may also be obtained from experiments of suddenly applied steady shear. Thus, in principle, we should be able to obtain W(y, t) from a series of experiments, and then proceed to check the BKZ theory by comparing measured and predicted stress for simple shearing histories. Of course, one can never realize experimentally these shearing histories exactly as assumed. From some of the experiments which approximated suddenly applied constant rate of shear history, discussed in section 6, we were able to get a rough approximation of the relaxation function. Applying corrections which will be discussed later using an iterative scheme, we recalculated a function W which is consistent with all our experiments which include measurements of viscosity as a function of the rate of shear, stress relaxation after shear (for different rates of shear), measurement of stress as a function of time for suddenly applied steady shear, and single step stress relaxation experiments. These results, we felt, justified the use of an expression for W(y, t) which can describe all our experiments and which is a special form of a more detailed expression consistent with the behavior of other materials and other deformations. 3. Experimental The data reported in this paper were obtained on a 10-percent solution of polyisobutylene (vistanex L-100 Enjay Chemical Co.) in cetane. A Weissenberg Rheogoniometer was used to shear the sample be tween a flat plate (7.5 cm diameter) and a cone such that the angle of the gap was 0.0268 radians. The cone was at the bottom and was connected to the driving shaft. The plate was at the top and was connected to a torsion bar which was used to measure the torque. For most of the experiments a torsion bar of 1/8-in diameter was used. The stress-time measurements and the dynamic response were recorded on an oscillograph. The chamber enclosing the cone-plate assembly was kept at a temperature of 25.0 °C ± 0.1 °C. Ambient temperature was controlled at 25.0 °C ±0.5 °C. An ambient temperature 3 degrees below the chamber temperature caused a noticeable change in the curve of viscosity versus rate of shear at high rates of shear. The zero shear viscosity at this temperature varies about 5 percent per degree. Periodically, degradation checks were made on samples of 10-percent PIB by taking viscosity versus rate of shear data. No changes were observed over a 2 year period. 4. Behavior at Small Deformations It is clear from eq (2.6) that the shear relaxation modulus, G(t), plays an important role in our curve fitting scheme. Since stress relaxation measurements cannot be carried out using deformations small enough to yield an infinitesimal modulus, this is ordinarily obtained by some sort of extrapolation procedure. An alternate method, adopted here, is to calculate a curve from measured values of the dynamic modulus, G' (w) [6]. We employed the conversion scheme given by Marvin [7], G(t)=G'(w)|1/w=t where G(t) is the shear stress relaxation modulus. 3 It is evident that there are many other forms which are consistent with our experimental data. Our motivation for choosing form (2.6) is its relative simplicity and its consistency with a strain potential function [1]. Furthermore, form (2.6) is useful in process control engineering since an immediate translation from the linear to the nonlinear behavior can be made as a 3 Notice that G(t), the shear relaxation modulus, is the limit of relaxation function Wy. t) as y goes to zero. 5. Single Step Stress Relaxation in Shear In section 2 the definition of a single step stress relaxation history in simple shear was given in which the step is instantaneous. In practice, it takes a finite time to reach the desired deformation, and it can be shown that the measured stress is an upper bound of the stress relaxation function. At long times the difference between the relaxation function and the bound becomes small. With a knowledge of the actual strain history, one can estimate this difference and obtain a better bound for the relaxation function. With the history W (y,t) = σ(ι) γ is an upper bound of W(y, t). Since - W(y, t) γ is a monotone nonincreasing function of t, the ratio of tG(t) sin wild In t (4.2) -x tive, tG(t) cos et d In t (4.3) a condition satisfied in the stress relaxation data reported here, it can be shown [8] that σ(t+t1)/y for tt1 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. 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 K 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). -22.2 sec 5.56 3.52 1.77 0.886 LOG (†, s) 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 perature. Since it took about 20 s to complete a run, the temperature variation was no more than ±0.5 °C at 25 °C. For these runs, due to the limitations of the transducer, we could monitor strains only up to y= 0.4. On repeating an experiment, different strain histories were invariably obtained (over the early part of the time scale); this was probably caused by differing positions of the gear teeth. So, from each series of repetitive runs, we selected the one with the least deviation from a 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) where y=ys and ỷ is the rate of shear. In figure 5 we show the data as obtained. The lines represent the prediction of (6.2) and (2.6) which assume a suddenly applied steady shear (for the rates of shear indicated), and the black solid circles are values calculated from (2.1) and (2.6) using the true history of the motion. The agreement is very good. The inertia of the upper platen and assembly was not corrected for. We estimated for y=22.2 s1 that the inertial effect caused an error of 1 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 y is given as rate of shear of 22.2 s-1. The open circles show the data obtained with a 0.125-in torsion bar. The solid circles are data obtained with a 0.25-in bar and the black triangles, with a 0.35-in bar. The histories of the motion of the cone for all the runs were almost identical, so that a direct comparison can be made of the effect of the motion of the upper platen. The maximum occurs at t=0.11 s for the 0.25-in and 0.35-in bars and at t=0.16 s for the 0.125-in bar. Y(T) = YT + Yo, Y(T) = Yo, T≤0 T≥ 0. (8.1) 00 · [* W (v(§ − 1), §) ( 1 + a In W a ln y dž. (8.2) 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 Differentiating (8.3) with respect to t we get (8.3) (8.4) B-50 at 23.9 °C |