# Preparation and Properties of Waterborne Polyurethane and SBS Composite-Modified Emulsified Asphalt
## 2 Materials and Methods
### Performance Test Methods
#### 2.2.4 Linear Amplitude Sweep Test  
The linear amplitude sweep (LAS) test evaluates the fatigue resistance of asphalt samples by applying cyclic loads with linearly increasing amplitudes. The LAS test is conducted using a dynamic shear rheometer (DSR) apparatus, utilizing standard parallel plates with a diameter of 8 mm and a plate spacing of 2 mm. The LAS testing method consists of two steps [25], adhering to the AASHTO TP 101-14 specification. The LAS test allows the quick determination of parameters A and B in the traditional strain-controlled fatigue equation. Parameter A represents the initial fatigue life, which indicates the asphalt binder's ability to resist fatigue damage before significant modulus reduction occurs. Parameter B represents the rate at which fatigue life decreases with increasing strain amplitude, reflecting the material's sensitivity to changes in strain level. Adhesives with strong fatigue resistance exhibit higher values of parameter A and lower absolute values of parameter B [26]. The traditional strain-controlled fatigue equation is defined as Equation (1). \[N_{f}=A\times\left(\gamma_{p}\right)^{-B} \tag{1}\]  
where  
\(N_{f}\) is the fatigue life;  
\(\gamma_{p}\) is the amplitude of the applied shear strain. Generally speaking, viscoelastic continuum damage model (VECD) mechanics [27] is used to calculate parameters \(A\) and \(B\). In VECD mechanics, the damage evolution of viscoelastic materials is characterized by [PERSON]'s work potential theory. There are two theoretical foundations for it [28], the first being the VECD method based on PSE ([PERSON]'s elastic-viscoelastic correspondence principle) [29]; the second method is VECD based on DE (dissipated energy density). The VECD method based on PSE has the advantage of ignoring viscoelastic effects and can effectively quantify the relationship between damage strength (S) and material integrity (C) through equations. Therefore, this study adopted the VECD mechanical method based on PSE for data analysis. Experiments were conducted at 25 \({}^{\circ}\)C, and fatigue parameters \(A\) and \(B\) were calculated using formulas. Then, the fatigue strength calculate based on the frequency sweep results and amplitude sweep results, as shown in Equation (2) below. \[D(t_{i})\cong\sum_{i=1}^{N}\left[\pi\gamma_{0}^{2}(C(t_{i})-C(t_{i-1}))\right] \frac{\alpha}{1+\alpha}\cdot(t_{i}-t_{i-1})\frac{1}{1+\alpha} \tag{2}\]  
where  
\(D(t)\) is the accumulated damage over time \(t\) (dimensionless);  
\(C(t)=\frac{|G^{*}|(t)|}{|G^{*}|_{initial}}\); \(|G^{*}|(t)\) is the complex modulus at time \(t\);  
\(|G^{*}|_{initial}\) is the initial value of the complex modulus  
\(\gamma_{0}\) is the strain (%);  
\(t\) is the testing time;  
\(\alpha=m^{-1}\) and \(m\) are the slope of the best-fit line on the logarithmic plot of storage modulus versus frequency. At the given time \(t\), we recorded the values of \(D(t)\) and \(C(t)\), which could be fitted into the following equation (Equation (3)):  
\[C(t)=C_{0}-C_{1}D(t)^{C_{2}} \tag{3}\]  
where \(C_{0}=1\), while \(C_{1}\) and \(C_{2}\) are curve fitting parameters. The damage value during destruction corresponds to the maximum peak shear stress, as shown in Equation (4). \[D_{f}=\left(\frac{C_{0}-C_{\text{atpeakstress}}}{C_{1}}\right)^{1/C_{2}} \tag{4}\]  
Thus, the fatigue equation can be expressed by Equation (5). \[N_{f}=A(\gamma_{\text{max}})^{-B} \tag{5}\]  
where  
\(\gamma_{\text{max}}\) is the maximum expected strain;  
\[A=\frac{f\times D_{f}\times(1+(1-C_{2})\alpha)}{(1+(1-C_{2})\alpha)\times(\pi C _{1}C_{2})\alpha};\]\(f\) is the loading frequency, which is 10 Hz here;  
\[B=2\alpha.\]  
The steps of the LAS test were as follows. The LAS test plan consisted of two consecutive types of testing. The first test was a frequency sweep experiment, aimed at obtaining the rheological properties of asphalt materials; the second test was an amplitude sweep experiment, aimed at measuring the damage characteristics of asphalt materials. The first step was the frequency sweep experiment.