Page 207 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 199

In terms of plasticity, the elasticity part may utilize the equivalent strain principle.
The yielding criterion may be revised accordingly.
Considering the yielding criterion:
σ = σ + RX
+
y (6-175)
or
f = σ − X − R −σ = 0
y (6-176)
Where s y is the initial yielding stress; R is the hardening part; and X is the back
stress or the kinematics hardening part.
It can be rewritten as:
σ
f = − X − R −σ = 0 (6-177)
1 − D y

or
D
+ )(1
σ = ( σ + RX − ) (6-178)
y
Which actually means the yield stress is proportionally reduced by a factor of (1-D).
In other words, it could also be written as:
+ )(1
D
σ = ( σ + RX − ) (6-179)
y
6.5.6 Assessments of the Damage Parameter
6.5.6.1 Direct Measurement
A direct measurement of the damage parameter is to use the RVE and following the
δ S
direct definitions D = D . Chapter 3 presents methods for such measurements. This
δ S
approach requires the direct measurements of the defects on the surfaces. More practi-
cal methods are those of the indirect methods. The following presents a brief summary
of these methods.
6.5.6.2 Young’s Modulus Degradation
Considering E = E(1 − D), one may obtain:
E
D =−1 (6-180)
E
Through measuring the degrading of the Young’s modulus, the damage parameter
can be obtained.

6.5.6.3 Mechanical Wave Velocity
The longitudinal wave velocity for the virgin material and the damaged materials are
respectively V = E 1−ν and V = E 1−ν , if the change in Poisson’s
2
2
L − L −
ρ ( 1+ν)( 1 2ν) ρ ( 1+ν)( 1 2ν)
ratio and density is negligible, one can obtain:
D =− E ≈ − V L 2 (6-181)
1
1
E V 2
L
It can be conveniently proved that the above equation is also valid if the longitudi-
nal wave velocity is replaced with shear wave velocity.