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... 

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.)... 

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... 

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. 

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]. 

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. 

In nylon, the Voigt average is closest to the experimentally observed data. It is interesting to note that both averaging schemes predict a maximum in the torsional modulus as a function of draw ratio. 

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