a screw dislocation interacting with an anisotropic elliptical nano-inhomogeneity with interface...

J ElastDOI 10.1007/s10659-014-9500-7

A Screw Dislocation Interacting with an AnisotropicElliptical Nano-Inhomogeneity with Interface Stressesin Anti-Plane Elasticity

Xu Wang · Peter Schiavone

Received: 17 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract An elegant, compact and rigorous analytical solution is derived to the problemof a screw dislocation interacting with an anisotropic elliptical elastic inhomogeneity withinterface stresses in the presence of remote uniform anti-plane shear stresses and uniformanti-plane eigenstrains imposed on the inhomogeneity. The screw dislocation can be locatedboth outside and inside the inhomogeneity. The internal inhomogeneity and the surroundingmatrix are monoclinic with the symmetry plane at x3 = 0. A modified anisotropic version ofthe Gurtin-Murdoch model of the surface/interface elasticity is incorporated into the elasticanalysis. It is clearly observed from our results that the elastic fields in the composite andimage force on the screw dislocation are size-dependent.

Keywords Anisotropic elliptical inhomogeneity · Screw dislocation · Surface elasticity ·Anisotropic elasticity · Anti-plane deformation

Mathematics Subject Classification 74B05 · 30E25

1 Introduction

Studies concerning anti-plane deformations of anisotropic elastic solids are fundamental tothe development of the theory of anisotropic elasticity [1]. In particular, the Green’s functionfor an anisotropic elliptical inhomogeneity under anti-plane deformations subjected to ananti-plane force and a screw dislocation has been rigorously and thoroughly constructed byTing [1]. In Ting’s analysis, the applied anti-plane force and screw dislocation can be located

X. WangSchool of Mechanical and Power Engineering, East China University of Science and Technology,130 Meilong Road, Shanghai 200237, Chinae-mail: [email protected]

P. Schiavone (B)Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building,Edmonton, Alberta, Canada T6G 2G8e-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

X. Wang, P. Schiavone

outside, inside or at the boundary of the elliptical inhomogeneity and the inhomogeneity istaken to be perfectly bonded to the surrounding matrix (i.e., tractions and displacements arecontinuous across the inhomogeneity-matrix interface). In the case when the elastic inho-mogeneity is nano-sized, however, it is well-known that traditional continuum models ofdeformation are inadequate to capture the significant contribution of an increasing surfacearea to volume ratio present at the nano-scale [2]. To answer this deficiency, conventionalcontinuum models have been enhanced by incorporating surface/interface elasticity into theelastic analysis of the composite structure. Among various theories of surface/interface elas-ticity, the Gurtin-Murdoch model and its modifications [3–5] have been successfully used toexplain many size-dependent phenomena at the nano-scale (see for example, [2, 6–20]). Inthe Gurtin-Murdoch model of interface elasticity [3] the displacement is assumed to be con-tinuous while the traction is taken to be discontinuous across the material interface. Morespecifically, the corresponding jumps in traction are simply the negative of the surface di-vergence of the surface stress tensor, which in turn can be expressed in terms of the surfacestrain tensor and residual surface tension [2–4, 21].

In this work, a modified anisotropic version of the Gurtin-Murdoch model of sur-face/interface elasticity is incorporated into the rigorous study of the Green’s functionsfor an anisotropic elliptical inhomogeneity subjected to anti-plane deformations arisingfrom a screw dislocation located either outside or inside the inhomogeneity. The elasticanisotropy of the bulk materials and the interface are taken into account within our theo-retical framework. In addition, uniform anti-plane stresses applied remotely and uniformanti-plane eigenstrains imposed on the inhomogeneity are also considered in our solution.We find an effective and elegant method to determine all the unknown coefficients includingthe constant terms appearing in the analytic functions.

2 Formulation

2.1 Bulk and Interface Elasticity

The equilibrium and constitutive equations of an elastically anisotropic bulk solid are givenby [1]:

σij,j = 0, σij = Cijklεkl, εij = 1

2(ui,j + uj,i), (1)

where i, j, k, l = 1,2,3;Cijkl are the elastic stiffnesses; σij and εij are, respectively, thestress and strain tensors in the bulk material; ui is the i-th component of the displacementvector; and δij is the Kronecker delta.

The equilibrium conditions on the interface incorporating surface/interface elasticity canbe expressed as [3–5, 15–17]

[σαjnj eα] + σ sαβ,βeα = 0 (tangential direction),

[σijninj ] = σ sαβκαβ (normal direction),

(2)

where α,β = 1,2;ni are the components of the unit normal vector of the interface; [∗]denotes the jump of the quantities across the interface; σ s

αβ is the surface stress tensor; andκαβ is the curvature tensor of the surface. If additionally, the effect of residual surface tensionis ignored, the constitutive equations on the anisotropic interface are given by

σ sαβ = ∂g

∂εsαβ

= Csαβωρε

sωρ, (3)

A Screw Dislocation Interacting with an Anisotropic Elliptical

where ω,ρ = 1,2, g is the deformation-dependent surface energy, εsαβ is the surface strain

tensor and Csαβωρ are the elastic stiffnesses of the interface which are constant along the

interface. To the authors’ knowledge, surface/interface anisotropy has been considered onlyrarely in the existing literature.

