Strain Hencky rate

Okamoto et al. [48] first conducted an elongation test on PP-based nanocomposites in the molten state at constant Hencky strain rate, so using elongation flow optorheo-metry [49]. They also attempted to control the alignment of the dispersed silicate... 

Regarding elongation-induced structure development, Figure 9.12(B) shows the Hencky strain rate dependence of the up-rising Hencky strain (r.t) ) — f.0 x ft) recorded for PLANC at 170 °C. The r.t) increases systematically with the eo- The lower the value of s0, the smaller the value ofs,lE. This tendency probably corresponds to the rheopexy of PLANC under slow shear flow. 

Fig. 12.26 True tensile stress-Hencky strain curves for resins C and E at Hencky strain rate of 20 s-1 and temperature of 170°C. [Reprinted by permission from E. G. Muliawan, S. G. Hatzikiriakos, and M. Sentmanat, Melt Fracture of Linear Polyethylene, Int. Polym. Process., 20, 60 (2005).]...

FIGURE 16.19 Cross-plot of SH from Figure 16.18 at the Hencky strain s = 0.3 vs. Hencky strain rate, e. See text. 

Figure 2.11 Stress growth function in elongation at constant Hencky strain rate, e, for linear and nonlinear viscoelastic melt - the latter shows the strain hardening (SH) see text.

Figure 2.14 Strain hardening in extensional flow of lightly-maleated PP with 4 wt% MMT-ODA at 150°C and Hencky strain rates e = 0.001 to 1.0s Inset SH at e = 0.5 is plotted against e (see broken vertical line in the main graph). Data from Okamoto et al. [385].

Stronger the rheological response. Nonlinear viscoelastic effects are also evident in elongational tests as an enhancement of strain hardening, most prominent at low Hencky strain rates. 

Hencky strain and Hencky strain rate in extension, respectively viscosity... 

Hencky strain rate Rate of elongational deeformation... 

Fig. 7.17. Tensile stress growth curves observed for PE at 150 °C, obtained for a constant Hencky strain rate eu =0.1 s . Data from Laun and Miinstedt [78]...

Fig. 7.19. Time dependent viscosities for shear and extensional flow, and as predicted by Lodge s equation of state. Calculations are performed for different Hencky strain rates en, assuming a single exponential relaxation modulus G(t) exp — t/r...

In order to understand better how this selection mechanism works we need a knowledge of elementary stress-extension curves, for example those which are measured for constant Hencky strain rates. If flow sets in locally and there the radius b of the sample begins to decrease, the extension A of the volume element at the centerline is given by... 

Fig. 8.7. Stress-extension curves measured for a sample of PE (M = 3.6 -10 ) at the indicated Hencky strain rates. Constant strain rates were realized by a registration of the strain at the location of a developing neck and a continuous readjustment of the applied tensile force, using an electronically controlled feedback circle. The broken line gives the cT22 (A)-curve measured for a poly(ethylene-co-vinylacetate)(27% vac-units, (j)c = 0.30). No strain rate dependence is observed for this rubbery material [85]...

A deformation of the central volume element with a constant Hencky strain rate, en, thus corresponds to an exponential time-dependence of the minimum radius in the flow zone... 

Figure 8.8 shows the relation between the plateau stress, cjy, and the Hencky-strain rate. It is well represented by the expression... 

Fig. 8.8. Dependence of the stress at the plateau, cTy, on the Hencky strain rates, extracted from the data in Fig. 8.7 [85]...

The stretching curve was measured with a fixed Hencky strain rate ch = 0.005s The shape of the curve is highly non-linear. It indicates a strain softening at a yield point, located at en 0.1. Later there follows a strain hardening, setting in at ch 0.6. 

Because step strain is not practical for melts, the experiment usually carried out to study uniaxial extension is start-up of steady simple extension at a constant Hencky strain rate e. The Hencky strain rate can be defined in terms of the length I of the sample as shown by Eq. 10.89. 

This strain rate is a measure of the speed with which material elements on a streamline are separated from each other. Note that both the velocity of the end of a sample and the sample length increase exponentially with time when the sample is subjected to a constant Hencky strain rate. 

But at a fixed distancez from az = 0 plane where = 0, the velodtyis constant and proportional to the Hencky strain rate ... 

We noted in Section 10.7.2 that the second-order fluid approximation for flows only marginally removed from the rest state indicates that the first and second normal stress differences are second order in the shear rate, so that the first and second normal stress coefficients Pj q and T z 0 approach non-zero limiting values at vanishing shear rate. The second-order approximation also predicts that the net stretching stress in uniaxial extension is second order in the Hencky strain rate, and this implies that the extensional viscosity approaches its limiting zero-strain-rate value 3t7o with a non-zero slope ... 

Strain-softening polymers are very prone to ductile failure in extension, and this poses a major challenge for the experimentalist. If there is a small variation in diameter along the sample, the resistance to further deformation will be reduced at that point, leading to instability and failure. This instability has been treated theoretically by McKinley and Hassager [158]. Because of this instability, it is difficult to continue an experiment to steady-state in such a material. This analysis [158] implies that if the stress passes through a smooth maximum before undergoing ductile failure, this maximum is the steady-state stress, and the extensional viscosity 7e (f) can be calculated. However, experience has shovm that it is difficult to be certain whether a maximum in the curve of rj t, e) versus time implies that steady state has been reached. If the tensile stress, rather than the Hencky strain rate, is held constant, the extensional creep compliance D t, steady state is achieved, the compliance becomes linear with time, and the intercept of this line with the vertical axis is the steady-state extensional creep compliance... 

Melt behavior has been studied using uniaxial (also called simple or tensile), biaxial, and planar extensional flows [9, Ch. 6]. However, only the first two of these are in general use and will be discussed here. A uniaxial extensional rheometer is designed to generate a deformation in which either the net tensile stress Tg or the Hencky strain rate e (defined by Eq. 10.89) is maintained constant. The material functions that can, in principle, be determined are the tensile stress growth coefficient / (f, ), the tensile creep compliance, andthetensile... 

We have seen that rheometers capable of accurate measiuements of extensional flow properties are limited to use at low Hencky strain rates, usually well below 10 s . In order to reach higher strain rates, the drawdown of an extruded filament ( melt spinning ) and the converging flow into an orifice die or capillary have been used to determine an apparent extensional viscosity . Since the stress and strain are not imiform in these flows, it is necessary to model the flow in order to interpret data in terms of material functions or constants. And such a simulation must incorporate a rheological model for the melt under study, but if a reliable rheological model were available, the experiment would not be necessary. This is the basic problem with techniques in which the kinematics is neither controlled nor known with precision. It is necessary to make a rather drastically simplified flow analysis to interpret the data in terms of some approximate material function. 

---