Hencky rate of extension

For solids one may choose either of the two drawing processes. For liquids a drawing with a constant Hencky rate of extension is to be preferred, because the original length included in A then becomes irrelevant, as is desired. 

Figure 9.18 refers to two other standard experiments. It depicts the results of stress growth experiments, conducted again on a polyethylene melt. The figure includes both measurements probing shear and tensile properties, thus facilitating a direct comparison. Curves show the building-up of shear stress upon inception of a steady state shear flow at zero time and the development of tensile stress upon inception of a steady state extensional flow. Measurements were carried out for various values of the shear rate 7 or the Hencky rate of extension ch ... 

Whereas experiments conducted with a constant rate of elongation yield a linear extension with time, those carried out with a constant Hencky rate result in an exponential time dependence of A... 

FIGURE 17.8 Fracture behavior of two cylindrical cheese samples subjected to uniaxial compression. Sample (1) concerns a hard and fairly brittle cheese, (2) a semihard green cheese that is quite extensible, (a) Stress a versus Hencky strain eH, resulting upon deformation at a strain rate W of 1CT2 s 1 the dotted lines indicate the strain at fracture fr. (b) Values of Sfr obtained at various W. (c) Fracture mode as seen in cross sections through the test pieces at the moment of maximum stress for (1) it is a vertical cross section, for (2) a horizontal one. 

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]...

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. 

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... 

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