Hencky strain measure

FIG. 16.21 Apparent or transient extensional viscosity of the round robin test fluid Ml as a function of Hencky strain, measured in many different devices (lames and Walters, 1993). The various instruments are spin line Binding et al. Ferguson and Hudson Ngyuen et al. horizontal spin line Oliver open siphon Binding et al. stagnation flow. Laun and Hingmann Schweitzer et al. contraction flow. Binding et al. ... 

A measure Which is frequently used in polymer melt constitutive equations is the Hencky strain measure, related to the stretch ratio by ... 

As mentioned above, it is far more difficult to measure extensional viscosity than shear viscosity, in particular of mobile liquids. The problem is not only to achieve a constant stretch rate, but also to maintain it for a sufficient time. As shown before, in many cases Hencky strains, e = qet, of at least 7 are needed to reach the equilibrium values of the extensional viscosity and even that is questionable, because it seems that a stress overshoot is reached at those high Hencky strains. Moreover, if one realises that that for a Hencky strain of 7 the length of the original sample has increased 1100 times, whereas the diameter of the sample of 1 mm has decreased at the same time to 33 pm, then it will be clear that the forces involved with those high Hencky strains become extremely small during the experiment. 

FIG. 15.24 Schematic representation of the four roller extensional rheometer, designed by Meissner (1972) to attain high Hencky strains. Two sets of rotary clamps are individually driven by two motors at constant rotation rates. The force in the sample is measured by a transducer F mounted on a leaf spring. From Barnes, Hutton and Walters (Gen Ref 1993). Courtesy Elsevier Science Publishers. 

The authors calculated the strain hardening intensity values as a measure of melt strength using the ratio of riE.noniinear/ HE,linear ( )- The slope of log versus Hencky strain defines the strain hardening intensity. As seen in Fig. 7.42 (a), effective improvements in melt strength required over 50 wt% PBSA which decreases the tensile modulus. 

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]...

This logarithmic strain measure is sometimes called the Hencky strain (1924). In the limit of small strain, it is the same as the usual ( engineering ) strain measure L/Lo — 1. 

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. 

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. 

Another end-separation method was recently described by Bach et al. [201]. Their filamentstretching rheometer (FSR) is an adaptation of a device originally developed for dilute polymer solutions [202,203 ] and later used for concentrated solutions [203,204]. As shown in Fig. 1024, the filament is formed from a small sample by stretching it between two cylindrical, steel fixtures to which it is attached by the direct adhesion of the sample to the metal. The strain in this device is not uniform, and the measurement is based on the portion of the filament midway between the end fixtures, where its diameter is a minimum. As a result, several preliminary experiments are required to establish an empirical relationship between the radius at this point, where the Hencky strain is -2[ln(R/Ro)] and the distance between the fixtures, L (t). The data for e < 1.0 are not reliable, but steady state was reached in experiments by Bach et al. for an LDPE and an LLDPE, making it possible to determine the extensional viscosity [201]. The results agreed with data from an RME. This technique has also been used to demonstrate extension thickening in linear polystyrene [204]. 

Extensional viscosity was measured at 170°C on a Sentmanat Extensional Rheometer (SER) fixture (Xpansion Instruments).[1] The SER is based on a dual drum system. It is designed as a fixture of a standard rotational rheometer which consists of a master and slave wind-up dmms coupled via intermeshing gears. A constant Hencky strain rate is obtained simply by setting a constant winding speed. The SER fits inside the environmental chamber of an Advanced Rheometric Expansion System (ARES) rheometer. Tests were carried out on strips cut out of a 0.5 mm thick compression molded sheet. Constant Hencky strain rates (1 and 10 s ) were applied and the time-dependent stress was determined from the measured torque and the sample time-dependent cross-section. The extensional viscosity, tie, was obtained by dividing the stress by the Hencky strain rate. 

Uniaxial elongation is also utilized to examine the nonlinearity in particular for polymeric materials. In the uniaxial elongation test, a material is stretched at a constant Hencky strain rates and the elongational stress cte is measured (cf. Figure 1 and eqn [2]). The nonlinearity detected by this test is characterized by... 

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