco q1 Þ
ð6:29Þ
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
The expression inside the square root in (6.28) must be positive. It is readily seen that
Q 1 < b < Q ;
if Q > 0;
Q < b < Q 1 ;
if Q < 0:
ð6:30Þ
Thus b can be arbitrarily prescribed subject to (6.30). The constants a and c are given by (6.28). Finally, we note that the conditions (6.27) on the elastic constants depend on c0 and not on cm (m P 1). This again reinforces the fact that the uniform stress solution for the inclusion and the matrix does not depend on the geometry of the inclusion since c0 simply corresponds to a rotation of the coordinate system. 7. Further discussion of the conditions for uniform stress inside the inclusion and the matrix The conditions (6.27) on the elastic constants are necessary for uniform shear stress inside the arbitrarily-shaped inclusion and in the surrounding matrix. In this section, we use a separate derivation to further illuminate the meaning of these conditions. When the shear stresses inside the inclusion (of arbitrary shape) and surrounding matrix are uniform, the condition of perfect bonding at the interface requires that the corresponding stress and strain fields be identical. The required conditions on the elastic constants arising from just this scenario have been discussed in a more general setting (not limited to antiplane shear) in [8] (see p. 390) and [10]. In the specific case of antiplane shear, the non-zero stresses are r32, r33, r31 and the nonzero strains are e32 and e31. The strain-stress laws in terms of the elastic compliances sab are
e11 ¼ s13 r33 þ s14 r32 þ s15 r31 ¼ 0; e22 ¼ s23 r33 þ s24 r32 þ s25 r31 ¼ 0; e33 ¼ s33 r33 þ s34 r32 þ s35 r31 ¼ 0; 2e32 ¼ s43 r33 þ s44 r32 þ s45 r31 ; 2e31 ¼ s53 r33 þ s54 r32 þ s55 r31 ; 2e12 ¼ s63 r33 þ s64 r32 þ s65 r31 ¼ 0:
ð7:1Þ
Solving for r33 from the third equation and substituting into the remaining equations yields
e11 ¼ s014 r32 þ s015 r31 ¼ 0; e22 ¼ s024 r32 þ s025 r31 ¼ 0; 2e32 ¼ s044 r32 þ s045 r31 ; 2e31 ¼ s054 r32 þ s055 r31 ; 2e12 ¼ s064 r32 þ s065 r31 ¼ 0;
ð7:2Þ
sa3 s3b s33
ð7:3Þ
where
s0ab ¼ sab
are the reduced elastic compliances. The strains e11, e22 and e12 in (7.2) vanish for any r32 and r31 if
s014 ¼ s015 ¼ s024 ¼ s025 ¼ s046 ¼ s056 ¼ 0:
ð7:4Þ
This is the counterpart of (2.6) for antiplane shear. It should be noted that (7.4) can hold without requiring any one of the 21 elastic compliances sab to vanish. Applying (7.2) to the materials inside and outside the inclusion leads to, noting that e32, e31, r32 and r31 are the same in both materials,
0 ¼ ½s044 r32 þ ½s045 r31 ;
0 ¼ ½s045 r32 þ ½s055 r31 ;
ð7:5Þ
where
½s0ab ¼ s0ab ðs0ab Þ0 :
ð7:6Þ
Consider the special case in which the applied shear is r32 = 0, r31 – 0. Eq. (7.5) holds if
½s045 ¼ ½s055 ¼ 0: s045
ð7:7Þ
s055
The fact that and in the two materials are identical does not necessarily mean that s45 and s55 are identical. If we require that the stress in the inclusion and the matrix be the same for any shear loading, we must have
½s044 ¼ ½s045 ¼ ½s055 ¼ 0: In terms of sab, (7.8) is
ð7:8Þ
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
