RD-E: 5600 Hyperelastic Material with Curve Input

The target of this example is to demonstrate how to use material test data for rubber hyperplastic materials.

Radioss supports both material test data input and parameter input for hyperelastic material laws. Depending on the deformation state and material test results available (uniaxial tension, biaxial tension or plane tension), different material models can be used in Radioss.

Options and Keywords Used

Ogden material law /MAT/LAW42 (OGDEN)
/MAT/LAW62 (VISC_HYP)
/MAT/LAW69
/MAT/LAW82
/MAT/LAW88
Arruda-Boyce material law /MAT/LAW92
Yeoh material law /MAT/LAW94 (YEOH)

Input Files

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

RD-E-5600_Hyperelastic_material.zip

Model Description

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

Units: mm, s, Mg, N, MPa

Table 1 shows curve input or parameter input available for hyperelastic material model.

Properties: /PROP/SOLID with, I_solid=24, I_smstr=10, and I_cpre=1.

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.

Recommend /PROP/SOLID for 8 nodes brick:

I_smstr=10
I_cpre=1 with I_solid=24.

Note: If hourglassing occurs, then I_solid=17 with I_frame=2 can be used.

Simulation Iterations

This example demonstrates how to use engineering stress versus strain test as curve input and parameter input different material laws. The hyperelastic data gathered by Figure 3 for 8 % sulfur rubber test data is used. Figure 3 shows the Treloar test data for the 3 strain states most important in characterizing a material, uniaxial tension, equal biaxial extension and pure shear (planar tension). ²

Table 1 summarizes which material laws have test data input and/or parameter input in Radioss for hyperelastic material model.

Table 1.
	Material Law	Test Data Input	Parameter Input
Ogden	LAW42		√
	LAW62		√
	LAW69	√
	LAW82		√
	LAW88	√
Arruda-Boyce	LAW92	√	√
Yeoh	LAW94		√

For the Ogden based laws, LAW69 will be used to curve fit the test data and extract material parameters that can be used in LAW42, LAW62, and LAW82. Next the test data will be used in LAW88 and LAW92. Finally, a fitting script will be used to extract the LAW94 material parameters.

Results

Ogden Model

Test Data Input with LAW69

Material LAW69 can define a hyperelastic and incompressible material which uses the Ogden, or Mooney-Rivlin material models. It is generally used to model incompressible rubbers, polymers, foams, and elastomers. Material parameters for the Ogden or Mooney-Rivlin material models are computed from the engineering stress-strain curve from uniaxial tension (positive strain) and compression (negative strain) tests. The uniaxial data from Treloar is tension only. However, for incompressible materials, uniaxial compressive data can be calculated from the equal biaxial tension test data using these formulas from ³.(1)

ε_{c} = \frac{1}{{(ε_{b} + 1)}^{2}} - 1

$ε_{c} = \frac{1}{{(ε_{b} + 1)}^{2}} - 1$

(2)

σ_{c} = σ_{b} {(1 + ε_{b})}^{3}

$σ_{c} = σ_{b} {(1 + ε_{b})}^{3}$

Where,

$ε_{c}$ $ε_{c}$: Uniaxial engineering compressive stain
$ε_{b}$ $ε_{b}$: Equal biaxial engineering tension strain
$σ_{c}$ $σ_{c}$: Uniaxial engineering compressive stress
$σ_{b}$ $σ_{b}$: Equal biaxial engineering tension stress

The engineering strain should be monotonically increasing going from a negative value in compression to a positive value in tension. In compression, the engineering strain should not be less than -1.0 since -100% strain is physically not possible. Fitting test data that represents the anticipated strain state of material will give the most accurate material parameters for a particular situation. Although a tension only uniaxial stress strain curve could be used, this example uses uniaxial compression and tension data to better fit for both compressive and tension loading conditions.

These LAW69 options are used in this example:

Number of material parameter ( $μ_{p}, α_{p}$ $μ_{p}, α_{p}$ ) pairs: N=2 or 3. (N > 3 is rarely used)
Engineering stress-strain curve from uniaxial compression/tension test set in fct_ID₁
$ν = 0.4997$ $ν = 0.4997$ : This value is to get the best fit for the incompressible rubber material but as described later, this high value causes a low time step and thus a value of 0.495 is typically recommended.

After running the Radioss Starter, the output file (*0000.out) contains the stress strain curve for the calculated material fit parameters and the original input data. The calculated fitted parameters, shear modulus, and bulk modulus are also included in the output.

