Reuss bulk modulus

Further properties include the isothermal bulk modulus (Kt), the thermal expansion coefficient (/ ) and the constant pressure heat capacity (Cp). The isothermal bulk modulus is calculated first (or from corrected elastic constants as discussed in Section 6.2) and is usually defined as the Reuss bulk modulus... 

It should be noted that for a polycrystal composed of cubic crystalhtes, the Voigt and Reuss approximations for the bulk modulus are equal to each other, as they should be since the bulk modulus represents a volume change but not shape change. Therefore, in a cube the deformation along the principal strain directions are the same. Hence, Eqs. 10.39 and 10.40 are equal and these equations also hold for an isotropic body. The... 

Use the relations in Tables 10.3 and 10.4 to derive the Voigt and Reuss approximations for the bulk modulus of an elastically isotropic polycrystalline aggregate composed of tetragonal monocrystals. 

Use the following values of the elastic-stiffness constants and the elastic-comphance constants (Kisi and Howard, 1998) for tetragonal zirconia monocrystals to determine the Voigt-Reuss-HiU averages for the Young s modulus, E, the shear modulus, G, and the bulk modulus, B. 

Bulk properties of an aggregate of stishovite crystals are also predicted to be substantially modified by the phase transition. These are obtained from the variations of the individual elastic constants using the average of Reuss and Voigt limits (Hill 1952, Watt 1979). The bulk modulus, K, is not sensitive to the transition but the shear modulus, G, is expected to show a large anomaly over a wide pressure interval (Fig. 17a). Consequently, the velocities of P and S waves should also show a large anomaly (Fig. 17b), with obvious implications for the contribution of stishovite to the properties of the earth s mantle if free silica is present (Carpenter et al. 2000a, Hemley et al. 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) Velocities of P (top) and S (lower) waves through a poly crystalline aggregate of stishovite. Circles indicate experimental values obtained by Li et al. (1996) at room pressure and 3GPa. After Carpenter et al. (2000a).

Hill s average bulk modulus Reuss average bulk modulus Voigt s average bulk modulus Molecular weight... 

It should be pointed out that temperatures of 300 K and 0 K are indicated in table 8.22 where the actual values were often 298 K and 4.2 K. Voigt, Reuss and Hill averages are given for the bulk modulus (K) and the shear modulus (G). Only Hill averages are given for Young s modulus ( ) and for Poisson s ratio (v) because they can be calculated from K and G values. Also included in table 8.22 are the three elastic anisotropy ratios (A, B and C ) and the Debye temperatures ( d) at absolute zero. The room temperature density (p) values used in the computation of the elastic properties are listed. 

In order to obtain clay bulk modulus law, we coiadder first, a media cmnposed by a fluid and clay hydrated layers. The clay laminated structure and clay layers / fluid attexnation imply consideting Reuss formula to express the modulus. 

Figure 3. Effective tensile modulus of dense alumina-zirconia composite ceramics Voigt bound (crosses slightly above the upper solid line calculated from the Voigt values of the effective shear and bulk moduli), approximate Voigt bound (according to the mixture rule, upper solid line), Reuss boimd (results of both calculations identical, crosses and lower solid curve), upper and lower Hashin-Shtrikman bounds (dashed curves) and values measured by the resonant frequency method for dense (porosity < 3 %) alumina-zirconia composite ceramics prepared by slip casting.

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