# Yeoh (/MAT/LAW94)

LAW94 is a hyperelastic material model that can be used to describe incompressible materials.

The strain energy density function of LAW94 only depends on the first strain invariant and is computed as:

$W=\sum _{i=1}^{3}\left[\underset{W\left({\overline{I}}_{1}\right)}{\underbrace{{C}_{i0}{\left({\overline{I}}_{1}-3\right)}^{i}}}+\underset{U\left(J\right)}{\underbrace{\frac{1}{{D}_{i}}{\left(J-1\right)}^{2i}}}\right]$

Where,
${\overline{I}}_{1}={\overline{\lambda }}_{1}^{2}+{\overline{\lambda }}_{2}^{2}+{\overline{\lambda }}_{3}^{2}$
First strain invariant
${\overline{\lambda }}_{i}={J}^{-\frac{1}{3}}{\lambda }_{i}$
Deviatoric stretch

The Cauchy stress is:

${\sigma }_{i}=\frac{{\lambda }_{i}}{J}\frac{\partial W}{\partial {\lambda }_{i}}$

## Material Parameters

For incompressible materials with $i$ =1 only and ${D}_{1}$ are input and the Yeoh model is reduced to a Neo-Hookean model.
${C}_{10},{C}_{20},{C}_{30}$
Material constants specify the deviatoric part (shape change) of the material
${D}_{1}$ , ${D}_{2}$ , ${D}_{3}$
Parameters specify the volumetric change of the material

These six material constants need to be calculated by curve fitting material test data. RD-E: 5600 Hyperelastic Material with Curve Input includes a Yeoh fitting Compose script for uniaxial test data. The Yeoh material model has been shown to model all deformation models, even if the curve fit was obtained using only uniaxial test data.

The initial shear modulus and the bulk modulus are:

$\mu =2\cdot {C}_{10}$

and

$K=\frac{2}{{D}_{1}}$

## Poisson's Ratio and Material Incompressibility

LAW94 is available only as an incompressible material model.

If ${D}_{1}$ = 0, an incompressible material is considered where, $\nu =0.495$ and ${D}_{1}$ is calculated as:

${D}_{1}=\frac{3\left(1-2v\right)}{\mu \left(1+v\right)}$

## References

1 Yeoh, O. H. "Some forms of the strain energy function for rubber." Rubber Chemistry and technology 66, no. 5 (1993): 754-771