Reuss-Voigt average

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... 

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. 

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]. 

MEASURED EXTENSIONAL AND TORSIONAL COMPLIANCES OF UNORIENTED POLYMERS COMPARED WITH REUSS AND VOIGT AVERAGE PREDICTIONS FOR AN AGGREGATE MODEL. 

Fkj. 2. Low density polyethylene. Variation of extensional / > transverse (Ego) and torsional (G) moduli with draw ratio (a) experimental, (b) aggregate model predictions. V = Voigt average, R = Reuss average. (Adapted from Ref. 9.)... 

Table 7.6 compares the measured compliances for isotropic samples of five polymers with the Reuss and Voigt average compliances calculated from measurements on highly oriented specimens. For polyethylene terephthalate and low-density polyethylene the measured isotropic compliances fall between the calculated boimds, suggesting that here molecular orientation could well be the principal factor that determines mechanical anisotropy. For nylon 6 6 the... 

Reuss average Voigt average Measured Reuss average Voigt average Measured... 

In particular polymers either the Reuss or the Voigt averages or a mean of the two lie closest to measured values. It is likely that these differences relate to details of the stress and strain distributions at a molecular level, which should in turn be related to the strueture. 

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]. 

For low-density polyethylene, the Voigt averaging scheme does not predict the anomalous behaviour. However, the Reuss average does, and therefore appears to describe the physical situation more closely. A similar conclusion was reached by Odajima and Maeda [60] who compared theoretical estimates of the Reuss and Voigt averages of single crystals of polyethylene with experimental values. 

Table 7. Effective elastic properties (elastic moduli and Poisson ratios) of polycrystalline alumina and tetragonal zirconia (t-Zr02), calculated from the components of the respective stiffness and compliance matrices [Wachtman et al. 1960, Kisi Howard 1998] Voigt boimd (subscript V), Reuss bound (subscript R) and Voigt-Reuss-Hill average (subscript VRH).

The real measured data are positioned between the two bounds. Therefore, as a representative value, the arithmetic mean of the two bound values is frequently used and called the Voigt-Reuss-Hill average... 

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/.]... 

Mavko et al. (1998) note f/te Voigt-Reuss-Hill average is used to estimate the effective elastic moduli of a rock in terms of its constituents and pore space . 

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).

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]...

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... 

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). 

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