Voigt average elastic constants

In the Voigt notation, the average elastic constants for three atomistic boxes are (accurate to approximately 0.1 GPa)... 

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). 

The results of such calculations for semi-crystalline polyethylene have been reviewed elsewhere [37]. A rather wide range of predicted values is obtained, due to the choice of force constants and also to sensitivity to detailed assumptions on the unit cell structure. In spite of these limitations the principal predictions for the elastic anisotropy are clear. These include the anticipated high values for C33 and the very low values for the shear stiffnesses C44, C55 and cee, which reflect the major differences between bond stretching and bond bending forces that control C33 and the intermolecular dispersion forces that determine the shear stiffnesses. It is therefore of value to compare such theoretical results with those obtained experimentally. Table 7.3 shows results for polyethylene where data for the orthorhombic unit cell at 300 K are used to calculate these constants for an equivalent fibre (Voigt averaging procedure see Section 7.5.2 below) compared with ultrasonic data for a solid sheet made by hot compaction. It can be seen that... 

Finally, it is of interest to compare the theoretical values for a uniaxially oriented sheet (calculated by averaging the stiffness values using the Voigt averaging scheme) with those obtained for a die-drawn rod and a sheet made by hot compaction of high modulus polyethylene fibres (Table 8.4). It can be seen that although, as expected, these materials have not reached full axial orientation so that the experimental values of C33 are much less than the theoretical value, the patterns of anisotropy are very similar, and some of the values for the other elastic constants are surprisingly close. 

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). 

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. 

---