Steady uniaxial extension flow

To solve Eq. (3-24) for a particular flow history, one must compute the tensor B and carry out the integration. An example of how this is done for start-up of steady uniaxial extension is given in Worked Example 3.1 at the end of this chapter. 

Extensional flows yield information about rheological behavior that cannot be inferred from shear flow data. The test most widely used is start-up of steady, uniaxial extension. It is common practice to compare the transient tensile stress with the response predicted by the Boltzmann superposition principle using the linear relaxation spectrum a nonlinear response should approach this curve at short times or low strain rates. A transient response that rises significantly above this curve is said to reflect strain-hardening behavior, while a material whose stress falls... 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

The most Important commercial blends of PE are those of LLDPE with LDPE (25, 26). The capillary flow data n (012) and B 8(012), Indicated (similar to HDPE/LDPE) PDB-type behavior (27-29). The latter authors also reported a PDB relation between melt strength and composition. Kecently (14, 15) these blende were studied under the steady state and dynamic shear flow as well as in uniaxial extension. A more detailed review of these results will be given in part 3 of this chapter. Like HDPE/LDPE blends, those of LLDPE/LDPE type are also consistently reported as immiscible. 

These remarks about reaching a steady state apply not only to uniaxial extensional flows, data for which appear in Figure 4.2.5, but for other extensional flows as well. Besides uniaxial extension, the two most important extensional flows are equal biaxial extension and planar extension. Kinematic tensors for these extensional flows were to have been found in Exercise 2.8.1. In uniaxial extension the material is stretched in one direction and compressed equally in the other two in equal biaxial extension the material is stretched equally in two directions and compressed in the third and in planar extension the material is stretched in one direction, held... 

In steady-state uniaxial extensional flows, the MLD and DEMG equations predict three regions of flow see Fig. 11.4. For monodisperse polymers, these regions are defined by the extension rate e relative to the two relaxation times Tj and T. For e < 1/, the extension rate is too... 

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. 

B and D for Steady Extension Consider three steady extensional flows (a) uniaxial, (b) equal biaxial, and (c) planar. Determine components of the tensors B and D for each and for their invariants. 

---