Page 153 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 145

Method L r e r
Eshelby method L 0 (matrix modulus) e o Prescribed uniform strain in matrix
–
Mori-Tanaka L 0 (matrix modulus) e o Average strain in matrix
– –
Self-consistent L (effective composite modulus) e Average strain of the composite
TABLE 5.1 Reference conditions of the three methods.

5.3.5 Self-Consistent Method
For the Self-consistent method (Qu and Ckerkaoui, 2006):
−
ˆ = ε
L = L and ε 0 (5-106)
ˆ
0
ε = ε + ε = ε + ε S * = T ε (5-107)
pt
i i i i
−1
(
T = [ I = S L L − L)] −1 (5-108)
i
i
i
σ = L ε = LT ε = LT M σ (5-109)
i i i i i i i
A = T ,and B = LT M (5-110)
i i i i i
N
L = L + ∑ c L − L T (5-111)
(
)
0 i i 0 i
i=1
− − N −
M = L + ∑ c M − M LT L −1 (5-112)
−1
)
(
i=1 i 0 i i i
ML = I
For spherical particles, there are the following predictions:
cK K − K )
(
K = K + 1 1 0 (5-113)
0 K + 3 (γ K − K)
1
μμ −
c ( μ )
μ = μ + 1 1 0 (5-114)
δ μ −
0 μ + 2 ( 1 μ)
5.3.6 Differential Schemes
The differential scheme ( Qu and Ckerkaoui, 2006) follows the philosophy of adding the
inclusions into the matrix with each time following the dilute solution but updating the
matrix.
ΔΩ
Lc +Δ c ) = Lc ) + 1 L ( 1 − Lc )) : A L(())c 1 (5-115)
(
(
(
(
1
1
1
1
1
Ω 0 + Ω 1 + ΔΩ 1
Lc +Δ c ) − Lc ) 1
(
(
1 1 1 = L ( − L c )) : A L c )) (5-116)
(
(
(
c Δ (1− c ) 1 1 1 1 1
1 1
dL c() 1
(
(
1 = L ( 1 − L c )) : A L c )) (5-117)
(
1
1
1
dc 1 (1− c )
1