Atherosclerotic plaque G&R

The G&R algorithm is based on the spatial velocity gradient $\mathbf{l}_g$ defined as to satisfy the homogenous stress hypothesis:

$\mathbf l_{g}={\sigma_{1}-\bar{\sigma}_{1}\over \bar{\sigma}_1}\left(\hat{\sigma}_{1}\mathbf n_1\otimes\mathbf n_1+\hat{\sigma}_{2}\mathbf n_{2}\otimes \mathbf n_{2}+\hat{\sigma}_{3}\mathbf n_{3}\otimes \mathbf n_{3}\right)$

Note

One day this algorithm will include several bio-chemo-mechanical interactions.

Exponential integration

See also [8] for the following algorithm details.

The spatial velocity gradient is defined as:

$\mathbf l_{g}=\dot{\mathbf F}\mathbf F^{-1}$

And the deformation gradient can be expressed as:

$\dot{\mathbf F}=\mathbf l_{g}\mathbf F_g$

And computed in terms of exponetial integration:

$\mathbf F_g=e^{\int_{0}^{t}\mathbf l_{g}\:d\tau}$

However, we need $\mathbf L_{g}$ , expressed in the reference configuration (we solve the formulation in the reference config). it can be expressed as:

$\mathbf L_{g}=\mathbf F_{e}\mathbf l_{g}\mathbf F_{e}^{-1}$

And the exponential integration can be expressed as:

$\mathbf F_{g}=\exp \left( \int_0^t \mathbf L_{g} \: d\tau \right)$

And exprssed in a discrete form as:

$\mathbf F_{g(n+1)}=\mathbf{F}_{e(n+1)}^{-1}\exp\left({\mathbf{l}_{g(n+1)}\Delta t} \right)\mathbf{F}_{e(n+1)} \mathbf{F}_{g(n)}$