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Voigt-Reuss constants

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Journal Article
Published: 01 January 2020
Russ. Geol. Geophys. (2020) 61 (1): 110–118.
...D.V. Peregudov Abstract —A general procedure is suggested for calculating the upper (Voigt) and lower (Reuss) bounds of the average elastic constants of an anisotropic medium from crystallographic directions. The elastic tensors of Hooke’s law can be expanded into irreducible representations...
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Variations of bulk properties derived from variations of the individual ela...
Published: 01 January 2000
Figure 17. Variations of bulk properties derived from variations of the individual elastic constants of stishovite. (a) Bulk modulus ( K ) and shear modulus ( G ). Solid lines are the average of Reuss and Voigt limits; the latter are shown as dotted lines (Voigt limit > Reuss limit). (b
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( a ) Variation of elastic <b>constants</b> through the tetragonal ↔ orthorhombic ...
Published: 01 February 2006
F igure 6. ( a ) Variation of elastic constants through the tetragonal ↔ orthorhombic transition in stishovite, based on expressions given in Carpenter et al. (2000) and the parameters listed in Table 1 . ( b ) Bulk and shear moduli calculated as the average of Voigt and Reuss limits
Journal Article
Published: 01 November 2006
American Mineralogist (2006) 91 (11-12): 1893–1900.
... to the uncertainty in mode assignment and poorer constraint of crystal orientation for the meridian platelet. Using the measured elastic constants, the Reuss and Voigt bounds, and Voigt-Reuss-Hill (VRH) average of the aggregate bulk and shear moduli were calculated and fit to the Eulerian finite strain equations...
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Journal Article
Journal: Geophysics
Published: 03 May 2013
Geophysics (2013) 78 (3): L27–L35.
... modulus in Gassmann’s equations. For small contrasts in mineral stiffness, the Brown-Korringa constants are approximately equal to the Voigt-Reuss-Hill average of the individual mineral bulk moduli. In a second approach, a multilayered spherical shell model is used to describe a rock where a particular...
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Calculated variations of bulk and shear moduli of polycrystalline SrTiO 3  ...
Published: 01 February 2007
F igure 8. Calculated variations of bulk and shear moduli of polycrystalline SrTiO 3 as a function of pressure at 295 K. These are average values of Voigt and Reuss limits, calculated from the variations of single-crystal elastic constants shown in Figure 7a .
Journal Article
Published: 01 August 2014
American Mineralogist (2014) 99 (8-9): 1562–1569.
... tensor was experimentally determined at ambient conditions by Brillouin scattering spectroscopy. The elastic modulii obtained from the Voigt-Reuss-Hill approximation using the elastic constants determined in this study are K S = 206 GPa and μ = 129 GPa, whereas the isotropic compressional and shear sound...
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Changes in selected physical properties through the α ↔ β transition as a f...
Published: 01 February 2006
expressions for Voigt and Reuss limits (dotted lines) from Watt and Peselnik (1980) and the calculated single crystal elastic constants shown in a . Filled circles are bulk and shear moduli obtained for a polycrystalline quartz sample at ambient conditions (M.A. Carpenter and T.W. Darling, unpublished data
Journal Article
Published: 01 April 2007
American Mineralogist (2007) 92 (4): 570–576.
.... The aggregate bulk modulus, shear modulus, and Poisson’s ratio are K S0 = 125.3(4) GPa, G 0 = 72.9(2) GPa, and σ 0 = 0.26(1) for the VRH (Voigt-Reuss-Hill) average, respectively. The maximum azimuthal anisotropy of zoisite is 22% for compressional velocity and 33% for shear velocity. The maximum shear splitting...
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Full elastic <b>constant</b> tensor vs. pressure: ( a ) three principal elastic co...
Published: 01 October 2015
Figure 4 Full elastic constant tensor vs. pressure: ( a ) three principal elastic constants, ( b ) six off-diagonal elastic constants, and ( c ) four shear elastic constants, ( d ) bulk and shear modulus vs. pressure. The Voigt and Reuss limits are indistinguishable for the bulk modulus at all
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Crossplot of normalized dry rock bulk modulus versus porosity with backgrou...
