First normal stress coefficient defined

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

Two other variables derivable from tensor analysis are the first normal stress, Ni, and the second normal stress, N2, which are related to the parameters often noted in polymer rheology studies, T [, and the first and second normal stress coefficients defined as... 

This flow is shown in Figure 2(a) where the velocity distribution is given by Vx = yy,Vy = 0,V2 = 0 and y = dv /dy is a constant. For this flow it is possible to measure a shear stress a first normal stress difference x — and a second normal stress difference These three quantities are in general strong functions of the shear rate y — dVx/dyl It is conventional to define three viscometric functions , namely the (non-Newtonian) viscosity rj (equation 1), the first normal stress coefficient Pi (equation 2) and the second normal stress coefficient 2 (equation 3), as follows... 

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]... 

Starting with cell model of creeping flow, Choi and Schowalter [113] derived a constitutive equation for an emulsion of deformable Newtonian drops in a Newtonian matrix. The authors characterized the interphase with an ill-defined interfacial tension coefficient, Vu, affecting the capillarity number, k = (Judfvu. The analysis indicated that depending on magnitude of /cy the emulsion may be elastic, characterized by two relaxation times. For the steady-state shearing, the authors expressed the relative viscosity of emulsions and the first normal stress difference as ... 

A capillary rheometer is a pressure-driven flow, the theme of this chapter, in contrast to the drag flows of Chapter 5. As Hagen first observed, when pressure drives a fluid through a channel, velocity is maximum at the center. The velocity gradient or shear rate and also the shear strain will be maximum at the wall and zero in the center of the flow. Thus all pressure-driven flows are nonhomogeneous. This means that they are only used to measure steady shear functions the viscosity and normal stress coefficients t] y), and Equations 5.1.1-5.1.3 define these functions, and Figure II.3 indicated how they are related to the other material functions. 

In start-up of steady simple shear, the measured stresses are divided by the imposed shear rate or its square to obtain the shear stress growth coefficient and the first normal stress growth coefficient, which are defined as follows ... 

The first and second normal-stress coefficients px and pz sjre defined in terms of the square of y because the normal stresses are even functions of shear rate ... 

Because the shear stress is an odd function of the shear rate and the normal stress differences are even functions, it is customary to define the viscosity function and the first and second normal stress coefficients as follows ... 

The first and second normal stress differences Ni and N2 can be expressed in terms of two coefficients, j/i and j/2, defined as follows ... 

When each component of one vector is linearly related to each component of another vector the coefficients of proportionality are the components of a second-rank tensor. Stress, defined as force per unit area, is the quotient of two vectors, and is an example of a second-rank tensor. Note that the condition defining positive in the direction of the outward normal means that a hydrostatic pressure must be negative. Strain is also a tensor, and in the most general case both stress and strain can be expressed in terms of nine tensor components. Stress and strain are both examples of second-rank tensors, which have nine components vectors, which have three components, are sometimes referred to as first-rank tensors single-valued scalars are zero rank tensors. 

Frederich established one of the first progressions [8]. The friction function which it defines results from on site tests. The fnction is a function of the rolling speed and the normal force applied to a wheel. Unfortunately, the relations of macroscopic causes (mechanical parameters) in microscopic effects (variation of the fnction) are not physically known. Therefore, Ohyama carried out laboratory experiments with layers injected into the contact where their shear stress were expressed as a function of the ratio of two coefficients static friction ps and kinematic pd [9]. The first coefficient was defined in the stick zone while the second was in the slip zone (Figure 4). Thus the total tangential force was the integral of the tangential stresses qi (x) and q2(x) of each zone. 

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