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. It is the simplest observer independent constitutive equation for viscoelasticity and further is able to reproduce first normal stresses. Thus, it constitutes one of the most fundamental models for rheology.
The model can be written as:
T+λ\stackrel{\nabla}{T
T
λ
\stackrel{\nabla}{T
\stackrel{\nabla}{T
v
η0
D
The model can be derived either by applying the concept of observer invariance to the Maxwell material or by two different mesoscopic models, namely Hookean Dumbells[1] or Temporary Networks.[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. [3]
For this case only two components of the shear stress became non-zero:
T12=η0
| \gamma |
T11=2η0λ{
| \gamma} |
2
| \gamma |
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 (
T11-T22
T22-T33
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.
For this case only two components of the shear stress became non-zero:
T12=η0
| \gamma | \left(1-\exp\left(- |
| t | |
| λ\right)\right) |
T11=2η0λ{
| \gamma} |
2\left(1-\exp\left(-
| t | |||
|
\right)\right)
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.
For this case UCM predicts the normal stresses
\sigma=T11-T22=T11-T33
| \sigma= |
| ||||
|
+
| |||||||||
|
| \epsilon |
The equation predicts the elongation viscosity approaching
3η0
| \epsilon |
\ll
| 1 | |
| λ |
| \epsilon |
infty=
| 1 | |
| 2λ |
| \epsilon |
-infty=-
| 1 | |
| λ |
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.