Command manual

Command list

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, $\rho$, $K$
$\mu$, $\lambda_L$, $a_0$, $a_1$, $\eta_\mathrm{max}$, $\dot{\gamma}_0$, $\xi$, $B$
$\sigma_0$, $Q$, $C$, $m$, $c_{dec}$, $\beta$, $W_c$
$b_0$, $b_1$, $b_2$, $\mu_B$

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 in Network A
$a_1$ Exponent driving damage in Network A
$\eta_\mathrm{max}$ Maximum damage in network A
$\dot{\gamma}_0$ Parameter in viscous flow law in Network B
$\xi$ Small constant in viscous flow rate law in Network B
$B$ Exponent in viscous flow law in Network B
$\sigma_0$ Viscous flow stress parameter in Network B
$Q$ Viscous hardening coefficient in Network B
$C$ Viscous hardening exponent in Network B
$m$ Viscous flow rate exponent in Network B
$c_{dec}$ Viscous decay parameter in Network B
$\beta$ Pressure dependency parameter for viscous flow in Network B
$W_c$ Failure parameter in Network B
$b_0$ Viscous flow stress parameter in Network B
$b_1$ Viscous flow stress exponent in Network B
$b_2$ Viscous flow stress exponent in Network B
$\mu_B$ Optional initial shear stiffness of Network B
default: not used

Description

Model for polymers. It is based the Bergstrom-Boyce model, but has an optional different treatment of viscous effects.

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, $J = \mathrm{det}\mathbf{F}$ and $\bar{\mathbf{C}_{dev}}$ is the deviatoric part of $\bar{\mathbf{C}}$:

$\displaystyle{\bar{\mathbf{C}} = J^{-2/3} \mathbf{F}^t \mathbf{F}}$

$\bar\lambda$ 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 visco-elastic (if $\mu_B>0$) or purely viscous (if $\mu_B=0$). The stress in the elastic network is defined as:

$\displaystyle{\mathbf{\sigma}_B = \frac{\mu_B}{J^e \bar\lambda^e} \cdot \frac{\mathcal{L}^{-1}\left( \bar\lambda^e / \lambda_L \right)} {\mathcal{L}^{-1}\left( 1 / \lambda_L \right)} \cdot \bar{\mathbf{C}}^e_{dev}}$

where $J^e = \mathrm{det}\mathbf{F}^e$ and $\bar{\mathbf{C}^e_{dev}}$ is the deviatoric part of $\bar{\mathbf{C}^e}$:

$\displaystyle{\bar{\mathbf{C}}^e = (J^e)^{-2/3} (\mathbf{F}^e)^t \mathbf{F}^e}$

$\bar\lambda^e$ is a measure of the elastic network stretch:

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

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:

$\displaystyle{ \dot{\bar{\epsilon}}_{dev} = \dot{\gamma}_0 ( \bar{\lambda}^v - 1 + \xi)^B \cdot \left( \frac{\sigma_B^{eff}}{\sigma_0 + Q\cdot(1 - \mathrm{exp}(-C \epsilon^{eff})} \right)^m \cdot \frac{\mathbf{\sigma}_B}{\sqrt{\mathbf{\sigma}_B : \mathbf{\sigma}_B}} }$
$\displaystyle{\bar\lambda^v = \sqrt{\frac{\mathrm{tr} \bar{\mathbf{C}^v}}{3}}}$

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

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

Alternative 2:

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