Average moduli and velocity of minerals

Mineral aggregate averages

VRH Bounds

• Voigt bound (upper): $K_\text{Voigt} = \sum n_i K_i$
• Reuss bound (lower): $K_\text{Reuss}^{-1} = \sum n_i K_i^{-1}$
• Voigt-Reuss-Hill Average: $K_\text{Voigt-Reuss-Hill} = K_\text{Voigt} + K_\text{Reuss}$

Hashin-Shtrikman (HS) Bounds

Polycrystalline averages

Bulk / shear moduli from elastic constants ($C_{ij}$) based on polycrystalline averages (Hill, 1952; Watt, 1987).

• $K_\text{V} = [(c_{11}+c_{22}+c_{33}) + 2(c_{12}+c_{23}+c_{31})]/9$
• $K_\text{R} = [(s_{11}+s_{22}+s_{33})+2(s_{12}+s_{23}+s_{31})]^{-1}$
• $K_\text{VRH} = (K_\text{V} + K_\text{R}) / 2$
• $G_\text{V} = [ (c_{11} + c_{22} + c_{33}) - (c_{12} + c_{23} + c_{31}) + 3 (c_{44} + c_{55} + c_{66}) ] / 15$
• $G_\text{R} = 15 / [4 (s_{11} + s_{22} + s_{33}) - 4 (s_{12} + s_{23} + s_{31}) + 3 (s_{44} + s_{55} + s_{66})]$
• $G_\text{VRH} = (G_\text{V} + G_\text{R}) / 2$

Acoustic velocity

• Compressional velocity: $V_\mathrm{P} = \sqrt{(K + \frac{3}{4} G)/ \rho}$
• Shear velocity: $V_\mathrm{S} = \sqrt{G / \rho}$
• $V_\phi = \sqrt{K / \rho}$
• $V_\mathrm{P}^2 = V_\mathrm{S}^2 + \frac{3}{4} V_\phi^2$

References

• Hill, R. (1952). The Elastic Behaviour of a Crystalline Aggregate. Proceedings of the Physical Society. Section A, 65(5), 349–354. https://doi.org/10.1088/0370-1298/65/5/307
• Watt, P. J. (1987). POLYXSTAL: A FORTRAN program to calculate average elastic properties of minerals from single-crystal elasticity data. Computers & Geosciences, 13(5), 441–462. https://doi.org/10.1016/0098-3004(87)90050-1
• Watt, P. J. (1979). Hashin‐Shtrikman bounds on the effective elastic moduli of polycrystals with orthorhombic symmetry. Journal of Applied Physics, 50(10), 6290–6295. https://doi.org/10.1063/1.325768
• MPtutorial2 (berkeley.edu)
• Rock physics - SEG Wiki
• Microsoft Word - CijExercise.docx (colorado.edu)
• The cij calculator cij.core.calculator module — Cij documentation (mineralscloud.github.io)