2.2 Complex Variable Formulation

We assume that the anisotropic bulk material is monoclinic with the symmetry plane atx3 = 0 so that the stress-strain relation for an anti-plane deformation is given by [1]

σ31 = C55u,1 + C45u,2,

σ32 = C44u,2 + C45u,1,(4)

where u = u3 and Cαβ is the contracted notation of Cijkl . The positive definiteness of thestrain energy density will require that

C44 > 0, C55 > 0, C44C55 − C245 > 0. (5)

For the special case of an orthotropic material with the orthotropy axes coinciding withthe reference axes, one has C45 = 0.

The equation of equilibrium is

σ31,1 + σ32,2 = C55u,11 + 2C45u,12 + C44u,22 = 0. (6)

The general solution of Eq. (6) can be expressed in terms of a single analytic functionf (zp) as

u = Im{f (zp)

}, zp = x1 + px2, (7)

where

p =−C45 + i

√C44C55 − C2

45

C44. (8)

The stresses σ31, σ32 and the stress function ϕ are given by [1, 22]

σ31 + pσ32 = iμ Im{p}f ′(zp), (9)

ϕ = μRe{f (zp)

}, (10)

where μ =√

C44C55 − C245, and the stresses σ31, σ32 are related to the stress function ϕ

through

σ31 = −ϕ,2, σ32 = ϕ,1. (11)

Let t3 be the anti-plane surface traction component on a boundary L. If s is the arc-lengthmeasured along L such that, when facing the direction of increasing s, the material is on theleft-hand side, it can be shown that [1]

t3 = −dϕ

ds. (12)


3 A Screw Dislocation Outside an Elliptical Inhomogeneity

Consider a domain inR2, infinite in extent, containing an elliptical elastic inhomogeneity.The linearly anisotropic elastic materials occupying both the inhomogeneity and the matrixare assumed to be homogeneous and monoclinic with the symmetry plane at x3 = 0 andwith the associated non-trivial elastic constants C

(1)

44 ,C(1)

45 ,C(1)

55 and C(2)

44 ,C(2)

45 ,C(2)

55 , respec-tively. The matrix is subjected to uniform anti-plane shear stresses σ∞

31 and σ∞32 at infinity.

Furthermore, uniform anti-plane eigenstrains ε∗13 and ε∗

23 are imposed on the inhomogeneityand a screw dislocation with Burgers vector bz is located at (x1, x2) = (x0

1 , x02 ) in the matrix.

We represent the matrix by the domain S2 : x21

a2 + x22

b2 ≥ 1, and assume that the inhomogene-

ity occupies the region S1 : x21

a2 + x22

b2 ≤ 1. The inhomogeneity-matrix interface is denoted by

L : x21

a2 + x22

b2 = 1. In what follows, the subscripts 1 and 2 (or the superscripts (1) and (2)) willrefer to the regions S1 and S2, respectively.

If we assume that the interface L is also monoclinic with the symmetry plane at x3 = 0and that it is a coherent one (i.e., εs

αβ = εαβ ), then it follows from Eqs. (2) and (3) that theinterface conditions along L can be specifically described by the equations

u1 + u∗ = u2, σ(1)

3n − σ(2)

3n = Cs44�s

(u1 + u∗), on L, (13)

where �s is the surface Laplacian operator, n is the outward unit normal to L, u∗ =(ε∗

13 − iε∗23)z + (ε∗

13 + iε∗23)z with z = x1 + ix2 is the additional displacement within the

inhomogeneity induced by uniform (stress-free) anti-plane eigenstrains ε∗13 and ε∗

23, andCs

44 = Cs2323.

Considering Eq. (12), the boundary conditions in Eq. (13) are equivalent to the following:

u1 + u∗ = u2, ϕ2 − ϕ1 = Cs44

d(u1 + u∗)ds

, on L, (14)

where s increases counter-clockwise around L.We first consider the following mapping function for an anisotropic inhomogeneity

z1 = ω1(ξ1) = 1

2(a − ip1b)ξ1 + 1

2(a + ip1b)ξ−1

1 . (15)

This mapping function maps the elliptical region with a cut in the z1-plane onto an annulus√|ρ| ≤ |ξ | ≤ 1, (ρ = a+ip1b

a−ip1b) in the ξ1-plane.

