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### MAT_BERGSTROM_BOYCE

###### Material properties
Attention: This command is in the beta stage and the format may change over time.
*MAT_BERGSTROM_BOYCE
"Optional title"
mid, $\rho$, $K$
$\mu$, $\lambda_L$, $a_0$, $a_1$, $\eta_\mathrm{max}$, $\dot{\gamma}_0$, $\xi$, $B$
$\sigma_0$, $Q$, $C$, $m$, $c_{dec}$, $\beta$, $W_c$
$b_0$, $b_1$, $b_2$

#### Parameter definition

VariableDescription
mid Unique material identification number
$\rho$ Density
$K$ Bulk modulus
$\mu$ Initial shear stiffness of (network A)
$\lambda_L$ Locking stretch
$a_0$ Parameter driving damage (network A)
$a_1$ Exponent driving damage (network A)
$\eta_\mathrm{max}$ Maximum damage (network A)
$\dot{\gamma}_0$ Parameter in viscous flow law (network B)
$\xi$ Small constant in viscous flow rate law (network B)
$B$ Exponent in viscous flow law (network B)
$\sigma_0$ Viscous flow stress parameter (network B)
$Q$ Viscous hardening coefficient (network B)
$C$ Viscous hardening exponent (network B)
$m$ Viscous flow rate exponent (network B)
$c_{dec}$ Viscous decay parameter (network B)
$\beta$ Pressure dependency parameter for viscous flow (network B)
$W_c$ Failure parameter (network B)
$b_0$ Viscous flow stress parameter (network B)
$b_1$ Viscous flow stress exponent (network B)
$b_2$ Viscous flow stress exponent (network B)

#### Description

Model for polymers. It is based the Bergstrom-Boyce model, but has a significantly different treatment of viscous effects (network B).

The rheological model consists of two parallell networks, here referred to as network A and network B.

The total stress in the material $\mathbf \sigma$ is the sum of stresses in the two networks.

$\mathbf{\sigma} = \mathbf{\sigma}_A + \mathbf{\sigma}_B$

Network A is hyperelastic with damage and its stress is defined as

$\displaystyle{\mathbf{\sigma}_A = (1 - \eta) \cdot \frac{\mu}{J \bar\lambda} \cdot \frac{\mathcal{L}^{-1}\left( \bar\lambda / \lambda_L \right)} {\mathcal{L}^{-1}\left( 1 / \lambda_L \right)} \cdot \bar{\mathbf{C}}_{dev} + K \cdot \mathrm{ln}(J) \mathbf{I}}$

where $\mu$, $\lambda_L$ and $K$ (bulk modulus) are material parameters. $\mathcal{L}^{-1}$ is the inverse of the Langevin function, $J = \mathrm{det}\mathbf{F}$, $\bar{\mathbf{C}} = J^{-2/3} \mathbf{F}^t \mathbf{F}$ and $\bar\lambda$ is a measure of the network stretch.

$\displaystyle{\bar\lambda = \sqrt{\frac{\mathrm{tr} \bar{\mathbf{C}}}{3}}}$

$\eta$ is the network damage and it evolves according to

$\displaystyle{\dot\eta = a_0 \cdot (\eta_{max} - \eta) \cdot \left( \frac{\bar\lambda}{\lambda_L} \right)^{a_1} \cdot \dot{\bar\lambda}}$

Network B is purely viscous. Two different relationships between viscous stress and deviatoric strain rate are offered.

Alternative 1:

The effective viscous stress $\sigma_B^{eff}$ is defined from the relationship:

$\displaystyle{ \dot{\bar{\epsilon}}_{dev} = \dot{\gamma}_0 ( \bar{\lambda}^v - 1 + \xi)^B \cdot \left( \frac{\sigma_B^{eff}}{\sigma_0 + Q\cdot(1 - \mathrm{exp}(-C \epsilon^{eff})} \right)^m \cdot \frac{\mathbf{\sigma}_B}{\sqrt{\mathbf{\sigma}_B : \mathbf{\sigma}_B}} }$

where $\dot{\bar{\epsilon}}_{dev}$ is a time averaged deviatoric strain rate:

$\displaystyle{ \dot{\bar{\epsilon}}_{dev} = \frac{1}{c_{dec}} \int_0^t \dot{\epsilon}_{dev} \mathrm{e}^{(\tau-t)/c_{dec}} \mathrm{d}\tau}$

Alternative 2:

$\displaystyle{ \sigma_B^{eff} = b_0 (\bar{\lambda} - 1)^{b_1} \cdot \left( \frac{ \dot{\bar{\epsilon}}_{dev}^{eff}}{\dot{\gamma}_0} \right)^{b_2} }$