Average extensional stress

The average extensional stress is obtained by integrating across the element so... 

Average extensional stress in Eq. (21) a Total stress tensor... 

Fig. 8.8 An ensemble-average extensional equivalent stress-strain curve of amorphous polypropylene, derived from axial extension and shear-flow ensembles of separate simulations. The smoothed broken line, average curve, drawn-in by eye, shows clear elasto-plastic behavior and a beginning yield phenomenon (from Mott et al. (1993) courtesy of Taylor and Francis).

The polymer is a branched polyethylene melt with = 1.55 x 10 and Mw/M = 11.9, flowing at 170 °C in a 3.3 1 planar contraction. The hnear viscoelastic properties and the nonlinear parameters for a four-mode PTT equation are shown in Table 10.1. Different values of e were used for each mode, but with a constant value of f = 0.08. These parameters provide a reasonable fit to the transient and steady-state shear and extensional data, although the nonlinear parameters for the two longest relaxation times cause small oscillations in startup of simple shear that are not observed experimentally using parameters that ehmi-nate the shear oscillations causes the calculated extensional stresses to be too low, and the contraction flow results are sensitive to the extensional stresses. The mean relaxation time was 1.74 s, the average velocity in the downstream channel was 7.47 mm/s, and the downstream channel half-width (the characteristic length) was 0.775 mm. The Weissenberg number based on downstream channel properties was therefore 16.8. 

Since, the applied external loads to the laminate are known it is more useful to know the average extensional strain e)ni and curvature k, in order to calculate individual ply stresses ... 

The required general equations for calculating extensional viscosity can be derived assuming first that the average stress an and the average extension rate e are related by a power law relationship (Steffe, 1996) ... 

Figure 2 shows the calculated distribution of average volumetric strain rate for the assumed stress field. The location of six points (0= 30° to 0 = 69°) where fluid pressure changes were evaluated and interval P4 of borehole FEX 95.002 are also shown. Simulation results, as well as analytical solutions, reveal that a pore-pressure increase will only occur at the four points located in the contracted zone of compressive strain rate (0 > 45° in Figure 2). However, the P4 interval (0 = 14° in Figure 2) is located in the zone of extensional strain rate, and therefore no pore pressure increase can occur in at that location for the assumed stress field.

The essential difficulty associated with this flow situation is that neither the stress nor the rate of strain can be calculated as a function of position from pressure-drop versus flow-rate data, which are the only measurements that can be made. As a consequence, only average quantities can be computed. Even so, some rather drastic assumptions have to be made. Cogswell assumed that the entrance pressure drop, Ap, could be separated into a shear contribution and an extensional contribution for free convergence. He derived expressions for the average exten-sional stress, o-g, and the average extension rate, e O.29) ... 

---