Rubber modeling

Deuterium NMR is very sensitive to orientational behavior and order there are a number of papers dealing with constrained polymeric networks. For example, 2H NMR (in both, solid state and solution) is used in the study of the orientational order generated in uniaxially strained rubbers as a function of the crosslink density. Two sets of rubbers (model end-linked silicone rubbers and randomly crosslinked diene networks) were investigated directly (on perdeuterated silicone labelled chains) and indirectly, via C6D6 as an NMR probe for diene rubbers 45). 

Chamberlain Dyson (1956) measured deposition of unattached 212Pb in a rubber model of the trachea and main bronchi. At an inspiratory flow of 20 1 min-1, corresponding to a ventilation rate of about 0.6 m3 h-1, the activity deposited per cm2 of surface in the bronchi was 0.18% of that entering the trachea. No difference was found in the deposition of unattached 218Po (RaA) compared with unattached 212Pb (ThB). 

Consider a generic atom [1 in the rubber model of Section I. From the virial Eq. (6), its nondimensional contribution Aa (P) to the stress caused by the nonbonded interaction through the potential u b(r) is... 

The simplest explanation is that there is a rubber-like network present and that this has a maximum extensibility due to the degree of entanglement, which is constant for a given grade of polymer and depends on its molar mass and method of polymerisation. This limiting extensibility is not to be confused with the limit of applicability of the affine rubber model for predicting orientation distributions discussed in section 11.2.1 because the limiting extension can involve non-affine deformation. 

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. 

Using only the first term is essentially equivalent to the Gaussian approximation of rubber elasticity theory developed in section 6.4.4. The form of (P2(cos9) versus X for the rubber model is then as shown... 

An important consequence of the affine assumption is that the rubber model in this simple form is applicable only up to Imax = because at this draw ratio the chains parallel to the draw direction in the undeformed material are fully extended in the deformed material and cannot extend further (see equation (3.5) and the sentence following it). Further extension of the material as a whole could thus take place only non-afiinely. Equation (11.6) shows that P2 cos0)) f X /(5n) for A > 3, so that the maximum value of (P2(cosP)) to which the simple theory applies is approximately 0.2, as fig. 11.2 shows. 

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... 

When a polymer is spun to form a fibre the molecular network in the melt is extended and subsequently frozen in the extended state. In commercial fibre production, heat setting or subsequent hot drawing above the glass-transition temperature may also be used, but for fibres that have only been spun and drawn simultaneously the distribution of orientations in the fibre should conform to the predictions of the rubber model. 

The affine rubber model and the stress-optical coefficient... 

The variation of birefringence with draw ratio for a set of uniaxially drawn samples of a certain polymer is found to be consistent with the simplest version of the affine rubber model when the draw ratio is less than 3.5. If the birefringence is 7.65 x 10 for draw ratio 3.0, calculate its value for a sample of draw ratio 1.5. If the birefringence for a very highly oriented sample is 0.045, what is the effective number of random links per chain ... 

Figure 1 S.F>. Failure ol lap joints (a) Fairbaim s picture of a riveted joint on the Conwy bridge, and (b) deformation of a rubber model adhesive lap joint.

The chemistry of the accelerated vulcanization of BR, SBR, and EPDM appears to have much in common with the vulcanization of natural rubber. Before the formation of cross-links, the rubber is first sulfurated by accelerator-derived polysulfldes (Ac-S -Ac) to give macromolecular, polysulfidic intermediates (rubber-Sx-Ac), which then form crosslinks (rubber-S -rubber). As in the case of natural rubber, the average length of a crosslink (its sulfidic rank, the value of x in the cross-link, rubber-Sx-rubber) increases with the ratio of sulfur concentration to accelerator concentration (S/Ac) used in the compounded rubber mix. However, in the case of BR or SBR, the cross-link sulfidic rank is not nearly as sensitive to S/Ac as it is in the case of natural rubber. Model compound studies of the vulcanization of EPDM (e.g., wherein ethylidenenorbomane was used as a model for EPDM) indicate that the polysulfidic rank of the EPDM cross-links probably responds to changes in S/Ac in a natural rubber-hke fashion. 

G. L. Bradley, P. C. Chang, and G. B. McKenna, Rubber Modeling Using Uniaxial Test Data J. Appl. Polym. Sci., in press. 

Mata, R, Boroschek, R., Oiler, S. C Barbat, A.H. 2007a. High damping rubber model for energy dissipating devices, Journal of Earthquake Engineering ll(2) 231-256. 

Comparison of Data with Calculations Based on Classical Rubber Models. . 84... 

One can then take the mesh size equal to this gives, via the rubber model, the plateau modulus ... 

The stress pattern at the end of a square-edged adhesive layer in a lap joint is shown in Fig. 21. This is again a plot of principal stresses, the interpretation of which was given in the discussion of the rubber model. The highest tensile stress exists at the corner of the adhesive... 

Fig. 20(a). Comparison between calculated and experimental displacements of the silicone rubber model. (The black crosses are the finite-element predictions of the intersections of the grid lines of the model.) (from Adams and Peppiatt,... 

Fig. 20(b). Principal stress pattern for silicone rubber model showing end effects (from Adams and Peppiatt, 1974). 

Another approach has been used by Tanaka and Ushiki. These authors used a simple Flory argument for the self-crosslinked molecule in solution. To account for the elastic and solution properties a combination of the Flory-Huggins lattice theory and phantom rubber model has been applied to the microgel. 

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