Eshelby’s problem for infinite, semi-infinite and two bonded semi-infinite laminated anisotropic thin plates Xu Wang & Peter Schiavone
Archive of Applied Mechanics ISSN 0939-1533 Volume 85 Number 5 Arch Appl Mech (2015) 85:573-585 DOI 10.1007/s00419-014-0931-1
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Author's personal copy Arch Appl Mech (2015) 85:573–585 DOI 10.1007/s00419-014-0931-1
O R I G I NA L
Xu Wang · Peter Schiavone
Eshelby’s problem for infinite, semi-infinite and two bonded semi-infinite laminated anisotropic thin plates
Received: 14 April 2014 / Accepted: 11 November 2014 / Published online: 22 November 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract We consider an Eshelby’s inclusion of arbitrary shape with prescribed uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. The inclusion has the same extensional, coupling and bending stiffnesses as the surrounding material. The boundary of the semi-infinite plate can be described by free, rigidly clamped and simply supported edges. We derive solutions of simple form by using the new Stroh octet formalism for the coupled stretching and bending deformations of anisotropic thin plates and the method of analytic continuation. In particular, real solutions of the far-field elastic fields induced by an inclusion of arbitrary shape are obtained. Specific examples of an elliptical inclusion in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite anisotropic plates are presented to demonstrate the obtained general solutions. Keywords Eshelby inclusion · Eigenstrain · Eigencurvature · Kirchhoff anisotropic plate · Stroh octet formalism · Interface
1 Introduction Eshelby’s problem of an arbitrarily shaped inclusion in which uniform or nonuniform eigenstrains are prescribed continues to be an interesting albeit challenging topic in micromechanics. In the following paragraphs, we present a brief review of recent relevant studies on the corresponding two-dimensional problems. In the context of isotropic plane elasticity, Ru [15,18] derived analytical solutions to Eshelby’s problem of an inclusion of arbitrary shape in a plane, half-plane and in one of two bonded half-planes using Muskhelishvili’s complex variable formulation [12] and the techniques of conformal mapping and analytic continuation. By means of the Stroh formalism, Ru [16,17] presented a simple method based on the construction of some auxiliary functions determined through conformal mapping to obtain analytical solutions for a two-dimensional inclusion of any shape in an anisotropic piezoelectric plane, half-plane and in one of two bonded dissimilar anisotropic piezoelectric half-planes. By using the line-source Green’s function solutions, polygonal-shaped inclusions in isotropic elastic and anisotropic piezoelectric planes and half-planes were addressed, respectively, by Kawashita and Nozaki [9] and Pan [13]. In addition, various numerical and analytic methods have been X. Wang School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China E-mail:
[email protected] P. Schiavone (B) Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, Alberta T6G 2G8, Canada E-mail:
[email protected] Tel.: 1 780 492 3638
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developed to obtain the stress field in a domain containing multiple elastic inhomogeneities (see for example, [10,11]). Studies dealing with inclusion problems in infinite isotropic homogeneous or laminated elastic plates have recently become increasingly important due to their increased use in commercial applications. For example, in many cases of practical interest, fibers are introduced into composite materials to improve the overall properties of the composite structure (e.g., improved strength, resistance to corrosion etc.). From the micromechanical point of view, composite materials reinforced with fibers are most often studied as thin plate structures containing inclusions. In the context of classical theories of bending of thin elastic plates, the presence of inclusions presents formidable challenges in modeling and analysis. Recently, however, Qin et al. [14], Beom [1] and Duong and Yu [8] have derived closed-form solutions for an infinite isotropic and homogeneous Kirchhoff plate containing elliptical and polygon-shaped inclusions with uniform eigencurvatures; Xu and Wang [20] have proved that the rotational symmetric inclusion with uniform eigencurvatures in an infinite isotropic and homogeneous plate possesses the interesting quasi-Eshelby property, while Beom and Earmme [2] and Beom and Kim [3] have derived closed-form solutions for the elastic fields within elliptical and cuboidal inclusions with eigenstrains in an infinite plate composed of thin layers of isotropic elastic materials. Each of these studies dealing with inclusion problems in infinite isotropic homogeneous or laminated plates have used methods based on influence functions or Green’s functions. In this paper, we continue the study of the influence of inclusions in plates and derive analytical solutions to the Eshelby problem of an inclusion of arbitrary shape with uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. Recently, Cheng and Reddy [4–7] developed an elegant Stroh octet formalism for the coupled bending and stretching deformations of inhomogeneous and laminated anisotropic thin elastic plates. This newly developed formalism retains almost all of the beautiful properties in the original Stroh sextic formalism for generalized plane strain problems [19]. We will adopt this new octet formalism together with the method proposed by Ru [16,17] to solve the Eshelby problem for Kirchhoff laminated anisotropic thin plates.
