Tube Strain-dependent

The function fj(y) represents the non-linear strain dependence of the deformed tube. The non-linear stress relaxation function in the reptation zone is thus... 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

It is of interest to think about relaxing the assumptions (i) and particularly (ii) introduced at the beginning of Sect. 6.1, although hard experimental tests for specific assumptions of the deformation of the tube constraint itself can never be confined to rheology alone, but will involve at least careful analysis of neutronscattering experiments [46,63]. Not only might the tube diameter depend on the local strain, the localising field described by the tube may well take on an anisotropy consistent with the symmetry of the bulk strain. For discussions of how tube variables deform with strain see [67,68]... 

Fig. 14. The strain dependent part of the shear stress relaxation function for different values of the tube constraint parameter z...

The deformation dependence of the stress in the Edwards tube model is the same as in the classical models [Eqs (7.32) and (7.33)] because each entanglement effectively acts as another crosslink junction in the network. Therefore, the Edwards tube model is unable to explain the stress softening at intermediate deformations, demonstrated in Fig. 7.8. The reason for the classical functional form of the stress strain dependence is that the confining potential is assumed to be independent of deformation. 

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)]... 

Manned et al. assumed that the volume of the tube remains constant by deformation, and derived a result which has the same time dependence as eqn (7.122) but different strain dependence. Though an experiment on PMMA seems to fit with the modified formula, caution is needed in accepting the modification since critical experiments need... 

It is the portion of the response curves around the maxima that are of primary interest in the characterization of nonlinear behavior, because this is where chain stretch has its most pronoimced effect. At longer times convective constraint release becomes dominant. Wagner etal. [23] used start-up of shear flow to evaluate the molecular stress function model for nonlinear behavior in which chain stretch and tube diameter are strain dependent. This theory was found to be suitable for describing an HDPE having a broad molecular weight distribution and an LDPE with random long-chain branching. 

The following additions are made to each tube of top agar the test article (or solvent control) in solution (10-200 pi), the test strain (100 pi) and, where necessary, S9 mix (500 pi). The test is carried out in the presence and absence of S9 mix. The exact volume of test article or solvent may depend on toxicity or solubility, as described in the preceding section. 

In the two classic viscometric deformations of simple shear and extension, the appropriate components of Q have very different behaviour. For small shear strains, the shear stress depends on the component Q which has the linear asymptotic form 47/15. This prefactor is the origin of tne constant v in the tube potential of Sect. 3.For large strains, however, Qxy 7 and therefore predicts strong shear-thinning. Physically this comes from the entanglement loss on re-... 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

It is not clear what the tube diameter is and what is its dependence on strain ... 

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