Stress Growth and Relaxation in Steady Shear

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  

At short times, when the deformation is small, we expect the shear stress growth coefficient to follow the prediction of the Boltzmann superposition principle. And if the shear rate is sufficiently low, the entire transient should be governed by this principle, and rf (t, y) becomes the same as r) t), which was shown as Eq. 4.8  

For start-up of steady-simple shear the rubberlike liquid model predicts that the shear stress is given Eq. 4.8. While h/2 is predicted to be zero, the first normal stress difference is  

the transient first normal stress coefficient is  

This is a curious result, as it indicates that a nonlinear property can be calculated from linear data, but it has been found to describe accurately the response of polymeric liquids at sufficiently low shear rates. The low-shear-rate limiting behavior indicated by the above equations, which involves monotone increasing functions, is always shown in plots of nonlinear data the nonlinear responses involve overshoots in the material functions, but should always start out at short times, when the strain is still small, by following the low-shear-rate, LVE curve. Then, as the shear rate increases, the nonlinear data fall below the linear envelopes at shorter and shorter times [44,45]. These features can be seen in Fig. 10.9, which shows the data of Menezes and Graessley [46] for shear and first normal stress difference in start-up of simple shear. The dashed lines are calculated from the linear spectrum using Eqs.4.8 and 10.49. As expected, the 

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