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Voigt-Reuss Bounds of Elastic Tensors of Crystal Systems
Cubic Systems
\[ B_V = B_R = \frac{(c_{11}+2c_{12})}{3} \] \[ G_V = \frac{(c_{11}-c_{12}+3c_{44})}{5} \] \[ G_R = \frac{ 5(c_{11}-c_{12})c_{44}}{4c_{44}+3(c_{11}-c_{12})} \] Mechanical Stability Criteria: \( c_{11} >0, c_{44}>0, c_{11}>|c_{12}|, (c_{11}+2c_{12})>0 \)Hexagonal Systems
\[ B_V = (1/9)[2(c_{11}+c_{12})+4c_{13}+c_{33}] \] \[ G_V = (1/30)(M + 12c_{44}+12c_{66}) \] \[ B_R = \frac{A^2}{M} \] \[ G_R = \left( \frac{5}{2}\right) \frac{ A^2 c_{44} c_{66} }{ 3B_V c_{44} c_{66} + A^2(c_{44}+c_{66})} \] \[ M = c_{11}+c_{12}+2c_{33}-4c_{13} \] \[ A^2 = (c_{11}+c_{12})c_{33}-2c_{13}^2 \] Mechanical Stability Criteria: \( c_{44}>0, c_{11}>|c_{12}|, (c_{11}+2c_{12})c_{33}>2c_{13}^2 \)Fiber-type Systems
These are not provided in Voigt-Reuss notation. There is an assumption in this model of quasi-isotropic nature. In dealing with fiber systems that are more cyrstalline in nature or possess axial anisotropy such as chirality, other models should be attempted. \[ B = \frac{-2c_{13}^2 + (c_{11}+c_{12})c_{33}}{c_{11}+c_{12}-4c_{13}+2c_{33}} \] \[ Y_{\parallel} = c_{33} - \frac{2c_{13}^2}{c_{11}+c_{12}} \] \[ Y_{\perp} = \frac{(c_{11}-c_{12})[c_{33}(c_{11}+c_{12})-2c_{13}^2]}{c_{11}c_{33}-c_{13}^2} \] \[ G = c_{44} \] \[ \sigma_{13} = \frac{c_{13}}{c_{11}+c_{12}} \] \[ \sigma_{12} = \frac{c_{33}c_{12}-c_{13}^2}{c_{33}c_{11}-c_{13}^2} \] Mechanical Stability Criteria: \(c_{11} > |c_{12}|\), \( (c_{11}+c_{12})c_{33} > 2c_{13}^2 \), \( c_{44} > 0 \), \( c_{66} > 0 \) And with symmetry in a fiber: \( c_{12} = c_{11} - 2c_{66} \)Tetragonal Systems
This works for classes 4mm, \(\bar{42}m\), 422, 4/mmm \[ B_V = (1/9)[2(c_{11}+c_{12}) + c_{33} + 4c_{13}] \] \[ G_V = (1/30)(M+3c_{11} - 3c_{12}+12c_{44}+6c_{66}) \] \[ B_R = \frac{A^2}{M} \] \[ G_R = \frac{15}{\left( \frac{18B_V}{A^2} \right) + \left( \frac{6}{(c_{11}-c_{12})} \right) + \frac{6}{c_{44}}+\frac{3}{c_{66}}} \] \[ M = c_{11} + c_{12}+2c_{33}-4c_{13} \] \[ A^2 = (c_{11}+c_{12})c_{33}-2c_{13}^2 \] Mechanical Stability Criteria: \( c_{11}>0, c_{33}>0, c_{44}>0, c_{66}>0, (c_{11}-c_{12})>0, (c_{11}+c_{33}-2c_{13})>0, [2(c_{11}+c_{12})+c_{33}+4c_{13}]>0 \)Orthorhombic Systems
\[ B_V = (1/9)[c_{11}+c_{22}+c_{33}+2(c_{12}+c_{13}+c_{23})] \] \[ G_V = (1/15)[c_{11}+c_{22}+c_{33}+3(c_{44}+c_{55}+c_{66}) - (c_{12}+c_{13}+c_{23})] \] \[ B_R = \frac{\Delta}{c_{11}(c_{22}+c_{33}-2c_{23})+c_{22}(c_{33}-2c_{13})-2c_{33}c_{12}+c_{12}(2c_{23}-c_{12})+c_{13}(2c_{12}-c_{13}) + c_{23}(2c_{13}-c_{23}) } \] \[ G_R = \frac{15}{4\frac{[c_{11}(c_{22}+c_{33}+c_{23})+c_{22}(c_{33}+c_{13}) + c_{33}c_{12}-c_{12}(c_{23}+c_{12}) - c_{13}(c_{12}+c_{13}) - c_{23}(c_{13}+c_{23})]}{\Delta} + 3\left[\left(\frac{1}{c_{44}}\right)+\left(\frac{1}{c_{55}}\right)+\left(\frac{1}{c_{66}}\right)\right]} \] \[ \Delta = c_{13}(c_{12}c_{23}-c_{13}c_{22})+c_{23}(c_{12}c_{13}-c_{23}c_{11}) + c_{33}(c_{11}c_{22}-c_{12}^2) \] Mechanical Stability Criteria: \( c_{11}>0, c_{22}>0, c_{33}>0, c_{44}>0, c_{55}>0, c_{66}>0, [c_{11}+c_{22}+c_{33}+2(c_{12}+c_{13}+c_{23})]>0, (c_{11}+c_{22}-2c_{12})>0, (c_{11}+c_{33}-2c_{13})>0, (c_{22}+c_{33}-2c_{23})>0 \)Monoclinic Systems
