Page 154 - Mechanics of Asphalt Microstructure and Micromechanics

146 Ch a p t e r Fiv e

For spherical particles:
dK + ( K − K )(3 K + 4μ ) = 0 (5-118)
1
dc (1 − c)(3 K + 4μ )
1
dμ + 5 ( − 1 3 K + 4 μ) = (5-119)
μμ μ )(
3
c 3
dc 1 ( − )[ K 3 μ + 2 μ +4 μ 2 μ + 3μ ] 0
1 1
5.3.7 Relationships Among the Four Methods (Qu and Ckerkaoui, 2006)
Low-concentration case
For the spherical particle cases, in the case of low concentrations, it can be proven that
the Mori-Tanaka method will reduce to the Elsheby solution.
cK − K )( 3 K + 4μ )
(
K = K + 1 1 0 0 0 (5-120)
0 3 K + 4μ
1 0
(
5c μμ − μ )( 3K + 4 μ )
μ = μ + 1 0 1 0 0 0 (5-121)
0 μ μ 2 2μ + 3μ
3K 0 3 μ + 2 1 +4 0 0 1
0
By expanding the self-consistent formulation around zero concentration and ne-
glecting the higher order terms, the following is produced (Qu and Ckerkaoui, 2006):
K = K + c K 1 () + c K 1 () + ... (5-122)
2
0 1 0 1 0
μ = μ + c μ 1 () + c 2 μ 1 () + ... (5-123)
0 1 0 1 0
It also reduces to the Eshelby solution.

High-concentration case
By allowing the concentration to Approach 1, the following predictions by the Eschelby
method are produced:
K 2( K − K + 4) μ K
K = 0 1 0 0 1 (5-124)
3 K + 4μ 0
1
μ 3[ K 7 μ − 2 μ +4 μ 8 μ − 3 μ ]
μ = 0 0 1 0 0 1 0 (5-125)
3K 0 3 μ + 2 μ 1 + +4μ 0 2μ 0 + 3μ 1
0
Obviously, it does not predict accurately. Both the Mori-Tanaka and the Self-consis-
tent methods predict it correctly.

Rigid inclusions
For rigid inclusions, the following can be assumed:
μ K
0 → 0, 0 → 0
μ K
1 1
μ → K →
μ 0, K 0
1 1