Remark The z-plane is mapped to the z1-plane via z1 = x11 + ix1

2 = x1 +p′1x2 + ip′′

1x2 withp′

1 p′′1 being the real and imaginary parts of p1. Consequently, the ellipse L in the z-plane

maps to the boundary L1 of the elliptical region in the z1-plane [23].We then consider another mapping function for the anisotropic matrix

z2 = ω2(ξ2) = 1

2(a − ip2b)ξ2 + 1

2(a + ip2b)ξ−1

2 . (16)

This mapping function maps the exterior of an elliptical region in the z2(= x1 +p2x2)-planeonto the exterior of the unit circle |ξ2| ≥ 1 in the ξ2-plane (the ellipse L in the z-plane mapsto the boundary L2 of the elliptical region in the z2-plane).


Thirdly, we consider the following conformal mapping function

z = ω(ξ) = R

(ξ + m

ξ

), (17)

where R = (a + b)/2 and m = (a − b)/(a + b),0 ≤ m < 1.In view of the fact that ξ1 = ξ2 = ξ on |ξ | = 1, we can first replace ξ1 and ξ2 by the

common variable ξ . When the analysis is complete, the complex variable ξ shall revert backto the corresponding complex variables ξ1 and ξ2 accordingly. For convenience and withoutloss of generality, we write f1(z1) = f1(ω1(ξ)) = f1(ξ) and f2(z2) = f2(ω2(ξ)) = f2(ξ).

In order to ensure that f1(z1) is analytic within the elliptical inhomogeneity, f1(ξ) shouldbe expanded in the following form

f1(ξ) = a0 ++∞∑

n=1

an

(ξn + ρnξ−n

),

√|ρ| ≤ |ξ | ≤ 1, (18)

where an (n = 0,1,2, . . . ,+∞) are unknown complex constants to be determined.The continuity condition of displacement across the interface |ξ | = 1 in Eq. (14)1 can

then be expressed as

f −2 (ξ) − f +

2 (1/ξ) = f +1 (ξ) − f −

1 (1/ξ) + 2(ε∗

23 + iε∗13

)ω(ξ) − 2

(ε∗

23 − iε∗13

)ω(1/ξ),

|ξ | = 1. (19)

Substitution of Eq. (18) into the above expression yields

f −2 (ξ) − f +

2 (1/ξ)

= a0 − a0 ++∞∑

n=1

[an − anρ

n + 2R(ε∗

23 + iε∗13

)δn1 − 2Rm

(ε∗

23 − iε∗13

)δn1

]ξn

−+∞∑

n=1

[an − anρ

n + 2R(ε∗

23 − iε∗13

)δn1 − 2Rm

(ε∗

23 + iε∗13

)δn1

]ξ−n,

|ξ | = 1. (20)

By applying Liouville’s theorem, we can then arrive at the following expression of f2(ξ)

f2(ξ) = k + i Im{a0} −+∞∑

n=1

[an − anρ

n + 2R(ε∗

23 − iε∗13

)δn1 − 2Rm

(ε∗

23 + iε∗13

)δn1

]ξ−n

+ bz

2πln(ξ − ξ0) + bz

2πln

1 − ξ0ξ

ξ+ i(a − ip2b)(σ∞

31 + p2σ∞32 )

2μ2 Im{p2} ξ

− i(a + ip2b)(σ∞31 + p2σ

∞32 )

2μ2 Im{p2} ξ−1, |ξ | ≥ 1, (21)

where k is a real constant, and

ξ0 = z(2)

0 +√

z(2)20 − (a2 + p2

2b2)

a − ip2b, z

(2)

0 = x01 + p2x

02 . (22)


By employing Eqs. (7) and (10), the interface condition (14)2 on the interface |ξ | = 1 canthen be expressed in terms of f1(ξ) and f2(ξ) as

μ2

Cs44

[f −

2 (ξ) + f +2 (1/ξ) − Γf +

1 (ξ) − Γ f −1 (1/ξ)

]

= ξf ′+1 (ξ) + ξ−1f ′−

1 (1/ξ) + 2(ε∗23 + iε∗

13)ξω′(ξ) + 2(ε∗23 − iε∗

13)ξ−1ω′(1/ξ)

|ξω′(ξ)| , |ξ | = 1,

(23)

where Γ = μ1/μ2.Substituting the above expressions (18) and (21) into Eq. (23), we obtain

|ω′(ξ)|a

[

b0 ++∞∑

n=1

(bnξ

n + bnξ−n

)]

= γ

⎡

⎢⎢⎢⎢⎣

+∞∑

n=1

n(an − anρ

n + 2R(ε∗

23 + iε∗13

)δn1 − 2Rm

(ε∗

23 − iε∗13

)δn1

)ξn

++∞∑

n=1

n(an − anρ

n + 2R(ε∗

23 − iε∗13

)δn1 − 2Rm

(ε∗

23 + iε∗13

)δn1

)ξ−n

⎤

⎥⎥⎥⎥⎦

,

|ξ | = 1, (24)

where γ = Cs44

aμ2is a size-dependent dimensionless parameter for the interface L, and

b0 = b0 = 2k − 2Γ Re{a0} + bz

πRe

{ln(−ξ0)

