## 3 Results and Discussion
### Discussion  
As can be seen from Tables 1 and 2 and Figure 5, the values of [PERSON]'s modulus obtained from quasi-static (Table 1) and ultrasonic (Table 2) measurements and dynamic mechanical analysis (Figure 5) differ significantly for the same composites. This can be due to essentially different characteristic frequencies used in these techniques for measurements of elastic characteristics of materials. As known, in polymeric materials and polymer-based composites the frequency dependence of elastic moduli plays a significant role. The frequency dependence of the dynamic elastic modulus is directly related to the loss modulus by the Kramers-Kronig relationship [54]. In amorphous thermoplastics, including polystyrene, the mechanical loss tangent weakly depends on the frequency in a very wide range from fractions of a Hz to several MHz [23; 24]. Therefore, according to the Kramers-Kronig relationship, the mechanical loss tangent, averaged on a logarithmic frequency scale from the frequency \(\omega_{1}\) to the frequency \(\omega_{2}\), equals [54]:  
\[\tan\delta=\frac{\pi}{2}\frac{\ln(E(\omega_{2})/E(\omega_{1}))}{\ln(\omega_{2} /\omega_{1})} \tag{1}\]  
For the materials used in this study, the ratio of the elastic modulus at the frequency of 2 MHz to that at 20 Hz is within the range of 1.3-1.5, which corresponds to the average value of the mechanical loss tangent \(\tan\delta=(35-55)\cdot 10^{-3}\) in the frequency range from 20 Hz to 2 MHz. These values agree well with those obtained by DMA in the range of 1-50 Hz (see Figure 5g-i). Figure 5: Frequency dependencies of the elastic modulus \(E^{\prime}\) (**a–c**), loss modulus \(E^{\prime\prime}\) (**d–f**) and mechanical loss tangent \(\tan\delta\) (**g–i**) of PS-based composites filled with fullerenes C60 (**a**,**d**,**g**), carbon nanotubes CNT (**b**,**e**,**h**) and graphene Gr (**c**,**f**,**i**), obtained by DMA. Figure 6 presents a graphical summary of the values of elastic moduli of PS-based composites filled with dispersed particles of carbon allotropes obtained in mechanical tests utilized. The comparison was made for samples containing 5 wt% of each type of carbon particle. DMA data are shown for the frequency of 5 Hz. As can be seen from Figure 6, spherical nanoparticles (fullerenes) provided a considerable difference in all elastic moduli obtained at static, dynamic, and ultrasonic tests. With anisometric particles (nanotubes) the difference in moduli obtained in three-point bending tests under static and dynamic loads decreased, and with the layered filler (graphene) it was practically absent. However, for both CNT and Gr, a considerable difference was still observed between the moduli determined in tensile and three-point bending tests and ultrasonic measurements. In general, elastic moduli obtained by nondestructive (ultrasonic) measurements showed higher values than those at tensile or three-point bending tests. Composites filled with graphene were characterized by higher elastic moduli of all types as compared to composites with other carbon nanoparticles. This finding correlates with the results by [PERSON] et al. [55], who showed that mechanical reinforcement with graphene was superior to that with carbon black or single-wall nanotubes. We note that as shown in our recent paper [34] and by other researchers [56; 57] hybrid nanofillers containing two carbon allotropes of different shapes can provide a synergistic effect and allow for obtaining better reinforcement of the material. Such fillers can form interconnected structures, improving the dispersion of filler particles and changing the crystallinity of the polymer matrix.