Starter Output File Example

    FITTING RESULT COMPARISON:

      UNIAXIAL TEST DATA

      NOMINAL STRAIN     NOMINAL STRESS(TEST)       NOMINAL STRESS(RADIOSS)
           -0.9495           -221.4526                   -220.7771
           -0.9449           -176.9213                   -175.8148
           -0.9396           -138.1769                   -139.3528
           -0.9289            -93.5350                    -93.1405
           -0.9150            -60.3881                    -61.2494
           -0.8911            -35.8292                    -35.2551
           -0.8387            -15.4000                    -15.5512
           -0.7343             -5.7900                     -5.7761
           -0.6457             -3.1912                     -3.2391
           -0.5041             -1.5128                     -1.5510
           -0.4173             -1.0147                     -1.0297
           -0.3056             -0.5855                     -0.5989
           .......             .......                    ........
-------------------------------------------
AVERAGED ERROR OF FITTING :       2.00%


     FITTED PARAMETERS FOR HYPERELASTIC_MATERIAL LAW 
      ----------------------------------------
     OGDEN LAW PARAMETERS:
     MU1 . . . . . . . . . . . . . . . . . .=-0.2397367723469    
     MU2 . . . . . . . . . . . . . . . . . .= -11.57584346215    
     MU3 . . . . . . . . . . . . . . . . . .=  11.57477242242    
     MU4 . . . . . . . . . . . . . . . . . .=  0.000000000000    
     MU5 . . . . . . . . . . . . . . . . . .=  0.000000000000    

     AL1 . . . . . . . . . . . . . . . . . .= -4.548308811208    
     AL2 . . . . . . . . . . . . . . . . . .=  5.714056272418    
     AL3 . . . . . . . . . . . . . . . . . .=  5.714110104590    
     AL4 . . . . . . . . . . . . . . . . . .=  0.000000000000    
     AL5 . . . . . . . . . . . . . . . . . .=  0.000000000000    

     GROUND-STATE SHEAR MODULE . . . . . . .= 0.5424499939537    
     BULK MODULUS. . . . . . . . . . . . . .=  903.9025065914

The test and Radioss data can then be plotted to compare how well the material law fits the input test data.

Then the calculated material parameters in the Radioss Engine solution are used.

Figure 5 shows the calculated stress strain curve from the 1 element model simulation matches the Treloar test data.

The engineering stress strain curves were calculated from the true stress and true strain in principal direction 1 (P1) from the animation output /ANIM/BRICK/TENS/STRAIN and /ANIM/BRICK/TENS/STRESS using these formulas.

$ε_{e} = λ - 1$ $ε_{e} = λ - 1$

$σ_{t r} = λ \cdot σ_{e}$ $σ_{t r} = λ \cdot σ_{e}$ , $ε_{t r} = ln (λ)$ $ε_{t r} = \ln (λ)$

$σ_{e} = \frac{σ_{t r}}{ε_{e} + 1}$ $σ_{e} = \frac{σ_{t r}}{ε_{e} + 1}$ , $ε_{e} = e_{t r}^{ε_{t r}} - 1$ $ε_{e} = e^{ε_{t r}} - 1$

Effect of Poisson’s Ratio and Bulk Modulus

In LAW69, the fitted Ogden parameters are based on assumption of incompressible material which means that

ν = 0.5

$ν = 0.5$ . However, in practice is not possible to use

ν = 0.5

$ν = 0.5$ because that would result in an infinite bulk modulus, an infinite speed of sound, and thus, an infinitely small solid element Time Step. (3)

K = μ \cdot \frac{2 (1 + ν)}{3 (1 - 2 ν)} = μ \cdot \frac{2 (1 + ν)}{3 (1 - 2 * 0.5)} = \infty

$K = μ \cdot \frac{2 (1 + ν)}{3 (1 - 2 ν)} = μ \cdot \frac{2 (1 + ν)}{3 (1 - 2 * 0.5)} = \infty$

The effect of different Poison’s ratio input can be seen in Figure 6. The largest difference in the results is at higher amounts of strain. The results will match the test data better when but this results in a time step that is 4 times lower than $ν = 0.495$ $ν = 0.495$ . Thus, to balance the computation time and accuracy it is recommended to use $ν = 0.495$ $ν = 0.495$ for incompressible rubber material.

The effect of Poisson’s ratio and bulk modulus are similar in other Ogden material law.

Test Data Input with LAW88

LAW88 supports the input of strain rate dependent loading curves and an unloading curve. In this law, the Ogden equation can be exactly fit by using the uniaxial test data. By using the uniaxial test data input and bulk modulus from in LAW69, LAW88 matches the test curve better than LAW69. If the bulk modulus is not available from test data, then LAW69 could be used to calculate the bulk modulus, which is then written to the Starter output file *0000.out, shown above.

