A Closed-Form Solution for the Eshelby Tensor and

Xiaoqing Jin Leon M. Keer e-mail: [email protected]

Qian Wang Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208

A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion From the analytical formulation developed by Ju and Sun [1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574], it is seen that the exterior point Eshelby tensor for an ellipsoid inclusion possesses a minor symmetry. The solution to an elliptic cylindrical inclusion may be obtained as a special case of Ju and Sun’s solution. It is noted that the closed-form expression for the exterior-point Eshelby tensor by Kim and Lee [2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503] violates the minor symmetry. Due to the importance of the solution in micromechanics-based analysis and planeelasticity-related problems, in this work, the explicit analytical solution is rederived. Furthermore, the exterior-point Eshelby tensor is used to derive the explicit closed-form solution for the elastic field outside the inclusion, as well as to quantify the elastic field discontinuity across the interface. A benchmark problem is used to demonstrate a valuable application of the present solution in implementing the equivalent inclusion method. 关DOI: 10.1115/1.4003238兴 Keywords: inclusion, exterior field, exterior-point Eshelby tensor, micromechanics

1

Introduction

According to Eshelby 关1兴 and Mura 关2兴, the term inclusion refers to a subdomain, ⍀, in a material with identical elastic moduli, where an eigenstrain, ␧ⴱij, is prescribed in ⍀ and null in the reminder of the material 共matrix兲. The inclusion problem is of fundamental importance to both mechanics and materials science and is still an active topic today 关3,4兴. Since eigenstrain encompasses a broad range of nonelastic strains such as plastic strain, thermal strain, and misfit strain, Mura 关5兴 pointed out that the eigenstrain method provides a unified treatment for a variety of subjects: 共1兲 stress fields caused by nonelastic strains, 共2兲 stress disturbances due to inhomogeneities, 共3兲 average material properties, 共4兲 cracks, voids, or rigid reinforcement in materials, etc. More recently, inclusion analysis has been successfully applied in modeling nanostructures 关6,7兴, such as quantum dots 共QDs兲 and quantum wires 共QWRs兲. In these tiny semiconductors, the unique properties of quantum structures are closely governed by the strain field induced by the lattice mismatch 共eigenstrain兲 between the QD/QWR and the surrounding matrix. The most remarkable result found by Eshelby 关1兴 is that when an ellipsoidal inclusion in an isotropic infinite space is subjected to uniform eigenstrain, the induced interior strain and therefore the stress fields are also uniform. This amazing result was expressed as a linear relation between the interior strain field and the interior eigenstrains. This relation is called Eshelby’s tensor and is given in terms of complete elliptic integrals. However, the expressions for the exterior elastic field appear to be more complicated. In this study, we utilize the formulation developed by Ju and Sun 关8兴 and derive the closed form solution for the exterior-point Eshelby tensor when the ellipsoid becomes an elliptic cylinder.

2 Exterior-Point Eshelby Tensor of an Ellipsoidal Inclusion Consider an isotropic infinite body containing an ellipsoidal inclusion, which is subjected to eigenstrain, ␧ⴱij, and is bounded by x ix i aI2

Journal of Applied Mechanics

共1兲

where aI is the semi-axis of the ellipsoid. For convenience, Mura’s summation convention 关2兴 has been adopted in Eq. 共1兲: repeated lowercase indices are implicitly summed over 1 to 3 while upper case indices take on the corresponding lower case number but are not summed. When the eigenstrain is uniform, Eshelby found that the induced interior strain field 共inside ⍀兲 and therefore the stress field are also uniform. The total strain ␧ij, i.e., the summation of elastic strain and eigenstrain, is related to the eigenstrain as ⴱ ␧ij = Sijkl␧kl

共2兲

where Sijkl for the interior field is a constant depending on the elastic moduli and the semi-axes of the ellipsoid and is called the Eshelby tensor. However, the exterior elastic field 共outside ⍀兲 is no longer constant. For the purpose of distinction, the exteriorpoint Eshelby tensor is denoted as Gijkl 关8兴 since the expression for Gijkl 关2,9兴 is much more complicated than that of Sijkl. On the contrary, for an inclusion of any polygonal shape, it might be interesting to note that both the interior and exterior fields can be represented in a unified formulation 关7,10兴. For any point x located outside the inclusion, an imaginary ellipsoid 共Fig. 1兲 is constructed as 关2,8兴 x ix i aI2

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 10, 2010; final manuscript received December 8, 2010; accepted manuscript posted December 13, 2010; published online February 15, 2011. Assoc. Editor: Pradeep Sharma.

