Extensional strain

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

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). 

From Fig. 5.3 it may be seen that this exceeds the shear rate for non-laminar flow (approximately 30 s ) so that the entry to this region would need to be streamlined. Fig. 5.3 also shows that the extensional strain rate, k, in the tapered entry region should not exceed about 15 s if turbulence is to be avoided. 

Similarly, under suitable constraints, deformation is possible in which only one extensional strain arises or is applied. Again, work is produced by the corresponding stress alone. Thus, because the work done is determined by the diagonal elements of the stiffness matrix, those elements must be positive, that is,... 

Note, in contrast to both an isotropic layer and a specially orthotropic layer, that extensional forces depend on shearing strain as well as on extensional strain. Also, the resultant shearing force, N y, depends on the extensional strains, and e, as well as on the shear strain, Similarly, the moment resultants all depend on both the bending curvatures, Kx and Ky, and on the twist curvature, k. ... 

However, in most experiments, the loads are applied, and the resulting deformations are measured, i.e., the deformations are the dependent variables, not the loads. Thus, the expressions for the middle-surface extensional strains and curvatures in terms of the force and moment resultants would be convenient. 

Note that the coefficients of thermal expansion affect only extensional strains, not the shearing strain. 

Fig. 2.8.8 (a) Implementation of four-roll mill device, (b) Speed image obtained across a horizontal slice by combining separate x- and y-velocity images, (c) Velocity profile taken across the x-axis of the cell showing the uniform extensional strain rate 8, such that x= ex (adapted from Ref. 15]). 

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... 

For a small compression/extensional strain k/a0 = E, the Young s modulus of the system, and ao is the volume occupied by the mass m, and so using the density of the system, p, we have... 

Dirac delta function in three dimensions extension rate extensional strain strain tensor (linear)... 

This work shows that high shear rates are required before viscous effects make a significant contribution to the shear stress at low rates of shear the effects are minimal. However, Princen claims that, experimentally, this does not apply. Shear stress was observed to increase at moderate rates of shear [64]. This difference was attributed to the use of the dubious model of all continuous phase liquid being present in the thin films between the cells, with Plateau borders of no, or negligible, liquid content [65]. The opposite is more realistic i.e. most of the liquid continuous phase is confined to the Plateau borders. Princen used this model to determine the viscous contribution to the overall foam or emulsion viscosity, for extensional strain up to the elastic limit. The results indicate that significant contributions to the effective viscosity were observed at moderate strain, and that the foam viscosity could be several orders of magnitude higher than the continuous phase viscosity. 

The theoretical analysis for two-dimensional foams and emulsions has recently been expanded to three dimensions [38], with Kelvin s minimal tet-rakaidecahedron as the unit cell. The system is subjected to a uniaxial extensional strain. As the elastic limit, or yield point, is approached, the cell shape tends towards a rhombic dodecahedron however, at the yield point, the shrinking quadrilateral faces of the polyhedron have finite (albeit small) area. 

Analyze lubricated squeezing flow to determine biaxial extensional viscosity (T)R), which is calculated from biaxial stress (cB) and biaxial extensional strain rate (eB). 

Also called uniaxial extension, in this type of deformation a rod-shaped sample is pulled in one direction keeping the other end fixed (Fig. 9a). The infinitesimal extensional strain, de, can be expressed as... 

Fig. 12.25 The Universal testing platform fitted on a rotational Rheometrics RDA II rheometer host station. The two counterrotating cylinders where the film is mounted cause the application of the extensional strain. [Reprinted by permission from E. G. Muliawan, S. G. Hatzikiriakos, and M. Sentmanat, Melt Fracture of Linear Polyethylene, Int. Polym. Process., 20, 60 (2005).]...

A very important diagram for fibres and yams is the stress—strain diagram, where the specific stress is plotted as a function of the elongation (extensional strain) in %. The curve starts at an elongation of zero and ends in the breaking point at the ultimate specific stress (=tensile strength or tenacity) and the ultimate elongation (=strain at break). 

Biaxial extensional flow is extensional flow in two directions with the same constant extensional strain rate qe. If the volume is constant then... 

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