Case of The Steady Shear
For this case only two components of the shear stress became non-zero:
and
where is the shear rate.
Thus, the upper-convected Maxwell model predicts for the simple shear that shear stress to be proportional to the shear rate and the first difference of normal stresses is proportional to the square of the shear rate, the second difference of normal stresses is always zero. In other words, UCM predicts appearance of the first difference of normal stresses but does not predict non-Newtonian behavior of the shear viscosity nor the second difference of the normal stresses.
Usually quadratic behavior of the first difference of normal stresses and no second difference of the normal stresses is a realistic behavior of polymer melts at moderated shear rates, but constant viscosity is unrealistic and limits usability of the model.
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