# Optimizing Viscoelastic Properties of Rubber Compounds for Ballistic Applications
## 3 Results  
### Model for Preselection of Materials  
In order to extrapolate viscoelastic properties of rubber compounds and thus determine the correlation between glass transition frequencies and temperatures, the Williams-Landel-Ferry (WLF) equation [18] can be used:  
\[log(\frac{1}{\beta_{T}})=\frac{-C_{1}\cdot(T-T_{0})}{C_{2}+(T-T_{0})} \tag{1}\]  
where \(\beta_{T}\) is the frequency shift factor, \(T_{0}\) is the reference temperature and \(C_{1}\) and \(C_{2}\) are empirical material and reference temperature-dependent parameters. They describe the influence of the free volume. \(C_{1}\) is defined as the reciprocal fractional free volume at the glass transition temperature \(T_{S}\), while \(C_{2}\) is the ratio of the free fractional volume at \(T_{S}\), and \(\alpha_{V}\) is the thermal expansion coefficient [19]. The standard procedure to identify glass transition temperatures using a DMA is to carry out a temperature sweep with a frequency of 1 Hz. The glass transition is at the maximum of the _tan_\(\delta\) peak. The parameter \(\beta_{T}\) is the quotient of the peak frequencies at both temperature \(T\) and the reference temperature \(T_{0}\):  
\[\beta_{T}=\frac{f_{T}}{f_{0}}. \tag{2}\]  
An example of an extrapolation of a _tan_\(\delta\) curve is given in Figure 3.  
Figure 3: Example of extrapolating a _tan_\(\delta\) curve to a different temperature using the shift factor \(\beta_{T}\). Figure 2: (**a**) Projectile M352 7.62 \(\times\) 51 AP. (**b**) Projectile 7.62 FSP. If a material for a specific damping application has to be selected, the glass transition frequency at the application temperature needs to be in the range of the load frequency. Referencing the VPAM guidelines [20], common projectiles have impact velocities \(v_{0}\) between 300 and 1000 m/s. Having armor thicknesses \(d\) of 5-16 mm and using  
\[f_{impact}=\frac{v_{0}}{d} \tag{3}\]  
leads to impact frequencies \(f_{impact}\) ranging from \(0.19\times 10^{5}\) to \(2.0\times 10^{5}\) Hz. The glass transition frequency \(f_{S}\) has to be in that range at room temperature in order to maximize the damping during ballistic impact. According to Equation (2) and using a peak frequency of 1 Hz at the reference temperature, shift factors range between \(0.19\times 10^{5}\) and \(2.0\times 10^{5}\). Incorporating these shift factors into Equation (1) and using the glass transition temperature \(T_{g}\) as \(T\) yields an equation that can be used to preselect materials by looking at their glass transition temperature:  
\[T_{g}=\frac{log\left(\frac{1}{f_{impact}}\right)\cdot\frac{C_{2}}{C_{1}}}{1+ log\left(\frac{1}{f_{impact}}\right)\cdot\frac{1}{C_{1}}}-T_{0} \tag{4}\]  
Equation (4) solved with both impact frequencies is plotted in Figure 4.  
The glass transition temperature of elastomers generally ranges between \(-100\) and \(0\)\({}^{\circ}\)C. By inserting the glass transition temperature of an elastomer into Equation (4), it is possible to calculate the material constants \(C_{1}\) and \(C_{2}\) that the material needs in order to show maximized damping in the given frequency range. By comparison of these values with those reported in the literature, it is possible to estimate if a material can be used as a damping material as long as the set of all three values (\(T_{g}\), \(C_{1}\) and \(C_{2}\)) is known. To find out if a material is suitable, it is necessary to mark a point in the diagram that corresponds to the three values of the material. If it is inside the marked range that is bordered by both graphs, then it will have its glass transition frequency in the ideal range. Having a glass transition temperature of \(-42\)\({}^{\circ}\)C, butyl rubber is a promising material with \(C_{1}\) = 4.78 and \(C_{2}\) = 81.1 K at a reference temperature of \(T_{0}\) = +10\({}^{\circ}\)C [21]. The literature does not mention the exact rubber grade  
Figure 4: Plots of Equation (4) using \(T_{0}\) = +10\({}^{\circ}\)C and impact frequencies of \(0.19\times 10^{5}\) and \(2.0\times 10^{5}\) Hz.