}, (25a)

bn = −(Γ + 1)an − (Γ − 1)ρnan +[

2Rm(ε∗

23 − iε∗13

) − 2R(ε∗

23 + iε∗13

)

+ i(a − ip2b)(σ∞31 + p2σ

∞32 )

μ2 Im{p2}]δn1 − bz

π

ξ−n0

n(n = 1,2, . . . ,+∞). (25b)

On the interface |ξ | = 1, |ω′(ξ)| can be expanded into the following convergent series[24]

|ω′(ξ)|a

=√

1 − h2 cos2 θ = I0 ++∞∑

n=1

I2n

(ξ 2n + ξ−2n

), ξ = eiθ , (26)

where h = √1 − b2/a2 (0 ≤ h < 1) is the eccentricity of the ellipse L, and

I2n = I2n =+∞∑

k=n

(−1)kCk1/2C

k−n2k

(h

2

)2k

(n = 0,1,2, . . .) (27)

with Cnα the binomial coefficient defined by

Cnα =

{α(α−1)···(α−n+1)

n! , if n �= 0,

1, if n = 0.(28)


For example, if h = 0.95, it can be calculated from Eq. (27) that: I0 = 0.7020, I2 =−0.1658, I4 = −0.0209, I6 = −0.0054, I8 = −0.0017 and I10 = −0.0006. In addition, ifthe series in Eq. (26) is truncated at n = 5, the maximum relative error of the expansion inEq. (26) for the range 0 ≤ θ < 2π is less than ±0.3 %. As another check, the coefficientsI2n (n = 0,1,2, . . .) in Eq. (27) indeed satisfy the following recurrence equation derived byShen et al. [25]

(1

2+ n

3

)I2(n+1) =

(1

2− n

3

)I2(n−1) + 2n(2 + b∗)

3b∗ I2n (n = 0,1,2, . . .),

where b∗ = h2/(1 − h2).Substitution of Eq. (26) into Eq. (24) yields[

I0 ++∞∑

n=1

I2n

(ξ 2n + ξ−2n

)][

b0 ++∞∑

n=1

(bnξ

n + bnξ−n

)]

= γ

⎡

⎢⎢⎢⎢⎣

+∞∑

n=1

n(an − anρ

n − 2Rm(ε∗

23 − iε∗13

)δn1 + 2R

(ε∗

23 + iε∗13

)δn1

)ξn

++∞∑

n=1

n(an − anρ

n − 2Rm(ε∗

23 + iε∗13

)δn1 + 2R

(ε∗

23 − iε∗13

)δn1

)ξ−n

⎤

⎥⎥⎥⎥⎦

,

|ξ | = 1. (29)

The product of the two series on the left-hand side of Eq. (29) can be further conciselyexpanded into the following:

[

I0 ++∞∑

n=1

I2n

(ξ 2n + ξ−2n

)][

b0 ++∞∑

n=1

(bnξ

n + bnξ−n

)]

=+∞∑

n=1

(+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

)

ξ 2n−1

++∞∑

n=0

(+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p

)

ξ 2n

++∞∑

n=1

(+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

)

ξ−2n+1

++∞∑

n=0

(+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p

)

ξ−2n. (30)

Insertion of Eq. (30) into Eq. (29) yields

+∞∑

n=1

(+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

)

ξ 2n−1

++∞∑

n=0

(+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p

)

ξ 2n


++∞∑

n=1

(+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

)

ξ−2n+1

++∞∑

n=0

(+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p

)

ξ−2n.

= γ

⎡

⎢⎢⎢⎢⎣

+∞∑

n=1

n(an − anρ

n − 2Rm(ε∗

23 − iε∗13

)δn1 + 2R

(ε∗

23 + iε∗13

)δn1

)ξn

++∞∑

n=1

n(an − anρ

n − 2Rm(ε∗

23 + iε∗13

)δn1 + 2R

(ε∗

23 − iε∗13

)δn1

)ξ−n

⎤

⎥⎥⎥⎥⎦

,

|ξ | = 1. (31)

Equating coefficients of like powers of ξ in Eq. (31), we finally arrive at the followingset of linear algebraic equations

I0b0 ++∞∑

p=1

I2pb2p ++∞∑

p=1

I2pb2p = 0,

+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p = 2nγ(a2n − a2nρ

2n),

+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

= (2n − 1)γ[a2n−1 − a2n−1ρ

2n−1 − 2Rm(ε∗

23 − iε∗13

)δ(2n−1)1 + 2R

(ε∗

23 + iε∗13

)δ(2n−1)1

]

(n = 1,2,3, . . . ,+∞).

