Reduced stress affine model

Note that, with this model, the reduced stress defined by Eq. (3) depends on the deformation X, whereas it did not for the simple affine model. 

Both the affine and the phantom network models predict that the reduced stress, [/ ], measured in uniaxial deformation is independent of the deformation ratio. However, it... 

For both the phantom and the affine networks, the reduced stress is calculated to be independent of deformation. However, stress-strain measurements carried out in uniaxial extension of dry and swollen networks have revealed departures from these predictions of simple models 5. These observations then gave rise to phenomenological equations like the Mooney-Rivlin expression, i.e. 

It is interesting to compare eq. (3.54) with the expressions obtained from the statistical theories (Fig. 3.20). According to both the affine network model and the phantom network model of James and Guth, the reduced stress remains constant and independent of strain, which is not the case for the Mooney-Rivlin equation. 

Fig. 3. Reduced stress as a fiinction of reciprocal extension ratio. The upper and lower horizontal lines represent results from affine and phantom network models, respectively. Circles show representative data from experiments, and the curves are from the constrained junction theory.

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

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