=1

+␭

=1

共3兲

where ␭ is the largest positive root of the above equation. To present the exterior-point Eshelby tensor Gijkl explicitly, Ju and Sun 关8兴 utilized the outward unit normal vector n៝ = n៝ 共n1 , n2 , n3兲 at

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1 1 I M = − J M 共␭兲 2 4␲ 1 1 I MN = − J MN共␭兲 J MN共␭兲 = 2 4␲

JM 共␭兲 =

共12兲

Note that the I-integrals, i.e., I M and I MN, can be expressed as standard elliptic integrals, which details are referred to Ref. 关2兴. It is seen from Eqs. 共4兲–共10兲 that the exterior-point Eshelby tensor possesses a minor symmetry Gijkl共x兲 = G jikl共x兲 = Gijlk共x兲

共13兲

Fig. 1 Schematic illustration of an ellipsoidal inclusion, the imaginary ellipsoid, and its outward unit normal vector n៝

Moreover, the interior-point Eshelby tensor Sijkl can be determined from Eqs. 共4兲–共6兲 by letting ␭ and n៝ vanish.

x on the imaginary ellipsoidal surface 共see Fig. 1兲. For ease of programming, Ju and Sun’s formulation is recast as follows:

3 Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion

Gijkl共x兲 = ␦ij␦kl ⫻

冋

冋

册

The results in the case of an elliptic cylindrical inclusion may be readily obtained from Sec. 2 by letting a3 → ⬁, and consequently

␯ 共2兲 Q共1兲共␭兲 + QIK 共␭兲 + 共␦ik␦ jl + ␦il␦ jk兲 1−␯ I

册

QI共1兲共␭兲 + Q共J1兲共␭兲 共2兲 + QIJ 共␭兲 + QI共3兲共␭兲共␦ijnknl 2

n3 = 0

+ ␦iln jnk兲 + Q共J3兲共␭兲共␦ jkninl + ␦ jlnink兲 + QK共3兲共␭兲共␦klnin j 共4兲 共5兲 + ␦kin jnl兲 + Qijkl + 关QIJKL 共␭兲 + Q共6兲共␭兲兴nin jnknl

共4兲

where Gijkl共x兲 is the exterior-point Eshelby tensor, ␯ is the Poisson’s ratio, and ␦ij is the Kronecker’s delta. The detailed expressions for the Q functions in 共4兲 are

共14兲

Moreover, for any exterior point x共x1 , x2兲 ␭ = 2 关x21 + x22 − a21 − a22 + 冑共x21 + x22 − a21 + a22兲2 + 4共a21 − a22兲x22兴 1

