Stress Overshoots

In the following sections examples of the application of this procedure to the analysis of specific phenomena such as wall slip and stress overshoot which affect polymeric flow processes are illustrated. 

Prediction of stress overshoot in the contracting sections of a symmetric flow domain... 

Figure 5.13 The predieted pattern of the stress overshoot in example 5.3.1...

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. 

Figure 6.23 Prediction of stress overshoot for different tube disengagement times. The shear rate used for the calculation was 5 s l...

For fast squeezing flow we would need a constitutive equation that accounts for the stress overshoot phenomenon. 

According to the Scott equation, plotting ln(ti/2) versus ln(l/FN) should give a straight line. This is what Leider (50) observed with a series of fluids in the Scott equation range. However, Leider and Bird (49) extended the analysis to include stress overshoot phenomena by using a semiempirical expression for the shear stress ... 

They also recommend the selection of parameters a and b so as to give the best fit for the stress overshoot data obtained for a constant shear-rate experiment. By following this procedure, good agreement between experiments and theory was obtained, as demonstrated in Fig. E6.14b. 

With regard to constitutive equations, White (13) notes that, in view of the short residence time of the polymer in the nip region (of the order of magnitude of seconds), it would be far more realistic to use a constitutive equation that includes viscoelastic transient effects such as stress overshoot, a situation comparable to that of squeezing flows discussed in Section 6.6. 

As mentioned above, it is far more difficult to measure extensional viscosity than shear viscosity, in particular of mobile liquids. The problem is not only to achieve a constant stretch rate, but also to maintain it for a sufficient time. As shown before, in many cases Hencky strains, e = qet, of at least 7 are needed to reach the equilibrium values of the extensional viscosity and even that is questionable, because it seems that a stress overshoot is reached at those high Hencky strains. Moreover, if one realises that that for a Hencky strain of 7 the length of the original sample has increased 1100 times, whereas the diameter of the sample of 1 mm has decreased at the same time to 33 pm, then it will be clear that the forces involved with those high Hencky strains become extremely small during the experiment. 

When the Ink Is allowed to rest In the Instrument for 15 minutes or more and then steady shear Is Initiated, there Is a significant stress overshoot (Figure 1). Subsequently, the stress level shows a significant time dependence for a period of time that depends on the experimental conditions but Is generally less than 10 seconds. After this Initial period the stress appears to level-off at what will be termed the short term steady flow value. If the steady shear Is maintained for long periods of time, however. It Is found that the stress Is not constant but shows a small and very slow decrease. For the range of conditions tested here, the stress, and therefore the viscosity, drops by about 15% In one hour (Figure 2). The decrease Is approximately linear In a log (n) vs log (time) plot. 

Alternatively, in transient flows the slip parameter can also be determined using the time position of the maximum of the experimental fimctions (tT for tangential stress and tN for normal stress). However, this can only be performed accurately if the stress overshoot is large enough to avoid uncertainties in these values and this can only be achieved at high shear rates. In this case, according to equations (50c) and (50d) ... 

One of the eharacteristics of viscoelastic foods is that when a shear rate is suddenly imposed on them, the shear stress displays an overshoot and eventually reaches a steady state value. Figure 3-43 illustrates stress overshoot data as a function of shear rate (Kokini and Dickie, 1981 Dickie and Kokini, 1982). The data can be modeled by means of equations which contain rheological parameters related to the stresses (normal and shear) and shear rate. One such equation is that of Leider and Bird (1974) ... 

Figure 3-43 Stress Overshoot Data of Several Foods as a Function of Shear Rate and Predicted Values by Leider and Bird Model (Dickie and Kokini, 1982).

Figure 3-44 Schematic Diagram for Phenomenological Analysis of the Recorded Stress Overshoot Data at a Constant Shear Rate (Elliott and Ganz, 1975).

Several studies were conducted on the stress overshoot and/or decay at a constant shear rate. Kokini and Dickie (1981) obtained stress growth and decay data on mayonnaise and other foods at 0.1, 1.0, lO.Oand 100 s . As expected from studies on polymers, shear stresses for mayonnaise and other food materials displayed increasing degrees of overshoot with increasing shear rates. The Bird-Leider empirical equation was used to model the transient shear stresses. 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

In transient shear flows starting from an isotropic distribution of fiber orientations, considerably higher viscosities will be initially observed, until the fibers become oriented. In Bibbo s experiments, t]r for isotropically oriented fibers is around 3.5 for v = 75. These viscosities can also be predicted reasonably well by semidilute theory and by simulations (Mackaplow and Shaqfeh 1996). Figure 6-25 shows the shear stress as a function of strain for a polyamide 6 melt with 30% by weight glass fibers of various aspect ratios, where the fibers were initially oriented in the flow-gradient direction. Notice the occurrence of a stress overshoot (presumably due to polymer viscoelasticity), followed by a decrease in viscosity, as the fibers are reoriented into the flow direction. 

Figure 5A Shear stress versus shear rate for an emulsion sample with a high solids content. The peak at low shear is reproducible and is due to oil separating from the emulsion onto the rotors. In rheometers that cannot measure at such low shear rates, this peak can be incorrectly attributed to stress overshoot...

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