2 Stroh octet formalism for laminated anisotropic thin plates In this section, we present the main results of the octet formalism developed by Cheng and Reddy [4–7]. Consider an undeformed plate of uniform thickness h in a Cartesian coordinate system (x1 , x2 , x3 ) in which the mid-plane of the plate is taken to be x3 = 0. The plate is composed of an anisotropic, linearly elastic material, possibly inhomogeneous and laminated in the thickness direction. Adopting the notation used in Cheng and Reddy [4–7], a repeated index implies, unless otherwise specified, summation over the range of the index with Greek indices ranging from 1 to 2, lowercase Latin indices from 1 to 3 and uppercase Latin indices ranging from 1 to 4. The general solution describing the deformation of the plate can be written in terms of generalized displacements and stress functions in the form [4–7] T ¯ u = u 1 u 2 ϑ1 ϑ2 = Af(z) + Af(z), T ¯ = φ1 φ2 ψ1 ψ2 = Bf(z) + Bf(z).
(1)
Here, u α and ϑα = −w,α (in which w is the transverse deflection) are the in-plane displacements and slopes on the mid-plane, φα and ψα are four stress functions, and A = a 1 a 2 a 3 a 4 , B = b 1 b2 b3 b4 , T f(z) = f 1 (z 1 ) f 2 (z 2 ) f 3 (z 3 ) f 4 (z 4 ) , z K = x1 + p K x2 , Im { p K } > 0, (K = 1, 2, 3, 4), with
N1 N2 N3 N1T
aK bK
= pK
aK bK
(2)
, (K = 1, 2, 3, 4),
where p K are the eigenvalues with associated eigenvectors a K and b K [4].
(3)
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The detailed definitions of the three 4 × 4 real matrices N1 , N2 and N3 can be found in [4]. In addition, N1 , N2 and N3 possess the following mathematical structure [7] ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ ∗ −1 ∗ ∗ ∗ ∗ 0 ∗ ∗ 0 ∗ ∗ ⎢∗ 0 ∗ ∗⎥ ⎢∗ ∗ 0 ∗⎥ ⎢∗ 0 ∗ ∗⎥ , N2 = ⎣ , N3 = ⎣ , (4) N1 = ⎣ 0 0 0 1⎦ 0 0 0 0⎦ 0 0 0 0⎦ ∗ 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ 0 ∗ ∗ where the asterisk * denotes a possible nonzero element. In addition, N2 and −N3 are symmetric and positive semi-definite [7]. The membrane stress resultants Nαβ , bending moments Mαβ , transverse shearing forces Rα and the modified Kirchhoff transverse shearing forces Vα can be expressed in terms of the derivatives of the four stress functions φα and ψα as Nαβ = − ∈βω φα,ω , Rα = −
Mαβ = − ∈βω ψα,ω −
1 ∈αβ ψω,ω , 2
1 ∈αβ ψω,ωβ , Vα = − ∈αω ψω,ωω , 2
(5)
where ∈αβ are the components of the two-dimensional permutation tensor. In addition, the in-plane stresses σαβ can be calculated as [4] ⎤ ⎡ σ11
˜ x3 Q ˜ x3 R ˜ ˜ Q R ⎢ σ21 ⎥ (z) + 2 > f (z) , (6) Re Af Re A < p ⎣σ ⎦ = 2 ˜ T ∗ T ˜ ˜ x3 T˜ 12 R x3 R T σ22 where < p∗ >= diag [ p1 , p2 , p3 , p4 ], and ˜ αβ = C˜ α1β1 , R ˜ αβ = C˜ α1β2 , T ˜ αβ = C˜ α2β2 , Q
(7)
with C˜ αβωρ the components of the reduced elastic tensor. The transverse stresses σα3 can then be obtained using the equilibrium equations σα3,3 + σαβ,β = 0 and the condition that σα3 = 0 at x3 = ±h/2. It is indicated that σ12 = σ21 in Eq. (6) [4]. Due to the fact that the two 4 ×4 matrices A and B satisfy the following normalized orthogonal relationship T T ¯ B A AA = I, (8a) ¯T B¯ T A B B¯ or equivalently
¯ AA ¯ BB
BT AT ¯T B¯ T A
= I,
(8b)
three real matrices S, H and L can be introduced as follows S = i(2ABT − I), H = 2iAAT , L = −2iBBT .