共15兲 and

Q共M1兲共␭兲 = J M 共␭兲 2兲 共␭兲 Q共MN

␳3 = 1

共5兲

1 1 = 关a2 J MN − J M 共␭兲兴 = 关a2 JNM − JN共␭兲兴 2共1 − ␯兲 N 2共1 − ␯兲 M 共2兲 共␭兲 = QNM

␳1 =

a1

冑a21 + ␭

␳2 =

a2

冑a22 + ␭

n1 =

m1

冑m21 + m22

n2 =

m2

冑m21 + m22 共16兲

共6兲 in which

␳1共␭兲␳2共␭兲␳3共␭兲 Q共M3兲共␭兲 = 关1 − ␳2M 共␭兲兴 2共1 − ␯兲 共4兲 =− Qijkl

+

共7兲 m1 =

冉

␳1共␭兲␳2共␭兲␳3共␭兲 ␦ikn jnl + ␦iln jnk + ␦ jkninl + ␦ jlnink 2 2␯ ␦klnin j 1−␯

共5兲 共␭兲 = QIJKL

冊

共8兲

J1共␭兲 =

␳1共␭兲␳2共␭兲␳3共␭兲 2 关␳I 共␭兲 + ␳2J 共␭兲 + ␳K2 共␭兲 + ␳L2 共␭兲兴 1−␯

␳1共␭兲␳2共␭兲␳3共␭兲 关␳m共␭兲␳m共␭兲 − 4␳2M 共␭兲nmnm − 5兴 2共1 − ␯兲 共10兲

wherein the above Eqs. 共7兲–共10兲

␳M 共␭兲 =

aM

冑a2M + ␭

共11兲

The functions J M and J MN in Eqs. 共5兲 and 共6兲 are related to the I-integrals of Ref. 关2兴 and J-functions of Ref. 关2兴 as follows: 031009-2 / Vol. 78, MAY 2011

m2 =

x2 +␭

a22

共17兲

Using the results given in Ref. 关2兴, the functions J M and J MN are

␳21␳2a2 a 1␳ 2 + a 2␳ 1

J2共␭兲 =

J12共␭兲 = J21共␭兲 =

共9兲 Q共6兲共␭兲 =

x1 +␭

a21

J11共␭兲 =

␳41␳2a2 共2a1␳2 + a2␳1兲 3a21 共a1␳2 + a2␳1兲2

␳22␳1a1 a 1␳ 2 + a 2␳ 1

␳31␳32 共a1␳2 + a2␳1兲2

J22共␭兲 =

共18兲

共19兲

␳42␳1a1 共2a2␳1 + a1␳2兲 3a22 共a1␳2 + a2␳1兲2 共20兲

J3共␭兲 = J33共␭兲 = J13共␭兲 = J31共␭兲 = J23共␭兲 = J32共␭兲 = 0 共21兲 In view of the minor symmetry property of Eq. 共13兲, the individual components of Eshelby tensor are given by the following matrix 关G兴: Transactions of the ASME

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关G兴 =

=

冤 冤

G1111 G1122 G1133 G1123 G1131 G1112 G2211 G2222 G2233 G2223 G2231 G2212 G3311 G3322 G3333 G3323 G3331 G3312 G2311 G2322 G2333 G2323 G2331 G2312 G3111 G3122 G3133 G3123 G3131 G3112 G1211 G1222 G1233 G1223 G1231 G1212 G1111 G1122 G1133

