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:

${\displaystyle \mathbf {T} +\lambda {\stackrel {\nabla }{\mathbf {T} }}=2\eta _{0}\mathbf {D} }$

where:

• ${\displaystyle \mathbf {T} }$ is the stress tensor;
• ${\displaystyle \lambda }$ is the relaxation time;
• ${\displaystyle {\stackrel {\nabla }{\mathbf {T} }}}$ is the upper-convected time derivative of stress tensor:

${\displaystyle {\stackrel {\nabla }{\mathbf {T} }}={\frac {\partial }{\partial t}}\mathbf {T} +\mathbf {v} \cdot \nabla \mathbf {T} -(\nabla \mathbf {v} )^{T}\cdot \mathbf {T} -\mathbf {T} \cdot (\nabla \mathbf {v} )}$

• ${\displaystyle \mathbf {v} }$ is the fluid velocity
• ${\displaystyle \eta _{0}}$ is material viscosity at steady simple shear;
• ${\displaystyle \mathbf {D} }$ is the deformation rate tensor.