(9)
In addition, H and L are positive definite matrices [5,7], and the following identities can also be easily derived [5,7,19] 2A < pα > AT = N2 − i(N1 H + N2 ST ), 2A < pα > BT = N1 + i(N2 L − N1 S), 2B < pα > BT = N3 + i(N1T L − N3 S), 2r A < 2r A < 2r B
B = [cos θ I − sin θ N1 (θ )] (I − iS) − isin θ N2 (θ )L, T
> AT = −i [cos θ I − sin θ N1 (θ )] H − sin θ N2 (θ )(I − iST ), > BT = i cos θ I − sin θ N1T (θ ) L − sin θ N3 (θ )(I − iS), > AT = cos θ I − sin θ N1T (θ ) (I − iST ) + isin θ N3 (θ )H,
(11)
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Fig. 1 An inclusion of arbitrary shape in an infinite laminated anisotropic plate
where < ∗ > is a 4×4 diagonal matrix in which each component is varied according to the Greek index α (from N1 (θ ) N2 (θ ) N1 N2 1 to 4), θ is the rotation angle about the x3 -axis [7], and N(θ ) = is related to N = N3 N1T N3 (θ ) N1T (θ ) by (see [6]) (cos θ I + sin θ N) [cos θ I − sin θ N(θ )] = I.
(12)
It is seen from the above that the Stroh octet formalism for anisotropic thin plates [4–7] retains almost all of the elegant properties in the Stroh sextic formalism for generalized plane strain deformations of anisotropic materials [19]. 3 An Eshelby inclusion in an infinite anisotropic plate As shown in Fig. 1, we first consider an infinite laminated anisotropic thin plate containing a subdomain of arbitrary shape which has the same extensional, coupling and bending stiffnesses as the surrounding material ∗ = ε ∗ (x ). In this discussion, the prescribed eigenstrains can and which undergoes in-plane eigenstrains εαβ αβ 3 vary only along the thickness direction of the laminated plate as in Beom and Earmme [2] and Beom and Kim [3]. Let S0 and S1 denote the subdomain and its supplement to the (x1 , x2 )-plane, and L the interface separating S0 and S1 . Throughout this work, the quantities in S0 and S1 will be identified by the subscripts 0 and 1. After some manipulations, the constitutive relations for the laminated anisotropic thin plate can be written as 0 0∗ ∗ Nαβ = Aαβωρ εωρ − χ εωρ ), + Bαβωρ (κωρ − χ κωρ 0 0∗ ∗ Mαβ = Bαβωρ εωρ − χ εωρ + Dαβωρ (κωρ − χ κωρ ), (13) where Aαβωρ = Q C˜ αβωρ , Bαβωρ = Q C˜ αβωρ x3 and Dαβωρ = Q C˜ αβωρ x32 are, respectively the extensional, h/2 0 coupling and bending stiffnesses with Q(· · · ) the integral operator defined by Q(· · · ) = −h/2 (· · · )dx3 ; εαβ and καβ are the in-plane strains and curvatures on the mid-plane of the plate; χ is equal to 1 if the field point is 0∗ and κ ∗ are the uniform mid-plane eigenstrains and eigencurvatures within S0 and to zero otherwise [13]; εαβ αβ ∗ = ε ∗ (x ) as imposed on S0 , and can be determined by given values of εαβ αβ 3 0∗ −1 QQ QQx3 QQε∗ (x3 ) ε = , (14) QQε∗ (x3 )x3 κ∗ QQx3 QQx32 ⎤ ⎤ ⎡ ∗ ⎤ ⎡ ∗ 0∗ ε11 κ11 ε11 (x3 ) 0∗ ⎦ , κ∗ = ⎣ κ ∗ ⎦ , ε∗ (x ) = ⎣ ε ∗ (x ) ⎦ , = ⎣ ε22 3 22 22 3 ∗ ∗ (x ) 0∗ 2ε12 2κ12 2ε12 3 ⎡
where ε0∗
(15)
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⎡
and
C˜ 11 ⎣ Q = C˜ 12 C˜ 16
C˜ 12 C˜ 22 C˜ 26
577
⎤ C˜ 16 C˜ 26 ⎦ . C˜ 66
(16)
