Page 240 - Mechanics of Asphalt Microstructure and Micromechanics

232 Ch a p t e r Sev e n

E − E *
*
D = 0 , defined as the loss of the norm of the complex modulus that takes place
E *
0
in a specimen during a test.
Bodin Model
Bodin, et al. (2004) proposed a non-local damage model to predict pavement fatigue
cracking, which was implemented in a finite-element code along with a self-adaptive
jump-in-cycle procedure for high-cycle fatigue computations.
The mathematical model used to describe mechanical damage is an elasticity-based
damage model for fatigue.

Fd ( )(β + )1 N ε β +1
Fd
N = crit with Fd ( ) = constannt, ( ) = ∑ a (7-89)
+
crit ε β +1 crit cycle β 1
a 1
–
Where d is the damage variable; e a is the amplitude of the equivalent strain over one
cycle; f (d) is the function of damage and F(d) is the scalar function of damage; and b is
a model parameter.
⎧ ⎡ − d ⎤ ⎪
⎫
⎪
α 3
fd() = Cd , and F d() = α ⎨ 1 − exp ⎢ ⎥ ⎬ (7-90)
α
⎬
1 α
⎪ ⎣ 2 ⎦ ⎪
⎩
⎭

Where C is the secant stiffness of the material; a 1 , a 2 and a 3 are the three model
parameters.
Lee Model
Based on the elastic-viscoelastic correspondence principle and continuum damage me-
chanics through mathematical simplifications, a fatigue performance prediction model
of AC was developed from a uniaxial constitutive model by Lee et al. (2000). The fatigue
model is similar to the phenomenological tensile strain-based fatigue model, a com-
parison between the two models yields the regression coefficients in the phenomeno-
logical model as functions of viscoelastic properties of the materials, loading condi-
tions, and damage characteristics. The fatigue model in terms of the total number of
loading cycles to failure N f , is given as:
fS ( ) p 1 −2α M fS ) 3 p −2α
(
N = f , 1 E * 1 ε 2 − α 1 + ∑ 3e E * 3 ε −2α 3 (7-91)
,
f Total α 0 0 α 0
p (. 0 125 IC C ) 1 i i=1 ⎡ R ⎤ 3
1 11 12 p 3 ⎣ 0 125I C. ( 2 + S C C) 31 32 ⎦
B

R
Where f is the loading frequency; S is the secant pseudostiffness; I is the initial
pseudostiffness; S m = internal state variables (or damage parameters) that account for
the effects of damage; S 1f = value of damage parameter S 1 at failure; p 1 = 1+ (1– C 12 ) C 1 ;
p 3 = 1+ (1– C 32 ) C 3 ; C ij = regression coefficients; C i = material constants (i = 1, 2, 3); e 0 is
the initial strain; and E * is the complex modulus.
7.10.5 Micromechanics-Based Model
Guddati et al. (2002) adopted a random truss lattice model to simulate the linear elastic
and viscoelastic deformation of homogeneous materials in axial compression and shear-
ing experiments. The linear elastic deformation and the stress field in heterogeneous