Non-affine tube model

The deformation dependence of the confining potential [Eq. (7.62)] results in a non-classical stress strain dependence of the non-affine tube model. The prediction of this model for the stress-elongation relation in tension is qualitatively similar to the Mooney-Rivlin equation [Eg. (7.59)]... 

The results in this section were all derived for unentangled networks. The Edwards tube model for entangled networks gives identical results with N replaced by N, the number of Kuhn monomers in an entanglement strand in the preparation state, because both entanglement strands and network strands are assumed to deform affinely in the Edwards tube model. If the Edwards tube model were correct, the universal relations [Eqs (7.91) and (7.92)] would still apply for entangled networks, since they are independent of N. However, the non-affine tube models predict that entangled networks will swell considerably more than the Edwards tube model predicts. 

Consider a non-affine tube model, with the number of monomers in the virtual chains changing with network deformation as , = A n,o, where ,o the number of monomers in virtual chains constraining fluctuations in the i direction in the undeformed state. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

In the non-affine slip-tube model, entanglements are represented by slip-rings that are attached to the elastic... 

Comparison of the non-affine slip-tube model [Eq. (7.65) using Gx and as adjustable parameters] with experiments, see M. Rubinstein and S. Panyukov, Macromolecules 35, 6670... 

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