Multi-mode Giesekus model

Summary

Multi-mode Giesekus Model (see Chapter 6 of [2]):

\[\begin{split}\boldsymbol \sigma &= \sum_{i=1}^n G_i \boldsymbol {A_i},\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac {1} {\tau_i} (\boldsymbol A_i - \boldsymbol I) - \dfrac {\alpha_i} {\tau_i} (\boldsymbol A_i - \boldsymbol I)^2,\end{split}\]
where for each mode \(i\):
  • \(G_i\): weight of mode \(i\)

  • \(\tau_i\): relaxation time of mode \(i\)

  • \(\alpha_i\): constant of proportionality mode \(i\)

  • Parameters
    • alpha_i \(\equiv \alpha_i\)