Published: 01 August 2020
Figure 6. Crossplot of normalized dry rock bulk modulus versus porosity with background contours of constant pore-space stiffness. Contours range between the Reuss and Voigt averages (red curves). Samples from well B are color coded with volume fraction of pore types: (a) stiff pores, (b
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( a ) Calculated variations of the bulk ( K ) and shear ( G ) moduli for a ...
Published: 01 February 2006
F igure 9. ( a ) Calculated variations of the bulk ( K ) and shear ( G ) moduli for a polycrystalline sample of SrTiO 3 . Softening as T → T c from above the transition point has been included, though calculated values are shown only down to T c +1 K. Reuss and Voigt limits are shown
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Bulk, shear, and isotropic Young’s moduli for polycrystalline SrTiO 3 , as ...
Published: 01 February 2007
) of Rehwald (1970) were used to calculate K , G (average of Voigt/Reuss limits) and E . Experimental values of the bulk modulus are consistent with calculated values for tetragonal SrTiO 3 , but values for the shear modulus are smaller than values calculated assuming that the only softening is due
Journal Article
Journal: Geophysics
Published: 01 January 2000
Geophysics (2000) 65 (5): 1355–1363.
... change in bulk volume ( v b ) in response to a change in confining pressure ( P c ) while pore pressure ( P p ) remains constant: C bc =- 1/v b ) · (∂ v b /∂ P c ) Pp . F IG . 6. Plot of α μ versus porosity ( ϕ ) for various modulus–porosity models: Voigt (V), Reuss (R), Wood (W...
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Journal Article
Journal: Geophysics
Published: 09 November 2021
Geophysics (2021) MR1–MR11.
... an even lower IF value. Figures  6 and 8 depict two extreme cases in which the pore fluids are mixed according to the Voigt upper bound and Reuss lower bound models. However, the real scenario might be more complex and the mixture modulus may be dependent on porosity. For a low-porosity chalk...
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Journal Article
Journal: Geophysics
Published: 01 January 2004
Geophysics (2004) 69 (2): 398–405.
... sensitivity in terms of a simplified gain function G . General Voigt-Reuss bounds and critical porosity limits constrain the equations and provide upper and lower bounds of the fluid-saturation effect on bulk modulus. The “D” functions are simplified modulus-porosity relations that are based on empirical...
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Journal Article
Published: 19 July 2019
Petroleum Geoscience (2020) 26 (3): 400–416.
..., but small enough to be considered as a material point (local equilibrium) at the reservoir scale. Voigt (1889) and Reuss (1929) bounds are widely accepted as the upper and lower boundaries of the effective elastic responses of an aggregate sample of heterogeneous materials. However, Voigt and Reuss...
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Predicted variations of single crystal (solid lines) and polycrystal (broke...
Published: 01 February 2007
) moduli are averages of the Reuss and Voigt limits calculated from the single-crystal elastic constants; the isotropic Young’s modulus, E , was calculated from these. Stars are room temperature values of E extracted from the experimental data of Harrison et al. (2003) for polycrystalline samples
Journal Article
Published: 01 May 2016
American Mineralogist (2016) 101 (5): 1228–1231.
...; K Voigt = 62.9(1.1), 64.4(0.6) GPa; G Reuss = 24.1(1), 24.5(1) GPa; and G Voigt = 36.1(5), 36.1(7) GPa. The small differences in moduli between the samples suggests that variations in composition and in state of Al, Si order only have minor effects on the average elastic properties of K-rich...
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Journal Article
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (2): 428–440.
... powders with mineral powders. In our data analysis, we evaluated three theories for inverting the unknown elastic constants of clays: the weighted Voigt-Reuss-Hill (VRH) average ( Voigt, 1928 ; Reuss, 1929 ; Hill, 1952 ), the modified (weighted) Hashin and Shtrikman (HS) average ( Hashin...
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