## Appendix B  
In this appendix, the Hashin and Shtrikman second-order bounds for n-phase materials are presented. The Hashin and Shtrikman [33] bounds of bulk modulus \(k\) are given by Equations (12) and (13) as:  
\[k^{HS-}=k^{1}+\frac{A_{1}}{1+\alpha_{1}A_{1}} \tag{12}\]  
\[k^{HS+}=k^{n}+\frac{A_{n}}{1+\alpha_{n}A_{n}} \tag{13}\]  
where \(\alpha_{1}\), \(\alpha_{n}\), \(A_{1}\) and \(A_{n}\) are expressed by Equations (14)-(17) respectively, as:  
\[\alpha_{1}\ =-\frac{3}{3k^{1}+4\mu^{1}} \tag{14}\]  
\[\alpha_{n}=-\frac{3}{3k^{n}+4\mu^{n}} \tag{15}\]  
\[A_{1}=\sum_{r=2}^{n}\frac{V_{V}^{r}}{\frac{1}{F^{r}-k^{1}}-\alpha_{1}} \tag{16}\]  
\[A_{n}=\sum_{r=1}^{n-1}\frac{V_{V}^{r}}{\frac{1}{F^{r}-k^{n}}-\alpha_{n}} \tag{17}\]  
with \(V_{V}^{r}\), volume fraction of phase \(r\) in \(R^{3}\). This way one can bound the homogenized bulk modulus \(\ k^{eff}\):  
\[k^{HS-}\leq k^{eff}\leq k^{HS+} \tag{18}\]  
Let us now consider the Hashin and Shtrikman [33] bounds of shear modulus \(\mu\):  
The lower bound is expressed by Equation (19),  
\[\mu^{HS-}=\mu^{1}+\frac{1}{2}\frac{B_{1}}{1+\beta_{1}B_{1}} \tag{19}\]  
And the highest bound is expressed by Equation (20), as  
\[\mu^{HS+}=\mu^{n}+\frac{1}{2}\frac{B_{n}}{1+\beta_{n}B_{n}} \tag{20}\]  
with the parameters \(\beta_{1}\), \(\beta_{n}\), \(B_{1}\) and \(B_{n}\) are defined by the expressions (21)-(24), respectively. \[\beta_{1}\ =-\frac{3(k^{1}+2\mu^{1})}{5\mu^{1}(3k^{1}+4\mu^{1})} \tag{21}\]  
\[k^{HS+}=k^{n}+\frac{A_{n}}{1+\alpha_{n}A_{n}} \tag{13}\]0  
\[k^{HS+}=k^{n}+\frac{A_{n}}{1+\alpha_{n}A_{n}} \tag{13}\]1  
\[k^{HS+}=k^{n}+\frac{A_{n}}{1+\alpha_{n}A_{n}} \tag{13}\]2The effective property \(\mu^{eff}\) is always surrounded by the two bounds, such that:  
\[k^{HS+}=k^{n}+\frac{A_{n}}{1+\alpha_{n}A_{n}} \tag{13}\]3  
In the case of two-phase composites, the bounds expressions of the bulk modulus k are expressed by Equations (101) and (102) and for by Equations (103) and (104) for the shear modulus:  
\[k^{HS-}=k^{1}+\frac{V_{V}^{2}}{\frac{1}{k^{2}-k^{1}}+\frac{3 \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(P_{2}\)} & \multicolumn{3}{c}{**Lower Bond HS\(-\)**} & \multicolumn{2}{c}{**DH Method**} & \multicolumn{2}{c}{**MS Method**} & \multicolumn{2}{c}{**Upper Bond HS+**} & \multicolumn{2}{c}{**Errors \%**} \\ \cline{2-11}  & \(k^{HS-}/k_{1}\) & \(\mu^{HS-}/\mu_{1}\) & \(k^{eff}/k_{1}\) & \(\mu^{eff}/k_{1}\) & \(\mu^{eff}/k_{1}\) & \(\mu^{eff}/k_{1}\) & \(\mu^{eff}/\mu_{1}\) & \(k^{HS+}/k_{1}\) & \(\mu^{HS+}/\mu_{1}\) & \(Err^{k}\) & \(Err^{\mu}\) \\ \hline
0.05 & 2.8475 & 3.0322 & 2.8498 & 3.0602 & 2.8497 & 3.0623 & 414.4485 & 558.1515 & 0 & 0.0689 \\ \hline
0.08 & 2.9963 & 3.2235 & 2.9997 & 3.2686 & 2.9997 & 3.2728 & 418.1047 & 561.7224 & 0 & 0.1290 \\ \hline
0.12 & 3.2119 & 3.5005 & 3.2166 & 3.5770 & 3.2166 & 3.5835 & 422.9860 & 566.4870 & 0 & 0.1837 \\ \hline
0.15 & 3.3881 & 3.7266 & 3.3938 & 3.8344 & 3.3938 & 3.8424 & 426.6517 & 570.0631 & 0 & 0.2103 \\ \hline
0.22 & 3.8571 & 4.3277 & 3.8654 & 4.5396 & 3.8654 & 4.5499 & 435.2209 & 578.4158 & 0 & 0.2277 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The two methods, DH and MS, normalized effective properties comparison for different volume fractions \(P_{2}\) and the contrasts \(C_{1}=50\), \(C_{2}=10\). The values of the normalized properties must be multiplied by \(10^{-2}\).