Affine deformation process

This results in a non-affine deformation process with the long chains taking up most of the elongation. This work by Mark et al clearly indicates the importance of the distribution of the molecular weight between crosslinks in determining the characteristics of a network. 

Shape Change of Structural Entities. In many cases the growing anisotropy is not only a phenomenon of rotating structural entities, but also goes along with a deformation of the structural entities themselves. This case will be studied here. Only affine deformations shall be discussed. In practice, such processes are observed while thermoplastic elastomers are subjected to mechanical load, but also while fibers are spun. 

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

Figure 8.1 5 The crystal orientation of polypropylene fibres and films as a function of the true strain in the deformation process draw temperatures ( ) 135 °C, (a) I 10 °C. The prediction of the pseudo-affine model is shown as a dashed line (from Samuels RJ. Structured Polymer Properties, Wiley, 1974).

Fig. 9 presents the modulus of PpPTA fibers as a function of the applied draw ratio for air-gap spinning at a very low winding tension. This figure demonstrates the importance of the spin-stretch stage in the spinning process. It wiU be shown here that these results are in agreement with the affine deformation model [230]. 

Calculations performed in [12] for the case of fixed elongation rates, ej(t) = = consti, indicate that at the beginning of the process chain deformation follows the macroscopic affine deformation mode... 

The affine deformation of the drops causes the drops to extend into long thin threads, which is referred to as fibrillation. This process continues until the local radii become so small that the Weber (Capillary) number starts to approach the critical Weber number. At this point the threads become unstable and disintegrate as a result of interfacial tension-driven processes the interfaces are now active. The most important mechanisms are the growth of Rayleigh disturbances in the midpart of the thread, end-pinching, retraction, and necking in the case of relatively short dumbbell-shaped threads. 

While all liquid media are viscoelastic, concentrated polymer solutions are characterized by an especially rich distribution of relaxation processes. In order to consider this phenomenon, the pure shear deformation will be employed. Consider an equilibrium concentrated polymer solution and identify the absolute location of every atom in the sample. Now deform the sample such that the new x-coordinate of each atom is increased by a constant multiple of its y-coordinate. This affine deformation might be difficult in the laboratory, but it can be executed for a computer sample of a concentrated polymer solution. Now allow the sample to evolve in time in contact with a heat bath. After a long time, the sample will reach equilibrium, with the new shape determined by the initial shear deformation. 

Fig. 44. Schematic of the stress relaxation process after a large step in deformation, (a) Equilibrium conformation of the tube prior to deformation, (b) Immediately after deformation, the primitive chain has been affinely deformed, (c) After the time Xr, the primitive chain retracts along the tube and recovers its equilibrium contour length (t=XR). (d) After the time Xd the primitive chain leaves the deformed tube by reptation (t=Xa). After Doi and Edwards (56), with permission.

Eigure 10.6 shows a sample result based on (10.2). The figure shows the variation of the aspect ratio of LCP fibers across the thickness of the mold cavity, at different injection speeds, melt temperatures (similar results were obtained for different mold temperatures). Here, it is assumed that the original aspect ratio of the fibers is unity, a fact home out by the results shown in Pig. 10.5. Based on prediction results such as those in Pig. 10.6 and, showing that the aspect ratio of the LCP fibers varies from large, near the skin, to small, in the core, and the fact that most of the LCP fibers near the surface are parallel to each other, we have developed a composite model that incorporates a variable fiber aspect ratio, the latter being dependent on process parameters such as injection speed and mold temperature. The model was based on the assumption of an affine deformation of the LCP domains, without interfacial reactions. 

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