Stress difference, normal

A consequence of finite deformations is the appearance of normal stresses in simple shearing deformations. Thus, even in steady-state simple shear flow (Fig. 1-16) where the rate of strain tensor (c/. equations 3 and 5) is 

Normal stresses in steady-state shear flow. 

The normal stresses c,/ cannot be specified on an absolute basis because of the arbitrary hydrostatic pressure in equation 59, but their differences are predicted by continuum mechanics and several molecular theories and can be measured. The primary and secondary normal stress differences are defined as (Th — an and 022 — an respectively. Data are often expressed in terms of the primary normal stress coefficient 

At low values of 7, the coefficients 1 and 2 approach the zero-shear-rate limits of j,o and 2,0 respectively. All the single-integral equations mentioned above predict for the limiting primary normal stress coefficient  

It will be shown in Chapter 3 that this relation is equivalent to = 2jj. The predictions of different nonlinear models for the magnitude of 4 2 vary considerably, but for most materials investigated thus far 2 is found to be less than 1 and of opposite sign. 

So far we have considered linear transport phenomena, in which the response is directly proportional to the circumstance causing the response. Polymer solutions, however, are fundamentally nonlinear, and show a wide variety of additional behaviors not expected from simple linear descriptions. These behaviors may be divided, somewhat crudely, into unusual flow behaviors arising from nonzero normal stress differences, time-dependent phenomena in which the system shows memory, so that the response to a series of forces depends on when they were applied, and several modern discoveries not discussed in more classical references. 

At some point, the constraints of time and space insist that the discussion be curtailed, so we here present a taxonomy of nonlinear viscoelastic phenomena, without the considerable quantitative analyses seen in prior chapters. The objective is to represent the range of observed phenomena and provide references that give entries into the literature. No effort has been made to give a thorough collection of published results. If Lord Rayleigh s critique - science is divided between quantitative measurement and stamp collecting - is invoked on this chapter, the stamps are indeed beautiful, but are likely to be more thoroughly quantitatively examined when readers become aware of their existence. 

A variety of the classical unusual flow effects seen with polymer solutions can be traced back to the normal stress differences N and N2. A full mathematical description becomes quite lengthy, and may be found in Graessley(l). The following remarks provide an exceedingly compressed description. 

Consider a small parallelepiped of fluid whose six faces are aligned with respect to the X, y, and z axes. For each axis, the fluid has two planes perpendicular to the axis and four planes parallel to the axis. The forces on the fluid within the parallelepiped are formally divided between body forces, e.g., gravity, whose 

We now introduce a stress tensor p, whose components pij are the contact forces across a surface, so for example the forces across a plane perpendicular to the X axis are 

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The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

Polyolefin melts have a high degree of viscoelastic memory or elasticity. First normal stress differences of polyolefins, a rheological measure of melt elasticity, are shown in Figure 9 (30). At a fixed molecular weight and shear rate, the first normal stress difference increases as MJM increases. The high shear rate obtained in fine capillaries, typically on the order of 10 , coupled with the viscoelastic memory, causes the filament to swell (die swell or... 

Fig. 9. First normal stress differences of polypropylene of different molecular weight and distribution (30) see Table 4 for key. To convert N /m to...

Description of normal stress measurements on a practical but complex material, paint, is available (150). More recent pubHcations (151—154) give the results of investigations of normal stress differences for a variety of materials. These papers and their references form a useful introduction to the measurement of normal stress differences. 

Fig. 22. Shear viscosity Tj and first normal stress difference (7) vs shear rate 7 for a low density polyethylene at 150°C (149), where (Q) — parallel plate ...

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. 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

Fig. 19. Shear stress and first normal stress difference plotted as a function of shear rate for different molar masses, and b at different concentrations of polystyrene in toluene... 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

The corresponding first normal stress difference N t) = tu(t) — t22( ) as predicted from Eq. 4-2... 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

When a viscoelastic liquid flows through a tube, the normal stress differences cause the liquid to be under an axial tension while a normal... 

A less well known effect occurs in open channel flow of a viscoelastic liquid, when the normal stress differences cause the free surface to bow upwards in the centre [Tanner (1985)]. 

Quotient of the first normal stress difference N ) and the square of the shear rate (y) in the limit of zero shear rate... 

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