½s44 ðs234 =s33 Þ ¼ ½s45 ðs34 s35 =s33 Þ ¼ ½s55 ðs235 =s33 Þ ¼ 0:
75
ð7:9Þ
Thus s44, s45 and s55 in the two materials need not be the same. In fact (7.9) can be satisfied without requiring any one of the 21 elastic compliances sab to be the same for the two materials (in the inclusion and the matrix). The above result applies equally to the more general case when there are several (more than one) inclusions of dissimilar materials and inclusion geometry provided (7.9) holds between any two corresponding materials. 8. Inclusion of arbitrary shape with imperfect bonding In this final section we consider the case of imperfect bonding between the inclusion and the matrix. Following [1,4], we adopt the imperfect interface model described in Section 1:
/jC ¼ /o jC ;
ð8:1Þ
d t3 jC ¼ /jC ¼ gðwÞðujC uo jC Þ; ds
ð8:2Þ
where g(w) > 0 is a real function of w. Three special cases should be noted. When g(w) = 0, the inclusion is a void (see Section 4). If g(w) is taken to be ‘infinite’, we must have ujC = uojC and the problem reduces to the case of perfect bonding discussed in Section 6. If g(w) is independent of w, the bonding is homogeneously imperfect. Otherwise, we refer to the interface as inhomogeneously imperfect. We assume that g(w) is bounded and non-zero. The solutions in the matrix and the inclusion are assumed to have the expressions given in (6.1) and (5.5) in which qo and qm (m P 1) are to be determined. Imposition of interface condition (8.1) leads to (6.13) and (6.14). It remains to consider the interface condition (8.2). On the interface, using (6.5) we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ ðdx1 Þ2 þ ðdx2 Þ2 ¼ jðwÞdw;
ð8:3Þ
where
jðwÞ ¼
1 p00
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ 0 p0 w _ 00 Þ2 þ ðw _ 00 Þ2 : ðp00 w
ð8:4Þ
Here the ‘dot’ denotes differentiation with respect to w. The second equality in (8.2) is
k
d /j ¼ lðuo jC ujC Þ; dw C
ð8:5Þ
where we have taken
gðwÞ ¼
l
ð8:6Þ
kjðwÞ
and k is an arbitrary positive constant. In the special case when the interface is an ellipse and the material in the matrix is isotropic, (8.6) recovers Eq. (6) in [5] except for a proportionality factor. Noting that Re[f(w)] = Im[if(w)], substitution of (6.3a,b) and (6.12b) into (8.5) yields
" 1
kIm iðq þ q1 Þe
iw
þi
M X
# imw
mqm e
( ) N X 00 iw 00 00 imw ¼ cIm ðiqo FÞco þ ðiqo þ FÞc1 e þ ðiqo þ FÞ cm e
m¼2
" 1 þ q1 Þeiw þ Im ðq
M X
# qm eimw :
m¼2
ð8:7Þ
m¼2
Equating coefficients of eiw and eimw we have 00
00
1 þ q1 Þ ¼ c½ðiqo FÞco þ ðiqo þ FÞc1 ðq 1 þ q1 Þ; ikðq ikmqm ¼ c
00 ðiqo
þ FÞcm qm ;
ðm P 2Þ:
ð8:8Þ ð8:9Þ
Eqs. (6.13), (6.14), (8.8) and (8.9) provide the equations for qo and qm (m P 1). Elimination of qm between (6.14) and (8.9) leads to 00
ð1 ikmÞðq0o þ iGÞcm ¼ cðiqo þ FÞcm ;
ðm P 2Þ: q0o
q00o
ð8:10Þ
Since G and F are real, (8.10) holds when (i) ¼ ¼ 0 or (ii) cm = 0 (m P 2). For Case (i) it means qo = 0. By (5.5), /o = uo = 0 in the inclusion, which is not an acceptable solution. For Case (ii) the mapping (3.1) consists of only the first two terms so that the interface is an ellipse. Moreover, cm = 0 (m P 2) means that qm = 0 (m P 2) from (8.9). Thus the solution in the matrix is given by (5.7a).