0

0

G1112

G2211 G2222 G2233

0

0

G2212

0

0

0

0

0

0

0

0

0

G2323 G2331

0

0

0

G3123 G3131

G1211 G1222 G1233

0

0

0 0 G1212

冥 冥

G1122共x兲 =

G1212共x兲 =

共22兲

共23兲

G2233共x兲 =

␯ 关J2共␭兲 − ␳1␳2n22兴 1−␯

共24兲

1

共25兲

G3131共x兲 = 2 关J1共␭兲 − ␳1␳2n21兴

1

共26兲

␯ ␳ 1␳ 2n 1n 2 G1233共x兲 = − 1−␯

共27兲

␳ 1␳ 2n 1n 2 关1 + 2␯ − 3␳21 + 共6␳21 + 2␳22 + T6兲n21兴 共29兲 2共1 − ␯兲

G2212共x兲 =

␳ 1␳ 2n 1n 2 关1 + 2␯ − 3␳22 + 共6␳22 + 2␳21 + T6兲n22兴 共30兲 2共1 − ␯兲

G1211共x兲 =

␳ 1␳ 2n 1n 2 关1 − 3␳21 + 共6␳21 + 2␳22 + T6兲n21兴 2共1 − ␯兲

共31兲

G1222共x兲 =

␳ 1␳ 2n 1n 2 关1 − 3␳22 + 共2␳21 + 6␳22 + T6兲n22兴 2共1 − ␯兲

共32兲

共1 − 2␯兲J1共␭兲 + 3a21J11共␭兲 2共1 − ␯兲 +

G2222共x兲 =

共8␳21

+

2共1 − ␯兲 +

G2211共x兲 =

+

␳1␳2n21 关2 + 2␯ − 6␳21 2共1 − ␯兲

T6兲n21兴

共1 − 2␯兲J2共␭兲 + 3a22J22共␭兲 共8␳22

+

共33兲 +

␳1␳2n22 关2 + 2␯ − 6␳22 2共1 − ␯兲

T6兲n22兴

共2␯ − 1兲J2共␭兲 + a21J12共␭兲 2共1 − ␯兲

共34兲 +

␳ 1␳ 2 关共1 − ␳22兲n21 2共1 − ␯兲

+ 共1 − 2␯ − ␳21兲n22 + 共4␳21 + 4␳22 + T6兲n21n22兴 Journal of Applied Mechanics

4共1 − ␯兲

␳22兲n21

+

共35兲

共36兲

␳ 1␳ 2 关共␯ 2共1 − ␯兲

+ 共␯ − ␳21兲n22 + 共4␳21 + 4␳22 + T6兲n21n22兴

共37兲

wherein the above Eqs. 共29兲–共37兲, 共38兲

The expressions derived in this section constitute explicit closedform solution to the exterior-point Eshelby tensor for an elliptic cylindrical inclusion. The indicial results, cf. Equations 共23兲–共37兲, demonstrate interesting permutations with respect to subindices 共1, 2兲. Furthermore, the interior-point Eshelby tensor can be deduced from Sec. 2 by setting ␭ = 0 and n1 = n2 = n3 = 0. The results 共cf. the Appendix兲 are identical to those of Mura 共see Eq. 共11.22兲 of Ref. 关2兴兲. It is of interest to point out that for any exterior point x located outside the elliptic cylindrical inclusion, the following identity holds: G1122共x兲 + G2211共x兲 − 2G1212共x兲 ⬅ 0

共39兲

On the other hand, for any interior point x located inside the elliptical cylindrical inclusion

G1122共x兲 + G2211共x兲 − 2G1212共x兲 ⬅

共28兲

G1112共x兲 =

G1111共x兲 =

␳ 1␳ 2 关共1 − ␳21兲n22 2共1 − ␯兲

T6 = ␳21 + ␳22 − 4␳21n21 − 4␳22n22 − 4

␯ 关J1共␭兲 − ␳1␳2n21兴 1−␯

␳ 1␳ 2 n 1n 2 2

+

共1 − 2␯兲关J1共␭兲 + J2共␭兲兴 + 共a21 + a22兲J12 −

G1133共x兲 =

G3123共x兲 = G2331共x兲 = −

2共1 − ␯兲

+ 共1 − 2␯ − ␳22兲n21 + 共4␳21 + 4␳22 + T6兲n21n22兴

where the zeros in Eq. 共22兲 are derived by substituting Eqs. 共14兲 and 共21兲 into 共4兲. After some algebraic simplification, the nonzero components on the right hand side of Eq. 共22兲 are obtained as the following closed-form solutions:

G2323共x兲 = 2 关J2共␭兲 − ␳1␳2n22兴

共2␯ − 1兲J1共␭兲 + a22J12共␭兲

4

2␯ − 1 1−␯

共40兲

Results and Discussions

4.1 Exterior Elastic Field. The exterior-point Eshelby tensor obtained in Sec. 3 may be utilized to drive the elastic field outside the cylindrical elliptic inclusion subjected to uniform eigenstrain. In what follows, the uniform eigenstrain components are considered to be symmetric ␧ⴱ23 = ␧ⴱ23 =

␥ⴱ23 2

␧ⴱ31 = ␧ⴱ31 =

␥ⴱ31 2

␧ⴱ12 = ␧ⴱ21 =

␥ⴱ12 2

共41兲

The total strain or equivalently the elastic strain outside the elliptic cylindrical inclusion maybe derived from Eq. 共2兲 by replacing Sijkl with Gijkl, and the final results are represented in the matrix form as MAY 2011, Vol. 78 / 031009-3

