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 model was proposed by James G. Oldroyd. The concept is named after James Clerk Maxwell.
Contents
- Case of the steady shear
- Case of start up of steady shear
- Case of the steady state uniaxial extension or uniaxial compression
- Case of small deformation
- References
The model can be written as:
where:
Case of the steady shear
For this case only two components of the shear stress became non-zero:
and
where
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 (
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.
Case of start-up of steady shear
For this case only two components of the shear stress became non-zero:
and
The equations above describe stresses gradually risen from zero the steady-state values. The equation is only applicable, when the velocity profile in the shear flow is fully developed. Then the shear rate is constant over the channel height. If the start-up form a zero velocity distribution has to be calculated, the full set of PDEs has to be solved.
Case of the steady state uniaxial extension or uniaxial compression
For this case UCM predicts the normal stresses
where
The equation predicts the elongation viscosity approaching
Case of small deformation
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.