In writing Eq. (16), we have adopted the contracted notations for C˜ αβωρ [4]. Even though the original ∗ = ε ∗ (x ) is in fact a three-dimensional one, it Eshelby’s inclusion with imposed in-plane eigenstrains εαβ αβ 3 reduces to two dimensional in the context of Kirchhoff plate theory. Here, it should be stressed that the midplane eigenstrains and eigencurvatures introduced in Eq. (13) are not the same as those in Beom and Earmme [2] and Beom and Kim [3]. Let u∗ be the additional generalized displacement vector ciated with the mid-plane eigenstrains and eigencurvatures given by ⎡ 0∗ 0∗ x ⎤ ε11 x1 + ε12 2 ⎥ ⎢ 0∗ 0∗ ⎢ ε12 x1 + ε22 x2 ⎥ ⎥ u∗ = ⎢ (17) ⎢ κ ∗ x1 + κ ∗ x2 ⎥ . 12 ⎦ ⎣ 11 ∗ x + κ∗ x κ12 1 22 2 The total generalized displacement vector and the generalized stress function vector should be continuous across the interface L between the inclusion and the surrounding material. Thus, we have u0 + u∗ = u1 , 0 = 1 , z ≡ x1 + ix2 ∈ L ,
(18)
which can be expressed in terms of the two analytic functions f0 (z) and f1 (z) as Af 1 (z) + Af1 (z) = Af 0 (z) + Af0 (z) + u∗ , Bf 1 (z) + Bf1 (z) = Bf 0 (z) + Bf0 (z),
z ∈ L.
Using the orthogonality relations in Eq. (8a, 8b), we can define a new vector function g(z) by f (z)+ < z α > c+ < Pα (z α ) > d, z ∈ S0 , g(z) = 0 f1 (z)− < Dα (z α ) − Pα (z α ) > d, z ∈ S1 ,
(19)
(20)
where z¯ α = Dα (z α ) along the interface L [16,17]. In addition, Dα (z α ) → Pα (z α ) + O(1) as |z α | → ∞. The two complex vectors c and d appearing in Eq. (20) are related to the mid-plane uniform eigenstrains and eigencurvatures by p¯ α 1 > BT ε∗1 − < > BT ε∗2 , p¯ α − pα p¯ α − pα 1 pα d=< > BT ε∗2 − < > BT ε∗1 , p¯ α − pα p¯ α − pα c=
c− < Pα (z α ) > d, f1 (z) =< Dα (z α ) − Pα (z α ) > d,
z ∈ S0 , z ∈ S1 .
(23)
0 = 1 (u By using Eq. (23)1 and the relationship that εαβ 2 α,β + u β,α ) and καβ = −w,αβ , the total mid-plane strains and total mid-plane curvatures within the inclusion S0 can thus be conveniently calculated as
0 0∗ ε11 = ε11 − 2i1T Re Ac + A < Pα (z α ) > d ,
0 0∗ ε22 = ε22 − 2i2T Re A < pα > c + A < pα Pα (z α ) > d ,
0 0∗ ε12 = ε12 − i2T Re Ac + A < Pα (z α ) > d − i1T Re A < pα > c + A < pα Pα (z α ) > d ,
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κ22
∗ κ11 = κ11 − 2i3T Re Ac + A < Pα (z α ) > d ,
∗ = κ22 − 2i4T Re A < pα > c + A < pα Pα (z α ) > d ,
∗ κ12 = κ12 − 2i4T Re Ac + A < Pα (z α ) > d ,
(24)
where i K is a unit 4-vector defined as (i K ) L = δ K L . with δ K L . being the Kronecker delta. Equation (24) can be more concisely expressed as 0 0∗ ∗ 0∗ ∗ εαβ = Sαβωρ εωρ + Tαβωρ κωρ , καβ = Jαβωρ εωρ + K αβωρ κωρ ,
(25)
where the tensors Sαβωρ , Tαβωρ , Jαβωρ , K αβωρ are usually functions of the coordinates x1 and x2 in view of the fact that Pα (z α ) is a function of z α . Using the identities in Eqs. (9) and (10), the asymptotic behavior of f1 (z) can be finally derived as A < (z α )−1 > B−1 N1T L − N3 S − iN3 ε∗1 − Lε∗2 + O < (z α )−2 > , as |z| → ∞ (26) f1 (z) = 4π where A is the area of the inclusion. It follows from the above asymptotic expansion and Eqs. (9) and (11) that the real-form expressions of the far-field behaviors of the generalized displacement and stress function vectors can then be obtained as ⎛ ⎞ −1 −1 3 A ⎝ sinθ N1 (θ )L−1 N3 − cos θ L N−1 ε∗ u= (27) + cos θ SL − sin θ N1 (θ )SL + sinθ N2 (θ ) (N3 S − N1T L) 1 ⎠ + O(r −2 ), 2πr + [cos θ S − sin θ N (θ )S + sinθ N (θ )L] ε∗ 1 2 2 T (θ ) + sinθ N (θ )SL−1 (N T L − N S) + sinθ N (θ )L−1 N ε∗ A cos θ I − sin θ N 3 3 3 3 1 1 1 + O(r −2 ), = 2πr − cos θ L − sin θ N1T (θ )L + sinθ N3 (θ )S ε∗2 (28) where we have written x1 = r cos θ, x2 = r sin θ . Equations (27) and (28) are valid for any mathematically degenerate material (in which there are repeated eigenvalues and less than four independent eigenvectors in Eq. (3) [4]) since the Stroh eigenvalues and eigenvectors are absent in these two expressions. 