a condition satisfied in the stress relaxation data reported here, it can be shown [8] that σ(t+t1)/y for t is a lower bound of W(y, t).5 An example of the two bounds and the function W(y, t) is shown in figure 3 for y=0.2 and t1 = 0.03 s. The values of W(y, t) are the values that we used to correlate our different experiments. We see that even for values of t=10 t, the upper bound is high. The average of the two bounds gives agreement to within 1 percent. In the example in figure 3, a very good approximation of W(y, t) can be obtained by averaging the time of the two bounds, viz, In our experiments the strains were induced through a shaft of the rheogoniometer normally used for sinusoidal deformation histories. The motion of the cone was monitored with a transducer. The deformations were obtained by using a springloaded lever arm to drive the shaft through part of a cycle. This arrangement allowed us to obtain strains up to y=1.8. By using only a part of the sinusoidal deformation, we could obtain a motion of the cone very close to a ramp function. The time required to reach the maximum strain was of the order of 0.01 to 0.05 s. K Unfortunately, in our experimental system there is another complication due to the motion of the upper platen. The true strain at any time t is more nearly σ(1) g(t) - where K is a constant depending on the stiffness of the torsion bar and the geometry of the cone and plate, and g(t) is the nominal strain at t calculated from the motion of the cone. With a stiffer torsion bar the error due to the motion of the upper plate would be smaller but small stresses could not be measured with enough precision. If the motion of the cone is monitored, one can calculate the error due to the motion of the plate. In our measurements the error in the stress at the early times was 6 percent. Figure 4 shows a plot of W(y, t) versus y in which both corrections were made. The isochrones at the early times are in very good agreement with eq (2.6) which corresponds with the lines. To the extent that the isochrones are parallel in this type of a plot, i.e.. superposition occurs by a vertical translation, one can justify the representation of W(y, t) as a product of the stress relaxation modulus and a function of shear, y, i.e., eq (2.6). 0(1+111/8 σ(1)/8 101, =22.2 sec ̄ FIGURE 5: The solid lines represent the stress calculated for a suddenly applied rate The open circles are experimental data, and the solid circles are calculated values using the true history of the motion. erature. Since it took about 20 s to complete a run, he temperature variation was no more than ±0.5 °C 25 °C. For these runs, due to the limitations of the transucer, we could monitor strains only up to y= 0.4. On epeating an experiment, different strain histories were variably obtained (over the early part of the time cale); this was probably caused by differing positions the gear teeth. So, from each series of repetitive uns, we selected the one with the least deviation from constant rate of shear. Using eq (2.1) with the history Our empirical form of W(y, t), eq (2.6), satisfies this condition at y=2.44. In table 2 we show the strains at which the maximum of the stress occurs from other data. Middleman's data [9] were estimated from the published graphs. The data of the more concentrated polyisobutylene solu(6.1) tions were obtained in our laboratory. The zero shear viscosities at 25 °C were 4460 poise for the 15.1 percent concentration and 17,760 poise for the 19.3percent concentration. Assuming the principle of reduced variables with respect to concentration to be valid [10], we see that the rates of shear reported in table 2 for the higher concentrations will be 8 times as high when reduced to 10-percent concentration, for which the zero shear viscosity is 540 poise.8 (6.2) ere y=ýğ and ỳ is the rate of shear. In figure 5 we ow the data as obtained. The lines represent the ediction of (6.2) and (2.6) which assume a suddenly plied steady shear (for the rates of shear indicated), d the black solid circles are values calculated from 1) and (2.6) using the true history of the motion. The reement is very good. The inertia of the upper platen d assembly was not corrected for. We estimated for =22.2 s-1 that the inertial effect caused an error of The position of this overshoot can be used to study the error caused by the motion of the upper platen. It is easy to show this error by experiments with torsion bars of different diameters. In figure 6 we show the data obtained from a 15.1 percent solution at a nominal If eq (2.6) is correct for all concentrations, the maximum of the stress must occur at the same value of y. The reduced rate of shear YR is given as rate of shear of 22.2 s-1. The open circles show the (8.1) In figure 8 and figure 9 we show the experimental results with the open circles and the calculated results with the solid circles. The experimental points fall slightly higher than the calculated values, as the motion of the upper platen would lead one to expect. At the limit of zero rate of shear, (8.2) reduces to (in cetane) B-50 at 23.9 °C |