Reuss average

Given the elastic tensor we can obtain Reuss average, isotropic bulk and shear moduli... 

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]...

As an example, it can be shown that the Reuss average compliance 53333 for a uniaxial sample consisting of an aggregate of transversely isotropic units is given in terms of the compliances Sy/y of the unit by... 

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. 

As an example, fig. 11.8 shows a comparison of experimental and predicted moduli for cold-drawn low-density polyethylene. The theoretical curves are calculated using the Reuss averaging scheme and the values of (/ 2(cos0)) and P4(cos0) are derived from NMR data for the crystalline regions only. The fit is now quite good. 

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.)... 

Reuss averaging bounds (Table 2), exceeding the mean predicted value by a ratio ranging only between M2 and M8. 

Like our earlier calculation for the bulk PLC, E3 and E[ can be obtained using either the Voigt or Reuss averaging scheme. The aspect ratio L/D can be estimated assuming that the deformation is affine [26,38] i.e. the deformation of the PLC domains is identical to that of the... 

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... 

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

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 orientation functions can be calculated on the pseudo-affine deformation scheme, and Figures 8.18-8.22 show that the aggregate model then predicts the general form of the mechanical anisotropy. It is particularly interesting that the predicted Reuss average curves for low-density polyethylene show the correct overall pattern, including the minimum in... 

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. 

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

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

The simplest case is a uniform distribution of the phases. In this case (isostress), Reuss average can be applied... 

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 ... 

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