Note: With LAW88, it is recommended to use uniaxial test data that includes both compression and tension.

If only uniaxial tension data is available, then LAW69 could be used to compress one element and extract the compressive stress strain data making sure to convert the true stress and strain to engineering stress and strain. Then, the compressive data could be included with the uniaxial tensile data and used in LAW88. This work around is probably only useful if LAW88 is being used with uniaxial tensile stress strain data at different strain rates.

Parameter Input with LAW42, LAW62 and LAW82

The Ogden material parameters calculated from LAW69 can be used as input in other Ogden material laws in Radioss: LAW42, LAW62 and LAW82.

For LAW42, the LAW69 fitted Ogden parameters can be taken from the Starter output and used as input in LAW42.
LAW42(4) $W = 2 \sum p = 1 \frac{μ_{p}}{α_{p}} ({¯ λ}_{1}_{p}^{α_{p}} + {¯ λ}_{2}_{p}^{α_{p}} + {¯ λ}_{3}_{p}^{α_{p}} - 3)$ $W = \sum_{p = 1}^{2} \frac{μ_{p}}{α_{p}} ({\bar{λ}}_{1}^{α_{p}} + {\bar{λ}}_{2}^{α_{p}} + {\bar{λ}}_{3}^{α_{p}} - 3)$
For LAW62 and LAW82, the Ogden parameters need to be converted from LAW69 fitted Ogden parameters because the Ogden model equation is different in LAW62 and LAW82.
Considering an Ogden model with N=2, the energy density function used in LAW42 and LAW82 are shown here for an incompressible material.

LAW62 (and LAW82)(5) $W = 2 \sum i = 1 \frac{2 μ_{i}}{α_{i}^{2}} ({¯ λ}_{1}_{i}^{α_{i}} + {¯ λ}_{2}_{i}^{α_{i}} + {¯ λ}_{3}_{i}^{α_{i}} - 3)$ $W = \sum_{i = 1}^{2} \frac{2 μ_{i}}{α_{i}^{2}} ({\bar{λ}}_{1}^{α_{i}} + {\bar{λ}}_{2}^{α_{i}} + {\bar{λ}}_{3}^{α_{i}} - 3)$

To convert the material input from LAW42 to LAW62 or LAW82, the following equations can be used:

LAW62: $α_{i}^{L A W 62} = α_{i}^{L A W 42}$ $α_{i}^{L A W 62} = α_{i}^{L A W 42}$; $μ_{i}^{L A W 62} = \frac{μ_{i}^{L A W 42} \cdot α_{i}^{L A W 42}}{2}$ $μ_{i}^{L A W 62} = \frac{μ_{i}^{L A W 42} \cdot α_{i}^{L A W 42}}{2}$
LAW82: $α_{i}^{L A W 82} = α_{i}^{L A W 42}$ $α_{i}^{L A W 82} = α_{i}^{L A W 42}$; $μ_{i}^{L A W 82} = \frac{μ_{i}^{L A W 42} \cdot α_{i}^{L A W 42}}{2}$ $μ_{i}^{L A W 82} = \frac{μ_{i}^{L A W 42} \cdot α_{i}^{L A W 42}}{2}$

Figure 9 shows the material fit parameters from LAW69 converted into LAW82 match the LAW69 and LAW42 results for Ogden order, N=2.

Example of LAW42 with N=2

#RADIOSS STARTER
/UNIT/1
UNIT FOR MAT
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/OGDEN/1/1
rubber LAW42
#              RHO_I
                1E-9
#                 Nu           sigma_cut           funIDbulk         Fscale_bulk         M     Iform
               .4997                   0                   0                   0         0         0
#               Mu_1                Mu_2                Mu_3                Mu_4                Mu_5
 1.2732565785698E-05    -0.2635330119696                   0                   0                   0
# blank card

#            alpha_1             alpha_2             alpha_3             alpha_4             alpha_5
      7.168617832124     -4.158214786551                   0                   0                   0
# blank card

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Example of LAW82 with N=2

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW82/1
rubber LAW82
#              RHO_I
                1E-9                   0
#        N                            Nu
         2                         .4997
#               Mu_i
0.000045637449070023   0.547913433558156                   0                   0                   0
#            Alpha_i
      7.168617832124     -4.158214786551                   0                   0                   0
#                D_i
                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Arruda-Boyce Model

The Arruda-Boyce material model (/MAT/LAW92) can accept either test data input or material parameter input. When using test data, there are three different test data inputs that can be used.