4 An Eshelby inclusion in a semi-infinite anisotropic plate In this section, we consider a semi-infinite laminated anisotropic thin plate (x2 > 0) containing a subdomain which has the same extensional, coupling and bending stiffnesses as the surrounding material and which ∗ = ε ∗ (x ), as illustrated in Fig. 2. In the context of Kirchhoff plate theory, undergoes in-plane eigenstrains εαβ αβ 3 the original three-dimensional Eshelby problem reduces to one of a two-dimensional inclusion of any shape 0∗ and eigencurvatures κ ∗ in a semi-infinite plate. Let S and S denote with uniform mid-plane eigenstrains εαβ 0 1 αβ the subdomain and its supplement to the upper semi-infinite plate, L the perfect interface separating S0 and S1 . If the auxiliary function vector g(z) defined by Eq. (20) is introduced, it can be easily verified that the introduced g(z) is analytic, continuous and single-valued everywhere in the whole upper half-plane including the point at infinity. Through satisfaction of the boundary condition on the edge of the semi-infinite plate, we can then obtain solutions g(z). In the following three subsections, we will derive solutions for a semi-infinite plate in each of the cases of a free, rigidly clamped and simply supported edge. 4.1 A semi-infinite plate with a free edge In Kirchhoff plate theory, the boundary condition for a free edge at x2 = 0 is given by N12 = 0, which is equivalent to
N22 = 0, V2 = 0, = 0, at x2 = 0.
M22 = 0,
(29) (30)
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Fig. 2 An inclusion of arbitrary shape in a semi-infinite laminated anisotropic plate
An argument for the equivalence between Eqs. (29) and (30) can be found in [4]. Condition (30) can be expressed in terms of f1 (z) as Bf 1 (z) + Bf1 (z) = 0, at x2 = 0. (31) By making use of the introduced auxiliary function vector g(z) defined in Eq. (20) and the method of analytic continuation [16], Eq. (31) can be easily solved to arrive at the following expressions of f0 (z) and f1 (z): 4 ¯ k d, ¯ z ∈ S0 , < P¯k (z α ) − D¯ k (z α ) >B−1 BI f0 (z) = − < z α > c− < Pα (z α ) > d + f1 (z) =< Dα (z α ) − Pα (z α ) > d +
4
k=1
(32) −1 ¯
< P¯k (z α ) − D¯ k (z α ) >B
¯ BIk d,
z ∈ S1 .
k=1
The remote asymptotic behavior of f1 (z) can be derived as f1 (z) = −
iA < (z α )−1 > B−1 N3 ε∗1 + O(< (z α )−2 >), as |z| → ∞, 2π
(33)
0∗ , ε 0∗ and κ ∗ do not contribute to the above asymptotic expansion. which indicates that the three components ε12 22 22 0∗ The reason why ε12 , the second component of ε∗1 , has no contribution to the expansion is due to the fact that the second column of N3 is zero [see Eq. (4)]. Using the identities in Eqs. (9) and (11), we can further derive real representations of the far-field expansions of u and :
A [cos θ I − sin θ N1 (θ )] L−1 N3 ε∗1 + O(r −2 ), πr A = sinθ N3 (θ )L−1 N3 ε∗1 + O(r −2 ). πr u=−
(34)
4.2 A semi-infinite plate with a rigidly clamped edge The boundary condition for a rigidly clamped edge at x2 = 0 is given by u 1 = 0, u 2 = 0, w = 0, ϑ2 = 0,
(35)
which is equivalent to u = 0,
at x2 = 0.
