Stress Relaxation after Cessation of Steady Shearing Flow

We next turn our attention to the relaxation of shear and normal stresses after a steady shearing flow is suddenly stopped. Before time t=0 the fluid is in a state of steady shear flow with a velocity profile vx = K0y, vy = 0, v2 = 0. After time t = 0 the fluid is completely motionless, but the shear and normal stresses decrease gradually to zero. This flow was analyzed for rigid dumbbells by Giesekus (12) and later by Bird, Warner, and Ramakka (10). Schremp, Ferry, and Evans (69), Benbow and Howells (2), and Huppler et al. (36) have measured the time-decay of the stresses in an experiment which closely approximates the above-described idealized problem. 

3) can be solved by the method of separation of variables and the solution must have the form  

6) through (10.8) are sketched in Fig. 6. All three sets of curves in Fig. 6 are decaying with an exponential rate e, /A in addition they exhibit discontinuities, which are not observed experimentally. 

Nonetheless there are some similarities between the results in Eqs. (10.6) and (10.7) and experimental curves. For example (a) all stresses relax more quickly as k0 increases, and (b) the ratio xyx — xyxs)+/ (tyx — xyxs) is always less than the corresponding ratio for the primary normal stress differences. 

Equations (10.7), (10.8), and (10.10) show that (xyy — t2z)+ = —(2/7) (xxx — xyy)+ for small Xk0. This result and the result in Eq. (6.3) suggest that the secondary normal stress difference might be easier to observe in an unsteady-state experiment than in a steady-state experiment. Note that, from Eqs. (10.6) and (10.10), the relation 

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After this rather extensive discussion of the experimental verification of eqs. (2.11), (2.20) and (2.22), where the interrelations of steady flow properties with dynamic properties and shear recovery are scrutinized, a few words should be added with respect to stress-relaxation after cessation of steady shear flow. As pointed out above, Lodge predicted a decrease of the extinction angle during stress relaxation (4). That this... 

Fig. 2.7. Drop off of doubled extinction angle 2"/ during stress-relaxation after cessation of steady shear flow according to Wales (59). Measurements on the melt of a high-density polyethylene (Marlex 6002) at a measurement temperature of 147° C. Shear rate of the steady shear flow q = 0.06 sec-1...

Stress Relaxation after Cessation of Steady Shearing Flow... 

In stress relaxation after cessation of steady shear flow, the elastic dumbbells give no dependence of the relaxation process on the steady-state shear rate, but the rigid dumbbells do. In addition the elastic dumbbells show the shear and normal stresses relaxing with exactly the same... 

Stress relaxation after cessation of steady shearing flow... 

Shear stress relaxation after cessation of steady shear flow (which includes Mooney stress relaxation [130] now described in ASTM D1646)... 

At very low shear rates, the normal stress coefficients 1,0 and 2,0 are also independent of 721 i.e., the normal stress differences are proportional to 721. At higher shear rates, 1 and 2 are observed to decrease. The course of stress relaxation after cessation of steady-state flow and the magnitude of the steady-state compliance J° are also strongly affected at high shear rates. In general, description of these phenomena requires more complicated constitutive equations than the single-integral models mentioned above. 

There are other more complicated experimental situations where viscoelastic behavior can also be predicted in terms of the relaxation and retardation spectra or other functions. These include deformations at constant rate of strain and constant rate of stress increase, stress relaxation after cessation of steady-state flow, and creep recovery or elastic recoil, all of which were mentioned in Chapter 1, as well as nonsinusoidal periodic deformations. In referring to stress a, strain y, and rate of strain 7, the subscript 21 will be omitted here although it is understood that the discussion applies to shear unless otherwise specified. 

In a similar test, the applied shear rate is suddenly reduced to zero. This process is called stress relaxation after cessation of steady-state flow. The decaying shear stress is measured as a function of time ... 

Figure 31.4 Stress relaxation curves after cessation of steady shear flow for the PET/PEN-clay systems (shear rate is 10 s ). Source Reproduced with permission from Calderas F, Sanchez-Solis A, Maciel A, Manero O. Macromol Symp 2009 283 354 [83], Copyright 2009 Wiley-VCH Verlag GmbH Co. KGaA.