In LAW92, if the curve input is used then the parameter lines can be left blank. Only one type of test data type can be considered at a time.

Example of LAW92 with Itype=1

#RADIOSS STARTER
#-  2. MATERIALS:
/UNIT/1
UNIT FOR MAT
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
rubber
#              RHO_I
            1.000E-9
#                 mu                   D                 LAM
                   0                   0                   0
#    IType    fct_ID                  NU
         1         3               .4997
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Then after running Radioss Starter, the output file (*0000.out) contains the Radioss fitted Arruda-Boyce curve and parameters.

-------------------------------------------
AVERAGED ERROR OF FITTING :      12.72%


     FITTED PARAMETERS FOR HYPERELASTIC_MATERIAL LAW 
      ----------------------------------------
      ARRUDA-BOYCE  LAW 
     MU . . . . . . . . . . . . . . . . . . . .= 0.3023683957840    
     D. . . . . . . . . . . . . . . . . . . . .= 3.8695518555789E-03
     LAM. . . . . . . . . . . . . . . . . . . .=  4.917777266862    
     GROUND-STATE SHEAR MODULE. . . . . . . . .= 0.3101754654817    
     BULK MODULUS . . . . . . . . . . . . . . .=  516.8557173143

When the Radioss Engine is ran, the fitted Arruda-Boyce parameters $μ, D, λ_{m}$ $μ, D, λ_{m}$ are used to compute the stress-strain behavior in hyperelastic material. Therefore, if the above fitted Arruda-Boyce parameter are used as LAW92 input, the results will be the same.

Example of LAW92 with Parameter Input

#RADIOSS STARTER
#-  2. MATERIALS:
/UNIT/1
UNIT FOR MAT
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
rubber
#              RHO_I
            1.000E-9
#                 mu                   D                 LAM
     0.3023683957840 3.8695518555789E-03      4.917777266862
#    IType    fct_ID                  NU

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Figure 13 shows the same results using curve input and parameter input in LAW92.

Note: For positive values of shear modulus,

μ

$μ$ , and Limit of stretch,

λ_{m}

$λ_{m}$ , the Arruda-Boyce material model is always stable.

When using test data input, use different Poisson’s ratio input leads to a different fitted

D

$D$ parameter. The

D

$D$ material parameter is used to calculate the bulk modulus,

K = \frac{2}{D}

$K = \frac{2}{D}$ .

: Fitted $D$ $D$ in LAW92 Curve Input
$ν = 0.495$ $ν = 0.495$: 6.4695283364675E-02
$ν = 0.4997$ $ν = 0.4997$: 3.8695518555789E-03

Only one test type (Itype) can be used in LAW92. Different Arruda-Boyce parameters are calculated for each different type of test data. If uniaxial test type is used in LAW92, it would describe the uniaxial deformation behavior better than biaxial or shear behavior. In order to better describe all the possible deformation behavior, an engineer can set proper weights for each

μ, D, λ_{m}

$μ, D, λ_{m}$ from three different test types.(6)

μ = \frac{1}{3} μ^{u n i a x i a l} + \frac{1}{3} μ^{e q u i b i a x i a l} + \frac{1}{3} μ^{p l a n a r}

$μ = \frac{1}{3} μ^{u n i a x i a l} + \frac{1}{3} μ^{e q u i b i a x i a l} + \frac{1}{3} μ^{p l a n a r}$

(7)

D = \frac{1}{3} D^{u n i a x i a l} + \frac{1}{3} D^{e q u i b i a x i a l} + \frac{1}{3} D^{p l a n a r}

$D = \frac{1}{3} D^{u n i a x i a l} + \frac{1}{3} D^{e q u i b i a x i a l} + \frac{1}{3} D^{p l a n a r}$

(8)

λ_{m} = \frac{1}{3} λ_{m}^{u n i a x i a l} + \frac{1}{3} λ_{m}^{e q u i b i a x i a l} + \frac{1}{3} λ_{m}^{p l a n a r}

$λ_{m} = \frac{1}{3} λ_{m}^{u n i a x i a l} + \frac{1}{3} λ_{m}^{e q u i b i a x i a l} + \frac{1}{3} λ_{m}^{p l a n a r}$

Yeoh Model

The Yeoh model (/MAT/LAW94 (YEOH)) currently only supports material parameter input. Thus, it is necessary to calculate the Yeoh material parameters outside of Radioss. For this example, a Compose script is provided to calculate the Yeoh parameters.

Note: The script can be found in the same folder as the input files for RD-E: 5600.

