## 1 Introduction
The method allows obtaining nanocomposites using conventional equipment, extruders, or special mixers. However, an account should be taken of potential deformation, destruction, and, in particular, aggregation of the filler. More even distribution of the filler can be achieved by utilizing a special extruder with two screws and several mixing zones. Despite numerous studies of composites filled with nanoparticles, no comprehensive studies have been published by now presenting data on the mechanical behavior of composites containing nanofillers of different shapes and chemical nature in different stress-strain states under static and dynamic loads. As mentioned in [12], no systematic studies have been performed to date that would compare the effect of aspect ratio, nanofiller purity, degree of functionalization, and type of functional group on composite properties. However, such thorough analysis is important for assessing the performance characteristics of the developed polymer composites under various conditions, and their resistance to external loads and can be helpful in the choice of the most appropriate filler for a specific application of the material. The behavior of a composite material under dynamic loads is determined by its viscoelastic properties, which can be expressed as the frequency dependence of elastic and loss moduli. At the same time, experimental data show that in polystyrene, polymethyl methacrylate, and other amorphous thermoplastics, the mechanical loss tangent weakly depends on frequency over a wide frequency range of more than 10 orders of magnitude [23; 24]. Viscoelastic properties play an important role in describing the dynamics of amorphous thermoplastics at different time scales. Therefore, analysis of the effect of nanoinclusions on the elastic properties of a composite at different frequencies is of considerable importance. In this work, in addition to commonly used standard mechanical tests, we performed measurements of the elastic properties of nanocomposite samples using ultrasonic and dynamic mechanical analysis. Our recent studies have shown that polymer materials and polymer-based composites exhibit nonlinear elastic properties, which lead to the appearance of rich nonlinear dynamics of the materials [25; 26; 27], which can be realized in particular in the formation of nonlinear strain solitary waves in waveguides made of these materials, see [28; 29; 30; 31] and references therein. Note that to date the nonlinear elastic properties of both polymer materials and polymer-based composites have been poorly studied. However, the data on the nonlinear elasticity of these materials are important for comprehensive analysis of their nonlinear dynamics which is based on an understanding of both static and dynamic mechanical properties [32; 33]. We have recently demonstrated that nonlinear elastic properties of polymer nanocomposites depend significantly on the type and concentration of nanoinclusions [25; 26; 34]. In the previous works [25; 35; 36; 37], we studied mechanical properties of polystyrene-based composites with dispersed fillers of different chemical nature: silicon oxide particles, aluminosilicates, carbon black, and carbon nanotubes. The most significant changes in material properties were achieved with carbonaceous fillers. Carbon nanofillers such as nanotubes and graphene are favorable due to their high mechanical strength and high aspect ratio. Carbon nanotubes were shown to have a tensile strength in the range of 50-150 GPa, 20 times higher than that of high-strength alloys [38], and the tensile modulus of 1 TPa [39; 40]. Graphene platelets have Young's modulus of about 1 TPa, ultimate strength of 130 GPa [41], and a high specific area of 2600 m\({}^{2}\)/g [42]. Recently, we performed an investigation of linear and nonlinear elastic properties of polystyrene-based composites with carbon nanoparticles of different dimensionality: zero-dimensional spherical fullerenes, one-dimensional carbon nanotubes, and two-dimensional graphene platelets and their binary mixtures [34].