The aim of a stress relaxation experiment is to observe how the stresses decay with time (i) after cessation of steady shear flow or (ii) after a sudden shearing displacement. In case (i) the fluid sample trapped in a small gap between two parallel plates is allowed to maintain constant shear rate that was started long before f = 0 so that all the transients during the stress growth period have evened out. Then at t = 0, the flow is stopped suddenly and the decay of the stress is monitored with respect to time till it becomes insignificant or dies out. The stress would relax monotonically to zero and more rapidly as the shear rate in the preceding steady shear flow is increased. 

FIG. 2-13. Relaxation of shear stress and primary normal stress difference after cessation of steady state flow, for a 4% solution of polystyrene with molecular weight 1.8 X 10 in chlorinated diphenyl as described in the text. Numbers refer to the shear rate preceding cessation of flow. 

The aim of a stress relaxation experiment is to observe how the stresses decay with time (1) after cessation of steady shear flow or (2) after a sudden shearing displacement. In case 1 the fluid sample trapped in a small gap between two... 

The nanocomposite PET-PEN/MMT clay was smd-ied under steady shear, instantaneous stress relaxation, and relaxation after cessation of steady flow [83]. Relaxation times of the slow mode in instantaneous stress relaxation were longer for the systems that have presumably permanent crosslinking networks (PET-PEN) or dynamic networks (PET-PEN-MMT). These results are consistent with those found in relaxation after cessation of flow (Fig. 31.4). Nanoclay addition somehow restricts the slow relaxation (due to polymer-particle interactions). The nanocomposite exhibits lower steady-state viscosity as compared to the polymer-matrix system. This is thought to be caused by polymer-polymer slipping, as revealed by the SEM observations (Fig. 31.5a and b). 

Figure 6.5 Maxwell viscoelastic model and its prediction of stress relaxation in a melt after the cessation of steady shear flow. t<, is the relaxation time.

The primary normal stress coefficient during steady-state shear flow is also related to the relaxation of shear stress after cessation of steady-state flow, as shown by combining equations 67 and 74 ... 

Measurements of normal stress differences during steady shear flow, and of normal stress growth approaching steady-state flow and stress relaxation after cessation of flow, provide additional information about nonlinear viscoelastic properties. The conventional identifications of the normal stresses for simple shear have been shown in Fig. 1-16 their orientations in several examples of experimental geometry are sketched in Fig. 5-5. 

Figure 9.38 Trace of first normal stress difference of a compression-molded 73/27 HBA/HNA copolyester specimen during transient and steady-state shear flow, and dnring the relaxation after cessation of steady-state shear flow. The normal stress before applying a sudden shear flow to the specimen is taken to be zero. (Reprinted from Han and Chang, Journal of Rheology 38 241. Copyright 1994, with permission from the Society of Rheology.)...

Figures 11.15 and 11.16 show the relaxation of shear stress O12 and first normal stress difference Ni, respectfully, for Titan at 340 ° C after cessation of steady flow with shear rate y = 6 s T Here, experimental data are shown by dots and fitting curve by dashed lines. The values ofboth O12 and during relaxation drop abruptly and reach zero at the time of around 4 s. The simulated relaxation curve for O12 has an excellent agreement with the experimental data, but this is not the case for Ni when t > 4 s. This disagreement is seemingly attributed to the experimental error because the final value of Ni during relaxation should reach zero.

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

System relaxation times have been determined from the relaxation of the stress after abrupt cessation of shear flow. Representative applications of the approach are found in Takahashi, etal. 9), who examined 355-3840 kDa polystyrenes in benzyl- -butylphthalate, at concentrations identified as showing dilute-solution behavior for the steady-state compliance Je and semidilute behavior for the zero shear viscosity. The stress relaxation after shear cessation, identified as the transient viscosity, decreased exponentially with time except at the shortest times studied, leading to an identification of an observed longest relaxation time Xm, whose c and M dependences were determined. 

The normal stresses present during the steady-state flow also relax after its cessation, and the course of the primary normal stress difference, (oi j — shear rate which precedes it. However, the normal stress difference relaxes more slowly than the shear stress. 

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