Page 155 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 147

The Eshelby method yields:
c 3( K + 4μ ) c 5 μ 3 K + 4μ
K = K + 1 0 0 , μ = μ + 1 0 0 0 (5-126a, b)
2
0 3 0 6 K + 2μ 0
0
The Mori-Tanaka method gives:
c 3( K + 4μ ) c 5 μ 3 K + 4μ
K = K + 1 0 0 , μ = μ + 1 0 0 0 (5-127a, b)
−
1
0 31 ( − c ) 0 6 K + 2μ )( 1 c )
0
0
1
And the Self-consistent method yields:
c 3( K + )μ c 5 μ 3 K + 4μ
K = K + 1 , μ = μ + 1 (5-128a, b)
0 3 0 6 K + 2μ)
These methods predict complex patterns and may not be correct.
Void inclusions
The Elshelby method yields:
cK 3( K + 4μ ) c 5 μ 3 K + 4μ
K = K − 1 0 0 0 , μ = μ − 1 0 0 0 (5-129a, b)
0 4μ 0 9K + 8μ
0 K 0 0
The Mori-Tanaka method gives:
c
cK 3( K + 4μ ) 5c μ 3 K + 4μ
K = K − 1 0 0 0 , μ = μ − 1 0 0 0 (5-130a, b)
0 K + 4μ 0 K + 4μ − 6 1− K + 2μ
c K
3 − 3 1( − ) 53 ( c )(
0 0 1 0 0 0 1 0 0
The Self-consistent method gives:
cK 3( K + 4 )μ c 5 μ 3 ( K + 4 ) μ
K = K − 1 0 , μ = μ − 1 0 (5-131a, b)
0 4μ 0 9 K + 8μ
5.3.8 The Composite Sphere Method (Qu and Ckerkaoui, 2006)
The general philosophy of the Composite Sphere method is illustrated in Figure 5.3.
The two-phase, three-phase, and four-phase model predictions are presented re-
spectively as follows.
cK − K )( 3 K + 4μ )
(
K = K = K + 1 1 0 0 0 (5-132a, b)
−
r 0 3 K + 4μ 0 + 1( − c )( K − K )
0
1
1
0
cK − K )( 3 K + 4μ )
(
K = K = K + 1 1 0 0 0 (5-133a, b)
r 0 3 K + 4μ + 3 1( − c )( K − K )
0 0 1 1 1 0
3
3
( K − K )( 3 K + 4μ )( b / )
c
K = K + e 0 0 0 (5-134a, b)
3
c
0 3 K + 4μ 0 + 3( K − K ))(1− b 3 / )
0
e
0
In the case of a multicoated inclusion problem, the prediction is as follows:
3
( R 3 / R )( K − K )(3 K + 4μ )
K = K + n−1 n n−1 n n n , n ≥ 2 (5-135a, b)
n n 3 K + 4μ + 331( − R 3 /R 3 )(K − K )
n n n − 1 n n − 1 n