Time-strain separability

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows ... 

Such a behavior of stress relaxation is called time-strain separability (Larson 1988). The factor h(y) is called the damping function. The empirical damping functions that are often used in polymer melts include the exponential function... 

Equation (9) has an empirical origin but a theoretical foundation can be proposed as follows. Indeed, quite a common assumption in many approaches of nonlinear viscoelasticity consists in considering time-strain separability (or factorability). Such an assumption readily means that the nonlinear relaxation modulus function G(t, y) can be separated into a time-dependent and a strain-dependent contributions, so that ... 

The damping function is expected to be independent of molecular weight. But since % increases with M, for a sufficiently broad molecular weight distribution, the retraction time of long molecules eventually overlap with the reptation times of the short chains leading to the failure of time-strain separability [7]. 

The study that produced the data shown in Fig. 10.2 [2] included several other solutions, and it was found that molecular weight and concentration had little effect on the damping function for cM around 5 lO. Todemonstrate the degree of time-strain separability, the data ofFig. 10.2 are replotted in Fig. 10.5 as relaxation modulus divided by the vertical shift factor required to superpose them, i.e., The superposibility is excellent for times longer than a... 

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