The upper-convected Maxwell (UCM) model is a generalisation of the Maxwell material for the case of large deformations using the upper-convected time derivative.
The concept is named after James Clerk Maxwell.
It is the simplest observer independent constitutive equation for viscoelasticity and further is able to reproduce first normal stresses.
[2] Even though both microscopic model lead to the upper evolution equation for the stress, recent work pointed up the differences when accounting also for the stress fluctuations.
) is proportional to the square of the shear rate, the second difference of normal stresses (
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.
The equation is only applicable, when the velocity profile in the shear flow is fully developed.
If the start-up form a zero velocity distribution has to be calculated, the full set of PDEs has to be solved.
The equation predicts the elongation viscosity approaching
(the same as for the Newtonian fluids) for the case of low elongation rate (
) with fast deformation thickening with the steady state viscosity approaching infinity at some elongational rate (
For the case of small deformation the nonlinearities introduced by the upper-convected derivative disappear and the model became an ordinary model of Maxwell material.