RD-V: 0210 Yeoh Hyperelastic Material

A one element model in tension and compression used as a test model and compare with analytical results.

The Yeoh model can be used to describe nearly impressible hyperelastic materials, like rubber. Set Yeoh parameter with /MAT/LAW94 and compare one element tension and compression Radioss results with analytical results.

Options and Keywords Used

/MAT/LAW94 (YEOH)

Input Files

Before you begin, copy the file(s) used in this problem to your working directory.

RD-V-0210_Yeoh_Hyperelastic_Material.zip

Model Description

Uniaxial tensile one brick element and compression one brick element with imposed displacement and fixed in other side only in X direction.

Units: mm, s, Mg, N, MPa

Properties: /PROP/TYPE14 (SOLID) with, I_solid=24, I_smstr=10, and I_cpre=1

Note: When using hyperelastic material laws, there are some recommended element property settings. When using solid elements, it is always better to mesh with 8 node /BRICK elements, if possible. If not, then /TETRA4 or /TETRA10 elements can be used. Recommended /PROP/SOLID for 8 nodes brick are, I_smstr=10, I_cpre=1, with I_solid=24. If hourglassing occurs, then I_solid=18 can be used.

With the input I_smstr=1, I_cpre=-1, Radioss will automatically determine the bast value according to the material and property used.

Simulation Iterations

Yeoh theory:
The Yeoh energy density is:
$W = \sum_{i = 1}^{3} [C_{i 0} {({\bar{I}}_{1} - 3)}^{i} + \frac{1}{D_{i}} {(J - 1)}^{2 i}]$
This example assumes an incompressible ( $J = 1$ ) hyperelastic material, then the strain energy density of Yeoh simplified, as:
$W = C_{10} {({\bar{I}}_{1} - 3)}^{1} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$
With
${\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2}$
Where,

${\bar{λ}}_{i}$

Deviatoric stretch with ${\bar{λ}}_{i} = J^{- \frac{1}{3}} λ_{i}$ and $λ_{i} = ε_{i} + 1$ .

$λ_{1}, λ_{2}, λ_{3}$

The stretch in principal direction 1, 2, 3.

$ε_{i}$

Engineer strain in principal direction I.

It shows only three parameters $C_{10}, C_{20}, C_{30}$ need to be defined for the (incompressible) Yeoh model.
Uniaxial test is used in this example, then:
$λ_{1} = λ$ and $λ_{2} = λ_{3} = λ^{- \frac{1}{2}}$
The Cauchy stress of Yeoh model is computed as:
$σ_{i} = \frac{λ_{1}}{J} \frac{\partial W}{\partial λ_{1}}$
In the uniaxial test, the Cauchy stress (true stress) in principal direction 1 is:
$σ_{1} = λ \frac{\partial W}{\partial {\bar{I}}_{1}} \frac{d {\bar{I}}_{1}}{d λ}$
The engineer stress is:
$σ_{1_e n g} = \frac{σ_{1}}{(ε_{1_e n g} + 1)}$
In this example, three material parameters are defined below:
$\begin{array}{l} C_{10} =0.18 \\ C_{20} {=-2e}^{-3} \\ C_{30} = 5 e^{- 5} \end{array}$
Figure 3. Engineer stress-strain curve for Poisson ratio $ν = 0.5$
/MAT/LAW94:
The hyperelastic model uses polynomial model and the Yeoh model is one of the polynomial models with only three parameters $C_{10}, C_{20}, C_{30}$ . In this example, viscous was not considered.

Results

In LAW94 Yeoh model, three parameters $C_{10}, C_{20}, C_{30}$ defined same as analytical case.

$\begin{array}{l} C_{10} =0.18 \\ C_{20} {=-2e}^{-3} \\ C_{30} = 5 e^{- 5} \end{array}$

D1 needs to be defined in the material card. If D1 is not defined in LAW94, then the default Poisson’s ratio 0.495 is used.

Then, 1/D1=17.94 is printed in the Starter output file.

This can be checked as follows:

G = 2 C_{10} =2*0 .18=0 .36

D 1 = \frac{3 (1 - 2 ν)}{G (1 + ν)} = \frac{3 (1 - 2 \times 0 .495)}{0 .36 (1 + 0 .495)} = 0.055741

\frac{1}{D 1} = 17.94

The analytical results are compared with the Radioss results below:

Table 1.
Poisson's Ratio	D1
$ν = 0.495$ (default value)	17.94011589315
$ν = 0.49999$ (to be as close to analytical value with $ν = 0.5$	8999.928000576

Both models return results very close to the analytical results.

The model with $ν =0 .495$ shows higher differences for the large tensile deformation (> 200%).

The time step and computation time is larger for the model with $ν = 0.49999$ due to the larger bulk modulus. The results of this model are closer to the analytical results.