The fitting script assumes an incompressible (

J = 1

$J = 1$ ) hyperelastic material. The strain energy density of Yeoh then becomes:(9)

W = C_{10} {({¯ I}_{1} - 3)}^{1} + C_{20} {({¯ I}_{1} - 3)}^{2} + C_{30} {({¯ I}_{1} - 3)}^{3}

$W = C_{10} {({\bar{I}}_{1} - 3)}^{1} + C_{20} {({\bar{I}}_{1} - 3)}^{2} + C_{30} {({\bar{I}}_{1} - 3)}^{3}$

Since the Yeoh model only depends on ${¯ I}_{1} = {¯ λ}_{1}^{2} + {¯ λ}_{2}^{2} + {¯ λ}_{3}^{2}$ ${\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2}$ , only the uniaxial test is needed to calculate the Yeoh material parameters. Here $λ_{1}, λ_{2}, λ_{3}$ $λ_{1}, λ_{2}, λ_{3}$ are the stretch in principal direction 1, 2, 3 and $λ_{i} = ε_{i} + 1$ $λ_{i} = ε_{i} + 1$ . $ε_{i}$ $ε_{i}$ is the engineer strain in principal direction $i$ $i$ .

Uniaxial Test

$λ_{1} = λ$ $λ_{1} = λ$ and $λ_{2} = λ_{3} = λ^{- \frac{1}{2}}$ $λ_{2} = λ_{3} = λ^{- \frac{1}{2}}$

Then with

{¯ λ}_{k} = J^{- 1 / 3} λ_{k}

${\bar{λ}}_{k} = J^{- 1 / 3} λ_{k}$ and

J = λ_{1} λ_{2} λ_{3}

$J = λ_{1} λ_{2} λ_{3}$ you get

{¯ I}_{1}

${\bar{I}}_{1}$ , as:(10)

${\begin{matrix} {\bar{λ}}_{1} = J^{- 1 / 3} λ_{1} = λ_{1}^{2 / 3} {(λ_{2} λ_{3})}^{- 1 / 3} = λ \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \\ {\bar{λ}}_{2} = J^{- 1 / 3} λ_{2} = λ_{2}^{2 / 3} {(λ_{1} λ_{3})}^{- 1 / 3} = λ^{- 1 / 2} \\ {\bar{λ}}_{3} = J^{- 1 / 3} λ_{3} = λ_{3}^{2 / 3} {(λ_{1} λ_{2})}^{- 1 / 3} = λ^{- 1 / 2} \end{matrix}$

$\Rightarrow {\bar{I}}_{1} = {\bar{λ}}_{1}^{2} + {\bar{λ}}_{2}^{2} + {\bar{λ}}_{3}^{2} = λ^{2} + λ^{- 1} + λ^{- 1} = λ^{2} + 2 λ^{- 1}$

And the Cauchy stress of Yeoh model is computed as:(11)

$σ_{i} = \frac{λ_{i}}{J} \frac{\partial W}{\partial λ_{i}}$

In uniaxial test the Cauchy stress in principal direction 1 is:(12)

$σ_{1} = \frac{λ_{1}}{λ_{1} λ_{2} λ_{3}} \frac{\partial W}{\partial λ_{1}} = \frac{1}{λ^{- 1 / 2} λ^{- 1 / 2}} \frac{\partial W}{\partial λ} = λ \frac{\partial W}{\partial λ}$

Run the Compose script, to get fitted Yeoh parameters

$C_{i 0}$ in command window:

Next, the curve fit $C_{i 0}$ values can be used in LAW94 and the results compared to the original test data. In the example, you have been using $ν = 0.4997$ to get a good fit to the test data. $D_{1}$ can then be calculated from $D_{1} = \frac{3 (1 - 2 v)}{μ (1 + v)}$ .

Where, $μ = 2 \cdot C_{10}$ .

Example Input of LAW94

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW94/1
plastic
#              RHO_I        
              1.0E-9 		
#Blank

#                C10                 C20                 C30
   0.184883390008739  -0.001996532878013   0.000047314869715
#                 D1                  D2                  D3
   0.003245938015181
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

References

Treloar, L. R. G. "Stress-strain data for vulcanised rubber under various types of deformation." Transactions of the Faraday Society 40 (1944): 59-70.

Miller, Kurt. "Testing Elastomers for Hyperelastic Material Models in Finite Element Analysis" Axel Products, Inc., Ann Arbor, MI (2017). Last modified April 5, 2017

http://www.axelproducts.com/downloads/TestingForHyperelastic.pdf

Axel Products, Inc. "Compression or Biaxial Extension" Ann Arbor, MI (2017). Last modified November 12, 2008

http://www.axelproducts.com/downloads/CompressionOrBiax.pdf