Stresses, growth

The application of a linearly ramped strain can provide information on both the sample elasticity and viscosity. The stress will grow in proportion to the applied strain. The ratio of the strain over the applied time gives the shear rate. Applying the Boltzmann Superposition Principle we obtain the following expression  

Now when we apply a shear rate to the sample, as t - oo the viscosity tends to rj(0). At short times the elasticity can be obtained by differentiating the viscosity versus time curve  

This is the stress relaxation function, so the slope plotted as a function of time provides us with G(t). Now in the limit of short times we find the exponential tends to unity  

The spectral relationships apply in a similar fashion to those for relaxation. For a viscoelastic liquid we have a viscosity that depends upon the spectrum and the relaxation time t  

For a viscoelastic solid the situation is more complex because the solid component will never flow. As the strain is applied with time the stress will increase continually with time. The sample will show no plateau viscosity, although there may be a low shear viscous contribution. This applies to both a single Maxwell model and one with a spectrum of processes  

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Keywords. Adherent cells. Chemical stress. Shear stress. Growth inhibition. Stress protection... 

It is difficult to predict these responses directly because they depend upon a wide range of instrumental properties in addition to the material properties of the sample. The onset of this behaviour can be explored through the use of a stress growth experiment. 

Figure 4.15 The stress growth function for a Maxwell model with a relaxation time tr...

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One major consequence of the Ml project was the development of a modified filament stretching instrument by Sridhar. In this device, the test sample is held horizontally between two Teflon discs and pulled equally at both ends at a programmable exponential rate such that a constant strain rate is achieved and the stress growth at a constant stretch rate is obtained (40). It appears though that the test sample has to adhere to the plates as the technique does not use aids to clamp samples. Consequently, it is not clear if the technique can be applied to products that are non-sticky or exhibit slip, which could be limiting factors for testing food products. 

A second device that is also able to generate truly extensional flow field has been developed by Meissner (41) utilizing the concept of rotary clamps. At a constant strain rate this instrument can measure stress growth and thus allow the steady state flow measurements. 

These results highlight the genuine need for carrying out steady state experiments for accurate measurement of dough extensional viscosity. At the present time, there is no other instrument available that can provide stress-growth profiles for doughs. 

Lodge rubberlike liquid (3.3-15) Constant = constant l/2 = 0 No predicts elongational stress growth rj1 (l, k) Yes... 

Finally, it has to be mentioned that stress growth experiments turn out to be very sensitive to the previous history (Moldenaers, 1996). 

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The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

Figure 13.5 Dependences of the reduced tensile stress growth coefficient ti (i,t)/ar at 170°C on reduced time fay and reduced strain rate iaj for a SAN resin (wAN = 0.264, Mw = 78 kg/mol, Mw/M = 2.8) during the startup of uniaxial extensions flow. Also illustrated (dashed curve) are dynamic shear viscosity data displayed as 3 r7 (c<, 170°C) versus or7 as suggested by Trouton [64]. Reproduced from L. Li, T. Masuda and M. Takahashi, J.Rheol., 34(1), 103(1990), with permission of the American Institute of Physics...

For strain rates lower than 8x10 s, it was found that the rheological behaviour is nearly linear viscoelastic Fig. 5 shows the tensile stress-growth function CT (0,t) = EXT]E (E,t) at 123°C for three different strain rates in the linear range after about 1000s, the stress reaches a... 

Figure 5. Tensile stress-growth function for sample SI at 123T.

Figure 8. Tensile stress-growth coefficient of sample SI at 123 °C...

Further tests have been carried out on sample SI at two higher strain rates. The stress-growth coefficient corresponding to these experiments is represented in Fig. 8, where the... 

Figure 11. Tensile stress-growth function of sample SI at 123°C. Symbols data of Fig. 5. Solid lines Rouse model.

Using the cone and plate geometry, stress growth experiments have also been performed using different temperatures and different shear rates. Correct tangential (X+(t,Y) Tj+(t,Y)) and normal stress (Ni(t,Y) Vi(t,Y)) data were... 

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