Page 136 - MODELING OF ASPHALT CONCRETE

114 Cha pte r F o u r

asymptotic mix stiffness values at cold temperatures. At warm temperatures
mastercurves deviated indicating that the mixture stiffness was affected by the applied
stress state, that is, stress to strength ratio and bulk stress q. At cold and intermediate
temperatures the measured strain levels stayed under 100 microsstrains for all applied
stress levels, but at high temperatures the high stress levels produced resilient strains up
to 1000 microstrains.
Model Development
Equation (4-19) shows the universal material model, also called the k -k model, proposed
1 3
by Witczak and Uzan (1988) for unbound materials
⎛ θ ⎞ ⎛ τ ⎞ k 3
k 2
oct
⎟ ⎜
a ⎜
E = ( k p ) ⎝ p ⎠ ⎝ p ⎠ ⎟ (4-19)
1
a
a
where E = resilient elastic modulus
p = atmospheric pressure
a
q = bulk stress = I (first stress invariant) (= s + s + s with s = principal
1 1 2 3 i
stress)
t = octahedral shear stress = √(2/3)√J (second stress invariant) {= [(√2)/3]s
oct 2 d
with s = deviatoric stress}
d
k = model coefficients
i
In a study conducted by Pellinen (2001) and Pellinen and Witczak (2002b) it was
hypothesized that the three separate mastercurves shown in Fig. 4-16 can be combined
to form a stress-dependent mastercurve of the mix using the approach suggested by
Witczak and Uzan for unbound materials. A proposed model form to fit a stress-
dependent mastercurve of a mix is described in Eqs. (4-20) and (4-21):
α − δ
6log|E ∗ |= δ + (βγ log( )) (4-20)
ξ
+
−
1 exp
⎡ ⎛ θ ⎞ ⎛ τ ⎞ k 3 ⎤
k 2
δ = ⎢(kp a ) ⎜ ⎟ ⎜ oct ⎟ ⎥ (4-21)
1
⎣ ⎝ p a ⎠ ⎝ p a ⎠ ⎦
∗
where log|E | = log of stress-dependent dynamic modulus,
p = atmospheric pressure
a
q = bulk stress
t = octahedral shear stress
oct
k = regression coefficients
i
a, b, g = regression coefficients for sigmoidal function
X = reduced frequency
The stress dependency has been incorporated into the equilibrium modulus
value, that is, parameter d in the mastercurve, instead of incorporating it into the
shift factors. This approach differs from the approach that Schapery (1969) has
proposed for constructing a mastercurve for nonlinear viscoelastic materials. He
incorporates the strain dependency to the reduced time by combining vertical
strain-dependent a (e) and horizontal time-dependent a (T) shift to a combined shift
e T
factor a (eT).
eT