# Effective Mechanical Properties of a Composite Material Reinforced by Oil Shale Ash Particles
## 5 Results and Discussion
### Results of the Epoxy-OSA Composite  
After a weight percentage of 30%, the growth rate of the modulus of elasticity slows down. This is very similar to what is described by [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON], [PERSON] et al. [16] and by [PERSON], [PERSON] and [PERSON] [15]. However, in the case of this experiment, the value of the elastic modulus continues to increase. Each data point is accompanied by an error bar representing the standard deviation of the measurements. The error bars indicate the variability in the experimental data, which can be attributed to factors such  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & & **Elastic Modulus (GPa)** \\ \hline \multirow{4}{*}{\(\begin{array}{c}\text{OSA}\\ \text{O  
as sample preparation, the randomness of the particle size distribution, and microbubbles that appeared during curing due to the effect of the OSA.  
Figure 6 presents the elastic modulus of epoxy composites reinforced with varying percentages of oil shale ash (OSA). The experimental data are compared with theoretical predictions from the rule of mixtures, the finite element method (FEM), Mori-Tanaka method, and Halpin-Tsai method. The blue circular markers represent the experimental elastic modulus values of the epoxy-OSA composite. The elastic modulus increases with the percentage of OSA in the composite, demonstrating an overall positive correlation between the OSA content and the composite's stiffness. Represented by the red line, the rule of mixtures predicts a linear increase in the elastic modulus as the OSA content increases. This model overestimates the experimental values across all OSA concentrations, indicating that the simple linear assumption of this model does not capture the complexities of the composite material's behavior. Depicted by the green line with square markers, the FEM's predictions show a more nuanced trend than that of the rule of mixtures. The FEM results are closer to the experimental data, especially at higher OSA concentrations, suggesting that this method effectively works for the interactions and distribution of OSA particles within the epoxy matrix. As shown by the yellow line, the Halpin-Tsai method's predictions are also closer to the experimental data than those of the rule of mixtures but are slightly lower than the FEM results. As shown by the purple line, the Mori-Tanaka method's predictions are also close to the experimental data but are closer to the FEM results. Up to a concentration of 30%, the FEM and the Mori-Tanaka method provide extremely close values. The Mori-Tanaka method provides a reasonable approximation of the composite's elastic modulus, particularly in the range of 0-18% OSA concentrations. For elastic modulus predictions, you can see Figure 6. The epoxy composite had a 25% OSA deviation between experimental data and the Mori-Tanaka (M-T) method, and we used an FEM numerical approach to reach 9%; between the experimental data and Halpin-Tsai method, it was 6%. For OSA concentration equal to 45%, deviation is 37.5% for Mori-Tanaka, 34% for FEM, and 31% for Halpin-Tsai.  
Technological limitation: starting with 40% epoxy/OSA material, it becomes brittle.  
Obtained data fit restrictions mentioned in [29]. The Mori-Tanaka approach deviation from the experiments is increasingly proportional to the concentration in the second power.  
Figure 6: Comparison of the experimental results with analytical calculations for the composite epoxy–OSA material.  
The M-T method works for relatively small OSA concentrations of OSA; a 16% deviation from the experimental data is 2.6%.