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冤冥 冤冥 冤 ␧11

␧ⴱ11

␧22

␧ⴱ22

␧33

␧23

= 关G兴

␧31 ␧12

=

H1111 =

␧ⴱ33

共47兲

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

H2222 =

G1111 G1122 G1133

0

0

G1112

G2211 G2222 G2233

0

0

G2212

0

0

0

0

0

0

0

0

0

G2323 G2331

0

0

0

0

G3123 G3131

0

0

G1211 G1222 G1233

0

G1212

冥冤 冥 ␧ⴱ11 ␧ⴱ22 ␧ⴱ33 ␥ⴱ23 ␥ⴱ31 ␥ⴱ12

共42兲 where the nonzero Gijkl components are given in Eq. 共23兲–共27兲. It is seen from Eq. 共42兲 that the total strain ␧33, which equals to the elastic strain e33, vanishes outside the inclusion. On the other hand, from the results for the interior field 关2兴, the total strain ␧33 also vanishes but the elastic strain e33 equals to −␧ⴱ33 inside the inclusion. According to Hooke’s law, the exterior stress field is ⴱ ␴ij = Cijkl␧kl = CijklGklmn␧mn

ⴱ ␴ij = Hijkl␧kl

共44兲

After the individual components of Hijkl are derived, Eq. 共44兲 may be equivalently represented in the matrix form as

冤冥 冤冥 冤 =

␮ 兵J2共␭兲 + 3a22J22共␭兲 + ␳1␳2n22关2 − 6␳22 + 共8␳22 + T6兲n22兴其 1−␯ 共48兲

␧ⴱ11 ␧ⴱ22 ␧ⴱ33 ␥ⴱ23 ␥ⴱ31 ␥ⴱ12

+ 共4␳21 + 4␳22 + T6兲n21n22兴其

共49兲

It can be shown that H1212 = H1122 = H2211. More importantly, it is noted that Hijkl possesses both minor and major symmetries 共50兲

Hijkl = H jikl = Hijlk = Hklij

Consequently, the matrix 关H兴 in Eq. 共45兲 is symmetric as opposed to the nonsymmetric matrix 关G兴 in Eq. 共42兲. 4.2 Discontinuities Across the Interface. By comparing the preceding exterior elastic field solution with the interior field listed in the Appendix, one notes that the stress and strain fields will suffer discontinuities across the interface between the inclusion and the matrix. The jump conditions at the interface are denoted as ⌬␧ij = ␧ij共out兲 − ␧ij共in兲

⌬␴ij = ␴ij共out兲 − ␴ij共in兲

共51兲

where the superscripts 共out兲 and 共in兲 represent quantities just outside and just inside the inclusion, respectively. It is then straightforward to obtain the jump in the total strain

