e Shelby Tensor
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Eshelby Tensor (from Chapter 2, Micromechanics of Defects in Solids, T. Mura, 1987) In class, we have shown that the strain inside inclusion is * kl ijkl ij S ε ε = where ijnm jimn ijmn S S S = = is called the Eshelby tensor. It is a function of the elastic properties of the solid and the geometry of the ellipsoid. Generally, elliptic integrals are involved to calculate the components of the Eshelby tensor. For isotropic solids, ( ) ( ) 1 11 2 1 1111 1 8 2 1 1 8 3 I I a S ν π ν ν π − − + − = ( ) ( ) 1 12 2 2 1122 1 8 2 1 1 8 1 I I a S ν π ν ν π − − − − = ( ) ( ) 1 13 2 3 1133 1 8 2 1 1 8 1 I I a S ν π ν ν π − − − − = ( ) ( ) ( ) 2 1 12 2 2 2 1 1212 1 16 2 1 1 16 I I I a a S + − − + − + = ν π ν ν π All other non-zero components are obtained by the cyclic permutation of . The components which cannot be obtained by the cyclic permutation are zero; for instance, ( 3 , 2 , 1 ) 0 1232 1223 1112 = = = S S S . Assume , 3 2 1 a a a > > ( )( ) ( ) ( ) { } k k a a a a a a a I , E , F 4 2 1 2 3 2 1 2 2 2 1 3 2 1 1 θ θ π − − − = ( )( ) ( ) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − − − − = k a a a a a a a a a a a a I , E 4 3 1 2 1 2 3 2 1 2 2 1 2 3 2 1 2 3 2 2 3 2 1 3 θ π where ( k , F ) θ and ( k , E ) θ are elliptic integrals defined as ( ) ( ) φ φ θ θ d sin 1 , F 2 1 0 2 2 − ∫ − = k k ( ) ( ) φ φ θ θ d sin 1 , E 2 1 0 2 2 ∫ − = k k ( ) 2 1 2 1 2 3 1 1 sin a a − = − θ , ( ) ( ) [ ] 2 1 2 3 2 1 2 2 2 1 a a a a k − − = 1
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Transcript of e Shelby Tensor
Eshelby Tensor (from Chapter 2, Micromechanics of Defects in Solids, T. Mura, 1987)
In class, we have shown that the strain inside inclusion is
*klijklij S εε =
where ijnmjimnijmn SSS == is called the Eshelby tensor. It is a function of the elastic properties
of the solid and the geometry of the ellipsoid. Generally, elliptic integrals are involved to calculate the components of the Eshelby tensor. For isotropic solids,
( ) ( ) 111211111 18
21183 IIaS
νπν
νπ −−
+−
=
( ) ( ) 112221122 18
21181 IIaS
νπν
νπ −−
−−
=
( ) ( ) 113231133 18
21181 IIaS
νπν
νπ −−
−−
=
( ) ( ) ( )2112
22
21
1212 11621
116IIIaaS +
−−
+−
+=
νπν
νπ
All other non-zero components are obtained by the cyclic permutation of . The
components which cannot be obtained by the cyclic permutation are zero; for instance,
( 3,2,1 )
0123212231112 === SSS .
Assume , 321 aaa >>
( )( ) ( ) ( ){ }kkaaaa
aaaI ,E,F4212
321
22
21
3211 θθπ
−−−
=
( )( )( ) ( )
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
−−= k
aaaaa
aaaaaaaI ,E4
31
2123
212
2123
21
23
22
3213 θπ
where ( k,F )θ and ( k,E )θ are elliptic integrals defined as
( ) ( ) φφθθ
dsin1,F21
0
22−
∫ −= kk
( ) ( ) φφθθ
dsin1,E21
0
22∫ −= kk
( ) 2121
23
1 1sin aa−= −θ , ( ) ( )[ ] 2123
21
22
21 aaaak −−=
1
The remaining I -functions follow from the identities:
π4321 =++ III
21131211 43 aIII π=++
1132312
2211
21 33 IIaIaIa =++
( ) ( )22
211212 aaIII −−=
The components of Eshebly tensor become elementary functions for special shapes of inclusions, as listed below.
Sphere ( ): aaaa === 321
34321 π=== III
2312312332211 54 aIIIIII π======
( )νν−
−===
11557
333322221111 SSS
( )νν−−
======115
15332222111133331122331122 SSSSSS
( )νν−
−===
11554
313123231212 SSS
or
( ) ( ) ( )jminjnimmnijijmnS δδδδννδδ
νν
+−
−+
−−
=115
54115
15 (see classnotes)
The stress inside the inclusion is
( ) ( ) ( )*33
*22
*1111 115
152115
15211516 ε
ννμε
ννμε
νμσ
−+
−−+
−−
−=
( )*1212 115
572 εννμσ−
−−=
All other stress components are obtained by the cyclic permutation of ( )3,2,1 .
