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2
CHAPTER 2
THEORY
2.1 DYNAMIC MODULUS (E *) dynamic modulus testing of HMA is normally conducted
using a uniaxially applied sinusoidal stress pattern as shown
For linear viscoelastic materials such as HMA, the stress- in Figure 1.
to-strain relationship under a continuous sinusoidal loading
is defined by its complex dynamic modulus (E*). This is a
complex number that relates stress to strain for linear 2.1.1. Dynamic Modulus: Master Curve
viscoelastic materials subjected to continuously applied
sinusoidal loading in the frequency domain. The complex In the Mechanistic-Empirical Pavement Design Guide
modulus is defined as the ratio of the amplitude of the sinu- developed in NCHRP Project 1-37A, the modulus of HMA
soidal stress (at any given time, t, and angular load at all levels of temperature and time rate of load is deter-
frequency, ), = 0 sin(t), and the amplitude of the sinu- mined from a master curve constructed at a reference
soidal strain = 0 sin(t - ), at the same time and fre- temperature (generally taken as 70°F). Master curves are
quency, that results in a steady-state response (Figure 1): constructed using the principle of time-temperature super-
position. The data at various temperatures are shifted with
ei t 0 sin(t ) respect to time until the curves merge into a single smooth
E* = = 0i( t - ) = (Eq. 1) function. The master curve of the modulus, as a function of
0e 0 sin(t - )
time, formed in this manner describes the time dependency
where of the material. The amount of shifting at each temperature
0 = peak (maximum) stress required to form the master curve describes the temperature
0 = peak (maximum) strain dependency of the material. In general, the modulus master
= phase angle, degrees curve can be mathematically modeled by a sigmoidal func-
= angular velocity tion described as
t = time, seconds
Log E * = + + (log tr )
(Eq. 3)
Mathematically, the dynamic modulus is defined as the 1+ e
absolute value of the complex modulus, or
0 where
E* = (Eq. 2) tr = reduced time of loading at reference temperature
0
= minimum value of E*
+ = maximum value of E*
For a pure elastic material, = 0, and it is observed that , = parameters describing the shape of the sigmoidal
the complex modulus (E*) is equal to the absolute value or function
dynamic modulus. For pure viscous materials, = 90°. The
The shift factor can be shown in the following form:
/ t
a(T ) = (Eq. 4)
tr
0 0
where
0 sin(t) a(T) = shift factor as a function of temperature
t = time of loading at desired temperature
0 sin(t - )
tr = time of loading at reference temperature
Figure 1. Dynamic (complex) modulus test. T = temperature