Shear stress-strain rate plots

Rheology is often studied in shear stress-strain rate plots (Figure 8.9). The variables are related through ... 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

The time dependency of the stress-strain rate relationship can be omitted for polymeric liquids in many practical situations. Now, let us consider Figure 22.6, which is a typical plot for viscosity in terms of shear rate for a polymer melt. Two different regions can be observed in the flgure. In the first region, which occurs at moderate low shear rate values, there is a smooth variation of polymer viscosity. In the second region, there is a more pronounced decrease of viscosity as shear rate is further increased. This section of the curve is often described mathematically by a power-law model that expresses the relationship between shear rate and the viscosity stated in Equation 22.9, as discussed in... 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

As a starting point it is useful to plot the relationship between shear stress and shear rate as shown in Fig. 5.1 since this is similar to the stress-strain characteristics for a solid. However, in practice it is often more convenient to rearrange the variables and plot viscosity against strain rate as shown in Fig. 5.2. Logarithmic scales are common so that several decades of stress and viscosity can be included. Fig. 5.2 also illustrates the effect of temperature on the viscosity of polymer melts. 

Fig. 19. a Predicted startup-flow in extension and shear for the constitutive equation derived for a pom-pom polymer with q=5. Stress divided by the strain-rate is plotted (so that early time response is rate-independent) against time after initiation of the flow for a range of dimensionless rates from 0.001 to 3. Both axes are logarithmic... 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

Recoverable shear strain measurements following the removal of shear stress after the sample has been sheared at a rate y0 collapse onto a single curve when the data are plotted against y0r [162],... 

With this data we can plot the stress-strain curves given in Figure 12.14. For the 1.5 /iim particles, the critical shear rate is small, so that only the high shear rate is shown on the plot of shear rate from 0 to 1000 sec But for the 0.3 p,m particles, the critical shear rate is 53 sec and we can see the shear thinning taking place over the shear rates 0 to 1000 sec in the plot. 

A plot of the stress-strain relationship for an ion-exchange ka-olinite also follows the Crossian rheology with very high low shear viscosities. In fact, the data show a yield stress at the lowest shear rate measured of y 1 sec and a high shear viscosity, which are both a function of the volume fraction of the clay, as shown in Figure... 

Figure 11.15 (a) Recoverable strain yr versus time for a 52.8% liquid-crystalline hydroxypropy-Icellulose solution in water at room temperature, 22°C. The curves from left to right were obtained by imposing stresses of 50, 125, 250, 500, 1000, and 2000 dyn/cm before recovery. These imposed shear stresses correspond to various values of the prior shear rate yq. (b) The same plotted against yot all curves fall within the indicated region between the curves for 50 and 2000 dyn/cm. (From Larson and Mead 1989, with permission from the-Joumal of Rheology.)... 

Melt rheometers either impose a fixed flow rate and measure the pressure drop across a die, or, as in the melt flow indexer, impose a fixed pressure and measure the flow rate. Equation (B.5) gives the shear stress, but Eq. (B.IO) requires knowledge of n to calculate the shear strain rate. It is conventional to plot shear stress data against the apparent shear rate y, calculated using n = 1 (assuming Newtonian behaviour). If the data is used subsequently to compute the pressure drop in a cylindrical die, there will be no error. However, if a flow curve determined with a cylindrical die is used to predict... 

Figure 7.20(a) gives the simulation results of the plastic response of this particular structure in a tensile-flow experiment at a given constant strain rate at 0 K and 300 K. The response is given as a deviatoric shear resistance (stress) shear resistance r at 0 K, plotted as a function of the total deviatoric shear strain y, where, in a formal application of a Tresca connection between tensile and shear response, a in shear is taken as half of the tensile deviatoric plastic resistance and y is twice the total uniaxial deviatoric strain (McClintock and Argon 1966). The initial quenched-in level of tp for this alloy is... 

Figure 9.4. Classification of rheological behaviour under steady-shear conditions, plotted as shear stress and viscosity versus strain rate (a) Newtonian (b) shear thickening (c) shear thinning (d) Bingham plastic (e) nonlinear plastic. (From ref. (4) with permission from Marcel Dekker Inc.)...

Figure 15.14 Strain rate versus stress plots, at (n = 1) —> shear thickening (n < 1) transition, temperatures between 1500 and 1600°C,fora Note that the transition stress a is SiAION (83.73% Si3N4, 7.77% AIN, 3.73% approximately independent of the temperature.

Fig. 6.26 Diagram of log 10 shear strain rate versus loglO shear stress/shear modulus curves obtained for temperatures of 1573, 1673 and 1773 K. Symbols represent experimental data, solid lines are best fit linear regression curves and dashed lines are data from Yoo [98] plotted using the value of the stress exponent n = 4.5 obtained in this study. The maximum error on the differential stress is ...

Sketch a plot of shear stress versus strain rate for (a) a dilatant fluid and (b) a pseudoplastic fluid. 

Figure 9.9 shows a classical data set by Meissner on the low-density polyethylene whose transient shear stress was shown in Figure 2.6. The tensile stress divided by the stretch rate is plotted versus time, together with three times the transient development of the zero-shear viscosity. The data deviate from a single curve at values of the strain (stretch rate multiplied by time) of about 2. There is a plateau at low stretch rates at a Trouton ratio of 3, but the plateau is followed by a sharp increase, and in general the tensile stress greatly exceeds three times the shear viscosity. (The shear viscosity for this polymer decreases with shear rate, so the deviation from three times the viscosity is greater than it would appear when the comparison is based only on the zero-shear viscosity.) The fact that the data lie above the band of three times the shear values at short times is probably an experimental artifact. A steady-state stress is not reached in these experiments, except perhaps at the highest and lowest stretch rates. An apparent steady state has been reported in other measurements.

The flow curve of a fluid (Figure 4-2) is obtained by plotting the shear stress (x. Pa) as a function of the shear rate (y, i.e., the change of shear strain per... 

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