(32)

The obtained linear algebraic equations in Eqs. (25b) and (32) are truncated at n = 2N ,with the truncated equations elegantly arranged into the following matrix form

y1 = −(Γ + 1)x1 − (Γ − 1)�ox1 + jbo + jσ + jε,

y2 = −(Γ + 1)x2 − (Γ − 1)�ex2 + jbe,(33)

Ay1 + Cy1 = γ�o(x1 − �ox1 − jε).

I0b0 + eT y2 + eT y2 = 0,

eb0 + Ay2 + By2 = γ�e(x2 − �ex2),

(34)

where

x1 =

⎡

⎢⎢⎢⎣

a1

a3...

a2N−1

⎤

⎥⎥⎥⎦

, x2 =

⎡

⎢⎢⎢⎣

a2

a4...

a2N

⎤

⎥⎥⎥⎦

, y1 =

⎡

⎢⎢⎢⎣

b1

b3...

b2N−1

⎤

⎥⎥⎥⎦

, y2 =

⎡

⎢⎢⎢⎣

b2

b4...

b2N

⎤

⎥⎥⎥⎦

, (35)

�o = diag[ρ ρ3 · · · ρ2N−1 ], �e = diag[ρ2 ρ4 · · · ρ2N ], (36)


jbo = −bz

π

⎡

⎢⎢⎢⎢⎢⎣

1ξ0

13ξ3

0...1

(2N−1)ξ2N−10

⎤

⎥⎥⎥⎥⎥⎦

, jbe = −bz

π

⎡

⎢⎢⎢⎢⎢⎣

12ξ2

01

4ξ40...1

2Nξ2N0

⎤

⎥⎥⎥⎥⎥⎦

,

jσ = i(a − ip2b)(σ∞31 + p2σ

∞32 )

μ2 Im{p2}

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

, jε = 2R[ε∗

23(m − 1) − iε∗13(m + 1)

]

⎡

⎢⎢⎢⎣

10...

0

⎤

⎥⎥⎥⎦

,

(37)

e =

⎡

⎢⎢⎢⎣

I2

I4...

I2N

⎤

⎥⎥⎥⎦

, �o = diag[1 3 · · · 2N − 1 ], �e = diag[2 4 · · · 2N ],

(38)

A = AT =

⎡

⎢⎢⎢⎢⎢⎣

I0 I2 · · · I2N−4 I2N−2

I2 I0 · · · I2N−6 I2N−4...

.... . .

......

I2N−4 I2N−6 · · · I0 I2

I2N−2 I2N−4 · · · I2 I0

⎤

⎥⎥⎥⎥⎥⎦

,

B = BT =

⎡

⎢⎢⎢⎢⎢⎣

I4 I6 · · · I2N I2N+2

I6 I8 · · · I2N+2 I2N+4...

.... . .

......

I2N I2N+2 · · · I4N−4 I4N−2

I2N+2 I2N+4 · · · I4N−2 I4N

⎤

⎥⎥⎥⎥⎥⎦

, (39)

C = CT =

⎡

⎢⎢⎢⎢⎢⎣

I2 I4 · · · I2N−2 I2N

I4 I6 · · · I2N I2N+2...

.... . .

......

I2N−2 I2N · · · I4N−6 I4N−4

I2N I2N+2 · · · I4N−4 I4N−2

⎤

⎥⎥⎥⎥⎥⎦

.

Substituting Eq. (33) into Eq. (34), we obtain[(Γ + 1)A + (Γ − 1)C�o + γ�o

]x1 + [

(Γ + 1)C + (Γ − 1)A�o − γ�o�o

]x1

= A(jbo + jσ + jε) + C(jbo + jσ + jε) + γ�ojε, (40)

b0 =(

Γ + 1

I0eT + Γ − 1

I0eT �e

)x2 +

(Γ + 1

I0eT + Γ − 1

I0eT �e

)x2 − eT jbe

I0− eT jbe

I0,

(41a)

eb0 − [(Γ + 1)A + (Γ − 1)B�e + γ�e

]x2 − [

(Γ + 1)B + (Γ − 1)A�e − γ�e�e

]x2

= −Ajbe − Bjbe. (41b)


Inserting Eq. (41a) into Eq. (41b) and eliminating b0, we will arrive at

[(Γ + 1)A + (Γ − 1)B�e − Γ + 1

I0eeT − Γ − 1

I0eeT �e + γ�e

]x2

+[(Γ + 1)B + (Γ − 1)A�e − Γ + 1

I0eeT − Γ − 1

I0eeT �e − γ�e�e

]x2

= I0A − eeT

I0jbe + I0B − eeT

I0jbe. (42)

