## 4 Viscoelastic Damage Model Considering the Damage Effect  
### The Establishment of Viscoelastic Damage Model Considering the Damage  
To obtain the viscoelastic characteristics for asphalt, [PERSON] model was used in many research studies. Lots of studies have demonstrated that asphalt is a viscoelastic fluid material, so asphalt's elasticity, viscoelasticity and flow limitation should be considered when the mechanical characteristics of asphalt are analyzed. [PERSON] model is a viscoelastic model that is suitable for analyzing asphalt's rheological behavior. [PERSON] model is made up of the Maxwell model and the Kelvin model, and is successfully applied to the viscoelastic analysis of asphalt and asphalt mixtures [32; 33; 34; 35].  
[PERSON] model is shown in Figure 10.  
The expression for the creep compliance of [PERSON] model is given by:  
\[J(t)=\frac{1}{E_{1}}+\frac{t}{\eta_{1}}+\frac{1}{E_{2}}\bigg{[}1-\exp\bigg{(}- \frac{t}{\tau}\bigg{)}\bigg{]} \tag{4}\]  
where \(\tau=\eta_{2}/E_{2}\); \(E_{1}\), \(\eta_{1}\), \(E_{2}\) and \(\eta_{2}\) are the viscoelastic parameters of [PERSON] model.  
However, from the point of view of the mechanics of damage, if asphalt is under a constant load in the bending creep test, lots of microdefects in the materials will occur [36]. Therefore, if the prediction  
Figure 10: Burgers’ model.  
of creep compliance is under the consideration of the damage factor in a special test temperature, the actual values and the fitted values have a very high fitting precision.  
It is known from the formation and distribution of material defects that the defects are discrete in the material. However, in the need of research, continuum damage mechanics make material defects continuous, and use damage variables to characterize the effect of defects on the material. Many studies suggest that the Weibull distribution is suitable for the characterization of the defects distribution within a material, and has been widely used to analyze the fatigue damage characterization of asphalt and asphalt mixtures [37; 38; 39; 40; 41]. If microdefects in BRA compound-modified asphalt obey the Weibull distribution, the microdefects' density function changing with time is under:  
\[f(t)=\frac{m}{n}(t-\gamma)^{m-1}\exp\left[-\frac{(t-\gamma)^{m}}{n}\right] \tag{5}\]  
where \(m\) and \(n\) are the material parameters obtained by the regression of data of experiments.  
\[\frac{dD(t)}{dt}=f(t) \tag{6}\]  
The results of integrating Equations (2) and (3) are calculated as follows:  
\[D(t)-D(\gamma)=\int_{\gamma}^{t}\frac{m}{n}(x-\gamma)^{m-1}\exp\left[-\frac{( x-\gamma)^{m}}{n}\right]dx \tag{7}\]  
If the initial damage factor \(D(\gamma)\) = 0, Equation (7) is simplified as:  
\[D(t)=1-\exp\left[-\frac{(t-\gamma)^{m}}{n}\right] \tag{8}\]  
where \(\gamma\) = critical time of defects starting to form, \(\gamma\) is considered as 0 in the test, and then the continuous damage factor is simplified as:  
\[D(t)=1-\exp\left(-\frac{t^{m}}{n}\right) \tag{9}\]  
\[\begin{array}{ll}\varepsilon(t)=&[1-D(t)]^{-1}\sigma_{0}J(t)\\ &=\exp\left(\frac{t^{m}}{n}\right)\sigma_{0}\Big{\{}\frac{1}{E_{1}}+\frac{t}{ \eta_{1}}+\frac{1}{E_{2}}\big{[}1-\exp\left(-\frac{t}{\tau}\right)\big{]} \Big{\}}\end{array} \tag{10}\]  
From Equation (9), the existence of defects increases the creep strain's growth. Thus, if the damage is considered, the model for the variation of creep compliance of viscoelastic material will be accurate.  
After putting \(\sigma_{0}\) to the left of Equation (10), [PERSON] constitutive mode considering defects in material is given by:  
\[\begin{array}{ll}J^{\prime}(t)&=\frac{\varepsilon(t)}{\sigma_{0}}\\ &=\exp\left(\frac{t^{m}}{n}\right)\Big{\{}\frac{1}{E_{1}}+\frac{t}{\eta_{1}}+ \frac{1}{E_{2}}\big{[}1-\exp\left(-\frac{t}{\tau}\right)\big{]}\Big{\}}\end{array} \tag{11}\]