(36)
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Condition (36) can be expressed in terms of f1 (z) as Af 1 (z) + Af1 (z) = 0, at
x2 = 0.
(37)
By making use of the introduced auxiliary function vector g(z) defined in Eq. (20) and the method of analytic continuation [16], Eq. (37) can be readily solved to arrive at the following expressions of f0 (z) and f1 (z): 4 ¯ k d, ¯ < P¯k (z α ) − D¯ k (z α ) >A−1 AI z ∈ S0 , f0 (z) = − < z α > c− < Pα (z α ) > d + f1 (z) =< Dα (z α ) − Pα (z α ) > d +
4
k=1
(38)
< P¯k (z α ) − D¯ k (z α ) >A
−1
¯ k d, ¯ AI
z ∈ S1 ,
k=1
where
I1 = diag 1
0
0
0 , I2 = diag 0
1
0
0 ,
I3 = diag 0
0
1
0 , I4 = diag 0
0
0
1 .
(39)
The remote asymptotic behavior of f1 (z) is found to be iA < (z α )−1 > A−1 L−1 N1T L − L−1 N3 S − SL−1 N3 ε∗1 − ε∗2 2π + O < (z α )−2 > , as |z| → ∞
f1 (z) = −
(40)
Using the identities in Eqs. (9) and (11), we can again derive (real) far-field expansions of u and (which we do not include here for sake of brevity).
4.3 A semi-infinite plate with a simply supported edge The boundary condition for a simply supported edge at x2 = 0 is given by N12 = 0 or u 1 = 0,
N22 = 0 or u 2 = 0, w = 0,
M22 = 0,
(41)
which is equivalent to Iu u + I = 0,
at x2 = 0,
(42)
where Iu and I are 4 × 4 diagonal matrices whose four diagonal elements are either one or zero, and satisfy the condition Iu + I = I, Iu I = 0. (43) The specific expressions of Iu and I can be found in [4]. Condition (42) can be expressed in terms of f1 (z) as (44) (Iu A + I B)f1 (z) + (Iu A + I B)f1 (z) = 0, at x2 = 0. Equation (44) can be easily solved to arrive at the following expressions of f0 (z) and f1 (z): f0 (z) = − < z α > c− < Pα (z α ) > d +
4
< P¯k (z α ) − D¯ k (z α ) >
k=1
¯ z ∈ S0 , ×(Iu A + I B)−1 (Iu A + I B)Ik d, 4 f1 (z) =< Dα (z α ) − Pα (z α ) > d + < P¯k (z α ) − D¯ k (z α ) > k=1
¯ z ∈ S1 . ×(Iu A + I B)−1 (Iu A + I B)Ik d,
(45)
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Fig. 3 An inclusion of arbitrary shape in one of two bonded semi-infinite laminated anisotropic plates
5 An Eshelby inclusion in one of two bonded semi-infinite anisotropic plates In this section, we consider two dissimilar semi-infinite laminated anisotropic thin plates perfectly bonded along the real axis. The upper semi-infinite plate contains a subdomain of any shape which has the same extensional, ∗ = coupling and bending stiffnesses as the surrounding material and which undergoes in-plane eigenstrains εαβ ∗ εαβ (x3 ), as illustrated in Fig. 3. In the context of Kirchhoff plate theory, the original three-dimensional Eshelby’s problem again reduces to one of a two-dimensional inclusion of any shape with uniform mid-plane