Uniaxial extensional flow, uniform

Figure 3.2 (Case 1) shows a simple uniaxial extensional flow created by the uniform stretching of a rectangular or a thin filament in the 1 direction. For this flow, 22 = - n /2, and because of the incompressibility assumption, 22 = 33. Thus, in Eq. 3.1-1, m = —0.5, giving the following rate of deformation matrix...

The uniaxial extensional flow, with regard to describing both the deformation and the resulting stresses, is uniform shear free flow, in which the strain rate is the same for every material element, and there is no relative... 

We now turn to uniform uniaxial extensional flow, with (cf. Equation 7.4a-b) the velocity field Wz = ke z, Wr = -5 K f. The velocity gradients are all constant in space, so it follows that the stresses are independent of spatial position. It is easily shown with this velocity field that Xn = 0 and xgg = Xrr. The total axial stress is then equal to the stress difference x z - trr (cf. Equation 7.9). The component equations for each dynamical mode are... 

If we want to find out how a fluid behaves under extension, we have to somehow grip and stretch it. Experimentally, this is much more difficult than the shear arrangement, especially if the fluid has a low viscosity. Earlier (see Section 5) we saw that it is possible to classify steady extensional flows under the categories of uniaxial, biaxial and planar flows. We will now examine uniaxial testing, since this mode is more commonly employed as a routine characterization tool. Here we encounter two approaches the first seeks to impart a uniform extensional field and back out a true material function, while the second employs a mixed flow field that is rich in its extensional component (e.g. converging flows) and use it to back out a measured property of the fluid which is somehow related to its extensional viscosity. 

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