Upper Convected Maxwell model

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.

The model can be written as:

where:

For this case only two components of the shear stress became non-zero:

and

where ${\displaystyle {\dot {\gamma }}}$ 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 (${\displaystyle T_{11}-T_{22}}$) is proportional to the square of the shear rate, the second difference of normal stresses (${\displaystyle T_{22}-T_{33}}$) 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.

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