## 3 Results and Discussion
### Tensile Properties
The corresponding increase of SCB-PE/1 GNPs w BM is 45%, suggesting that the reinforcing effect of the GNPs is significant even at lower filler loadings due to the effect of the ball-milling process. The elastic modulus' profound enhancement of SCB-PE/GNPs nanocomposites raises the interest to further study their elastic behavior, applying various models used to predict Young's modulus of composite materials. These models take into account several factors through various assumptions (interfacial adhesion, filler dispersion, aspect ratio and shape of the particles, aggregates' concentration etc.). Numerous equations were fitted to the experimental data for the micromechanical modeling to compare the different predictions for the prepared materials' elastic modulus. The first two models are the parallel (Voight) and series (Ruess) geometrical models and are described by Equations (3) and (4), respectively (rule of mixture method) [36]. In these models, the hypothesis is that the composite consists of two different piled-up phases \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Elastic Modulus (MPa)** & **Tensile Strength at Yield (MPa)** & **Tensile Strength at Break (MPa)** & **Elongation at Break (\%)** \\ \hline neat SCB-PE & 650 \(\pm\) 37 & 15 \(\pm\) 1.3 & 21.3 \(\pm\) 5.2 & 623 \(\pm\) 126 \\ \hline SCB-PE/0.5 GNPs no BM & 790 \(\pm\) 42 & 19.6 \(\pm\) 0.6 & 29.6 \(\pm\) 6.3 & 710 \(\pm\) 152 \\ \hline SCB-PE/1 GNPs no BM & 810 \(\pm\) 42 & 19.5 \(\pm\) 1 & 21.9 \(\pm\) 3.1 & 590 \(\pm\) 91 \\ \hline SCB-PE/2.5 GNPs no BM & 880 \(\pm\) 45 & 20.1 \(\pm\) 1 & 22.3 \(\pm\) 2.2 & 589 \(\pm\) 88 \\ \hline SCB-PE/5 GNPs no BM & 1140 \(\pm\) 51 & 22.6 \(\pm\) 0.8 & 21.8 \(\pm\) 3.2 & 421 \(\pm\) 123 \\ \hline SCB-PE/0.5 GNPs w BM & 830 \(\pm\) 58 & 20.8 \(\pm\) 0.6 & 26.4 \(\pm\) 3.7 & 646 \(\pm\) 110 \\ \hline SCB-PE/1 GNPs w BM & 960 \(\pm\) 65 & 21.8 \(\pm\) 1 & 27.2 \(\pm\) 3.6 & 583 \(\pm\) 87 \\ \hline SCB-PE/2.5 GNPs w BM & 1110 \(\pm\) 55 & 23.1 \(\pm\) 0.4 & 28.7 \(\pm\) 2.5 & 424 \(\pm\) 103 \\ \hline SCB-PE/5 GNPs w BM & 1290 \(\pm\) 68 & 25.3 \(\pm\) 1.2 & 28.3 \(\pm\) 1.3 & 342 \(\pm\) 121 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elastic modulus, tensile strength at yield and break, and elongation at break values of neat SCB-PE, SCB-PE/GNPs no BM and SCB-PE/GNPs w BM nanocomposites. in parallel and series geometry. The Voight model uses the assumption of the isostrain condition where the two phases experience the same strain while the stress is additive. The following equation describes the parallel model:  
\[E_{c}=E_{f}V_{f}+E_{m}V_{m}, \tag{3}\]  
where \(E_{c}\), \(E_{f}\) and \(E_{m}\) are the elastic moduli of the composite, the filler and the matrix, respectively and \(V_{f}\) and \(V_{m}\) are the volume fractions of the filler and the matrix. In the Reuss model, the hypothesis is that the composite's phases carry the same external load and the strain is additive. Equation (4) gives the predicted elastic modulus values according to this model:  
\[\frac{1}{E_{c}}=\frac{V_{f}}{E_{f}}+\frac{V_{m}}{E_{m}}, \tag{4}\]  
[PERSON] et al. introduced a combination of series-parallel and parallel-series two-phase models [21]. In this concept, the main assumption is that the polymer composite consists of a homogenous rigorous discontinuous phase and a homogeneous continuous matrix phase. The series-parallel (Takayanagi I) equation is the following:  
\[\frac{1}{E_{c}}=\frac{\varphi}{\lambda E_{f}+(1-\lambda)E_{m}}+\frac{1-\varphi }{E_{m}}, \tag{5}\]  
while in the parallel-series (Takayanagi II) model, the modulus of a polymer composite is described by the following equation:  
\[E_{c}=\lambda\left(\frac{\varphi}{E_{f}}+\frac{1-\varphi}{E_{m}}\right)^{-1}+ (1-\lambda)E_{m}, \tag{6}\]  
where the product of the parameters \(\varphi\) and \(\lambda\) equals to the volume fraction of the filler phase. These parameters are related to the degree of series and parallel coupling of the system. [PERSON] et al.