Network draw ratio

Van Ruiten and co-workers [45] studied the drawability and attainable mechanical properties of PA 4,6 yarn using true stress-strain curves. The concept of the molecular network was applied to an analysis of these fibres and yarns and the mechanical properties of yarns with different draw ratios was evaluated in terms of the network draw ratio, which was determined by matching true stress-strain curves. The validity of the molecular network concept for these yarns and its suitability for predicting fibre drawability was assessed. A method for predicting the maximum attainable tenacity of drawn yarns under given drawing conditions from precursor mechanical properties is proposed. 

The Concept of the True Stress-True Strain Curve and the Network Draw Ratio... 

Figure 12.33 Curve matching procedure to obtain the network draw ratios for polyester fibres. (Reproduced with permission from Long, S.D. and Ward, I.M. (1991) Shrinkage force studies of oriented polyethylene terephthalate. ]. Appl. Polym. Sci., 42, 1921. Copyright (1991) John Wiley Sons, Inc.)...

Recent research has also addressed two issues. First, to what extent does the macroscopic measured draw ratio reflect the network draw ratio, and whether slippage occurs at a molecular level. Although curve matching attempts to resolve this problem it has also been instructive to invite more sophisticated molecular studies, for example by scanning near-field optical microscopy [97]. 

The discussions of the true stress-true strain curve and the network draw ratio are clearly consistent with the concept of a molecular network, which provides a physical understanding of what is known to engineers as strain hardening and has also been shown to be relevant to slow crack propagation (see Section 13.6.3). 

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. 

It has been shown that the anisotropy depends on the orientation of the diagonals of indentation relative to the axial direction 14). At least two well defined hardness values for draw ratios A. > 8 emerge. One value (maximum) can be derived from the indentation diagonal parallel to the fibre axis. The second one (minimum) is deduced from the diagonal perpendicular to it. The former value is, in fact, not a physical measure of hardness but responds to an instant elastic recovery of the fibrous network in the draw direction. The latter value defines the plastic component of the oriented material. 

For a simple rubber-like network, the maximum draw ratio varies with the number of statistical chain segments between crosslinks as... 

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. 

The stress-strain relationships of elastomeric (rubbery) networks at low extension (or draw) ratios (k, which is the length of the deformed specimen divided by the length of the initial undeformed specimen) can be described in terms of Equation 11.37 [29], which is the simplest possible constitutive equation for the deformation of an isotropic incompressible medium. 

Figure 11.13. Comparison of the predictions of two models for the stress-strain behavior of elastomeric networks. There are 100 Kuhn segments between adjacent network junctions in this particular example. Stress is denoted by a, shear modulus by G, and draw ratio by X.

Figure 11.14. Empirical master curve for ultimate stress Gmax of a crosslinked elastomer (normalized by crosslink density and by absolute temperature) as a function of maximum draw ratio Amax-Vn", where n is the average number of Kuhn segments between network junctions.

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

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