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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