eigenstrains 0∗ and eigencurvatures κ ∗ in one of two bonded dissimilar semi-infinite plates. Let S and S denote the εαβ 0 1 αβ subdomain and its supplement to the upper semi-infinite plate, L the perfect interface separating S0 and S1 and S2 the lower semi-infinite plate. All quantities in S0 , S1 and S2 will be denoted by the subscripts 0, 1 and 2 or the superscripts (0), (1) and (2). The continuity conditions of the generalized displacement and stress function vectors across the perfect interface x2 = 0 can be expressed in terms of the two analytic functions f1 (z) and f2 (z) as A1 f1 (z) + A1 f1 (z) = A2 f2 (z) + A2 f2 (z), on x2 = 0 B1 f1 (z) + B1 f1 (z) = B2 f2 (z) + B2 f2 (z),
(46)
which can be easily solved by means of analytic continuation [17] to arrive at the following expressions for f0 (z), f1 (z) and f2 (z): f0 (z) = − < z α > c− < Pα (z α ) > d + ×B−1 1
¯ ∗ L−1 − I B¯ 1 Ik d, ¯ 2M 1
f1 (z) =< Dα (z α ) − Pα (z α ) > d +
4
4
< D¯ k (z α ) − P¯k (z α ) >
k=1
z ∈ S0 ,
(47)
¯ ∗ L−1 − I B¯ 1 Ik d, ¯ 2M < D¯ k (z α ) − P¯k (z α ) > B−1 1 1
z ∈ S1 ,
k=1
(48) f2 (z) = 2
4
−1 < Dk (z α∗ ) − Pk (z α∗ ) >B−1 2 M∗ L1 B1 Ik d,
z ∈ S2 ,
(49)
k=1
where the vectors c and d are given by Eq. (20) with the eigenvalues pα and the matrix B pertaining to the upper semi-infinite plate, and M∗ is the 4 × 4 Hermitian matrix defined by ¯ −1 + M−1 = L−1 + L−1 + i(S1 L−1 − S2 L−1 ), M∗−1 = M 1 2 1 2 1 2
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The far-field expressions of N12 , N22 , M22 and V2 along the real axis can be derived as ⎤ ⎡ N12 T ∗ −1 (1)T A 1 (1) (1) −1 ∗ ˜ 2 − LL ˜ ˜ ⎣ N22 ⎦ ∼ i 1 i2 i4 ε1 + O Lε N1 L1 − N3 S1 − WL = 1 1 N3 2 π x1 x13 M22 2 A ˜ ∗ ˜ −1 (1)T (1) ˜ −1 N(1) ε∗1 N1 L1 − N3 S1 − WL V2 ∼ = − 3 i3T Lε 2 − LL1 3 1 π x1 1 , |x1 | → ∞, and x2 = 0 +O x14 where
and
(50)
,
(51)
−1 ˜ = D−1 WL, ˜ , W L˜ = D − WT D−1 W
(52)
D = L1−1 + L2−1 , W = S1 L1−1 − S2 L2−1 .
(53)
In addition, we find that f1 (z) =
A ¯ −1 L−1 N(1)T L1 − N(1) S1 + iN(1) ε∗1 − L1 ε∗2 − iN(1) ε∗1 N < (z α )−1 > B−1 1 3 3 3 1 1 2π + O < (z α )−2 > , as |z| → ∞
(54)
A (1)T (1) (1) −1 −1 N1 L1 − N3 S1 − iN3 ε∗1 − L1 ε∗2 < (z α∗ )−1 > B−1 2 N L1 2π +O(< (z α∗ )−2 >), as |z| → ∞
(55)
and f2 (z) =
Again, the identities in Eqs. (9) and (11) can be used to derive real forms of the far-field expansions of u and in the upper and lower semi-infinite laminated anisotropic thin plates.
6 Examples In order to illustrate the application of the general solutions derived in the previous three sections, we present the solution of an elliptical inclusion in each of the cases of an infinite, semi-infinite and in one of two bonded semi-infinite anisotropic thin plates.