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T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
In summary, the shear stress inside the inclusion cannot be uniform in the presence of imperfect bonding unless the inclusion is an ellipse. This result generalizes to this class of anisotropic materials the same result obtained in [1] for isotropic materials (in the absence of an eigenstrain inside the inclusion). We now employ (5.7a,b) or (5.8a–d), for the solution in the matrix and the inclusion. Imposition of (8.1) leads to (5.10a,b). Imposition of (8.5) gives
kRe½ðq1 þ q1 Þ sin w þ iðq1 q1 Þ cos w ¼ cIm½qo ða cos w þ po b sin wÞ Im½ðq1 þ q1 Þ cos w þ iðq1 q1 Þ sin w:
ð8:11Þ
Equating coefficients of cosw and sinw yields
k½ðq1 Þ00 þ q001 ¼ caq00o ðq1 Þ00 q001 ;
ð8:12aÞ
k½ðq1 Þ0 þ q01 ¼ cbðp00o q0o þ p0o q00o Þ ðq1 Þ0 þ q01 :
ð8:12bÞ
Eqs. (5.10a,b) and (8.12a,b) provide four equations for the real and imaginary parts of qo and q1. It should be noted that (8.12a,b) reduce to (5.11a,b) when k = 0. Elimination of q01 between (5.10a) and (8.12b) gives 00
0
2ðq1 Þ0 ¼ ½ð1 þ kÞa þ cbpo q0o þ cbpo q00o : Elimination of
q001
ð8:13Þ
between (5.10b) and (8.12a) yields 0
1 00
00
2ðq Þ ¼ ð1 þ kÞbpo q0o þ ½ca þ ð1 þ kÞbpo q00o :
ð8:14Þ
Eqs. (8.13) and (8.14) can be solved for q0o and q00o . The result is 00 0 b q0o ¼ 2f½ca þ ð1 þ kÞbpo ðq1 Þ0 cbpo ðq1 Þ00 g= D;
b q00o ¼ 2fð1 þ kÞbpo ðq1 Þ0 þ ½ð1 þ kÞa þ cbpo ðq1 Þ00 g= D; 0
00
ð8:15Þ
where
b ¼ ð1 þ kÞca2 þ ½ð1 þ kÞ2 þ c2 abp00 þ ð1 þ kÞcb2 p p D o o: o q0o
q00o
q01
ð8:16Þ
q001
With and given by (8.15) and are computed from (5.10a) and (5.10b). We conclude with an interesting observation. In [4], in the case of isotropic elasticity, the authors proved that, for antiplane shear deformations, under the assumption of uniform remote loading, the stress field inside a circular inclusion is uniform even when the interface is homogeneously imperfect (characterized by a constant interface function). This surprising result is in sharp contrast to the results obtained by Hashin [6] and Gao [11] for the corresponding problems in three-dimensional and plane elasticity, respectively, where, in each case, it is shown that the stress field inside the circular inclusion is intrinsically non-uniform. It is of interest to ask whether the result in [4] continues to hold in the more general case of anisotropic elasticity. In fact, for the elliptic inclusion, using (5.2) in (8.3) leads to
jðwÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 b þ ða2 b Þ sin w
ð8:17Þ
so that the interface function (8.6) is given by
l
gðwÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 2 k b þ ða2 b Þ sin w
ð8:18Þ
Clearly, uniform stress inside the elliptic inclusion requires that the interface function be inhomogeneous (depends on w). However, in the case of a circular inclusion, a = b so that g(w) is independent of w and uniformity of stress inside the inclusion is again achieved in the presence of a homogeneously imperfect interface. 9. Conclusions We consider an elastic inclusion of arbitrary shape embedded inside an infinite anisotropic elastic medium (matrix) subjected to linear antiplane deformations under the assumption of uniform loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical. Acknowledgement This work is supported by the Natural Sciences and Engineering Research Council of Canada. The authors are indebted to a reviewer whose constructive comments have greatly improved the paper.
T.C.T. Ting, P. Schiavone / International Journal of Engineering Science 48 (2010) 67–77
77
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