冤 冥冤 ⌬␧11

D1111 D1122 D1133

0

0

D1112

⌬␧22

D2211 D2222 D2233

0

0

D2212

0

0

D3312

⌬␧33

⌬␧23

=

D3311 D3322

⌬␧31

⌬␧12

0

0

0

0

D2323 D2331

0

0

0

D3123 D3131 0

D1211 D1222 D1233

0

0 0 D1212

冥冤 冥 ␧ⴱ11

␧ⴱ22

␧ⴱ33

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

共52兲

where

H1111 H1122 H1133

0

0

H1112

H2211 H2222 H2233

0

0

H2212

0

0

H3312

H3311 H3322

0

0

0

0

H2323 H2331

0

0

0

0

H3123 H3131

0

0

H1211 H1222 H1233

0

H1212

冥冤 冥 ␧ⴱ11

D1111 =

n21共␯ − 1 − n22兲 1−␯

D2222 =

D1122 = −

n21共␯ − 1 + n21兲 1−␯

D2211 = −

␧ⴱ22

␧ⴱ33

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

共45兲

D1112 =

n1n2共␯ − 1 + n21兲 1−␯

H1112 = H1211 = 2␮G1211共x兲

H2212 = H1222 = 2␮G1222共x兲

H3312 = H1233 = 2␮G1233共x兲

H2331 = H3123 = 2␮G3123共x兲

H3311 = H1133 = 2␮G1133共x兲

H3322 = H2233 = 2␮G2233共x兲

H3131 = 2␮G3131共x兲

031009-4 / Vol. 78, MAY 2011

␯n22 1−␯

D1233 = −

␯ n 1n 2 1−␯

D2331 = D3123 =

n 1n 2 2

D2323 = −

n22 2

D3131 = −

n21 2

n1n32 1−␯

D1222 = −

n2n31 1−␯

D2233 = −

D1212 =

␯ − 1 + 2n21n22 2共1 − ␯兲 共53兲

H1212 = 2␮G1212共x兲 共46兲

n22共␯ − 1 + n22兲 1−␯

n1n2共␯ − 1 + n22兲 1−␯

␯n21 1−␯

D1211 = −

n22共␯ − 1 − n21兲 1−␯

D2212 =

D1133 = −

where

H2323 = 2␮G2323共x兲

␮ 兵a2J12共␭兲 − J2共␭兲 + ␳1␳2关1 − ␳22n21 − ␳21n22 1−␯ 1

H1122 = H2211 =

共43兲

where Cijkl = ␭␦ij␦kl + ␮共␦ik␦ jl + ␦il␦ jk兲, ␮ is the shear modulus, and Láme constant ␭ = 2␮␯ / 共1 − 2␯兲. Analogous to Eq. 共2兲, the tensorial equation 共43兲 may be rewritten as

␴11 ␴22 ␴33 = 关H兴 ␴23 ␴31 ␴12

␮ 兵J1共␭兲 + 3a21J11共␭兲 + ␳1␳2n21关2 − 6␳21 + 共8␳21 + T6兲n21兴其 1−␯

Moreover, the stress jump is Transactions of the ASME

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冤冥 ⌬␴11

⌬␴22

⌬␴33

⌬␴23

=

2␮ 1−␯

⌬␴31

⌬␴12

冤

n42

n21n22

␯n22

0

0

− n1n32

n21n22 ␯n22

n41 ␯n21

␯n21

0

0

− n31n2

1

0

0

− ␯ n 1n 2

0

0

0

1−␯ 2 n 2 1

0

0

0

−

1−␯ n 1n 2 2

− n1n32 − n31n2 − ␯n1n2

0

−

1−␯ n 1n 2 2

0

1−␯ 2 n 2 2

0

0

n21n22

The above Eqs. 共52兲 and 共54兲 are useful for evaluating the elastic field just outside the inclusion when the stress just inside the inclusion is known since the interior filed solution 共see the Appendix兲 may be much simpler. Equation 共54兲 might also be of help in computing the stress concentration factors just outside an elliptical inclusion or inhomogeneity. Furthermore, it is interesting to note that the jump conditions, Eqs. 共52兲 and 共54兲, are explicitly presented solely in terms of the material constants and the outward unit normal of the interface, as first evidenced by Hill 关11兴. 4.3 Benchmark Example. Finally, as a benchmark problem, the stress field of a plane strain elliptical cavity 共Fig. 2兲 subjected to a remote uniform isotropic loading, i.e., ␴⬁11 = ␴⬁22 = ␴0, may be solved by utilizing the present solution in view of the equivalent inclusion method 共EIM兲 关1,2兴. The detailed implementation of the EIM is omitted here for simplicity since this is a standard procedure and has been discussed in details in Ref. 关2兴. The nonvanishing equivalent eigenstrain components in this case are found to be ␧ⴱ11 =

1 − ␯ a2 ␴0 ␮ a1

␧ⴱ22 =

1 − ␯ a1 ␴0 ␮ a2

共55兲

Accordingly, the stress field outside the cavity is

␴11 = ␴0 + H1111␧ⴱ11 + H1122␧ⴱ22 ␴22 = ␴0 + H2211␧ⴱ11 + H2222␧ⴱ22 ␴12 = H1211␧ⴱ11 + H1222␧ⴱ22 ␴33 = ␯共␴11 + ␴22兲

共56兲

Using the results of Eqs. 共47兲–共49兲 and 共55兲 on the x-axis outside the elliptical cavity 共兩x1兩 ⬎ a1 , x2 = 0兲, the stresses of Eq. 共56兲 may be simplified as

冥冤 冥 ␧ⴱ11

␧ⴱ22

␧ⴱ33

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

␴11 = ␴0 ␴22 = ␴0

共54兲

x1共␻22 − a22兲

␻32 x1共␻22 + a22兲

␻32

␴33 = 2␯␴0

x1 ␻2

共57兲

where

␻2 = 冑x21 − a21 + a22

共58兲

It is seen that the above results, Eqs. 共57兲 and 共58兲, agree with the known solution 关12兴.