Equations (40) and (42) can be solved quite simply to arrive at the two N -dimensionalvectors x1 and x2 as

x1 = (G−12 G1 − G−1

1 G2)−1(G−1

2 h1 − G−11 h1),

x2 = (G−14 G3 − G−1

3 G4)−1(G−1

4 h2 − G−13 h2),

(43)

where

G1 = (Γ + 1)A + (Γ − 1)C�o + γ�o, G2 = (Γ + 1)C + (Γ − 1)A�o − γ�o�o,

G3 = (Γ + 1)A + (Γ − 1)B�e − Γ + 1

I0eeT − Γ − 1

I0eeT �e + γ�e,

G4 = (Γ + 1)B + (Γ − 1)A�e − Γ + 1

I0eeT − Γ − 1

I0eeT �e − γ�e�e,

h1 = A(jbo + jσ + jε) + C(jbo + jσ + jε) + γ�ojε,

h2 = I0A − eeT

I0jbe + I0B − eeT

I0jbe.

(44)

Once x2 has been obtained using Eq. (43)2, the real constant b0 can then be uniquely deter-mined using Eq. (41a). From Eq. (25a) we find that Re{a0}− k

Γ= − b0

2Γ+ bz

2πΓRe{ln(−ξ0)}.

If we set k = Im{a0} = 0 as in [1], the only remaining unknown constant a0 can then bedetermined as a0 = a0 = − b0

2Γ+ bz

2πΓRe{ln(−ξ0)}.

By using the Peach-Koehler formula, the image force acting on the dislocation in thepresence of remote uniform stresses and imposed uniform eigenstrains can be written con-cisely as

ip2F1 − iF2

= μ2bz Im{p2}ω′

2(ξ0)

+∞∑

n=1

n(an − anρn)

ξn+10

− μ2b2z Im{p2}ω′′

2(ξ0)

4π [ω′2(ξ0)]2

+ μ2b2z Im{p2}

2πξ0(ξ0ξ0 − 1)ω′2(ξ0)

+ ibz[ξ 20 (a − ip2b)(σ∞

31 + p2σ∞32 ) + (a + ip2b)(σ∞

31 + p2σ∞32 )]

2ξ 20 ω′

2(ξ0)

+ 4Rμ2bz Im{p2}[(1 − m)ε∗23 − i(1 + m)ε∗

13]2ξ 2

0 ω′2(ξ0)

, (45)

where F1 and F2 are the x1 and x2 components of the image force.


In the physical plane, f1(z1) within the elliptical inhomogeneity is given explicitly as

f1(z1) = d0 +2N∑

n=1

dnzn1,

x21

a2+ x2

2

b2≤ 1, (46)

where the coefficients u1 = [d1 d3 · · · d2N−1

]Tand u2 = [

d2 d4 · · · d2N

]Tcan be sepa-

rately determined by

u1 = M−11 x1, u2 = M−1

2 x2, (47)

where M1 and M2 are two N × N upper triangular matrices, and are given by

M1 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(a−ip1b

2 )ρ0C01 (

a−ip1b

2 )3ρ1C13 (

a−ip1b

2 )5ρ2C25 · · · (

a−ip1b

2 )2N−1ρN−1CN−12N−1

0 (a−ip1b

2 )3ρ0C03 (

a−ip1b

2 )5ρ1C15 · · · (

a−ip1b

2 )2N−1ρN−2CN−22N−1

0 0 (a−ip1b

2 )5ρ0C05 · · · (

a−ip1b

2 )2N−1ρN−3CN−32N−1

......

.... . .

...

0 0 0 · · · (a−ip1b

2 )2N−1ρ0C02N−1

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(48)and

M2 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(a−ip1b

2 )2ρ0C02 (

a−ip1b

2 )4ρ1C14 (

a−ip1b

2 )6ρ2C26 · · · (

a−ip1b

2 )2NρN−1CN−12N

0 (a−ip1b

2 )4ρ0C04 (

a−ip1b

2 )6ρ1C16 · · · (

a−ip1b


0 0 (a−ip1b

2 )6ρ0C06 · · · (

a−ip1b


......

.... . .

...

0 0 0 · · · (a−ip1b

2 )2Nρ0C02N

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(49)Furthermore, once dn (n = 1,2, . . . ,2N) have been determined from Eq. (47), the con-

stant term d0 can be uniquely obtained as

d0 = a0 −N∑

n=1

(a − ip1b

2

)2n

ρnCn2nd2n. (50)

Thus the internal stress field inside the elliptical inhomogeneity can be derived as

ip1σ32 + iσ31 = μ1 Im{p1}2N∑

n=1

ndn(x1 + p1x2)n−1,

x21

a2+ x2

2

b2≤ 1. (51)

