## Introduction

The Bergstrom-Boyce (BB) model is an advanced material model that I developed for predicting the time-dependent, large-strain behavior of elastomer-like materials. The model has been shown to be accurate for both traditional engineering rubbers, and soft biomaterials. The version of the BB-model that is discussed here is the PolyUMod version of the BB-model. This model is part of the PolyUMod library, and has Material Model (MM) id=5.

The model uses the following material parameters:

Parameter Name | Description |
---|---|

mu | Shear modulus of network A |

lambdaL | Locking stretch |

kappa | Bulk modulus |

s | Relative stiffness of network B |

xi | Strain adjustment factor |

C | Strain exponential |

tauBase | Flow resistance |

m | Stress exponential |

tauCut | Normalized cut-off stress for flow |

**General notes about the material parameters:**

- There is no need to search for
`lambdaL`

unless the experimental data contains large strains. - The parameter
`m`

should not be larger than 20 in order to get good convergence in a FE simulation. - Having a very small
`tauBase`

compared to stiffness can cause convergence problems.

Here is an exemplar set of stress-strain predictions from the BB-model created by MCalibration.

## BB-model in MCalibration

The material model can be solved using the internal MCalibration solver, or using any of the supported FE solvers. It is recommended to use the MCalibration native solver when possible since it runs significantly faster. If `kappa=0`

, the applied load is uniaxial, and MCalibration is set as the solver, then a fast incompressible solver will be used to calculate the stress-strain response.

## Model Theory

In the Bergstrom-Boyce (BB) model the applied deformation gradient is acting on two parallel macromolecular networks: \(\mathbf{F} = \mathbf{F}_A = \mathbf{F}_B\), see the following rheological representation.

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)} \,

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)} \,

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:

\[

\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} \\

= \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} \\

= \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} \\

= \mathbf{L}_B^e + \mathbf{F}_B^e \mathbf{L}_B^v (\mathbf{F}_B^e)^{-1} \\

= \mathbf{L}_B^e + \tilde{\mathbf{L}}_B^v,

\]

where

\[

\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.

\]

To make the unloading unique, prescribe \(\tilde{\mathbf{W}}_B^v \equiv 0\). 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{dev[\boldsymbol{\sigma}_B]}{\tau} = \frac{dev[\boldsymbol{\sigma}_B]}{|| 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:

\[

\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,\\

\Rightarrow\qquad \dot{\mathbf{F}}_B^v = \dot{\gamma}_B^v \left(\mathbf{F}_B^e\right)^{-1}

\frac{dev[\boldsymbol{\sigma}_B]}{|| dev[\boldsymbol{\sigma}]_B ||_F}

\mathbf{F}_B^e \mathbf{F}_B^v.

\]

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,

\]

\(R(x) = (x + |x|) / 2\) is the ramp function, \(\hat{\tau}_{\mathit{cut}}\) is a cut-off stress below which no flow will occur, and where \(\dot{\gamma}_0 \equiv 1\)/s is a constant introduced to ensure dimensional consistency,

\[

\overline{\lambda_B^v} = \sqrt{\frac{tr[\mathbf{b}_B^v]}{3}}.

\]

is the viscoelastic chain stretch. The effective stress driving the viscous flow is:

\[

\tau = || dev[\boldsymbol{\sigma}_B] ||_F =

\sqrt{ tr \left[ \boldsymbol{\sigma}_B’ \boldsymbol{\sigma}_B’ \right] }.

\]

## References

- My Ph.D. Thesis
- Mechanics of Materials, Vol. 33, pp. 523–530, 2001
- J. Mech. Phys. Solids., Vol. 46, pp. 931–954, 1998
- 100 Questions and answers about the BB-model
- Chapter 8 in my Mechanics of Polymer Book

## My Ph.D. Thesis Abstract

The mechanical behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Although these features of the rubbery constitutive response are well-recognized and important to its function, few models attempt to quantify these aspects of response. Based on a detailed experimental investigation a new constitutive model for the time-dependence of unfilled elastomers has been developed. The foundation of the model is that the mechanical behavior can be decomposed into two parts: an equilibrium network corresponding to the state that is approached in long time stress relaxation tests; and a second network capturing the non-linear rate-dependent deviation from the equilibrium state. The time-dependence of the second network is further assumed to be governed by the reptational motion of molecules having the ability to significantly change conformation and thereby relaxing the overall stress state. To model the behavior of particle filled elastomers the newly developed constitutive framework is then extended to include filler interactions by amplification of the first strain invariant. In an effort to examine some of the assumptions that are common in the constitutive modeling of particle filled elastomers, a detailed series of micromechanical models were constructed using two- and three-dimensional finite element simulations. The results indicate that the effect of filler particles can be accurately predicted using stochastic three-dimensional simulations suggesting that successful modeling mainly requires a rigorous treatment of the composite nature of the microstructure and not molecular level concepts such as alteration of mobility or effective crosslinking density in the elastomeric phase of the material. A direct comparison between the new model and experimental data for a number of different elastomers suggest that the new framework successfully captures the observed behavior.

Jorgen Bergstrom, Ph.D. Thesis Abstract