5

Concluding Remarks

The inclusion problem is of fundamental importance to both mechanics and materials science and is the reason for this note. The formulation presented in this work may be conveniently enforced by computer programming, and Eqs. 共39兲 and 共40兲 can be served as benchmarks during the computer code development. The results given by Kim and Lee 关13兴, however, contain fundamental errors since their expressions lack the minor symmetry of Eq. 共13兲, and except for G1133共x兲 and G2233共x兲, all the other components in Ref. 关13兴 are found to be in error. Furthermore, their interior field is not in agreement with Mura’s results 共see Eq. 共11.22兲 of Ref. 关2兴兲. It is also seen that the exterior-point Eshelby tensor is directly related to the elastic field outside an elliptic cylindrical inclusion. The present solution may be integrated with the equivalent inclusion method to provide a powerful alternative in solving certain plane elliptic inhomogeneity problems. This work also demonstrates that the exterior-point Eshelby tensor is of minor symmetry but the fourth-rank tensor corresponding to stress field is of both major and minor symmetries. By utilizing the present solution, the jump conditions of the elastic field across the interface between the inclusion and the surrounding matrix may be established in explicit forms, which are represented in terms of only material constants and the outward unit normal of the interface.

Acknowledgment The results reported in this work were obtained in the course of a research project sponsored by the Timken Co. and U.S. Army TACOM. X.J. is grateful to Dr. Liz Fang for valuable discussions and encouragement. The authors would also like to acknowledge Prof. Jiann-Wen Woody Ju and Prof. Lizhi Sun for the helpful communication.

Appendix: Interior Elastic Field of an Elliptic Cylindrical Inclusion With Uniform Eigenstrain Fig. 2 A plane strain elliptical cavity subjected to a remote uniform isotropic loading

Journal of Applied Mechanics

The interior-point Eshelby tensor Sijkl can be deduced from Eqs. 共16兲–共38兲 by setting ␭ = 0 and n1 = n2 = n3 = 0. Considering the MAY 2011, Vol. 78 / 031009-5

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symmetric eigenstrain of Eq. 共41兲, the nonvanishing components of Sijkl can be used to derive the interior total strain components, which are presented in the matrix form as follows:

