Page 224 - Mechanics of Asphalt Microstructure and Micromechanics

216 Ch a p t e r Sev e n

And the viscoelastoplastic continuum damage model became (the total strain):
⎛ σ ⎞
d ⎜ CS ⎠ ⎟ ⎛ 1/(p +1) 1/(p +1)
ε = E R∫ ξ D − ' ⎝ () ξ d + ' ⎜ p + ⎞ 1 ⎟ ⎟ (∫ ξ σξ ) (7-16)
q
d
ξ ξ )
(
T 0 ξ d ' ⎝ Y ⎠ 0

Logically, the major step for calibrating the above VECD or VEPCD models is to char-
acterize the damage function and damage parameter C(S m ) and S m . Through more than
20 years continuing efforts, Kim’s group has established the models, the corresponding
methods to calibrate models, and numerical tools through FEM implementation.
7.3 Disturbed State Models
The concept of disturbed state is based on the assumption that the response of a dam-
aged material can be interpolated between two extreme conditions, the relative intact
(RI) condition and the fully adjusted (FA) condition through a scalar damage parameter
D (could be a tensor) (Desai, 2001, 2007, 2009). In the case of the stress-strain relation-
ship, it can be represented as:
i
a
c
dσ = (1 − D dσ +) i Ddσ + dD σ −( c σ ) (7-17)
˜ ˜ ˜ ˜ ˜
The above equation can be considered as the incremental format of the following
static formulation with the consideration that D is an evolving state parameter.
i
a
σ = (1 − D) σ + D σ c (7-18)
˜ ˜ ˜
c
a
c
i
dσ = (1 − D C dε +) i i DC dε + dD σ −( c σ ) (7-19)
˜ ˜ ˜ ˜ ˜ ˜ ˜
dσ = C DSC dε (7-20)
a
˜ ˜ ˜
where a represents any states observed
i and c represent the corresponding parameters at RI and FA states,
respectively
s and e = stress and strain vectors, respectively
˜ ˜
C , C = compliance tensor or stress-strain matrix at corresponding
i
c
˜ ˜
RI and FA states
D = disturbance, a similar concept to damage parameter
dD = increment or rate of D
C DSC = compliance tensor for the disturbed state
˜
Taking the plasticity model as an example, the format of the yield function could be
modified as:
F = J − −α J + γ J )( 1 − β S ) −0 5 = 0 (7-21)
n
.
2
(
2 D 1 1 r
–
where J 2D = J 2D /p a = dimensionless second invariant (J 2D ) of the deviatoric
2
stress tensor S ij
p a = atmospheric pressure constant
–
J 1 = (J 1 + 3R)/p a = dimensionless first invariant of the total stress tensor s ij
R = parameter proportional to cohesion
3
S r = stress ratio J 3 D J ⋅ − 2 D 2