Page 152 - Mechanics of Asphalt Microstructure and Micromechanics

144 Ch a p t e r Fiv e

5.3.3 Eshelby Dilute Solution
Therefore the Eshelby method (Qu and Ckerkaoui, 2006) (see Figure 5.2) will have:
ˆ
L = L and ε 0 0 (5-91)
ˆ = ε
0 0
ˆ ˆ
ˆ
ε = [( L − L S + L ]( L − L ε ) ˆ 0 (5-92)
−1
*
)
r r 0 r 0 r 0
T = [I S L (L − L ˆ )] −1 (5-93)
+
ˆ
ˆ ˆ −1
r r 0 r 0
N
0 ∑
L = L + ∑ c L − L T = L + N c L − L )[ I S M L − I)] −1 (5-94 )
+
)
(
(
(
0 r r 0 r r r 0 r 0 r r
r=1 r=1

− N
M = M + ∑ c M −( L LT M (5-95)
)
0 i i 0 i i 0
i=1
It should be noted that:
I
ML ≠ .
For spherical inclusions, the formulations are as follows:
cK − K )( 3 K + 4μ )
(
K = K + 1 1 0 0 0 (5-96)
0 3 K + 4μ
1 0
(
5c μμ − μ )( 3K + 4 μ )
μ = μ + 1 0 1 0 0 0 (5-97)
0 3K ( 3 μ + 2 μ + 4) μ (2μ + 3μ )
2
0 0 1 0 0 1
5.3.4 Mori-Tanaka Method
The Mori-Tanaka Method (Qu and Ckerkaoui, 2006):
ˆ
ˆ = ε
L = L and ε 0 (5-98)
0 0 0
ˆ
S = S
r r (5-99)
N
L = L + ∑ c L − L A (5-100)
(
)
0 i i 0 i
i=1
N
ii i ⎢∑
L = ∑ c LT ⎡ N c T ⎤ −1 (5-101)
k k ⎥
r=0 k ⎣ =0 ⎦
+
(
)
L = ( c L T + c L T c T + c T ) −1 = ( c L + c L T ))(cI c T ) − 1
0 0 0 1 11 0 0 11 00 1 1 1 0 1 1 (5-102)
− N ⎡ N ⎤ −1
ii ⎢∑
M = ∑ c T c L T (5-103)
k k ⎥
k
i=0 ⎣ k=0 ⎦
ML = I
For spherical particles the predictions are:
cK − K )( 3 K + 4μ )
(
K = K + 1 1 0 0 0 (5-104)
0 3 K + 4μ + 3 1( − c )( K − K )
0 0 1 1 0 0
5c μμ − μ )( 3K + 4 μ )
(
μ = μ + 1 0 1 0 0 0 (5-105)
0 5 μ 3K + 4 μ +6 1( − c )(μ − μ )( K + 2μ )
c
0 0 0 1 1 0 0 0