Page 223 - Mechanics of Asphalt Microstructure and Micromechanics

Models for Asphalt Concrete 215

By introducing the strain energy density function to include terms related to dam-
age W = (e ij , S m ) and the dissipated energy due to damage growth W s = W s (S m ). They
expressed the stress-strain relationships as σ = ∂ W and the damage evolution laws
ε ∂
ij
∂W ∂W ij
as − = s
∂S ∂S
m m
where s ij = stresses
e ij = strains
S m = internal state variables (or damage parameters)
In analogy to the elasticity situations, the pseudostrain energy density function
R
R
R
W = W (e , S m ) was introduced. Correspondingly, the stress-strain relationship and the
damage evolution law becomes:
∂W R ⎛ ∂W R ⎞ α m
σ = R S m =− ⎟ (7-10a, b)
⎜
ε ∂ ⎝ ∂S m ⎠
·
where S m = damage evolution rate
a m = material-dependent constants related to the viscoelasticity
of the material
Through these manipulations, the viescoelastic stress-strain relationship with dam-
age becomes:
⎛ σ ⎞
d ⎜ CS ⎠ ⎟
ε = E R∫ ξ D − ⎝ () τ d (7-11)
ξ τ)
(
ve 0 τ d
Following the strain hardening viscoplastic model proposed by (Uzan, 1996; Seibi
et al., 2001),
σ
g()
ε = (7-12)
VP η
vp
·
Where e VP is the viscoplastic strain rate at g(0) = 0; h vp is viscosity. By introducing
σ
the power law of viscosity, ε = g() . Integration of this equation results in:
VP
A ε p
vp
p + 1 t
ε p+1 = ∫ g() (7-13)
σ dt
vp A 0
or
⎛ p + ⎞ 1 1/( p+1) t ) 1/( p+1)
ε = ⎜ ⎟ (∫ g() (7-14)
σ dt
vp ⎝ A ⎠ 0

Through adopting g(s) = Bs following Uzan (1996), Perl et al. (1983), and Kim et
q
al. (1997), and coupling the A and B into a coefficient Y, they obtained the viscoplastic
strain expression as:
⎛ p + ⎞ 1 1/( p+1) ξ ) 1/( p+1)
ε = ⎜ σ ξ d (7-15)
q
vp ⎝ Y ⎠ ⎟ (∫ 0