Reuss average elastic constants

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).

There are two possible sources for the discrepancies between measured and predicted elastic constants when the orientation averages used are determined from models for molecular orientation the models may be incorrect and neither Voigt nor Reuss averaging may be appropriate. In order to examine the second of these possibilities more directly it is necessary to determine the orientation averages experimentally. 

Many ceramics are used in a random polycrystalline form and thus, it is useful to be able to predict the elastic constants from those of the single crystals. The approaches outlined in the last two sections are used for this procedure by considering the random polycrystal as an infinite number of phases with all possible orientations. For example, Voigt and Reuss used a technique based on averaging the stiffness or compliance constants and obtained upper and lower bounds. The Voigt upper bounds for the bulk (B) and shear (/i) moduli of the composite can be written as... 

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]. 

For an isotropic aggregate, the stiffness averaging procedure had been proposed by Voigt, ° and the compliance averaging procedure by Reuss, many years previously. Each had been used to compare the elastic constants of single crystals with those of an isotropic aggregate of single crystals (see for example Ref. 12). 

Averaging the compliance constants defines the elastic properties of the isotropic aggregate in terms of 33 and s. This is called the Reuss average [76]. Averaging the stiffness constants defines the elastic properties of the aggregate in terms of C33 and C44. This is... 

The aggregate model predicts only that the elastic constants should lie between the Reuss and Voigt average values. In polyethylene terephthalate, it is clear that the experimental compliances lie approximately midway between the two bounds. For cold-drawn fibres, it has been shown that this median condition applies almost exactly [87]. 

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). 

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. 

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