Shear stress decay

The Bird-Leider model predicted well the maximum and the steady state stresses as well as the times at which they occurred, but its prediction of the shear stress decay was not satisfactory (Dickie and Kokini, 1982). 

Kokini and Dickie (1981) found that the Bird-Leider equation provided moderately good predictions for peak shear stresses (trmax) and the corresponding times (tmax), but the prediction of shear stress decay was poor. They suggested that a series of relaxation times, as opposed to a single exponential term, would be needed for mayonnaise and other materials. 

The quantity rj/G represents the melt relaxation time to, the time in which shear stress decays to He (34%) of its original value. The relaxation time depends on the molecular mass and the melt temperature, factors which... 

After a constant shear strain is created in a previously relaxed material, the resulting shear stress decays with ensuing time to zero for a viscoelastic liquid and to a finite equilibrium value for a viscoelastic solid. The shear stress relaxation modulus (Pa or dynes/cm )... 

Tobolsky and his coworkers made extensive efforts to characterize the stress relaxation characteristics of elastomers, notably polyisobutylene. The stress would decay over time to zero at a rate dependent on temperature and molecular weight (Fig. 2). They expressed the relaxation through a series of exponentials or a spectrum of relaxation times. Consider the shear stress decay a t) following an shear imposed strain yo- This may be used to define a shear relaxation modulus G i) through... 

Out-of-phase component of complex viscosity Shear stress growth coefficient Shear stress decay coefficient Tensile stress growth coefficient Tensile stress decay coefficient... 

The interfacial shear stress decays within 2-3 fiber diameters from the fiber end when the end is bonded to the matrix. 

The shear stress decay coefflcient r (t) evidently describes the relaxation properties of the viscoelastic samples. 

There is also a shear stress decay coefficient Tf t,y) that describes stress relaxation following the cessation of steady simple shear at a time f = 0. Data for this material function were reported by Menezes and Graessley [17], but it has been little used as it has not turned out to be a sensitive probe of nonlinear behavior. We note that so-called step strain in reality involves a very high, but not infinite, shear rate and is therefore a special case of relaxation following start-up of steady-simple shear. However, the most interesting nonlinear features of the response, which are the result of chain stretch, are lost when the shear rate is not high and the shearing time are not very short. 

Viscosity T), first normal-stress coefficient /i, second normal-stress coefficient /2 Shear stress growth coefficient T1+, first normal-stress growth coefficient t t[, second normal-stress growth coefficient /J Shear stress decay coefficient Tj", first normal-stress decay coefficient /f, second normal-stress decay coefficient /j Shear creep compliance J... 

Time-dependent effects ate measured by determining the decay of shear stress as a function of time at one or more constant shear rates (Fig. 7)... 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Fig. 7. Decay of shear stress during steady shear at various shear rates. Determination of zero-time shear stresses or yield stresses and equiUbrium shear...

With turbulent flow, shear stress also results from the behavior of transient random eddies, including large-scale eddies which decay to small eddies or fluctuations. The scale of the large eddies depends on equipment size. On the other hand, the scale of small eddies, which dissipate energy primarily through viscous shear, is almost independent of agitator and tank size. 

Kumar and Clifton [31] have shock loaded <100)-oriented LiF single crystals of high purity. The peak longitudinal stress is approximately 0.3 GPa. Estimates of dislocation velocity are in agreement with those of Flinn et al. [30] when extrapolated to the appropriate shear stress. From measurement of precursor decay, inferred dislocation densities are found to be two to three times larger than the dislocation densities in the recovered samples. 

To answer questions regarding dislocation multiplication in Mg-doped LiF single crystals, Vorthman and Duvall [19] describe soft-recovery experiments on <100)-oriented crystals shock loaded above the critical shear stress necessary for rapid precursor decay. Postshock analysis of the samples indicate that the dislocation density in recovered samples is not significantly greater than the preshock value. The predicted dislocation density (using precursor-decay analysis) is not observed. It is found, however, that the critical shear stress, above which the precursor amplitude decays rapidly, corresponds to the shear stress required to disturb grown-in dislocations which make up subgrain boundaries. 

