Shear limiting value

There are three important points about Equation (3.47). Firstly the viscosity is the low shear limiting value, rj(0), indicating that we may expect some thinning as the deformation rate is increased. The reason is that a uniform distribution was used (ensured by significant Brownian motion, i.e. Pe < 1) and this microstructure will change at high rates of deformation. Secondly there is a difference between the result for shear and that for extension. Thirdly the equation is only accurate up to cp < 0.1 as terms of order 3 become increasingly important. If we write the equation in the form often used for polymer solutions we have for Equation (3.47 a) ... 

In pi actice, loads are not necessarily uniformly distributed nor uniaxial, and cross-sectional areas are often variable. Thus it becomes necessary to define the stress at a point as the limiting value of the load per unit area as the area approaches zero. Furthermore, there may be tensile or compressive stresses (O,, O, O ) in each of three orthogonal directions and as many as six shear stresses (t, , T ). The... 

At the thin film limit, the hydrodynamic pressure will approach a distribution that is consistent with the pressure between the two solid surfaces in dry static contact, while the shear stress experienced by the fluid film will reach a limiting value that is equal to the shear strength of a boundary film. 

If the intrinsic viscosity is large (i.e., greater than about 4 deciliters per gram), the viscosity is likely to be appreciably dependent on the rate of shear in the range of operation of the usual capillary viscometer. Measurements in a viscometer permitting operation at a series of rates of shear extending to very low rates are then required in order to extrapolate nsp/c to its limiting value at a shear rate of zero. Extrapolation to infinite dilution does not eliminate the effect on this ratio of a dependence on shear rate. 

Figure 3 shows plots of p versus shear rate at three different temperatures for the same latex (20% w/w latex A) at full coverage with PVA. These curves are typical of a pseudoplastic system showing a reduction of n with increasing shear rate, 7 p reaches a limiting value at If > 50 s l. It is also clear from fig. 3 that at 7 < 10 s-- -, n increases rapidly with reduction in 7. Comparison with nQ values obtained from the creep curves would indicate the p should increase very steeply with reduction of 7, in the low shear rate region (p is the limit of p as Y+0). °... 

The intrinsic viscosity is the Einstein value [rj] = 2.5 and the packing fraction cpm(0) is that in the low shear limit. As the volume fraction approaches the maximum packing fraction, the viscosity rapidly... 

For this to happen we know that f(y) = y in the low shear limit. As the shear stress is increased we also know that we want our viscosity to fall so we need to multiply our strain by a damping function that reduces from unity at low strains to a lesser value at high strains. A good candidate for... 

A plot of rj(o) versus the log of reduced stress shows a linear slope between the high and low shear limits. We can use this feature with our master curve to define another value of b. This slope is given by... 

If we extrapolate this slope toward low rates, where the tangent equates with the low shear viscosity and assume the Peclet number here is unity we eventually obtain a value of b = 2.55. This defines the Peclet number as unity, at a stress somewhere just after the curvature of the viscosity curve deviates from the low shear limit. This seems quite an appropriate reference system. By setting the Peclet number to the appropriate value of b we can determine the variation in packing fraction with stress between the high and low shear limits to the viscosity ... 

When the extrusion is a profile with sections of different thickness the shear rates and die swell will be higher in the thinner sections than elsewhere—so drawing again may be of limited value. The thinner sections may have a shorter die parallel (to ensure that the rate of extrusion is constant over the entire cross-section), and this will alter die swell even further. 

For the films and conditions we have used, the transmission line and lumped element models give indistinguishable results. Fitting of the data of Fig. 13.7 yields G as a function of time. These values increase at short times (due to nucleation phenomena) to long time limiting values of G = 1.9 x 106 dyne cm-2 and G" = 3.0 x 108 dyne cm-2. These values of the shear modulus components show that, in dichloromethane, the PVF film is a very rubbery polymer in which there is considerable viscoelastic loss when the film thickness exceeds 1 p.m. 

In order to elucidate the correlation method it may be recalled that the viscosity 77 approaches asymptotically to the constant value r c with decreasing shear rate q. Similarly, the characteristic time t approaches a constant value xQ and the shear modulus G has a limiting value G0 at low shear rates. Bueche already proposed that the relationship between 77 and q be expressed in a dimensionless form by plotting 77/r]0 as a function of qx. According to Vinogradov, also the ratio t/tq is a function of qxQ. If the zero shear rate viscosity and first normal stress are determined, then a time constant x0 may be calculated with the aid of Eqs. (15.60). This time constant is sometimes used as relaxation time, in order to be able to produce general correlations between viscosity, shear modulus and recoverable shear strain as functions of shear rate. 

If values of rja are now plotted as a function of strain rate (Figure 13-62), it can be seen that although there is a dramatic decrease in apparent viscosity at high values of y, at low strain rates the apparent viscosity is essentially constant. Obviously, if you want to compare the rheological properties of different types of polymers, it is this strain rate-independent parameter that would be most useful, as it would presumably be a characteristic property of the polymer. This limiting value is called the zero shear-rate viscosity, rjm. 

For an unvulcanized polydimethylsiloxane, the biaxial viscosity was approximately six times the shear viscosity over the biaxial extensional rates from 0.003 to 1.0 s (Chatraei et al., 1981), a result expected for Newtonian fluids, that is, the relationship between the limiting value of biaxial extensional viscosity (j, ) at zero strain rate and the steady zero-shear viscosity (i o) of a non-Newtonian food is ... 

In the case of real substances, the response to the shear stress involves an instantaneous deformation of the Hookean type followed by gradual increase of the shear strain with time. If the strain at long times approaches a limiting value e q, the substance is considered a solid. However, if at long times the strain is a linear function of time, the substance is considered a liquid. Schematic representations of these responses are given in Figures 5.5a and 5.5b for real solids and liquids. 

Extrapolation of the H values for p = 1 and p = 0.86 g cm yields the limiting values for an ideal PE crystal He 150-180 MPa) and an ideal PE amorphous matrix Ha 1 MPa), respectively. It is noteworthy that the extrapolated value obtained for He in PE almost coincides with the theoretical value of ultimate shear stress Xy given in Table 2.1. The experimental H values given in the literature evidently correspond to materials with p mostly deviating from unity. 

---