Overshooting relaxation

Whereas a) could be accounted for by a pore model like that in Section 5.1, an "overshooting" relaxation behaviour as indicated by the solid line in Fig. 8 cannot be brought into agreement with the relaxation behaviour of the pore network of (5.1), the reason lying in the fact that there is only one single relaxation time T for (5.1) as shown already in Section 2.2 and as given by (5.6). Thus, the relaxation following a steplike increase of the external Na -concentration will always be a monotomic function of time as indicated by the dotted line in Fig. 8. 

The essential point of the model which Lindemann has suggested for saturation and overshooting relaxation is the assumption that the Na -pores have two sites or entries for Na one for transport of Na across the membrane by a pore mechanism and a second one which closes the pore if occupied by a Na -ion. The network of this model simply reads... 

Fig. 2 shows the dynamic response of stack voltage to the step changes of various applied current densities. Like the former case of applied current pulses, the response exhibits the overshooting and relaxation which is caused by the methanol oxidation kinetics on the catalyst surface. The steady state stack voltage was found to be the same for both pulse and step loads with the same current density. 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

The most surprising result is that such simple non-linear relaxation behaviour can give rise to such complex behaviour of the stress with time. In Figure 6.3(b) there is a peak termed a stress overshoot . This illustrates that materials following very simple rules can show very complex behaviour. The sample modelled here, it could be argued, can show both thixotropic and anti-thixotropic behaviour. One of the most frequently made non-linear viscoelastic measurements is the thixotropic loop. This involves increasing the shear rate linearly with time to a given... 

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

Stratton,R. A., Butcher,A.F. Stress relaxation upon cessation of steady flow and the overshoot effect of polymer solutions. J. Polymer Sci. Polymer Phys. Ed. 11,1747-1758 (1973). 

No overshoot and linear limits in transient stress growth. Linear relaxation modulus in step shear strain. 

A more precise comparison can be made by looking at the birefringence changes along the flow axis (Pig. 34). The birefringence increase in the reservoir is similar for both models and veiy close to experimental values, as already observed with results from other numerical computations [27,30,65]. The relaxation along the die land is very consistent for the mPTT model, where the stresses do not relax totally before the channel exit. Results with the GOB model indicate a less realistic relaxation, vdth a local overshoot near the exit, which is not observed experimentally. 

Although this experimentally observed scaling behavior is correctly predicted by the Doi-Ohta theory, the shape of the transient response curve—in particular, the overshoot and undershoot in the shear stress—are not predicted. This implies that the relaxation expressions chosen by Doi and Ohta, Eqs. (9-46) and (9-47), are inaccurate. This is not very surprising, since Eqs. (9-46) and (9-47) were chosen rather arbitrarily from many possible forms that satisfy the scaling relationship. Optical microscopy suggests that the overshoot and undershoot are caused by elongation of droplets followed by their breakup (Takahashi and Noda 1995). Vinckier et al. (1997) have shown that the stress growth after start-up or... 

Metastable liquid will relax by fast vapourization. The process should be adiabatic, but often can be approximated as an isothermal process. After the vapourization, the pressure will jump up to the corresponding vapour pressure. Actually that pressure can be overshoot but here we neglect this... 

This transition differs from the one discussed in Section Ill.l.(i) insofar as one of the steady states is a focus (which is, of course, only possible in a two-variable system), and the current does not monotonically increase but overshoots its stationary value, toward which it slowly relaxes. Also, here the increase in current density is accompanied by an accelerated front, whereas the delayed relaxation of the steady state occurs on a spatially quasi-homogeneous electrode. 

However expansions of the type of Eq. (20.4) would not be able to explain such phenomena as stress relaxation ( 10) or the overshoot phenomenon in stress growth ( 11). It might prove fruitful to explore alternate methods of obtaining constitutive equations which would not be of the form of Eq. (20.4), i.e. stress = function of y and its derivatives. After all, time derivatives of stress could appear [as in Eq. (4.23)], or stress could be given as an integral over the strain history. This question of the connection between dumbbell models and continuum mechanics has been studied extensively in a series of papers by Giesekus (3/). 

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