Page 473 - Mechanics of Asphalt Microstructure and Micromechanics

APPENDIX1

Eshelby’s Tensor (S) for Special Cases

This appendix is abstracted from (Nemat-Nasser and Hori, 1998). It refers to the Eshelby
tensor components for some specific cases. The general case is the ellipsoids embedded
with semi-principal axes (a i ) coincide with the coordinate axes (x 1 , x 2 , x 3 ) (Figure A.1).

FIGURE A.1 Ellipsoid coaxial x
with the cartesian 3
coordinates
a 3

x a 1 a
1 2

x
2
General form of the components for a 1 > a 2 > a 3 are:
−
2
S = 3 a I + 12ν I
1111 π 1 11 π 1
81−ν( ) 81−ν( )
−
2
S = 1 a I − 12ν I
1122 π 2 12 1
81−ν( ) 81−ν)
π(
−
+
2
2
S = 1 ( a + a I + 12ν I ( + I )
)
1212 π 1 2 12 π 1 2
16 1−ν( ) 16 1−ν( )
The I i and I ij integrals are given by:
4π aa a
I = 12 3 {( k − (
F θ, )
E θ,, )}k
/
2
2
1 a − 2 a − 21 2
( a )( a )
1 2 1 3
4π aa a aa − a ) ) /
2
2 12
(
I = 12 3 { 2 1 3 − E θ k
(, )}
3 a − 2 a − 21 2
2
2
/
( a )( a ) aa
2 3 1 3 13
I 1 + I 2 + I 3 = 4π
and
4π I − I
3I + I + I = , 3aI + aI + aI = 3III = 2 1
2
2
2
,
2
11 12 13 2 1 11 2 12 3 13 1 12 a − 2
a a
1 1 2
465