# Arruda-Boyce (/MAT/LAW92)

LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions.

It assumes that the chain molecules are located on the average along the diagonals of the cubic in principal stretch space.

## Material Parameters

The strain energy density function is:(1)
$W=\underset{W\left({\overline{I}}_{1}\right)}{\underbrace{\mu \sum _{i=1}^{5}\frac{{c}_{i}}{{\left({\lambda }_{m}\right)}^{2i-2}}\left({\overline{I}}_{1}^{i}-{3}^{i}\right)}}+\underset{U\left(J\right)}{\underbrace{\frac{1}{D}\left(\frac{{J}^{2}-1}{2}+\mathrm{ln}\left(J\right)\right)}}$

The material constant, ${c}_{i}$ are:

${c}_{1}=\frac{1}{2},\text{\hspace{0.17em}}{c}_{2}=\frac{1}{20},\text{\hspace{0.17em}}{c}_{3}=\frac{11}{1050},\text{\hspace{0.17em}}{c}_{4}=\frac{19}{7000}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{c}_{5}=\frac{519}{673750}$
${\overline{I}}_{1}={\overline{\lambda }}_{1}^{2}+{\overline{\lambda }}_{2}^{2}+{\overline{\lambda }}_{3}^{2}$
First strain invariant
${\lambda }_{i}$
${i}_{th}$ principal engineering stretch
A material with LAW92 can be defined in two different ways:
• Parameter Input

Shear modulus, bulk modulus and strain stretch ( $\mu ,D,{\lambda }_{m}$ )

Where, only the above 3 parameters with clear physical meaning are necessary to define the material.

$\mu$ is shear modulus at zero strain.(2)
$D=\frac{2}{K}$
Where,
$K$
Bulk coefficient at zero strain
${\lambda }_{m}$
Defines the limit of stretch
Also called locking stretch. It specifies the beginning of the hardening phase in tension (locking strain in tension). Default = 7.0.
In parametric input, Poisson’s ratio is computed as:(3)
$\nu =\frac{3K-2\mu }{6K+2\mu }$
• When using function input, Poisson ratio $\nu$ and Itype must be defined. Itype defines which type of engineering stress strain test data that is being used as input.

## Poisson's Ratio and Material Incompressibility

If function input is defined, then parameters $\mu ,D,{\lambda }_{m}$ are ignored and Radioss will calculate the material constant by fitting the input function. A nonlinear least squares algorithm is used to fit the Arruda-Boyce parameters by Radioss. The curve fitting is performed using the assumption that Poisson’s value is close to 0.5, which means the material is incompressible. Similar to the other hyperelastic material models, Poisson ratio values closer to 0.5 result in high bulk modulus and a lower timestep. For a good balance between incompressibility and a reasonable timestep, a Poisson’s ratio value of 0.495 is recommended.

The material fitting information can be found in the Starter output file (*0000.out).
The fitting error and fitted material parameters are printed in the Starter output file.

## Viscous (Rate) Effects

/VISC/PRONY must be used with LAW92 to include viscous effects.

## References

1 Arruda, E. M. and Boyce, M. C., 1993, “A three-dimensional model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, 41(2), pp. 389–412