Stress shear, distribution curves

From the shear stress distribution curve in Figure 5.4 we can see that the highest stress is at the ends of the bond and by increasing the joint overlap there is no significant change in the strength of the bond. 

At the subsequent time instances not shown here, the pressure gradient becomes positive again, so the medium moves to the right again. Further, the described medium motion is periodically repeated. It should be noted that, even for this frequency, a small lag takes place between the motion and the pressure change the velocity in the duct does not equal exactly to zero for the zero pressure gradient at the time moments close to 6. The shear stress distributions in the duct follow basically to the stationary law, by reducing linearly from a certain stress on the wall r0(f) to zero on the duct axis the curve bends near the walls are very small. 

In brittle plastics, with a typical stress vs. strain curve as shown in (b), stress increases linearly from the center to the ends of the joint (a). In elastic materials, with a typical stress vs. strain curve as shown in (d), the shear stress distribution is non-linear (c), and stress is distributed over a larger area near the ends of the joints. ... 

FIGURE 34.3 Probability distribution for cell adhesion. NIH 3T3 fibroblasts were detached from glass using a radial-flow chamber. Squares correspond to the fraction of cells that resisted detachment as a function of the applied shear stress. The curve is a best fit of the integral of a normal distribution function to the data. The characteristic measure of cell adhesion, T50, is 129 dyn/cm for this test. 

A CaC03-filled PP melt is found to obey the relationship given by Eq. (12.4), in which Pis 2.5 x 10 Pa, A is 4.5 x lO Pa s, and n = 0.5 at 200 °C. (a) Sketch the viscosity versus shear rate curve, (b) Sketch velocity and shear stress distributions in a long cylindrical tube through which this material is extruded, (c) What will the pressure drop be when this material is extruded in a long flat film die (60 cm wide, 0.2 cm thick, and 10 cm long) at a volumetric flow rate of 100 cm /s at 200 C ... 

Studies of melt flow properties of polypropylene indicate that it is more non-Newtonian than polyethylene in that the apparent viscosity declines more rapidly with increase in shear rate. The melt viscosity is also more sensitive to temperature. Van der Wegt has shown that if the log (apparent viscosity) is plotted against log (shear stress) for a number of polypropylene grades differing in molecular weight, molecular weight distribution and measured at different temperatures the curves obtained have practically the same shape and differ only in position. 

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

An empirical method to cope with the effect of molecular-weight distribution was proposed by Van der Vegt (1964). Fie determined viscosities of several grades of polypropylenes with different Mw and MMD as a function of the shear stress ash- A plot °f V/Vo vs. the product tvdv = t1wQ proved to give practically coinciding curves. This generalised curve has been reproduced in Fig. 15.21. 

In contrast, Figure 95 shows a plot of the response to shear stress of polymers made at a high temperature in the solution process [407,521]. Once again, the reaction time was varied to produce polymers in different yields. The reaction temperature was also varied to produce polymers of varying MW. Under these conditions, the catalyst exhibited full activity immediately. The response to shear stress is in essence an approximation of the slope of the melt viscosity curve. A high response to shear stress indicates rheological breadth, which can derive from breadth in the MW distribution, or from LCB. In this case, the MW distribution was quite ordinary and did not vary with time. Therefore, the response to shear stress can serve as a surrogate for the level of LCB. 

For the curves in Fig. 11.7, the value of n that gives the best representation of the experimental curves varies from for the lowest Reynolds number to jq for the highest Reynolds number. Prandtl selected j as the best average, deducing Prandtl s power velocity distribution rule. This is not an exact rule, because if it were a general rule, then all the curves in Fig. 11.7 would be identical. Furthermore, it cannot be correct very near the wall of the tube, because there it predicts that dVIdy is infinite and hence that the shear stress is infinite. Nonetheless, it is widely used because it is simple and, as we will see in Sec. 11.5, because it gives useful results. 

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