# Multi-mode Upper Convected Maxwell Model¶

## Summary¶

Multi-mode Upper Convected Maxwell model (see Chapter 1 of [2]):

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