## 3 Results and Discussion
### Tensile Properties
introduced the percolation filler concept in [PERSON]'s model, considering the filler-filler interactions and describing a percolating network of filler's inclusions in a continuous phase such as a polymer matrix [27]. The elastic modulus of the composite is described, in this case, based on the elastic moduli of each component. The equation that describes the elastic behavior of the composite is the following:  
\[E_{f}=\frac{\left(1-2\psi+\psi V_{f}\right)E_{m}E_{f}+\left(1-V_{f}\right) \psi E_{f}^{2}}{\left(1-V_{f}\right)E_{f}+\left(V_{f}-\psi\right)E_{m}}, \tag{7}\]  
where  
\[\psi=\left\{ \begin{array}{cc}0,&V_{f}<V_{c}\\ V_{f}\frac{(V_{f}-V_{c})}{(1-V_{c})},&V_{f}\geq V_{c}\end{array} ,\right. \tag{8}\]  
and  
\[V_{c}=\frac{0.7}{r}, \tag{9}\]  
where \(V_{c}\) is the percolation threshold of the filler in the polymer matrix, the parameter b takes a value 0.4 in the case of a 3-D structure and \(r\) is the aspect ratio of the filler particle. When \(\psi\) = 0 the Ouali model is identical to the series model. Despite introducing the percolation concept, [PERSON]'s equation fails to describe the composites' elastic modulus behavior when the filler particles form agglomerates in the matrix volume. In case there are filler aggregates in the polymer, considering that the composite consists of three phases, the elastic moduli and volume fractions of the matrix, the dispersed filler and the agglomerated filler are taken into account in order to rearrange the two-phase rule of mixture and Takayanagi series and parallel models. In order to calculate the volume fraction of each component, an f-switching function has been proposed [37]:  
\[f\Big{(}V_{f}\Big{)}=\left\{ \begin{array}{cc}0,&V_{f}<V_{c}\\ 1-e^{-A\left(\frac{V_{f}}{V_{c}}-1\right)},&V_{f}\geq V_{c}\end{array} \right., \tag{10}\]  
where \(A\) is an adjustable parameter. Using this function, the volume fractions of the dispersed and aggregated filler can be calculated according to the following equations:  
\[V_{f}^{agg}=f\Big{(}V_{f}\Big{)}\,V_{f}, \tag{11}\]  
\[V_{f}^{dis}=\Big{(}1-f\Big{(}V_{f}\Big{)}\Big{)}\,V_{f}, \tag{12}\]  
while the elastic moduli of the dispersed and aggregated filler are calculated using the equations:  
\[E_{agg}=E_{f}\Big{(}V_{f}-V_{c}\Big{)}^{c}, \tag{13}\]  
\[E_{dis}=\frac{1+(h-1)V_{f}^{2/3}}{1+(h-1)(V_{f}^{\frac{2}{3}}-V_{f})}, \tag{14}\]  
where \(h=E_{\mu}/E_{m}\) and \(c\) the percolation exponent. Using the above, the modified parallel model is described by the following equation:  
\[E_{c}=V_{f}^{agg}E_{agg}+V_{f}^{dis}E_{dis}+(1-V_{f}^{agg}-V_{f}^{dis})E_{m}, \tag{15}\]  
The modified series model is written as:  
\[E_{c}=\frac{E_{agg}E_{dis}E_{m}}{V_{f}^{agg}E_{dis}E_{m+}V_{f}^{dis}E_{agg}E_ {m}+(1-V_{f}^{agg}-V_{f}^{dis})E_{agg}E_{dis}}, \tag{16}\]  
The modified Takayanagi I (series-parallel) and II (parallel-series) models are described by the following equations, respectively:  
\[E_{c}=\frac{\Big{(}1-V_{f}\Big{)}E_{m}E_{agg}+\Big{(}V_{f}-\psi\Big{)}E_{dis} E_{agg}}{\Big{(}1-V_{f}\Big{)}\psi E_{m}+\psi\Big{(}V_{f}-\psi\Big{)}E_{dis}+(1- \psi\Big{)}^{2}E_{agg}}, \tag{17}\]  
and  
\[E_{c}=\frac{\psi\Big{(}1-V_{f}\Big{)}E_{dis}E_{agg}+\psi\Big{(}V_{f}-\psi \Big{)}E_{m}E_{agg}+(1-\psi)^{2}E_{m}E_{dis}}{\Big{(}1-V_{f}\Big{)}E_{dis}+ \Big{(}V_{f}-\psi\Big{)}E_{m}}, \tag{18}\]  
[PERSON]'s model was also applied to the experimental elastic modulus data of the prepared SCB-PE/GNPs w BM nanocomposites. [PERSON]'s approximation considers that the reinforcement effect derives from the presence of rigorous inclusions in the matrix without taking into account the filler-filler and filler-matrix interactions [25; 38] when the nanocomposite consists of the polymer matrix and randomly oriented fibers.