Normal stress ratio

Unfortunately, cannot be easily determined analytically, and is often either estimated using methods from the field of soil mechanics (from the normal stress ratio, K, or simply adjusted to a value which produces good fits for the experimental data. The forwarding force f, at the barrel wall is positioned in the plane of the barrel surface. This force is adjusted using as found in Eq. A5.1 and results in Eq. A5.2. This is because is the local pressure in the channel. Referring to Fig. A5.4, the forces are defined as follow ... 

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... 

The second normal stress difference has been found to be negative with a magnitude less than that of Wj. It is quite sensitive to assumptions used in deriving a tube model of rheological behavior. A useful material function is the normal stress ratio it,f) defined as follows ... 

The normal stress ratio is predicted by the DE model to be independent of time during a relaxation experiment. In fact, any separable constitutive equation of the BKZ type predicts this result, which implies that ... 

Figure 10.8 Average values of the normal stress ratio over times from 0.1 to 2 s during stress relaxation as a function of the stress ratio, which is assumed to be equal to the strain (Eq. 10.38). Data for a polystyrene solution (circles) and for a polyisoprene melt (triangles) are shown. Also shown are the DE predictions with (dashed) and without (solid) the lA assumption, which show that N2 is the property most affected by the lA simplification. From Olson etal. [41 ].

This relation, which Venerus et al. [22] call the Osaki-Lodge equation, has been found to be valid for entangled polystyrene solutions [22,42,43]. Another quantity of interest is the normal stress ratio defined as ... 

The second normal stress difference is usually reported in relation to the first normal stress difference by use of a shear-rate dependent normal stress ratio ... 

Figure 6 Nonlinear viscoelastic normal stress ratio parameter for the different coextruded formulations...

Even if no perceptible motion occurs (see later, however), application of a force leads to microdisplacements of one surface relative to the other and, again, often a large increase in area of contact. The ratio F/W in such an experiment will be called since it does not correspond to either the usual ns or can be related semiempirically to the area change, as follows [38]. We assume that for two solids pressed against each other at rest the area of contact Aq is given by Eq. XII-1, A W/P. However, if shear as well as normal stress is present, then a more general relation for threshold plastic flow is... 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

A representative measure of rubbery elasticity of a material may be two quantities dimensionless ratio (ct/t) and characteristic relaxation time 9 = ct/2ty. According to the data of works [37, 38] when fibers are introduced into a melt, ct/t increases (i.e. normal stresses grow faster than stresses) and 0 also increases on a large scale, by 102-103 times. However, discussing in this relation the papers published earlier, we noted in the paper cited that the data were published according to which if fibers were used as a filler (as in work [37]), 9 indeed increased [39], but if a filler represented disperse particles of the type Ti02 or CaC03, the value of 0 decreased [40],... 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Hooke s Law It is the ratio of normal stress to corresponding strain (straight line) for stresses below the proportional limit of the material. 

Dynamic similarity occurs in two geometrically similar units of different sizes if all corresponding forces at counterpart locations have a constant ratio. It is necessary here lo distinguish between the various types of force inertial, gravitational, viscous, surface tension and other forms, such as normal stresses in the case of viscoelastic non-Newtonian liquids. Some or all of these forms may be significant in a mixing vessel. Considering... 

Velocity profiles are demonstrated from Figs. 6 and 7. Figure 6 shows the effects of the viscosity-to-elasticity ratio at negative and positive normal stress gradients. Figure 7 shows the effects of various normal stress gradients on velocity profiles at the value of = 0.6. No general conclusions can be drawn... 

Here a is the normal stress equal to the ratio of the force F to the elementary surface dS, provided that this force has only the component normal to dS ... 

A criterion has been developed based on the tensile normal stress and the anisotropic tensile strength on arbitrary planes about the crack tip. It is assumed that the crack grows along the plane on which the stress ratio is maximum (Buezek and Herakovich, 1985). A maximum stress criterion is also proposed (Gupta et al., 1993) for a crack which deflects along the interface... 

Elastic behavior of liquids is characterized mainly by the ratio of first differences in normal stress, Ai, to the shear stress, t. This ratio, the Weissenberg number Wi = Ai/r, is usually represented as a function of the rate of shear y. 

B Ratio of vertical shear stress at wall to vertical normal stress at wall... 

---