Normal stress coefficient secondary

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

The cone-plate rheometer. The cone-plate rheometer is often used when measuring the viscosity and the primary and secondary normal stress coefficient functions as a function of shear rate and temperature. The geometry of a cone-plate rheometer is shown in Fig. 2.47. Since the angle Oo is very small, typically < 5°, the shear rate can be considered constant throughout the material confined within the cone and plate. Although it is also possible to determine the secondary stress coefficient function from the normal stress distribution across the plate, it is very difficult to get accurate data. 

Concentrated emulsions can exhibit viscoelasticity, as can gelled foams and some suspensions. Compared with the previous equations presented, additional coefficients (including primary and secondary normal stress coefficients) are needed to characterize the rheology of viscoelastic fluids [376,382]. 

We note that the primary normal stress coefficient P 1 is positive, whereas the secondary normal stress coefficient P2 is negative, but with a lot of scatter in the data. It is difficult to measure (r22 — T33) and its value is in doubt, but the ratio — (tn — X22)/ x22 — T33) appears to be about 0.1. 

Additionally, primary and secondary normal stress coefficients and j/2 are defined by the respective relations... 

The cone-plate rheometer is used when measuring the viscosity and the primary and secondary normal stress coefficient functions as a function of shear rate and... 

The viscosity function rj (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients ij/, and respectively, are the three viscometric functions which completely determine the state of stress in any rheologically steady shear flow. They are defined as follows ... 

The elasticity of polymer melts is manifested through two material functknis, namely, the primary normal stress coefficient ii and the secondary normal stress coefficient ijia. The secondary normal stress coefficient is not as well duuacter-ized as the primary normal stress coefficient due to its small magnitude. The primary normal stress measurements are themselves difficult and require highly... 

Fundamental rheological quantities such as shear viscosity, primary and secondary normal stress coefficients, elongational viscosity and complex viscosity are introduced. Most molten polymers show a reduction in shear viscosity with increasing shear rate (pseudoplasticity). At low shear rates, however, the shear viscosity is practically independent of shear rate. This viscosity value is called the zero-shear-rate viscosity (rjo)- At higher shear rates, the viscosity ( ) decreases with shear rate (y) according to a power law expression t] = where K and n are... 

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