Reuss average modulus

EmVf + EfVm and is a Reuss average modulus (see Section 8.6.1). 

Fig. 30. Young s modulus in the 3-direction E3 for semicrystalline PTTs with different crystallinity. The solid and dashed lines represent the Voigt and Reuss averages, respectively [106]...

Steady-state creep rate Shear modulus Hill s average shear modulus Reuss average shear modulus Voigt s average shear modulus Planck s constant... 

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

For any mixture, Voigt s average gives the upper bound of the effective elastic modulus and Reuss average gives the lower bound. 

A patchy saturation is given when fluids are not uniformly mixed. Effective modulus values cannot be estimated from Reuss averaging (lowest level) an upper limit is given by Voigt s averaging ... 

If, however, one assumes uniform stress throughout the same nontextured polycrystal a similar averaging procedure can be performed over the elastic-compliance tensor using the corresponding nine elastic compliance constants Sn, S12, S33, S44, S55, Sss, S12, S23, and S31. This is known as the Reuss approximation (Reuss, 1929), after Endre Reuss (1900-1968), and it yields the theoretical minimum of the elastic modulus. 

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).

We have also used the elastic constants to calculate a Voigt-Reuss-Hill average shear modulus of 178 1 GPa. The value is significantly smaller than the 196 8 calculated by Cohen[2], but is close to the 184 GPa measured by Yeganeh-Haeri et a/[33]. 

The stiffness of a DMO composite of AI2O3-AI with 22% alloy and 4% porosity (231 GPa) was modeled successfully [60] by assuming the metal and the ceramic skeleton to deform equally in series and in parallel, that is, by taking a Reuss-Voigt average. The additional effect of isolated pores could be included by using empirical expressions derived from data on porous alumina. The elastic modulus... 

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. 

G p Gp and G are Voigt, Reuss.and Hill averages of the shear modulus computed from... 

The elastic moduli of a polycrystalline material can be approximately estimated from the elastic constants of the single crystal. First, the maximum and minimum values of the moduli are obtained using Voigt s and Reuss s approximations, respectively. The method in which the average value of the maximum and minimum values is adopted is called Hill s approximation, and the estimated values of Young s modulus are known to be in good agreement with the measured ones (Anderson, 1963). 

FIGURE 6.26 Voigt, Reuss, Voigt-Reuss-Hill average, and Hashin-Shtrikman bounds for com-pressional modulus as function of porosity. Input parameters are quartz = 11 GPa, )Tma = 44GPa, and water 4fl=2.2GPa. [An Excel worksheet you find in the folder Elastic-mechanical, bound models on the website (see also Section 11.2) http /A>ooksite.elsevier.com/ 9780081004043/.]... 

Voigt and Reuss equations in general describe the upper and lower limit of the compressional modulus for a mixture. Domenico (1976) notes that Voigt s averaging gives the best estimate when patch sizes are large with respect to the seismic wavelength. 

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