Bergstrom-Boyce Model

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The BB-model is a useful material model for predicting the non-linear viscoelastic response of elastomer-like materials. The BB-model is already a native built-in material model in Abaqus, ANSYS, LS-DYNA, MSC.Marc, and Adina, but the PolyUMod implementation of this model supports additional element types, temperature effects, and failure models.

Parameters for Bergstrom-Boyce Model

IndexSymbolNameUnitDescription
1\(\mu\)muStressShear modulus of network A
2\(\lambda_{lock}\)lambdaL-Locking stretch
3\(\kappa\)kappaStressBulk modulus
4\(s\)s-Relative stiffness of network B
5\(\xi\)xi-Strain adjustment factor
6\(C\)C-Strain exponential
7\(\tau_{base}\)tauBaseStressFlow resistance
8\(m\)m-Stress exponential
9\(\tau_{cut}\)tauCut-Normalized cut-off stress for flow

Model Theory

The details of the BB-model are described in the Mechanics of Solid Polymers book. This section only summarizes the key features of the model.


In the Bergstrom-Boyce (BB) model the applied deformation gradient is acting on two parallel macromolecular networks: \(\mathbf{F} = \mathbf{F}_A = \mathbf{F}_B\).
The deformation gradient acting on network B is further decomposed into elastic and viscoelastic components: \( \mathbf{F}_B = \mathbf{F}_B^e \mathbf{F}_B^v\).

The response of network A is given by the eight-chain model:
$$
\boldsymbol{\sigma}_A = \frac{\,\mu\,}{J \overline{\lambda^*}} \,
\frac{\mathcal{L}^{-1}\left(\overline{\lambda^*}/\lambda_L\right)}{\mathcal{L}^{-1}\left(1/\lambda_L\right)} \,
\text{dev}[\mathbf{b}^*] + \kappa (J-1) \mathbf{I}.
$$

The stress on network B is also given by the eight-chain model, but with a different effective shear modulus:
$$
\boldsymbol{\sigma}_B = \frac{s\,\mu\,}{J_B^e \overline{\lambda^{e*}_B}} \,
\frac{\mathcal{L}^{-1}\left(\overline{\lambda^{e*}_B}/\lambda_L\right)}{\mathcal{L}^{-1}\left(1/\lambda_L\right)} \,
\text{dev}[\mathbf{b}^{e*}_B] + \kappa (J_B^e-1) \mathbf{I},
$$
where s is a dimensionless material parameter specifying the shear modulus of network B relative to network A, and \(\overline{\lambda_B^{e*}}\) is the chain stretch in the elastic part of Network B.
Using this representation the total Cauchy stress is given by
$$
\boldsymbol{\sigma} = \boldsymbol{\sigma}_A + \boldsymbol{\sigma}_B.
$$
The velocity gradient on network B, \(\mathbf{L}_B = \dot{\mathbf{F}}_B \mathbf{F}_{B}^{-1}\), can be decomposed into elastic and viscous components:
$$
\begin{align}
\mathbf{L}_B &= \left[\frac{d}{dt}\left(\mathbf{F}_B^e \mathbf{F}_B^v \right)\right] \left(\mathbf{F}_B^e \mathbf{F}_B^v \right)^{-1} \notag \\
&= \left[ \dot{\mathbf{F}}_B^e \mathbf{F}_B^v + \mathbf{F}_B^e \dot{\mathbf{F}}_B^v \right]
\left(\mathbf{F}_B^v\right)^{-1} \left(\mathbf{F}_B^e\right)^{-1} \notag \\
&= \dot{\mathbf{F}}_B^e \left(\mathbf{F}_B^e\right)^{-1} +
\mathbf{F}_B^e \dot{\mathbf{F}}_B^v \left(\mathbf{F}_B^v\right)^{-1} \left(\mathbf{F}_B^e\right)^{-1} \notag \\
&= \mathbf{L}_B^e + \mathbf{F}_B^e \mathbf{L}_B^v (\mathbf{F}_B^e)^{-1} \notag \\
&= \mathbf{L}_B^e + \tilde{\mathbf{L}}_B^v,
\end{align}
$$
where
$$
\begin{align}
\mathbf{L}_B^v &= \dot{\mathbf{F}}_B^v \left(\mathbf{F}_B^v\right)^{-1} = \mathbf{D}_B^v + \mathbf{W}_B^v, \\
\tilde{\mathbf{L}}_B^v &= \tilde{\mathbf{D}}_B^v + \tilde{\mathbf{W}}_B^v.
\end{align}
$$
The rate of viscous deformation of network B is constitutively prescribed by:
$$
\tilde{\mathbf{D}}_B^v = \dot{\gamma}_B(\boldsymbol{\sigma}_B,\mathbf{b}_B^{e*}) \, \mathbf{N}_B^v,
$$
where
$$
\mathbf{N}_B^v = \frac{\text{dev}[\boldsymbol{\sigma}_B]}{\tau} = \frac{\text{dev}[\boldsymbol{\sigma}_B]}{|| \text{dev}[\boldsymbol{\sigma}]_B ||_F}.
$$
and \(\tau\) is the effective stress driving the viscous flow.
The time derivative of \(\mathbf{F}_B^v\) can be derived as follows:
$$
\begin{align}
\tilde{\mathbf{L}}_B^v &= \dot{\gamma}_B^v \mathbf{N}_B^v, \\
\Rightarrow\qquad \mathbf{F}_B^e \dot{\mathbf{F}}_B^v \left(\mathbf{F}_B^v\right)^{-1} \left(\mathbf{F}_B^e\right)^{-1} &=
\dot{\gamma}_B^v \mathbf{N}_B^v, \notag\\
\Rightarrow\qquad \dot{\mathbf{F}}_B^v &= \dot{\gamma}_B^v \left(\mathbf{F}_B^e\right)^{-1}
\frac{\text{dev}[\boldsymbol{\sigma}_B]}{|| \text{dev}[\boldsymbol{\sigma}]_B ||_F}
\mathbf{F}_B^e \mathbf{F}_B^v.
\end{align}
$$

The rate-equation for viscous flow is given by:
$$
\dot{\gamma}_B^v = \dot{\gamma}_0 \left(\overline{\lambda_B^v} – 1 + \xi \right)^C \,
\left[ R\left( \frac{\tau}{\tau_{\mathit{base}}} – \hat{\tau}_{\mathit{cut}} \right) \right]^m,
$$
where \(\dot{\gamma}_0 \equiv 1\)/s is a constant introduced to ensure dimensional consistency,
\(R(x) = (x + |x|) / 2\) is the ramp function, \(\hat{\tau}_{\mathit{cut}}\) is a cut-off stress below which no flow will occur, and
$$
\overline{\lambda_B^v} = \sqrt{\frac{\text{tr}[\mathbf{b}_B^v]}{3}} .
$$
is the viscoelastic chain stretch.
The effective stress driving the viscous flow is:
$$
\tau = || \text{dev}[\boldsymbol{\sigma}_B] ||_F =
\sqrt{ \text{tr} \left[ \boldsymbol{\sigma}_B’ \boldsymbol{\sigma}_B’ \right] }.
$$