Page 194 - MODELING OF ASPHALT CONCRETE

172 Cha pte r Se v e n

This variable then tracks the behavior, or state, of the given element throughout
loading. The formulation used in this research is shown as

m
(
ε R n+ ) 1 = 1 ⎡ ⎢ η n+1 + ∑ η n+1 ⎤ ⎥ (7-11)
E R ⎣ 0 i=1 i ⎦
where h and h are internal state variables for the elastic response and for the specific
0 i
Maxwell element i at time step n + 1, respectively. Definitions of these variables are
given by Eqs. (7-12) and (7-13), respectively:
η n+ 1 = E ε ( n+ 1 − ε ) and (7-12)
0
0 ∞
−Δ t −Δ t
η i n+1 = e ρ i η + E e 2 ρ i ε ( n+1 − ε ) (7-13)
n
n
i
i
Equation (7-11) is a remarkably efficient solution technique for pseudostrains. For
comparative purposes, a data set of 4000 points requires approximately 100 seconds to
analyze using Eq. (7-9), but requires only 1.5 seconds if analyzed by Eq. (7-11).
Work Potential Theory
Figure 7-6(a) shows typical stress-pseudostrain hysteresis loops at different numbers of
cycles in the controlled-stress cyclic test. Relatively high-stress amplitude is used to
induce significant damage in the specimen. Unlike the negligible damage case seen in
Fig. 7-4(b), change in the slope of each σ − ε cycle (i.e., reduction in the pseudostiffness
R
of the material) can be observed from this figure due to the damage incurred in the
specimens.
The effect of damage on pseudostiffness can also be seen in the monotonic data
shown in Fig. 7-6(b). In this figure, the stress-pseudostrain curves deviate from the line
of equality as damage grows. Also, the onset of this deviation occurs at different times
as the rate of loading changes, indicating the presence of the rate-dependent damage
mechanism.
Based on these observations, the following uniaxial versions of constitutive equations
are presented for linear elastic and linear viscoelastic bodies with and without damage.
They also show how models of different complexity may evolve from simpler ones.

ε
Elastic body without damage: σ = E (7-14)
Elastic body with damage: σ = CS E( ) ε (7-15)
m
Viscoelastic body without damage: σ = E ε R (7-16)
R
Viscoelastic body with damage: σ = CS E( m ) R ε R (7-17)

where C(S ) indicates that C is a function of damage parameters S . The function C(S )
m m m
represents the changing stiffness of the material due to growing damage. Equation (7-17)
results from Eqs. (7-14), (7-15), and (7-16). The form of Eq. (7-17) is also supported by the
observations made in Fig. 7-6; that is, the pseudostiffness changes as the damage grows. To
determine an analytical representation of the damage function C, the work potential theory,
the continuum damage mechanics principle developed by Schapery (1990), is adopted.