Gaussian elasticity theory

Evaluation of Non-Gaussian Elasticity Theories. There are now numerous theories of rubberlike elasticity which use non-Gaussian distribution functions to take into account limited... 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

The role of chain entangling in cross-linked elastomers is an old issue which has not yet been settled. The success of Flory s new rubber elasticity theory 0-5) in describing some of the departures from the simple Gaussian theory has acted as a strong catalyst for new work in this area. 

The chains of typical networks are of sufficient length and flexibility to justify representation of the distribution of their end-to-end lengths by the most tractable of all distribution functions, the Gaussian. This facet of the problem being so summarily dealt with, the burden of rubber elasticity theory centers on the connections between the end-to-end lengths of the chains comprising the network and the macroscopic strain. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

In Chapter III, section 1 the elasticity theory for ideal Gaussian networks is reviewed. Since in actual practice dry rubbery networks virtually never adhere to these theories — even in the range of moderate strains — experimental verification of the theory has been undertaken... 

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. 

For a network of uniform length chains. Lake and Thomas substituted for L with an Equation predicted from rubber elasticity theory. They also derived an alternate expression for L for a network of random Gaussian chains. The two expressions differ only by a small numerical constant. Making either substitution, and rearranging terms, it can be shown that... 

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... 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

In the development of rubber elasticity theory (Section 9.7.1), it will be shown that the restoring force,/, on a chain or chain portion large enough to be Gaussian, is given by... 

The rubber in a blown-up balloon is stretched in a biaxial fashion. Derive the force-strain relationship under the assumption that the rubber follows the Gaussian statistical theory of rubber elasticity. 

At the same temperature the stress/strain response, as conventionally measured, appears to follow the Gaussian kinetic theory of rubberlike elasticity, though careful experiments under biaxial conditions show that this is not true for strains less than lO. ... 

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. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

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