LOG (8,5-1)

FIGURE 10. Variation of ý σ(t)dt as a function of y using dif ferent torsion bars, where σ(t) is the measured stress after the cessation of steady shear.

simple form that we used to represent the function W(y, t) seems to describe the behavior adequately, at least within the range of shearing deformations of our experiments. The separation of W(y, t) into a product of two functions, one of time and one of strain, i.e., W(y, t) =G(t)WR(Y), may be an oversimplification. Although a more complicated form of W(y, t) gave us better agreement, we felt that the uncertainty in our experimental data did not justify its use.

Our purpose in this paper is not to establish a unique form of Wy, t), but rather to show how a wide variety of measurements from different shear histories can be related. Indeed, we know that the behavior of plasticized PVC is not consistent with a W(y, t) which is a product of a function of strain and a function of time. Experiments on normal stresses are more critically dependent on the values of W(y, t) at long times and large strains. For instance, the first normal stress difference, σ(t)-2(t), for a suddenly

then the maximum of σ11(t)-σ(t) will occur at time tm for which

a In W (ytm, tm) -2.

a ln y

For longer times σ11(t)σ22(t) will tend to level to a lower value. This behavior of the first normal stress difference is roughly similar to that of the shear stress. There are clear differences however: a comparison of eq (9.1) with eq (6.2) shows that the overshoot for the shearing stress, if it occurs, must do so at an earlier time than tm. It may well be possible, however, to find a form of W(y, t) compatible with the shear stress data of this paper which never satisfies the condition (9.2) for any value of y.

10. References

[1] Bernstein, B., Kearsley. E. A., and Zapas. L. J., Trans. Soc Rheol. 7, 391-410 (1963).

[2] Zapas, L. J., and Craft, T., J. Res. Nat. Bur. Stand. (U.S.). 69A (Phys. and Chem.), No. 6, 541-546 (1965).

[3] Broadbent, J. M., Kaye, A., Lodge, A. S., and Vale, D. G.. Nature 217, 55 (1968).

[4] Bernstein, B.. Acta. Mechanica, II, No. 4, 329-354 (1966). [5] Bernstein, B., and Fosdick, R. L.. Rheologica Acta 9, 106 (1970). [6] Ferry, J. D., Viscoelastic Properties of Polymers (John Wiley, New York, 1961).

[7] Marvin, R. S., Phys. Rev. 86, 644 (1952).

[8] Zapas, L. J., and Marvin, R. S., A Correction for a Non-Zero Loading Time in Stress Relaxation (in preparation). [9] Middleman, S., Trans. Soc. Rheol. 13, 123 (1969). [10] DeWitt, T. W., Markovitz, H., Padden, F. J., and Zapas, L. J.. J. Colloid Sci. 10, 174-188 (1955).

(Paper 75A1-648)

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, viz:

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 between a flat plate (7.5 cm 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],