Multi-mode Upper Convected Maxwell Model

Summary

Multi-mode Upper Convected Maxwell model (see Chapter 1 of [4]):

\[\begin{split}\boldsymbol \sigma &= \sum_{i=1}^n G_i \boldsymbol A_i\\ \dfrac {\mathrm D \boldsymbol A_i} {\mathrm D t} &= \boldsymbol \kappa \cdot \boldsymbol A_i + \boldsymbol A_i\cdot \boldsymbol \kappa ^T - \dfrac 1 {\tau_i} (\boldsymbol A_i - \boldsymbol I)\end{split}\]

• Functions

  • Analytical solution in shear

    \[\eta^+(t) = \sum_{i=1}^n G_i \tau_i (1 - \exp(-t/\tau_i))\]

  • Analytical solution in uniaxial extension

    \[\eta^+_\mathrm E (t) = \dfrac 1 {\dot\varepsilon} \sum_{i=1}^n G_i (A_{xx, i}(t) - A_{yy, i}(t))\]

    with

    \[\begin{split}A_{xx, i}(t) &= \dfrac{ 1 - 2 \dot\varepsilon\tau_i \exp(-(1 - 2 \dot\varepsilon\tau_i) t / \tau_i) } {1 - 2 \dot\varepsilon\tau_i }\\ A_{yy, i}(t) &= \dfrac{ 1 + \dot\varepsilon\tau_i \exp(-(1 + \dot\varepsilon\tau_i)t/\tau_i) } { 1+ \dot\varepsilon\tau_i}\end{split}\]

  where for each mode \(i\):

  • \(G_i\): weight of mode \(i\)

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

• Parameters

  [none]