# Optimizing Viscoelastic Properties of Rubber Compounds for Ballistic Applications
## 3 Results
### Time-Temperature Superposition of Butyl Rubber  
In order to set up a model for extrapolating viscoelastic behavior to higher frequencies, a conventional butyl rubber compound was analyzed through dynamic mechanical analysis. First, a temperature sweep (Figure 5a) was carried out to obtain the glass transition temperature of the rubber compound. Afterwards, frequency sweeps above \(T_{g}\) (Figures 5b and 6) were conducted to characterize the temperature dependence of the glass transition. In both experiments the dynamic moduli \(E^{\prime}\) and \(E^{\prime\prime}\) were measured for a given temperature or frequency range to examine their temperature and frequency dependency. The peak of \(tan\)\(\delta\), which is the ratio of \(E^{\prime}\) and \(E^{\prime\prime}\), was used to determine the glass transition temperature and frequency. According to the time-temperature superposition principle, the shift factors \(\beta_{T}\) (Table 3) can be calculated using Equation (2). The temperature of \(-50\)\({}^{\circ}\)C was chosen as the reference temperature \(T_{0}\). As expected, the peak frequency of \(tan\)\(\delta\) shifts to higher frequencies for higher temperatures. Figure 5: (**a**) Temperature sweep (1 Hz) of butyl rubber from \(-100\) to \(-20\)\({}^{\circ}\)C. (**b**) Frequency sweep of butyl rubber, \(T=-50\)\({}^{\circ}\)C. Storage modulus \(E^{\prime}\), loss modulus \(E^{\prime\prime}\) and loss factor \(tan\)\(\delta\) are plotted against temperature (**a**) and frequency (**b**). Figure 6: Frequency sweeps of butyl rubber: (**a**) \(T=-40\)\({}^{\circ}\)C and (**b**) \(T=-30\)\({}^{\circ}\)C. Storage modulus \(E^{\prime}\), loss modulus \(E^{\prime\prime}\) and loss factor \(tan\)\(\delta\) are plotted against frequency. Using the WLF Equation (1), an equation system consisting of two equations for the temperatures \(-40\) and \(-30\)\({}^{\circ}\)C can be established for determining the material and reference temperature-dependent constants \(C_{1}\) and \(C_{2}\):  
\[\Rightarrow C_{1}=8.96\quad C_{2}=45.73\quad K. \tag{5}\]  
Equation (1) shows that the temperature at which the glass transition occurs at 1 Hz can be calculated by using the room temperature as the new reference temperature \(T^{\prime}\). Keeping in mind that constants \(C_{1}\) and \(C_{2}\) also depend on the reference temperature, it is necessary to extrapolate the new constants \(C^{\prime}{}_{1}\) and \(C^{\prime}{}_{2}\) for the room temperature by using  
\[C^{\prime}_{2}=C_{2}+\left(T^{\prime}_{0}-T_{0}\right) \tag{6}\]  
and  
\[C^{\prime}_{1}=\frac{C_{1}\cdot C_{2}}{C^{\prime}_{2}}, \tag{7}\]  
which lead to  
\[\Rightarrow C^{\prime}_{1}=3.39\quad C^{\prime}_{2}=120.73\quad K. \tag{8}\]  
It is now possible to determine the temperature range that corresponds to the frequencies induced during ballistic impact. The plot of Equation (1) including the new values for \(C^{\prime}{}_{1}\), \(C^{\prime}{}_{2}\) and \(T^{\prime}\) is shown in Figure 7a. With decreasing armor thickness and increasing projectile velocity, the temperature that corresponds to the impact frequency decreases. By comparison with the temperature sweep of butyl rubber (Figure 7b), it can be concluded that frequencies induced during ballistic impact are located in the area of maximized damping of the material. Even for high projectile velocities (1000 m/s) and very thin armors (5 mm), the temperature of \(-49\)\({}^{\circ}\)C corresponds to a _tan_\(\delta\) of 0.7, which still is 84% of the maximum tan \(\delta\) of the analyzed rubber compound. A _tan_\(\delta\) of around 0.1 can be seen as the minimum value necessary for a damping material [22]. The experimental values of the manufactured butyl rubber are much higher, which leads to the assumption that it could be a well-suited damping material for ballistic protection.