## 3 Parameter Determination
### Identification of Damage Parameters  
The Linear Amplitude Sweep (LAS) test has been approved by AASHTO as a standard method estimating damage tolerance of asphalt binders. The LAS test is a cyclic torsion test conducted in the dynamic shear rheometer (DSR) that uses increasing loading amplitudes to accelerate damage. In order to evaluate the damage of asphalt mastic, the same testing method in LAS test is applied. The same sample used to determine the viscoelastic model parameters in Section 3.1 is prepared for the LAS test. Two types of testing are performed in succession. The first one is the frequency sweep designed to obtain information on the rheological properties, and the second one is the amplitude sweep intended to measure the damage characteristics of the material. The Kinexus lab+ rotational rheometer produced by Kinexus Rotational Rheometer (Produced by Malvern Panalytical Ltd., Royston, United Kingdom) is applied to conduct the two types of testing. The frequency sweep test is performed at the selected temperature and applies oscillatory shear loading at constant amplitude over a range of loading frequencies to determine the damage parameter \(\alpha\) in Equation (12). For this test method, the frequency sweep test employs an applied load of 0.01% strain over a range of frequencies from 0.01 Hz to 30 Hz at 20 \({}^{\circ}\)C. Data is sampled at the following 13 frequencies selected from the controller software: 0.01 Hz, 2.57 Hz, 5.14 Hz, 7.71 Hz, 10.27 Hz, 12.83 Hz, 15.39 Hz, 17.95 Hz, 20.52 Hz, 23.08 Hz, 25.66 Hz, 28.22 Hz, and 30.78 Hz. Complex shear modulus \(|\mathrm{G}^{\mathrm{s}}|\) and phase angle \(\delta\) are recorded at each frequency as shown in Figure 2.  
In order to determine the damage parameter \(\alpha\) in Equation (12) from frequency sweep test, data for the dynamic modulus\(|G*|(\alpha)\) and phase angle \(\delta(\omega)\)for each frequency is converted to storage modulus \(G^{\prime}(\omega)\) by Equation (10). Then, A best-fit straight line is applied to a plot with log \(\omega\) on the horizontal axis and log \(G^{\prime}\) (\(\omega\)) on the vertical axis using Equation (11) as shown in Figure 3. Thus, the value obtained for \(m\) is recorded and the value of \(\alpha\) is obtained by Equation (12). In this test, the value of \(\alpha\) is: \(\alpha=1/m=1/0.3319=3.013\). The second test is run at temperature 20 \({}^{\circ}\)C using oscillatory shear in strain control mode at a frequency of 5 Hz. The loading scheme consists of a continuous oscillatory strain sweep. Strain is increased linearly from 0 to 1% over the course of 25,000 cycles of loading for a total test time of 5000 s. Peak shear strain and peak shear stress are recorded every 5 load cycles (1 s), along with phase angle \(\delta\) and complex shear modulus \(|G^{\prime}|\). Figure 4 shows the complex shear stress and shear modulus extracted from amplitude sweep test for asphalt mastic at temperature 20 \({}^{\circ}\)C. The stress-strain diagram shows that the complex shear stress grows with a higher rate until the peak point as the strain increases, where the peak stress occurs at 0.7% strain level. And then the stress decreases after the stress-strain peak point. The shear modulus versus strain diagram shows that the shear modulus decreases with a higher rate until 0.7% strain level. After this region the shear modulus decreases with a constant slope. Figure 2: Complex shear modulus and phase angle from frequency sweep test. In the pre-peak region, micro-cracking evolves and accumulates as the strain increases. In the post-peak region, the micro-cracking leads to a separation or debonding after the stress-strain peak point. In linear viscodamage theory domain, the failure behavior is not considered. Therefore, we assume that the maximum value of the rate of the damage density occurs at the peak point of the stress-strain diagram and considered the inflection point of stress-strain diagram as the strain at which the maximum nominal stress occurs.