# Bergstrom-Boyce (/MAT/LAW95)

This law is a constitutive model for predicting the nonlinear time dependency of elastomer like materials. It uses a polynomial material model for the hyperelastic material response and the Bergstrom-Boyce material model to represent the nonlinear viscoelastic time dependent material response.

This law is only compatible with solid elements.

## Material Parameters

The same polynomial strain energy density formulation is used for the hyperelastic components in both networks. In Network B, it is scaled by a factor ${S}_{b}$ . The strain energy density is then written for the hyperelastic component of the network.

and

- ${\overline{I}}_{1}={\overline{\lambda}}_{1}^{2}+{\overline{\lambda}}_{2}^{2}+{\overline{\lambda}}_{3}^{2}$
- ${\overline{I}}_{2}={\overline{\lambda}}_{1}^{-2}+{\overline{\lambda}}_{2}^{-2}+{\overline{\lambda}}_{3}^{-2}$
- ${\overline{\lambda}}_{i}={J}^{-\frac{1}{3}}{\lambda}_{i}$
- ${C}_{ij}$ and ${D}_{i}$
- Material parameters

The hyperelastic component the Cauchy stress is computed as:

$\sigma ={\overline{\sigma}}_{A}+{\overline{\sigma}}_{B}$

Since ${W}_{B}={S}_{b}\cdot {W}_{A}$ , then ${\overline{\sigma}}_{B}={S}_{b}\cdot {\overline{\sigma}}_{A}$ and total stress is $\sigma =(1+{S}_{b})\cdot {\overline{\sigma}}_{A}$ .

- Yeoh:
$j=0$

Where, ${C}_{10},{C}_{20},{C}_{30}$ are not zero. - Mooney-Rivlin:
$i+j=1$

Where, ${C}_{10}$ and ${C}_{01}$ are not zero and ${D}_{2}={D}_{3}=0$ .

- Neo-Hookean:
Only ${C}_{10}$ and ${D}_{1}$ are not zero.

- ${C}_{ij}$ and ${D}_{i}$
- Material parameters which can be calculated by completing a curve fit for quasi-static material test data.

RD-E: 5600 Hyperelastic Material with Curve Input, contains a curve fit example for Mooney-Rivlin and Yeoh material models. ${D}_{1}$ can be calculated from the bulk modulus or left blank.

The initial shear modulus and the bulk modulus are computed as:

and

If the bulk modulus of the material is known, ${D}_{1}$ can be calculated, or if ${D}_{1}$ =0, an incompressible material is assumed.

## Viscous (Rate) Effects

The effective creep strain rate in Network B is given by:

- $\tilde{\lambda}=\sqrt{\frac{{\overline{I}}_{1}}{3}}$
- ${\overline{\sigma}}_{B}$
- Effective stress in Network B.
- $A,\text{\hspace{0.17em}}\xi ,\text{\hspace{0.17em}}M,\text{\hspace{0.17em}}C$ , and ${\tau}_{ref}$
- Input material parameters.

The material constants
$A$
,
$M$
and
$C$
are limited to a specific range of real values as
defined in the Reference Guide. If limited data is available, a trial
and error method ^{1} could be used to determine these
constants. Start with the default values of
$\text{\hspace{0.17em}}\xi ,\text{\hspace{0.17em}}M,\text{\hspace{0.17em}}C$
,
${S}_{b}$
=1.6; and
$A$
=5. Next, compare model predictions with experimental
data for at least one strain rate and adjust
$A$
to get a fit for the strain rate data.

## References

^{1}Bergström, J. S., and M. C. Boyce. "Constitutive modeling of the large strain time-dependent behavior of elastomers." Journal of the Mechanics and Physics of Solids 46, no. 5 (1998): 931-954