The shear stress is greatest at the ends of the fiber and decays to zero somewhere along it. The tensile... 

The experiment here is a small rapid shear-strain at time zero - after this the shear stress in a viscoelastic liquid will not vanish instantaneously, but decay as a characteristic function with time. When normalised by the strain to yield the dimensions of modulus, this is G(f). 

The non-aqueous HIPEs showed similar properties to their water-containing counterparts. Examination by optical microscopy revealed a polyhedral, poly-disperse microstructure. Rheological experiments indicated typical shear rate vs. shear stress behaviour for a pseudo-plastic material, with a yield stress in evidence. The yield value was seen to increase sharply with increasing dispersed phase volume fraction, above about 96%. Finally, addition of water to the continuous phase was studied. This caused a decrease in the rate of decay of the emulsion yield stress over a period of time, and an increase in stability. The added water increased the strength of the interfacial film, providing a more efficient barrier to coalescence. 

A particular conclusion from this theoretical analysis is that, if a crack has faces that are separated by a thin layer of fluid, so that normal components of traction and displacement are transmitted across the crack but the faces are free with regard to shear components of traction and displacement, then there will be a scattered wave however thin the fluid layer is. This is perhaps not surprising. A Rayleigh wave can exist only because solids can support both longitudinal and shear waves, and the greater part of the displacement in a Rayleigh wave is shear in character ( 6.3). Of course, liquids can support shear stress over a short distance. In a liquid of viscosity r/, and density po, at a frequency o) the amplitude of a shear wave decays by a factor e over a distance... 

Stress Decay at the Termination of Steady Shearing Flow (356-360)... 

At sufficiently low shear rates the shear stress should decay at the termination of steady state shear flow according to the equation from linear viscoelasticity ... 

Accordingly, plots of a t)/a(0) vs t from different shear rates should superimpose. Experimentally the curves do not superimpose when the shear rate is in the non-Newtonian region, the initial rate of relaxation being increasingly more rapid for higher shear rates. The normal stress decays more slowly than shear stress, but behaves similarly with respect to the effect of previous shearing flow in the non-Newtonian region. 

For shear strains greater than approximately 2 the ratio cr(r)/> 0 for a concentrated polystyrene solution was reduced at all observable times. For the large strains, relaxation proceeded more rapidly at short times, but at longer times the residua] stress decayed with about the same time dependence as that in the linear viscoelastic region. 

The slower rise and decay of normal stress transients compared to shear stress arises quite simply and directly from the polydispersity of relaxation times (78), and probably has no direct bearing on entanglement mechanisms per se. Likewise, the depression of the superimposed moduli at low frequencies follows from rather non-specific continuum models, the loss moduli by Eqs.(8.49) and (8.50) from the simple fluid model, and the storage moduli from the following properties of the more specific but still quite general BKZ model (366,372) ... 

Figure 6.4. Shear and pipeline flow data of a thixotropic Pembina crude oil at 44.5°F. (a) Rheograms relating shear stress and rate of shear at several constant durations of shear (Ritter and Govier, Can. J. Chem. Eng. 48, 50S (1970)]. (b) Decay of pressure gradient of the fluid flowing from a condition of rest at 15,000 barrels/day in a 12 in. line [Ritter and Batycky, SPE Journal 7, 369 (1967)].

At higher shear rates, Watanabe and Kotaka (1983) observed thixotropy, i.e. stress decay increasing as a function of shear rate, in PS-PB diblocks in dibutyl phthalate (DBP), which is a selective solvent for PS.The fact that the flow crossed over from plastic to viscous non-Newtonian on increasing the shear rate indicated the breakdown of the micellar lattice structure, rather than of the individual micelles. This was confirmed by parallel measurements on a cross-linked PB-PS system, where stress decay and recovery were also observed. Thus the... 

Creep measurements involve the application of a constant stress (usually a shearing stress) to the sample and the measurement of the resulting sample deformation as a function of time. Figure 9.6 shows a typical creep and recovery curve. In stress-relaxation measurements, the sample is subjected to an instantaneous predetermined deformation and the decay of the stress within the sample as the structural segments flow into more relaxed positions is measured as a function of time. 

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