Extensional rate uniaxial

Extensional flow (also called elongational flow) is defined as a flow where the velocity changes in the direction of the flow dvi/ dxy in contradistinction with shear flow where the velocity changes normal to the direction of flow (dv1/dx2). In uniaxial flow in the x1 direction the extensional rate of strain is defined as ... 

Transforming this deviatoric shear-rate expression, eq. (10.7), into a uniaxial extensional rate expression gives... 

The uniaxial extensional rate may be constant or vary in the Xi direction of flow. When s is constant, i.e. when the axial velocity is proportional to Xj, the resulting flow is steady uniaxial extensional flow. In such a flow situation, a cylindrical rod of length I is stretched along its axis according to the following equation ... 

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

A method for measuring the uniaxial extensional viscosity of polymer soHds and melts uses a tensile tester in a Hquid oil bath to remove effects of gravity and provide temperature control cylindrical rods are used as specimens (218,219). The rod extmder may be part of the apparatus and may be combined with a device for clamping the extmded material (220). However, most of the mote recent versions use prepared rods, which are placed in the apparatus and heated to soften or melt the polymer (103,111,221—223). A constant stress or a constant strain rate is appHed, and the resultant extensional strain rate or stress, respectively, is measured. Similar techniques are used to study biaxial extension (101). 

Extensional flow describes the situation where the large molecules in the fluid are being stretched without rotation or shearing [5]. Figure 4.3.3(b) illustrates a hypothetical situation where a polymer material is being stretched uniaxially with a velocity of v at both ends. Given the extensional strain rate e (= 2v/L0) for this configuration, the instantaneous extensional viscosity r e is related to the extensional stress difference (oxx-OyY), as... 

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...

For this simple uniaxial extensional flow to be steady, the instantaneous rate of change of the 1 direction length (/) must be constant... 

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. 

Birefringence setups can be designed to characterize molten materials undergoing isothermal homogeneous flow. The ranges of strains and strain rates also often coincide with those of rheometers, and consequently may be limited relative to those used in fabrication. Similarly, time-temperature superposition approaches may be used to expand the rate window. State-of-the-art setups suitable for rapid screening of new materials with research-scale quantities (5-20 g) are available for shear flow [72] and startup of uniaxial extensional flow [73,74]. 

M.Takahashi, T.Isaki, T.Takigawa, T.Masuda, Measurement of biaxial and uniaxial extensional flow of polymer melts at constant strain rates, J. Rheol. 31 (1993), 827-846. 

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 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...

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 uniaxial extensional viscosity rj(s) and the viscometric functions rj(y) and ki(y), predicted by the Doi-Edwards model for monodisperse melts, are shown in Fig. 3-32. The Doi-Edwards model predicts extreme thinning in these functions the high-shear-rate asymptotes scale as 17 oc oc y , and4 i oc The second normal... 

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 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... 

Figure 5.1 5 Tensile stress vs. tensile strain rate (log scales) for the uniaxial extensional flow of LDPE, ethylene propylene copolymer and PMMA,...

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