冤冥 冤冥 冤 ␧ⴱ11 ␧ⴱ22 ␧ⴱ33 ␥ⴱ23 ␥ⴱ31 ␥ⴱ12

␧11 ␧22 ␧33 ␧23

= 关S兴

␧31 ␧12

S1111 S1122 S1133

0

0

0

S2211 S2222 S2233

0

0

0

=

0

0

0

0

0

0

0

0

0

S2323

0

0

0

0

0

0

S3131

0

0

0

0

0

0

S1212

冤冥 冤冥 冤 ␴11 ␴22 ␴33 = 关T兴 ␴23 ␴31 ␴12

=

冥冤 冥 ␧ⴱ11

␧ⴱ22

␧ⴱ33

␧ⴱ11

␧ⴱ22

␧ⴱ33

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

T1111 T1122 T1133

0

0

0

T2211 T2222 T2233

0

0

0

T3311 T3322 T3333

0

0

0 0

0

0

0

T2323

0

0

0

0

0

T3131

0

0

0

0

0

0

T1212

T1111 = −

␮ a1共2a1 + a2兲 1 − ␯ 共a1 + a2兲2

T2222 = −

where

冋

册

冋

册

a21 + 2a1a2 1 a1 S2222 = 2 + 共1 − 2␯兲 2共1 − ␯兲 共a1 + a2兲 a1 + a2

␧ⴱ33

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

where

T1122 = T2211 = T1212 = −

a22 + 2a1a2 1 a2 + 共1 − 2␯兲 2共1 − ␯兲 共a1 + a2兲2 a1 + a2

␧ⴱ22

共A3兲

␥ⴱ23 ␥ⴱ31 ␥ⴱ12

共A1兲

S1111 =

冥冤 冥 ␧ⴱ11

T1133 = T3311 = − S2323 = − ␮

2␮␯ a1 1 − ␯ a1 + a2

a2 a1 + a2

␮ a2共2a2 + a1兲 1 − ␯ 共a1 + a2兲2

a 1a 2 ␮ 1 − ␯ 共a1 + a2兲2

T2233 = T3322 = −

S3131 = − ␮

a1 a1 + a2

2␮␯ a2 1 − ␯ a1 + a2

S3333 = −

2␮ 1−␯ 共A4兲

It is noted again that the above matrix 关T兴 is symmetric but 关S兴 is not.

References

冋

册

冋

册

a22 1 a2 S1122 = − 共1 − 2␯兲 2共1 − ␯兲 共a1 + a2兲2 a1 + a2

S2211 =

a21 1 a1 − 共1 − 2␯兲 2共1 − ␯兲 共a1 + a2兲2 a1 + a2

S1212 =

S1133 =

冋

a21 + a22 1 − 2␯ 1 + 2共1 − ␯兲 2共a1 + a2兲2 2

册

a2 ␯ 1 − ␯ a1 + a2

S2233 =

a1 ␯ 1 − ␯ a1 + a2

a1 2共a1 + a2兲

S3131 =

a2 2共a1 + a2兲

S2323 =

共A2兲

Accordingly, the stress components inside the cylindrical elliptic inclusion may be given as

031009-6 / Vol. 78, MAY 2011

关1兴 Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proc. R. Soc. London, 241共1226兲, pp. 376– 396. 关2兴 Mura, T., 1993, Micromechanics of Defects in Solids, 2nd ed., Kluwer Academic, Dordrecht. 关3兴 Liu, L. P., 2010, “Solutions to the Periodic Eshelby Inclusion Problem in Two Dimensions,” Math. Mech. Solids, 15共5兲, pp. 557–590. 关4兴 Zou, W. N., He, Q. C., Huang, M. J., and Zheng, Q. S., 2010, “Eshelby’s Problem of Non-Elliptical Inclusions,” J. Mech. Phys. Solids, 58共3兲, pp. 346– 372. 关5兴 Mura, T., 1988, “Inclusion Problems,” Appl. Mech. Rev., 41共1兲, pp. 15–20. 关6兴 Maranganit, R., and Sharma, P., 2005, “A Review of Strain Field Calculations in Embedded Quantum Dots and Wires, ” Handbook of Theoretical and Computational Nanotechnology, M. Rieth and W. Schommers, eds., American Scientific, Stevenson Ranch, CA, pp. 1–44, Chap. 118. 关7兴 Jin, X., Keer, L. M., and Wang, Q., 2009, “New Green’s Function for Stress Field and a Note of Its Application in Quantum-Wire Structures,” Int. J. Solids Struct., 46共21兲, pp. 3788–3798. 关8兴 Ju, J. W., and Sun, L. Z., 1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574. 关9兴 Eshelby, J. D., 1959, “The Elastic Field Outside an Ellipsoidal Inclusion,” Proc. R. Soc. London, 252共1271兲, pp. 561–569. 关10兴 Jin, X., Keer, L. M., and Wang, Q., 2010, “Analytical Solution for the Stress Field of Eshelby’s Inclusion of Polygonal Shape,” ASME, New York, pp. 487–489. 关11兴 Hill, R., 1961, “Discontinuity Relations in Mechanics of Solids,” Progress in Solid Mechanics, II, I. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, pp. 245–276. 关12兴 Maugis, D., 2000, Contact, Adhesion, and Rupture of Elastic Solids, Springer, Berlin. 关13兴 Kim, B. R., and Lee, H. K., 2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503.

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