Page 474 - Mechanics of Asphalt Microstructure and Micromechanics

466 A p p e n d i x O n e — Eshelby’ s T ensor (S) for Special Cases

Where F and E are the elliptic integrals of the first and second kind, and
a 2 − a 2 a 2 − a 2
/
/
}
}
θ = arcsin{ 1 3 12 k , = arcsin{ 1 2 12
2 2
a 2 a − a 2
1 1 3
For sphere (a 1 = a 2 = a 3 = a)
−
5ν − 1 45ν
S = δδ + ( δδ +δδδ )
ijkl ij kl ik jl il jk
15 1 ( −ν) 15 1 ( −ν)
For elliptical cylinder (a 3 → )
2
1 a + 2 a a a
−
S 1111 = { 2 12 + ( 12 ) ν 2 }
a
21−( ν ) ( a + a ) 2 a + a
1 2 1 2
2
1 a + 2 a a a
−
S = { 1 12 + ( 12 ) ν 1 }, S 0
2222 ν a + 2 3333
21−( ) ( a ) a + a a
1 2 1 2
1 a 2 a
−
S = { 2 − ( 12 ) ν 2 }
1122 ν 2 a +
21− ) ( a + a ) a
(
1 2 1 2
1 2ν a
S = 1
2233 ) ν a +
21−( a
1 2
1 a 2 a
−
S = { 1 − ( 12 ) ν 1 }
2211 ν 2 a +
(
21− ) ( a + a ) a
1 2 1 2
1 2ν a
S = 2 , S = S = 0
1133 ) ν a + 3311 3322
21−( a
1 2
−
2
1 a + a 2 12ν
S = { 1 2 + }
1212 21− ) ν 2( a + a ) 2 2
(
1 2
S = a 1 , S = a 2
2323 a + 3131 a +
2( a ) 2( a )
1 2 1 2
For penny shape (a 1 = a 2 >> a 3 )
−
−
π( 13 8ν) a π(112ν) a
1
S = S = 3 , S =− 3
1111 2222 3333
(
32 1−ν) a 41−ν) a
(
1 1
πν 1) a πν − )1 a
8 −
(
(
2
S = S = 3 , S = S = = 3
1122 2211 1133 2233
(
32 1−ν) a ( 81 −ν) a
1 1
8 −
πν 1) a πν − )1 a
(
(
2
S = S = 3 , S = S = = 3
1122 2211 1133 2233
(
32 1−ν) a ( 81 −ν) a
1 1
−
(
)
π( 78ν) a 1 π πν − 2 a
S = 3 , S = S = { 1+ 3 }
1212 3131 2323 ( −
(
32 1−ν) a 2 41 ν) a
1 1
Reference
Nemat-Nasser, S. and Hori, M. (1998). Micromechanics: Overall Properties of Heterogeneous
Materials. North-Holland.