Page 159 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 151

Va’ Va’
Vc

Vc Vv Vm Vv Vm

Va’ Va’s
Vc Vv Vm Va’p Vmp Vvp
Vms Vvs

FIGURE 5.5 Schematic representation of four alternate versions of modifi ed Hirsch model
(Christensen, 2003).

(c) Dispersed version
E = VcEa + (1 − Vc) 11 m ( Va Ea +' m VmEm ) 1 / m (5-154)
m
− /
c
Einstein equation by Shashidhar and his associates:
⎡ 1 + AVf ' ⎤
Em = ⎢ + ⎥ Eb (5-155)
')
⎣ 1 − 1( CVf Vf ' ⎦
where A = K E − 1
K E = generalized Einstein coefficient
Vf = volume fraction filler in mastic = Vf/Vm
2
C = (1-Vf max )/Vf max
Vf max = maximum volume fraction of filler in mastic

Generalized law of mixtures:

Em = ( Vf Ea +' n Vb Eb ) 1 / n (5-156)
n
'

where Vb = volume fraction binder in mastic = Vb/Vm
Eb = modulus of binder
n = exponent with values from −1 to +1