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]