# Effective Mechanical Properties of a Composite Material Reinforced by Oil Shale Ash Particles
## 3 Mechanical Properties of the Epoxy-Resin-OSA Composite
### Analytical Calculation for the Epoxy-Resin-OSA Composite
#### 3.2.3 Mori-Tanaka Method  
The Mori-Tanaka analytical model is based on the Eshelby solution. [PERSON] and [PERSON] calculated elastic moduli for fibers and dispersed resistant composites with an isotropic matrix based on the Mori-Tanaka method [42; 43; 44]. The modulus of elasticity equation according to the Mori-Tanaka method is obtained as follows:  
\[E_{c}=\frac{E_{m}}{1+\frac{V_{p}\cdot(A1+2\cdot v_{m}\cdot A2)}{A}}, \tag{6}\]  
where  
* \(v_{m}\)--[PERSON]'s ratio of the matrix (epoxy resin);
* \(A,\ A1,\ A2\)--coefficients found in [44] are calculated using Equations (18)-(23).  
The Eshelby constants depend on the shape of the filler. In this case, the filler particle is assumed to be spherical, and the constants are for spherical filler particles [45]  
\[S_{1111}=S_{2222}=S_{3333}=\frac{7-5\cdot v_{m}}{15\cdot(1-v_{m})}, \tag{7}\]  
\[S_{1122}=S_{2233}=S_{3311}=S_{1133}=S_{2211}=S_{3322}=\frac{5\cdot v_{m}-1}{1 5\cdot(1-v_{m})}, \tag{8}\]  
\[S_{1212}=S_{2323}=S_{3131}=\frac{4-5\cdot v_{m}}{15\cdot(1-v_{m})}, \tag{9}\]  
\[B_{1}=V_{p}\cdot D_{1}+D_{2}+\big{(}1-V_{p}\big{)}\cdot(D_{1}\cdot S_{1111}+2 \cdot S_{2211}), \tag{10}\]  
\[B_{2}=V_{p}+D_{3}+\big{(}1-V_{p}\big{)}\cdot(D_{1}\cdot S_{1122}+S_{2222}+S_{2 233}), \tag{11}\]  
\[B_{3}=V_{p}+D_{3}+\big{(}1-V_{p}\big{)}\cdot(S_{1111}+(1+D_{1})\cdot S_{2211}), \tag{12}\]  
\[B_{4}=V_{p}\cdot D_{1}+D_{2}+\big{(}1-V_{p}\big{)}\cdot(S_{1122}+D_{1}\cdot S_ {2222}+S_{2233}), \tag{13}\]  
\[B_{5}=V_{p}\cdot D_{3}+\big{(}1-V_{p}\big{)}\cdot(S_{1122}+S_{2222}+D_{1}\cdot S _{2233}), \tag{14}\]  
\[D_{1}=1+2\cdot\big{(}_{p}-_{m}\big{)}/\big{(}\lambda_{p}-\lambda_{m}\big{)}, \tag{15}\]  
\[D_{2}=\big{(}\lambda_{m}-2\cdot_{m}\big{)}/\big{(}\lambda_{p}-\lambda_{m}\big{)}, \tag{16}\]  
\[D_{3}=\lambda_{m}/\big{(}\lambda_{p}-\lambda_{m}\big{)}, \tag{17}\]  
where  
* \(S_{ijkl}\)--Eshelby's constant;
* \(\lambda_{m}\); \(n\)--Lame constants for the matrix (epoxy resin);
* \(\lambda_{p}\); \(p\)--Lame constants for the filler (OSA particles);
* \(B_{x}\); \(D_{y}\)--constants from [44].  
\begin{table}
\begin{tabular}{c c} \hline  & **Elastic Modulus \(E_{c}\) (GPa)** \\ \hline
**0F 0\%** & 3.24 \(\times\) 10\({}^{9}\) \\
**10F 10\%** & 3.51 \(\times\) 10\({}^{9}\) \\
**20F 20\%** & 3.84 \(\times\) 10\({}^{9}\) \\
**30F 30\%** & 4.24 \(\times\) 10\({}^{9}\) \\
**40F 40\%** & 4.72 \(\times\) 10\({}^{9}\) \\
**50F 50\%** & 5.34 \(\times\) 10\({}^{9}\) \\ \hline \end{tabular}
\end{table}
Table 10: The results of the Halpin–Tsai method.  
\[A_{1}=D_{1}\cdot(B_{4}+B_{5})-2\cdot B_{2}, \tag{18}\]  
\[A_{2}=(1+D_{1})\cdot B_{2}-(B_{4}+B_{5}), \tag{19}\]  
\[A_{3}=B_{1}-D_{1}\cdot B_{3}, \tag{20}\]  
\[A_{4}=(1+D_{1})\cdot B_{1}-2\cdot B_{3}, \tag{21}\]  
\[A_{5}=(1-D_{1})/(B_{4}-B_{5}), \tag{22}\]  
\[A=2\cdot B_{2}\cdot B_{3}-B_{1}\cdot(B_{4}+B_{5}). \tag{23}\]  
The results obtained using the Mori-Tanaka method are shown in Table 11.  
A similar approach used for short fiber elastic matrix composites can be found in [46; 47; 48; 49; 50; 51].