Rubbers deformation

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

In rubber deformation, the ratio of the area of one loaded surface to the free area is termed the shape factor . 

The adhesion of cord, textile or metal, to rubber is a specialised measurement in that virtually all interest centres on tyres and to some extent belting. Most static tests consist essentially of measuring the force to pull a cord out of a block of rubber into which it has been vulcanised and it is apparent that the result is critically dependent on the efficiency with which the test piece was moulded. The measured force is also dependent on the amount that the rubber deforms during the test. 

The extremely high energy dissipation in the craze layer, 5 X 108 ergs/gram, would lead to an adiabatic temperature rise of about 24°C during crack/craze propagation, which is insufficient to cause most matrix polymers to pass through their T0 at usual environmental temperatures. Introduction of rubber particles can only lower this temperature rise since rubber secant modulus is very low and rubber deformation does not exceed the 300% or so of the deformed matrix. 

Rubber deformation occurs at nearly constant volume. The work of deformation dW is equal to the free energy dF at constant volume ... 

Figure 3.12 shows a shear strain y imposed on a rubber block containing a network chain with an end-to-end vector r. It is assumed that the crosslink deformation is affine with the rubber deformation, meaning that the components of r change in proportion to the rubber block dimensions. The components ty and are unchanged, but r becomes... 

As rubbers deform without change of volume, the extension ratios in the two lateral directions are 1/VX. Rubber elasticity theory leads to the prediction that the true stress (the force divided by the deformed cross-sectional area) is given by... 

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... 

For the polymer considered in the previous section the birefringence measurements and the stretching or shrinkage took place at different times the birefringence was measured in the frozen-in state of orientation. It is, however, possible to measure the birefringence of a real rubber when it is still under stress at a temperature above its glass-transition temperature. This provides a simultaneous test of the predictions of the rubber deformation theory for both orientation and stress. 

The Guch—Joule Effect and Thermal Aspects of Rubber Deformation... 

As the two smooth spherical surfaces approached each other, within a few micrometers of contact, the familiar Newton s ring pattern could be seen in the narrow gap between the smooth surfaces. Then, as the rubber lenses were moved still nearer, a sudden jump of the rubber was observed and the black contact spot grew rapidly to a large size as the rubber deformed and spread under the influence of molecular adhesion (Fig. 3.12(c)). The appearance of this was very similar to the liquid drop spreading over a glassy polymer surface. To get the rubber lenses apart, a tensile force had to be applied to overcome this molecular adhesion. It is this cracking apart of adhering surfaces which we consider next. 

Rubber deforms at constant volume, and so has a Poisson s ratio of 0.5000 at small... 

The theory of rubber elasticity explains the relationships between stress and deformation in terms of numbers of active network chains and temperature but cannot correctly predict the behavior on extension. The Mooney-Rivlin equation is able to do the latter but not the former. While neither theory covers all aspects of rubber deformation, the theory of rubber elasticity is more satisfying because of its basis in molecular structure. 

At temperatures well above the second order transition temperature, the rate at which the chain segments move is greater than the applied deformation rate and so the rubber deforms satisfactorily. As the temperature is lowered, the molecular deformation rate approaches that of the applied deformation until two rates are equal, at which point the rubber can undergo brittle fracture. This means the rubber has little low-temperature flexibility. 

Rubber-toughened epoxy (10% rubber), deformation test [18] ... 

The experimental data about rubbers deformation are usually interpreted within the frameworks of the high-elastieity entropic theory [1-3], elaborated on the basis of assumptions about high-elastic polymers incompressibility (Poisson s ratio V = 0.5) and polymer chains Gaussian statistics. As it is known [4], the Gaussian statistic is characteristic only for the networks, prepared by chains concentrated solution curing, in the case of their compression or weak (draw ratio A, < 1.2) tension. For such stmctures the fractal dimension d = 2 and in case of v = 0.5 the following classical expression was obtained [3] ... 

Elasticity theory for solids is formulated with the assumption that strains remain small. One then finds the linear relations between stress and strain as they are described by Hook s law. For rubbers, deformations are generally large and the linear theory then becomes invalid. We have to ask how strain can be characterized in this general case and how it can be related to the applied stress. 

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