Pseudo-affine deformation

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

Fig. 3a. P200 and P400 as a function of draw ration for the pseudo-affine deformation scheme (uniaxially oriented sample) b P20o and P400 as a function of draw ratio X for the rubber network affine deformation scheme (N = 6, uniaxially oriented sample). Reproduced from Journal of Polymer Science by permission of the publishers, John Wiley Sons Incs (C)...

Such considerations appear to be very relevant to the deformation of polymethylmethacrylate (PMMA) in the glassy state. At first sight, the development of P200 with draw ratio appears to follow the pseudo-affine deformation scheme rather than the rubber network model. It is, however, not possible to reconcile this conclusion with the temperature dependence of the behaviour where the development of orientation reduces in absolute magnitude with increasing temperature of deformation. It was proposed by Raha and Bowden 25) that an alternative deformation scheme, which fits the data well, is to assume that the deformation is akin to a rubber network, where the number of cross-links systematically reduces as the draw ratio is increased. It is assumed that the reduction in the number of cross-links per unit volume N i.e. molecular entanglements is proportional to the degree of deformation. 

Pseudo-affine deformation scheme 96, 97 Pseudohexagonal (rotator) phase 67 Pseudorotator phase 67 Pulsed NMR techniques 30... 

Several models have been proposed that attempt to describe the development of orientation in drawn polymers. The pseudo-affine deformation model... 

An interesting comparison has been drawn between the prediction of pseudo-affine deformation and affine deformation of a rubbery network [3] which showed that the orientation distribution functions (cos " /3) grow at a rate which increases with increasing draw ratio in contrast to the behavior of the pseudo-affine scheme. 

Fig. 11. 3 The variation of P2 = (P2(cos e)) and P4 = Pi(cos ff)) with X according to the pseudo-affine deformation scheme. Full curves, P2 (upper) and P4 dashed curves are for affine deformation as in fig. 11.2.

Fig. 11.6 Birefringence plotted against draw ratio for a series of drawn samples of low-density polyethylene. The broken curve shows values calculated according to the pseudo-affine deformation scheme. (Adapted by permission of I. M. Ward.)...

Fig, 7. Variation of second moment with draw ratio for oriented low density polyethylene (curves show results for angles y between Hq and draw direction), (a) experimental airve (b) theoretical curve based on pseudo-affine deformation scheme (after McBrierty and Ward),... 

Fig, 9. Comparison of cosf A and cos A obtained for oriented PVC % and oriented PMMA O Lower and upper curves show predicted relationship according to the affine rubber elasticity model and the pseudo-affine deformation scheme respectively. 

In a somewhat later study of mechanical anisotropy, Hennig and Kausch-Blecken von Schmeling have both independently considered the application and possible development of the aggregate model. Kausch reviewed the applicability of compliance and stiffness averaging predictions for several polymers. He noted that the compliance averaging predictions with the pseudo-affine deformation scheme were close to the experimentally observed behaviour for nylon 66, Dacron and regenerated... 

In a further paper Hennig showed that the linear compressibility results for polyvinyl chloride stretched in the temperature range 90-120 C fitted this relationship very well, with values of cos" 0 calculated from the draw ratio using the pseudo-affine deformation scheme. It was also found... 

The pseudo-affine deformation scheme gives a reasonable approximation to experimental data for low-density polyethylene (Figure 7.15(b)), nylon, polyethylene terephthalate and polypropylene, despite ignoring the distinction between crystalline and disordered regions. It thus provides a basis for extending Ward s aggregate model to predict the compliance constants s y and stiffiiess constants c y of the partially oriented polymer in terms of the constants sy and Cy for the anisotropic elastic unit (in practice, those of the most highly oriented sample obtainable). 

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 amorphous polymers, Ward et al. [96] and Kausch [88] and later Rawson and Rider [95] are in agreement that the mechanical anisotropy can be discussed very satisfactorily by the aggregate model. Moreover, the development of anisotropy with draw ratio can often be described by the pseudo-affine deformation scheme [94]. 

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