6.1 An elliptical inclusion in an infinite anisotropic plate We consider an elliptical inclusion in an infinite anisotropic plate. The inclusion occupies the elliptical region: S0 : x12 /a 2 + x22 /b2 ≤ 1 . In this case Dk (z), Pk (z) and Dk (z) − Pk (z) can be explicitly determined as ! a 2 + | p k |2 b 2 i ( pk − p¯ k ) ab 2 2 2 b2 , z + z − a + p k 2 2 a 2 + pk b 2 a 2 + pk b 2 a − i p¯ k b Pk (z) = z, a − i pk b i ( p¯ k − pk ) ab ! Dk (z) − Pk (z) = . z + z 2 − a 2 + pk2 b2 Dk (z) =
(56) (57) (58)
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By substituting Eqs. (56–58) into Eq. (23), we arrive at the following explicit expressions for f0 (z) and f1 (z): i p¯α b > d , z ∈ S , f0 (z) = − < z α > c+ < a− 0 a−i pα b (59) ( p¯ α − pα )ab i √ f (z) =< > d, z∈S . 1
zα +
z α2 −(a 2 + pα2 b2 )
1
which clearly indicates that the stress resultants and bending moments are uniform within the elliptical inclusion. In particular, the uniform rigid body rotation within the elliptical inclusion is given by 12 =
1 (u 1,2 − u 2,1 ) 2
= i1T Re A < pα > c + A < pα
" " a − i p¯ α b a − i p¯ α b > d − i2T Re Ac + A < >d . a − i pα b a − i pα b
(60)
6.2 An elliptical inclusion in a semi-infinite anisotropic plate We consider an elliptical anisotropic plate x2 >0. The inclusion occupies the inclusion in a semi-infinite elliptical region: S0 : x12 /a 2 + (x2 − h)2 /b2 ≤ 1 , h > b >0. In this case Dk (z), Pk (z) and Dk (z) − Pk (z) are explicitly determined as ! a 2 + | p k |2 b 2 i( pk − p¯ k )ab Dk (z) = (z − p h) + p ¯ h + (z − pk h)2 − (a 2 + pk2 b2 ), (61) k k a 2 + pk2 b2 a 2 + pk2 b2 a( pk − p¯ k ) a − i p¯ k b z− h, (62) Pk (z) = a − i pk b a − i pk b i( p¯ k − pk )ab ! (63) Dk (z) − Pk (z) = . z − pk h + (z − pk h)2 − a 2 + pk2 b2 Equations (32), (38) and (45), now lead us to the following explicit expressions for f0 (z) and f1 (z): f0 (z) = − < z α > c− < +
4
ϒ ϒIk d, z ∈ S0 , 2 2 z α − p¯ k h + (z α − p¯ k h) − a 2 + p¯ k b2
i ( p¯ α − pα ) ab ! >d z α − pα h + (z α − pα h)2 − a 2 + pα2 b2 4
ϒ ϒIk d, z ∈ S1 . 2 2 z α − p¯ k h + (z α − p¯ k h) − a 2 + p¯ k b2
(64)
where ϒ = B for a free edge, ϒ = A for a rigidly clamped edge and ϒ = Iu A + I B for a simply supported edge. It is observed from the above expression of f0 (z) that the internal field inside the elliptical inclusion is no longer uniform due to the influence of the nearby edge. 6.3 An elliptical inclusion in one of two bonded semi-infinite anisotropic plates We consider an elliptical inclusion in the upper anisotropic plates. The one of two bonded semi-infinite inclusion occupies the elliptical region: S0 : x12 /a 2 + (x2 − h)2 /b2 ≤ 1 , h > b >0. Thus, Dk (z), Pk (z) and Dk (z) − Pk (z) are still determined by Eqs. (61–63). Consequently, by substituting the above specific expressions into Eqs. (47–49), we find f0 (z), f1 (z) and f2 (z) as follows: f0 (z) = − < z α > c−
d a − i pα b a − i pα b
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+
4
B 1 1 z α − p¯ k h + (z α − p¯ k h)2 − a 2 + p¯ k2 b2 (65)
f1 (z) =
d z α − pα h + (z α − pα h)2 − a 2 + pα2 b2 4
B−1 1 1 z α − p¯ k h + (z α − p¯ k h)2 − a 2 + p¯ k2 b2 (66)
f2 (z) = 2
4 k=1
B2 M∗ L1 B1 Ik d, z ∈ S2 . 2 ∗ ∗ 2 2 z α − pk h + z α − pk h − a + pk b
(67)
It is clear that the influence of the lower semi-infinite plate renders the internal field within the elliptical inclusion non-uniform.
7 Conclusions By using the newly developed Stroh octet formalism [4–7] and techniques adapted from [16,17], we have successfully derived solutions in simple form for an Eshelby inclusion with uniform mid-plane eigenstrains and eigencurvatures in infinite, semi-infinite and in one of two perfectly bonded semi-infinite laminated anisotropic thin plates. In particular, the far-field elastic fields induced by an Eshelby inclusion of any shape are obtained in real form. Acknowledgments We are grateful to referees for valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11272121), Innovation Program of Shanghai Municipal Education Commission, China (Grant No. 12ZZ058) and the Natural Sciences and Engineering Research Council of Canada.
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