Rubber high-elasticity theory

As it is known [83], a glassy polymers behavior on cold flow plateau (part III in Fig. 4.17) is well described within the frameworks of the rubber high-elasticity theory. In Ref [39] it has been shown that this is due to mechanical devitrification of an amorphous polymers loosely packed matrix. Besides, it has been shown [82, 84] that behavior of polymers in rubber-like state is described correctly under assumption, that their structure is a regular fractal, for which the identity is valid ... 

Thus the results obtained within the frameworks of the rubber high-elasticity theory suggest that the macromolecular entanglement network in semi-crystalline polymers is... 

The density of the chemical crosslinking nodes network was determined according to the rubber high-elasticity theory [10] ... 

Rubber-like Elasticity Theory and Highly Crosslinked Epoxies.120... 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. 

In this chapter, AFM palpation was introduced to verify the entropic elasticity of a single polymer chain and affine deformation hypothesis, both of which are the fundamental subject of mbber physics. The method was also applied to CB-reinforced NR which is one of the most important product from the industrial viewpoint. The current status of arts for the method is still unsophisticated. It would be rather said that we are now in the same stage as the ancients who acquired fire. However, we believe that here is the clue for the conversion of rubber science from theory-guided science into experiment-guided science. AFM is not merely high-resolution microscopy, but a doctor in the twenty-first century who can palpate materials at nanometer scale. 

Priss LS (1980) The theory of high elasticity and birefringence of rubber. Int J Polym Mater... 

In conclusion, it can be said that the theory can well describe the development of the gel structure. The correlation between the equilibrium modulus and sol fraction is very good so that the sol fraction can alternatively be used for determination of the concentration of EANC s if an accurate and precise determination of conversion meets with difficulties. It is to be recalled here that the Gaussian rubber elasticity theory does not apply to highly crosslinked networks of usual stoidiiometric systems. When a good theory is available, the calculated value of taking possibly into account the topological limit of the reaction will be ne ed. 

The condition of swelling equilibrium can be calculated by means of two theoretical approaches. It is assumed that the chemical potential of mixing for a network is the same as the chemical potential of mixing an uncross-linked polymer of high molecular weight and of the same structure as the network polymer. The mixing term can be described by means of the Flory-Huggins (FH) equation. The calculation of the elastic deformation term is based on the rubber elasticity theory (RET). 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

Neither the uniform strain model nor the uniform stress model is appropriate for this microstructure. Consequently, the elastic moduli of polyurethanes lie between the limits set by Eqs (4.11) and (4.12). For a network chain of Me = 6000, the rubber elasticity theory of Eq. (3.20) predicts a shear modulus of about 0.4 MPa. The hard blocks will have the typical 3GPa Young s modulus of glassy polymers. Increases in the hard block content cause the Young s modulus to increase from 30 to 500 MPa (Fig. 7.13). For automobile panel applications it is usual to have a high per cent of hard blocks so that the room temperature flexural modulus is 500 MPa. 

Returning now to the left part of the curve, for a 350% extension it has been shown that, at about —70° C, it suddenly turns in the same direction as for hard materials like metals. Referring to p. 661 it may be remembered that —70° C is the temperature T, at which rubber freezes, loosing its high-elastic properties. In accordance with the above developed theories, the conditions for ideal high-elasticity are now no longer present, because the molecules are not sufficiently movable. At about —70° C the kinetic (or entropic) elasticity is therefore transformed into potential elasticity. 

In Fig. 2 and 3 the dependences a on generalized stress for studied rubbers, corresponding to the equations (13) and (14), are shown. As can be seen, in case of composites the linearity of these dependences is violated, i.e., at least, the filled rubbers behaviour does not corresponded to high-elasticity classical theory, that is assumed above. Differently speaking, filled rubbers are impossible to consider as ideal, for which internal energy change MJ is equal to zero in deformation process. 

Figure 3. The dependence of stress a on generalized strain (X-X ) at temperatures 250 (1) and 380 K (2) for rubber SKI-3, filled by technical carbon. The shaded lines are shown the dependencies assumed by high-elasticity classical theory...

In Fignre 5, the storage modulus of a typical cross-linked rubber is compared to that of a PSA made from a mixture of the rubber and tackifier. The rubber has a low which is why it is soft and mbbery at room (or use) temperature, but its modulus is too high for it to be a PSA. Addition of tackifier decreases the modulus below the Dahlquist criterion and allows for the mixture to be a PSA. The effect of dilution of the rubber network with a tackifier can be predicted by rnbber elasticity theory nsing the equation below (31,32). The effect of fillers can also be predicted. 

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