The affine rubber deformation scheme

In this scheme it is assumed that, at the temperature of drawing or other deformation, the polymer is in the rubbery state, either as a true rubber containing chemical cross-links or as a polymer containing tangled amorphous chains, in which the entanglements can behave like cross-link points. Once deformation has been accomplished, the polymer is cooled down into the glassy state before the stress is removed, so that the orientation is frozen in . 

The simplest version of the rubber model makes the assumption of an affine deformation when the polymer is stretched the cross-link points move exactly as they would if they were points in a completely homogeneous medium deformed to the same macroscopic deformation (see section 6.4.4, fig. 6.12). The following additional assumptions are also made in the simplest form of the theory. 

For simpKdty, only uniaxially oriented samples are considered here, which means that Aj = A2 = y/lfh hence only A3, which will be called A, the draw ratio, need be specified. 

When the polymer is drawn, two effects contribute to the orientation of the random links  

The calculation of N(6), the distribution of orientations of the random links with respect to the draw direction, therefore involves three steps (see fig. 11.1)  

---

Fig. 11.2 P2 = (P2(cos ff)) plotted against the draw ratio A according to the simplest form of the affine rubber deformation scheme (the first term of equation (11.6)). Curves from left to right are for n =10, 36 and 100, respectively, and each curve is plotted up to A = v7 .

When (P2(cos 6)) is calculated from equation (11.8) it is found to depend on X as shown in fig. 11.3. The variation of Pn cos,6)) with X is quite similar except that it is slightly concave upwards below Xva2 and (P4(cos0)) has a somewhat lower value than (P2(cos0)) for a given X, as shown in fig. 11.3. Unlike the curves for the affine rubber deformation scheme, which are different for different values of n, these curves have no free parameters, so that they are the same for all polymers. The shapes contrast strongly with those for the rubber model, being concave to the abscissa, whereas the latter are convex. The curves for the affine rubber... 

---