Elliptic cylinder ( ): ∞→3a
2
( )21
21
4aaaI+
=π
, ( )21
12
4aaaI+
=π
, 03 =I
( )22112
4aa
I+
=π
, 1221
1143 Ia
I −=π
, 1222
2243 Ia
I −=π
, 0332313 === III
11323 IIa = , , 223
23 IIa = 033
23 =Ia
( ) ( )( )
⎭⎬⎫
⎩⎨⎧
+−+
++
−=
21
22
21
2122
1111 212121
aaa
aaaaaS ν
ν
( ) ( )( )
⎭⎬⎫
⎩⎨⎧
+−+
++
−=
21
12
21
2121
2222 212121
aaa
aaaaaS ν
ν
03333 =S
( ) ( )( )
⎭⎬⎫
⎩⎨⎧
+−−
+−=
21
22
21
22
1122 21121
aaa
aaaS ν
ν
( ) 21
12233
2121
aaaS+−
=ν
ν
03311 =S
( ) 21
21133
2121
aaaS+−
=ν
ν
( ) ( )( )
⎭⎬⎫
⎩⎨⎧
+−−
+−=
21
12
21
21
2211 21121
aaa
aaaS ν
ν
03322 =S
( ) ( ) ⎭⎬⎫
⎩⎨⎧ −
+++
−=
221
2121
221
22
21
1212ν
ν aaaaS
( )21
12323 2 aa
aS+
=
( )21
23131 2 aa
aS+
=
The stress inside the inclusion is
3
( ) ( )*33
21
1
*22
21
22
21
22*
1121
22
21
2122
11
12
12
21
εν
μν
ενμε
νμσ
aaa
aaa
aaa
aaa
aaaaa
+−−
⎟⎟⎠
⎞⎜⎜⎝
⎛
+−
+−+⎟
⎟⎠
⎞⎜⎜⎝
⎛
++
++
+−−
=
( ) ( )*33
21
2
*11
21
12
21
21*
2221
12
21
2121
22
12
12
21
εν
μν
ενμε
νμσ
aaa
aaa
aaa
aaa
aaaaa
+−−
⎟⎟⎠
⎞⎜⎜⎝
⎛
+−
+−+⎟
⎟⎠
⎞⎜⎜⎝
⎛
++
++
+−−
=
*33
*22
21
2*11
21
133 1
212
12 ε
νμε
νμνε
νμνσ
−−
+−−
+−−=
aaa
aaa
( )*122
21
2112 1
2 ενμσ
aaaa+−
−= , *23
21
223 2 εμσ
aaa+
−=
*31
21
131 2 εμσ
aaa+
−=
Penny shape ( ): 321 aaa >>=
132
21 aaII π== , 132
3 24 aaI ππ −=
313
22112 43 aaII π==
21
1
32
32312313 343 a
aaIIII ⎟⎟⎠
⎞⎜⎜⎝
⎛−====ππ
313
22211 43 aaII π== , 2
333 34 aI π
=
( ) 1
322221111 132
813aaSS π
νν
−−
== , 1
33333 41
211aaS π
νν
−−
−=
( ) 1
322111122 132
18aaSS π
νν−−
== , ( ) 1
322331133 18
12aaSS π
νν−−
==
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−
−==
1
333223311 8
1411 a
aSS πν
νν
ν
( ) 1
31212 132
87aaS π
νν−
−= , ⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
+==1
323231313 41
2121
aaSS π
νν
4
ννπ
νν
−+
−−
==141
21
1
32211 a
aSS kkkk , 1
333 21
211aaSkk
πνν
−−
−=
When , 03 →a
021 == II , π43 =I
012 =I , 2223 4 aI π= , 2
131 4 aI π=
02211 == II , 343323 π=Ia
21
31312323 == SS
νν−
==133223311 SS
13333 =S
All other 0=ijklS
Flat ellipsoid ( ): 321 aaa >>>
( ) ( ){ }22
21
321
EF4aa
kkaaI−
−=
π
( ) ( ) ( ){ }22
21
32
2
32
EF4E4aa
kkaaa
kaI−
−−=
ππ
( )2
33
E44a
kaI ππ −=
( ) ( ) ( )( ) ( )22
21
22
213223
12EF8E4
aaaakkaaakaI
−−−−
=ππ
( ) ( ) ( )( ) ( )22
22
213223
23EF4E84
aaakkaaakaI −−+−
=πππ
( ) ( ) ( )( ) ( )21
22
213223
31EF4E44
aaakkaaakaI −−−−
=πππ
23
33 34a
I π=
where and ( )kF ( )kE are the complete elliptic integrals of the first and the second kind,
5
respectively,
( ) ( ) φφπ
dsin1E212
0
22∫ −= kk
( ) ( ) φφπ
dsin1F212
0
22−
∫ −= kk
( ) 21
22
21
2 aaak −=
Oblate spheroid ( 321 aaa >= ):
( ) ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−== −
21
21
23
1
3
1
31232
321
321
21 1cos2aa
aa
aa
aaaaII π
13 24 II −= π
122211 III ==
( )21
23
3121
1321
12 441
aaII
aI
aI
−−
−=−=ππ
21
23
312313 aa
IIII−−
== , 1323
33 243 Ia
I −=π
Prolate spheroid ( ): 321 aaa =>
( ) ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
−== −
3
1121
23
21
3
1232
321
231
32 cosh12aa
aa
aa
aaaaII π
21 24 II −= π , 22
21
1212 aa
III−−
= , 1221
11 243 Ia
I −=π
233322 III ==
22
21
12232
222
43aaIII
aI
−−
−−=π
, ( )22
21
1222
23 4 aaII
aI
−−
−=π
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