This is standard orientation with the diad parallel to x2. \[ B_V = \frac{1}{9}(c_{11} + c_{22} + c_{33}+ 2(c_{12}+c_{13}+c_{23})) \] \[ G_V = \frac{1}{15}\left(c_{11} + c_{22} + c_{33}+ 3(c_{44}+c_{55}+c_{66})-(c_{12}+c_{13}+c_{23})\right) \] \[ B_R = \frac{\Omega}{a(c_{11}+c_{22}-2c_{12})+b(2c_{12}-2c_{11}-c_{23}) + c(c_{15}-2c_{25})+d(2c_{12}+2c_{23}-c_{13}-2c_{22}) + 2e(c_{25}-c_{15})+f} \] \[ G_R = \frac{15}{\frac{4[a(c_{11}+c_{22}+c_{12})+b(c_{11}-c_{12}-c_{23})+c(c_{15}+c_{25}) + d(c_{22}-c_{12}-c_{23}-c_{13})+e(c_{15}-c_{25})+f]}{\Omega} + 3\left[\frac{g}{\Omega} + \frac{(c_{44}+c_{66})}{(c_{44}c_{66}-c_{46}^2)}\right]} \] \[ a = c_{33}c_{55}-c_{35}^2 \] \[ b = c_{23}c_{55}-c_{25}c_{35} \] \[ c = c_{13}c_{35}-c_{15}c_{33} \] \[ d = c_{13}c_{55}-c_{15}c_{35} \] \[ e = c_{13}c_{25}-c_{15}c_{23} \] \[ f = c_{11}(c_{22}c_{55}-c_{25}^2) - c_{12}(c_{12}c_{55}-c_{15}c_{25}) + c_{15}(c_{12}c_{25}-c_{15}c_{22}) + c_{25}(c_{23}c_{35}-c_{25}c_{33}) \] \[ g = c_{11}c_{22}c_{33}-c_{11}c_{23}^2-c_{22}c_{13}^2 - c_{33}c_{12}^2 + 2c_{12}c_{13}c_{23} \] \[ \Omega = 2[c_{15}c_{25}(c_{33}c_{12}-c_{13}c_{23}) + c_{15}c_{35}(c_{22}c_{13}-c_{12}c_{23}) + c_{25}c_{35}(c_{11}c_{23}-c_{12}c_{13})] - [c_{15}^2(c_{22}c_{33}-c_{23}^2)+c_{25}^2(c_{11}c_{33}-c_{13}^2) + c_{35}^2(c_{11}c_{22}-c_{12}^2)]+gc_{55} \] Mechanical Stability Criteria: \( c_{11}>0, c_{22}>0, c_{33}>0, c_{44}>0, c_{55}>0, c_{66}>0, \\ [c_{11}+c_{22}+c_{33}+2(c_{12}+c_{13}+c_{23})]>0, (c_{33}c_{55}-c_{35}^2)>0, (c_{44}c_{66}-c_{46}^2)>0, (c_{22}+c_{33}-2c_{23})>0,\\ [c_{22}(c_{33}c_{55}-c_{35}^2)+2c_{23}c_{25}c_{35}-c_{23}^2c_{55}-c_{25}^2c_{33}]>0,\\ \{2[c_{15}c_{25}(c_{33}c_{12}-c_{13}c_{23})+c_{15}c_{35}(c_{22}c_{13}-c_{12}c_{23}) + c_{25}c_{35}(c_{11}c_{23}-c_{12}c_{13})] - [c_{15}^2(c_{22}c_{33}-c_{23}^2)+c_{25}^2(c_{11}c_{33}-c_{13}^2)+c_{35}^2(c_{11}c_{22}-c_{12}^2)]+c_{55}g\}>0 \)Isotropic Systems
For completeness, these are provided. Note there is no Voigt-Reuss averaging scheme. \[ Y = \frac{3 c_{12} c_{44} + 2 c_{44}^2}{c_{12} + c_{44}} = \frac{(c_{11}-c_{12})(c_{11}+2c_{12})}{c_{11}+c_{12}} \] \[ G = c_{44} \] \[ B = \frac{1}{3}\left(c_{11}+2c_{12}\right) \] \[ \sigma = \frac{c_{12}}{c_{11}+c_{12}} \] Subject to the constraint: \( c_{44} = \frac{1}{2}(c_{11}-c_{12}) \)
Adapted from Wu et al. Physical Review B, 76, 054115 (2007)
and from Holm et al. Physical Review B, 59, 12777 (1999)
and from Koski et al. Nature Materials, 12, 262 (2013)