It is observed from the above results that: (i) remote uniform stresses and uniform eigen-strains imposed on the inhomogeneity will induce only odd indices of ai and di , and will notinduce even indices of ai and di ; (ii) the dislocation will induce both the odd and even indicesof ai and di ; (iii) there exists no correspondence principle for the internal stress field betweenthe remote loading and the imposed eigenstrains as in the case of an elliptical inhomogene-ity with a spring-type imperfect interface [25]; (iv) due to the existence of the non-zerointerface parameter γ , the internal stress field inside the elliptical inhomogeneity induced


by remote uniform loading and imposed uniform eigenstrains is intrinsically non-uniform;(v) the elastic fields in the two-phase composite as well as the direction and magnitude ofthe image force acting on the dislocation are both size-dependent values characterized by γ .We note that: (i) when γ = 0, our solution reduces to that of Ting [1]; (ii) when both theinhomogeneity and the matrix are elastically isotropic, our solution recovers the solutions in[13, 14].

4 A Screw Dislocation Inside the Elliptical Inhomogeneity

We now consider the case in which the composite system is loaded only by a screw disloca-tion located at (x1, x2) = (x0

1 , x02 ) inside the inhomogeneity.

In this case, f1(ξ) can be expanded into the following form

f1(ξ) = a0 ++∞∑

n=1

an

(ξn + ρnξ−n

) + bz


2πln

ξ − ρξ−10

ξ,

√|ρ| ≤ |ξ | ≤ 1,

(52)where

ξ0 = z(1)

0 +√

z(1)20 − (a2 + p2

1b2)

a − ip1b, z

(1)

0 = x01 + p1x

02 . (53)

By imposing the continuity condition of displacement across the interface |ξ | = 1, wearrive at the following expression of f2(ξ):

f2(ξ) = k + i Im{a0} −+∞∑

n=1

(an − anρ

n)ξ−n + bz


2πln

ξ − ρξ−10

ξ, |ξ | ≥ 1,

(54)where k is real. We can also set k = Im{a0} = 0, as in [1].

Substituting Eqs. (52) and (54) into the following interface condition on |ξ | = 1

μ2

Cs44

[f −

2 (ξ) + f +2 (1/ξ) − Γf +

1 (ξ) − Γ f −1 (1/ξ)

] = ξf ′−2 (ξ) + ξ−1f ′+

2 (1/ξ)

|ξω′(ξ)| , |ξ | = 1,

(55)we arrive at the following relationship

|ω′(ξ)|a

[

b0 ++∞∑

n=1

(bnξ

n + bnξ−n

)]

= γ

[bz

π+

+∞∑

n=1

n

[(an − anρ

n) + bz

2πn

(ξ n

0 + ρnξ−n0

)]ξn

++∞∑

n=1

n

[(an − anρ

n) + bz

2πn

(ξn

0 + ρnξ−n0

)]ξ−n

]

, |ξ | = 1, (56)

where the size-dependent parameter γ is again that introduced in Sect. 3, and

b0 = b0 = −2Γ a0, (57a)


bn = −(Γ + 1)an − (Γ − 1)anρn + bz(Γ − 1)(ξ n

0 + ρnξ−n0 )

2πn(n = 1,2,3, . . . ,+∞).

(57b)

Substituting Eq. (30) into Eq. (56) and equating coefficients of like powers of ξ , we arriveat the following linear algebraic equations

I0b0 ++∞∑

p=1

I2pb2p ++∞∑

p=1

I2pb2p = γ bz

π,

+∞∑

p=0

I2|n−p|b2p ++∞∑

p=1

I2(n+p)b2p = 2nγ

[a2n − a2nρ

2n + bz

2π(2n)

(ξ 2n

0 + ρ2nξ−2n0

)],

+∞∑

p=1

I2|n−p|b2p−1 ++∞∑

p=1

I2(n+p−1)b2p−1

= (2n − 1)γ

[a2n−1 − a2n−1ρ

2n−1 + bz

2π(2n − 1)

(ξ 2n−1

0 + ρ2n−1ξ−(2n−1)

0

)]

(n = 1,2,3, . . . ,+∞).

(58)

These linear algebraic equations in Eqs. (57b) and (58) are truncated at n = 2N , and theresulting equations are elegantly arranged into the following matrix form

y1 = −(Γ + 1)x1 − (Γ − 1)�ox1 + (Γ − 1)j1,

y2 = −(Γ + 1)x2 − (Γ − 1)�ex2 + (Γ − 1)j2,(59)

Ay1 + Cy1 = γ�o(x1 − �ox1 + j1),

I0b0 + eT y2 + eT y2 = γ bz

π,

eb0 + Ay2 + By2 = γ�e(x2 − �ex2 + j2),

(60)

where x1,x2,y1,y2,�o,�e,�o,�e,A,B,C and e have been defined by Eqs. (35), (36),(38) and (39), and

j1 = bz

2π

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

ξ0 + ρξ−10

ξ30 +ρ3 ξ−3

03

...

