Normal Stress Differences in Steady Simple Shear

2 Normal Stress Differences in Steady Simple Shear 

From the basic axioms of continuum mechanics, it is possible to show that for a simple fluid subjected to a perfectly smooth (infinitely diflererrtiable) strain history the stress terrsor is 

Another symbol sometimes used in place of Pj is 6. The rubberlike liquid model (Eq. 10.5), predicts that Pj is independent of shear rate and related to the linear relaxation modulus  

Experimental observations indicate that at very low shear rates Pj does, indeed, become independent of shear rate and that its value is given by Eq. 10.62. Its limiting, low-shear-rate value is assigned the symbol Pi g. Thus, although the Boltzmann superposition principle indicates that is zero, Pj q appears to be a material constant of linear viscoelasticity. By use of Eqs. 4.31 and 10.62 it can be related to other material constants of linear viscoelasticity  

We recall from Chapter 5 that for an entangled, monodisperse polymer is independent of molecular weight. Thus, since the viscosity is proportional to M , Eq. 10.63 implies that  

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