Uniaxial extension steady

When the strain rate, s, is maintained constant, the deformation obtained is called steady simple extension or steady uniaxial extension (Dealy, 1982) and the extensional viscosity, is related to the normal stress difference ... 

Figure 1.14 Uniaxial extensional viscosity rj (open symbols) and shear viscosity r] (closed and half-closed symbols) as functions of time after start-up of steady uniaxial extension or steady shearing for Melt I. (From Meissner, J. Appl. Polym. Sci. 16 2877, Copyright 1972. Reprinted by permission...

Figure 3.2 Trouton ratio, Tr, of uniaxial extensional viscosity to zero-shear viscosity jq after start-up of steady uniaxial extension at a rate of 1 sec i for a Boger fluid consisting of a 0.185 wt% solution of flexible polyisobutylene (Mu, = 2.11 x 10 ) in a solvent composed mostly of viscous polybutene with some added kerosene (solid line). The dashed line is a fit of a multimode FENE dumbbell model, where each mode is represented by a FENE dumbbell model, with a spring law given by Eq. (3-56), without preaveraging, as described in Section 3.6.2.2.I. The relaxation times were obtained by fitting the linear viscoelastic data, G (co) and G"(cu). The slowest mode, with ri = 5 sec, dominates the behavior at large strains the best fit is obtained by choosing for it an extensibility parameter of = 40,000. The value of S — = 3(0.82) n/C(x, predicted from the...

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 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. ... 

Figure 3.39 Uniaxial extensional viscosity rj as a function of time following start-up of steady uniaxial extension at the extension rates e indicated. Data are shown for an unbranched polystyrene (PS I), a high-density polyethylene with short, unentangled side branches (HOPE I), and two low-density polyethylenes (LDPE III and lUPAC A), with long side branches. (From Laun 1984, with permission from the Universidad Nacional Autonoma de Mexico.)------------------------------...

From Eq. (3-24) for a rubber-like liquid, assuming a single relaxation time r and modulus G, calculate formulas for the extensional viscosity as a function of time after start-up of steady uniaxial extension at extension rate e. 

The most efficient mechanism of drop breakup involves its deformation into a fiber followed by the thread disintegration under the influence of capillary forces. Fibrillation occurs in both steady state shear and uniaxial extension. In shear (= rotation + extension) the process is less efficient and limited to low-X region, e.g. X < 2. In irrotatlonal uniaxial extension (in absence of the interphase slip) the phases codeform into threadlike structures. 

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. 

The behavior of LLDPE blends at constant rate of stretching, e, was examined at 150°C. The results are shown In Fig. 13 for Series I and II as well as in Fig. 14 for Series III. The solid lines In Fig. 13 represent 3n calc values computed from the frequency relaxation spectrtmi by means of Equation (36), while triangles Indicate the measured in steady state 3n values at y = 10 2 (s ), I.e. the solid lines and the points represent the predicted and measured linear viscoelastic behavior respectively. The agreement Is satisfactory. The broken lines In Fig. 13 represent the experimental values of the stress growth function In uniaxial extension, nE 3he distance between the solid and broken lines Is a measure of nonlinearity of the system caused by strain hardening, SH. 

Material functions must however be considered with respect to the mode of deformation and whether the applied strain is constant or not in time. Two simple modes of deformation can be considered simple shear and uniaxial extension. When the applied strain (or strain rate) is constant, then one considers steady material functions, e.g. q(y,T) or ri (e,T), respectively the shear and extensional viscosity functions. When the strain (purposely) varies with time, the only material functions that can realistically be considered from an experimental point of view are the so-called dynamic functions, e.g. G ((D,y,T) and ri (a), y,T) or E (o),y,T) and qg(o),y, T) where the complex modulus G (and its associated complex viscosity T] ) specifically refers to shear deformation, whilst E and stand for tensile deformation. It is worth noting here that shear and tensile dynamic deformations can be applied to solid systems with currently available instruments, whUst in the case of molten or fluid systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear viscoelasticity, relevant to processing. 

Thus eq. 2.2.9 gives the velocity gradient tensor L for steady uniaxial extension. When this tensor operates on displacement vectors imbedded in the liquid, it generates the velocity field for steady uniaxial extension. 

For steady uniaxial extension, eq. 2.2.19 gives the components of 2D. Using these components with eq. 2.3.2 gives the stresses in a Newtonian fluid... 

This important result demonstrates the value of the tensor form of Newton s viscosity law. It is directly analogous to the result in Chapter 1, that the tensile modulus is three times the shear modulus, eq. 1.5.11. The three times rule for viscosity in steady uniaxial extension is often called the Trouton ratio. We see it holds true at low rates for the polymer melt in Figure 2.1.3. The following examples give applications of the Newtonian model to more complex deformations. Further examples appear at the end of the chapter. Bird, et al. (1987, Chapter 1) or any other good fluid mechanics book contains many worked Newtonian examples. 

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... 

Calculate the predictions of the upper-convected Maxwell equation in (a) start Up of steady shear and (b) steady state uniaxial extension for arbitrary shear rate y and extension rate e, and compare these predictions with those for the Newtonian and second-order fluids. 

Substituting the velocity gradient for uniaxial extension, eq. 2.2.9, into eq. 4.3.7, we readily find that at steady state... 

Calculate, using eq. 4.3.18, the growth of the extensional viscosity after start-up of a steady uniaxial extension. Compare the steady... 

For steady uniaxial extension Frankel and Acrivos (1970) report... 

They assumed that the parameter j8 is constant and found that a value of -0.27 fitted their planar extension data. While it is now understood that the normal stress ratio is a function of strain, it is shown in Section 10.4.5 that it does approach a specific limiting value as y 0. Based on data for one melt in step shear and start-up of steady uniaxial extension, Wagner and Demarmels proposed the following empirical relationship for the damping function with... 

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. 

Stress growth coefficient in uniaxial extension at strain rates from 0.001 to 30 s" for an LDPE (lUPAC A) at 150 "C. As the strain rate increases,the data rise above the linearenvelope curve at progressively shorter times. The steady-state value (extensional viscosity) first Increases with strain rate and then decreases. From MQnstedt and Laun [1521-... 

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... 

Figure 11.25 also compares the predictions of a refined version of the pom-pom model by McLeish et al. [97] with these data for start-up of steady uniaxial extension as well as data for start-up of steady shear. The theory shows good agreement with these data, although some adjustments had to be made to parameter values based on the linear viscoelastic data for this H polymer. 

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. 

---