ξ2N−10 +ρ2N−1 ξ

−(2N−1)0

2N−1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

, j2 = bz

2π

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

ξ20 +ρ2 ξ−2

02

ξ40 +ρ4 ξ−4

04

...

ξ2N0 +ρ2N ξ−2N

02N

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (61)

Substituting Eq. (59) into Eq. (60), we obtain

G1x1 + G2x1 = [(Γ − 1)A − γ�o

]j1 + (Γ − 1)Cj1, (62)

b0 =(

Γ + 1

I0eT + Γ − 1

I0eT �e

)x2 +

(Γ + 1

I0eT + Γ − 1

I0eT �e

)x2 − (Γ − 1)eT

I0j2


− (Γ − 1)eT

I0j2 + γ bz

πI0, (63a)

eb0 − [(Γ + 1)A + (Γ − 1)B�e + γ�e

]x2 − [

(Γ + 1)B + (Γ − 1)A�e − γ�e�e

]x2

= −[(Γ − 1)A − γ�e

]j2 − (Γ − 1)Bj2, (63b)

where G1 and G2 have been defined in Eq. (44).Inserting Eq. (63a) into Eq. (63b) and eliminating b0, we will arrive at

G3x2 +G4x2 =[(Γ −1)A− (Γ − 1)eeT

I0−γ�e

]j2 + (Γ −1)

(B− eeT

I0

)j2 + γ bz

πI0e, (64)

where G3 and G4 have been defined in Eq. (44).Equations (62) and (64) can be solved to arrive at the two N -dimensional vectors x1 and

x2 as

x1 = (G−1

2 G1 − G−11 G2

)−1(G−1

2 q1 − G−11 q1

),

x2 = (G−1

4 G3 − G−13 G4

)−1(G−1

4 q2 − G−13 q2

),

(65)

where

q1 = [(Γ − 1)A − γ�o

]j1 + (Γ − 1)Cj1,

q2 =[(Γ − 1)A − (Γ − 1)eeT

I0− γ�e

]j2 + (Γ − 1)

(B − eeT

I0

)j2 + γ bz

πI0e.

(66)

Once x2 has been determined by using Eq. (65)2, the real constant b0 can be uniquelydetermined by using Eq. (63a). Consequently the only remaining unknown real constant a0

can be obtained from Eq. (57a) as a0 = − b02Γ

.In the physical plane, f1(z1) within the elliptical inhomogeneity can be explicitly given

by

f1(z1) = d0 +2N∑

n=1

dnzn1 + bz

2πln

(z1 − z

(1)

0

),

x21

a2+ x2

2

b2≤ 1, (67)

where the coefficients u1 = [d1 d3 · · · d2N−1

]Tand u2 = [

d2 d4 · · · d2N

]Tcan also be sep-

arately determined by using Eq. (47) now with x1 and x2 being given by Eq. (65).Once dn (n = 1,2, . . . ,2N) have been determined, the constant term d0 can be uniquely

obtained as

d0 = a0 −N∑

n=1

(a − ip1b

2

)2n

ρnCn2nd2n − bz

2πln

a − ip1b

2. (68)

When the screw dislocation is located inside the inhomogeneity and there is no otherloading on the composite, the image force on the dislocation can be simply given by

ip1F1 − iF2 = μ1bz Im{p1}2N∑

n=1

n(z(1)

0

)n−1dn. (69)

which is size-dependent.


We again remark that when γ = 0 (the case of a perfect interface), our solution for ascrew dislocation inside an anisotropic elliptical inhomogeneity simply reduces to that ofTing [1].

5 Conclusions

We have successfully obtained elegant representations of Green’s functions for an anisotro-pic elliptical inhomogeneity with interface stresses. In summary, we have considered fourtypical loading cases: (i) remote uniform loading; (ii) uniform eigenstrains imposed onthe inhomogeneity; (iii) a screw dislocation located in the matrix; (iv) a screw dislocationpresent inside the inhomogeneity. The novelties of the solution lie in that: (i) all of the co-efficients I2n(n = 0,1,2, . . .) in the expansion of the real function |ω′(eiθ )| are determinedalgebraically (see Eq. (27)); (ii) the product of the two series on the left-hand side of Eq. (29)is concisely expanded into the right-hand side of Eq. (30); (iii) simple matrix algebra is usedto resolve all the unknown coefficients appearing in the analytic functions f1(ξ), f1(z1) andf2(ξ). The analytical results in Sects. 3 and 4 clearly indicate that the elastic fields and im-age force on the screw dislocation are influenced by the size-dependent parameter γ . Thecorrectness of the present solution is verified by comparison with existing solutions.

Acknowledgements Two reviewers’ comments are highly appreciated. This work is supported by the Na-tional Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from theNatural Sciences and Engineering Research Council of Canada (Grant No. 155112).

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