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. 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:

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

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 and the gradient of a vector follows the convention ${\displaystyle (\nabla {\mathbf {v} })_{ij}=\partial _{i}v_{j}}$.
• ${\displaystyle \eta _{0}}$ is material viscosity at steady simple shear;
• ${\displaystyle \mathbf {D} }$ is the deformation rate tensor.

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]