Page 156 - Mechanics of Asphalt Microstructure and Micromechanics

148 Ch a p t e r Fiv e

a

b

a a
b b

Composite Homogeneous

FIGURE 5.3 Composite-sphere model (Qu and Ckerkaoui, 2006).

5.3.9 Lower and Upper Bounds (Qu and Ckerkaoui, 2006)
The Minimum Potential Energy theorem and Minimum Complementary Energy theo-
rem are used to predict the bounds of the effective moduli.
Π[] = 1 V ∫ L u u dv − S ∫ p u ds (5-136)
0
u
i 2 ijkl , k l i j , σ i i
Π [σ ] = 1 V ∫ M σ σ dv − Su ∫ u σ n ds (5-137)
0
c ij 2 ijkl kl ij i ij j
Where V is the total volume and S u is the portion of the boundary of V where dis-
placement u i is prescribed.
0
The Voigt Upper Bounder and Reuss Lower Bound can be represented as follows:
ˆ −
2εσ σ ˆ M R σ ˆ ≤ 2U ≤ ε 'L ε (5-138)

R V
L ≤ L ≤ L
(5-139)
For spherical particles, it results in the following bounds:
KK
0 1 ≤ K ≤ K + c K −( K )
K − c K −( K ) 0 1 1 0 (5-140)
1 1 1 0
μμ
0 1 ≤ μ ≤ μ + c ( μ − μ ) (5-141)
μ − c ( μ − μ ) 0 1 1 0
0
1
1
1
The Hashin-Shtrikman Variational Principle states that when L is negative semi-
p
definite, among all the symmetric second-order tensors, the true solution to the stress
polarization tensor t ˆ renders the following minimum: