# Bergstrom-Boyce Model

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 the 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 by J. Bergstrom. This section only summarizes the key features of the model theory.

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:
$$\tau = || \text{dev}[\boldsymbol{\sigma}_B] ||_F = \sqrt{ \text{tr} \left[ \boldsymbol{\sigma}_B’ \boldsymbol{\sigma}_B’ \right] }.$$

# Exemplar Model Prediction

The following figure shows exemplar stress-strain predictions from the BB-model in uniaxial tension at different strain rates.