Page 206 - Mechanics of Asphalt Microstructure and Micromechanics

198 Ch a p t e r S i x

In the case where micro voids exist on the surface, which cannot take the loads, the
effective stress is defined as:
F
σ =
−
SS
D (6-167)
F σ
σ = or σ =
S(1 − S D ) 1 − D
S (6-168)
Obviously, if the force is tangent to the normal, the above definition is also valid.
Nevertheless, in either case, the definitions provide only nominal measurements. For
example, when the pore size is very small, it may be able to take load.
6.5.4 Effective Modulus
If there is no damage D = 0, Hooke’s law will give the strain:
σ
ε =
e
E
In the case with damage 0 < D < 1, one may have the effective strain as:
σ
ε = (6-169)
e
−
11 E( 1 D)

ε = ε = − νε e (6-170)
e
e
22 33 11
It should be noted that the above equations are approximate, as the modulus and
the Poisson’s ratio is also a function of the damage parameter. Obviously, the effective
strain thus defined may not be the same as the true effective strain measured. The de-
gree of deviation is a measure of the validity of CMD. By using the effective modulus
concept, one can define the effective modulus as:
E = E − D) (6-171)
1
(

And the effective strain is thus defined as:
σ
ε = (6-172)
e
11 E
6.5.5 Strain Equivalency Principle
Lamaitre (1971) postulated a strain equivalency principle. It states that “the constitutive
equation of a damage material can be derived using the same way as for virgin mate-
rial except that the usual stress is replaced by the effective stress.”
In the elasticity case the effective strains for virgin and damaged materials are cor-
respondingly defined as:
σ σ
ε = ε = (6-173)
e E e E
One may assume that the strain decompositions are still valid:
ε = ε + ε
e p (6-174)