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.


Figure 1.

Options and Keywords Used

Input Files

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

Model Description

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



Figure 2. Problem Description

Units: mm, s, Mg, N, MPa

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

Properties: /PROP/SOLID with, Isolid=24, Ismstr=10, and Icpre=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:
  • Ismstr=10
  • Icpre=1 with Isolid=24.
Note: If hourglassing occurs, then Isolid=17 with Iframe=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


Figure 3. Test Data from Treloar [1944]
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 = 1 ( ε b + 1 ) 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaadaqa daqaaiabew7aLnaaBaaaleaacaWGIbaabeaakiabgUcaRiaaigdaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0IaaGym aaaa@4379@ (2) σ c = σ b ( 1 + ε b ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamOy aaqabaGcdaqadaqaaiaaigdacqGHRaWkcqaH1oqzdaWgaaWcbaGaam OyaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@43FA@
Where,
ε c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaaaaa@391A@
Uniaxial engineering compressive stain
ε b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaaaaa@391A@
Equal biaxial engineering tension strain
σ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaaaaa@3936@
Uniaxial engineering compressive stress
σ b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaaaaa@3936@
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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaamiCaaqabaGccaGGSaGaeqySde2aaSbaaSqaaiaadcha aeqaaaaa@3D9C@ ) 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@ : 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.


Figure 4. LAW69 Fitted Curve with N=2 and N=3 for Odgen Model
Then the calculated material parameters in the Radioss Engine solution are used.


Figure 5. LAW69 Results from a 1 Element Model with N=2 and N=3

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyzaaqabaGccqGH9aqpcqaH7oaBcqGHsislcaaIXaaa aa@3E91@

σ t r = λ σ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamiDaiaadkhaaeqaaOGaeyypa0Jaeq4UdWMaeyyXICTa eq4Wdm3aaSbaaSqaaiaadwgaaeqaaaaa@432E@ , ε t r = ln ( λ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamiDaiaadkhaaeqaaOGaeyypa0JaciiBaiaac6gadaqa daqaaiabeU7aSbGaayjkaiaawMcaaaaa@415C@

σ e = σ t r ε e + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamyzaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaBaaa leaacaWG0bGaamOCaaqabaaakeaacqaH1oqzdaWgaaWcbaGaamyzaa qabaGccqGHRaWkcaaIXaaaaaaa@4391@ , ε e = e ε t r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyzaaqabaGccqGH9aqpcaWGLbWaaWbaaSqabeaacqaH 1oqzdaWgaaadbaGaamiDaiaadkhaaeqaaaaakiabgkHiTiaaigdaaa a@41C2@

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3C34@ . However, in practice is not possible to use ν = 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3C34@ 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 2 ( 1 + ν ) 3 ( 1 2 ν ) = μ 2 ( 1 + ν ) 3 ( 1 2 * 0.5 ) = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9iabeY7aTjabgwSixpaalaaabaGaaGOmamaabmaabaGaaGymaiab gUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaaiodadaqadaqaaiaaig dacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaaiabg2da9iab eY7aTjabgwSixpaalaaabaGaaGOmamaabmaabaGaaGymaiabgUcaRi abe27aUbGaayjkaiaawMcaaaqaaiaaiodadaqadaqaaiaaigdacqGH sislcaaIYaGaaiOkaiaaicdacaGGUaGaaGynaaGaayjkaiaawMcaaa aacqGH9aqpcqGHEisPaaa@5C83@

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGynaaaa@3DB5@ . Thus, to balance the computation time and accuracy it is recommended to use ν = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGynaaaa@3DB5@ for incompressible rubber material.

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


Figure 6. Effect of Poisson’s Ratio in Results and in Time Step
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.


Figure 7. Results of LAW69 and LAW88 Compared with Test Data
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.


Figure 8. Test Data Extended in Compression
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 = p = 1 2 μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0ZaaabCaeaadaWcaaqaaiabeY7aTnaaBaaaleaacaWGWbaabeaa aOqaaiabeg7aHnaaBaaaleaacaWGWbaabeaaaaGcdaqadaqaamaana aabaGaeq4UdWgaamaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGa eqySde2aaSbaaWqaaiaadchaaeqaaaaakiabgUcaRmaanaaabaGaeq 4UdWgaamaaBaaaleaacaaIYaaabeaakmaaCaaaleqabaGaeqySde2a aSbaaWqaaiaadchaaeqaaaaakiabgUcaRmaanaaabaGaeq4UdWgaam aaBaaaleaacaaIZaaabeaakmaaCaaaleqabaGaeqySde2aaSbaaWqa aiaadchaaeqaaaaakiabgkHiTiaaiodaaiaawIcacaGLPaaaaSqaai aadchacqGH9aqpcaaIXaaabaGaaGOmaaqdcqGHris5aaaa@59B3@
  • 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 = i = 1 2 2 μ i α i 2 ( λ ¯ 1 α i + λ ¯ 2 α i + λ ¯ 3 α i 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaaSaaaeaacaaIYaGaeqiVd02aaSbaaSqaaiaadMga aeqaaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOWaaWbaaSqabe aacaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaaikdaa0Ga eyyeIuoakmaabmaabaGafq4UdWMbaebadaWgaaWcbaGaaGymaaqaba GcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGPbaabeaaaaGccqGH RaWkcuaH7oaBgaqeamaaBaaaleaacaaIYaaabeaakmaaCaaaleqaba GaeqySde2aaSbaaWqaaiaadMgaaeqaaaaakiabgUcaRiqbeU7aSzaa raWaaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacqaHXoqydaWgaa adbaGaamyAaaqabaaaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaaaaa @5ADB@
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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaaiaadYeacaWGbbGaam4vaiaaiAdacaaIYaaa aOGaeyypa0JaeqySde2aa0baaSqaaiaadMgaaeaacaWGmbGaamyqai aadEfacaaI0aGaaGOmaaaaaaa@44BE@
μ i L A W 62 = μ i L A W 42 α i L A W 42 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda qhaaWcbaGaamyAaaqaaiaadYeacaWGbbGaam4vaiaaiAdacaaIYaaa aOGaeyypa0ZaaSaaaeaacqaH8oqBdaqhaaWcbaGaamyAaaqaaiaadY eacaWGbbGaam4vaiaaisdacaaIYaaaaOGaeyyXICTaeqySde2aa0ba aSqaaiaadMgaaeaacaWGmbGaamyqaiaadEfacaaI0aGaaGOmaaaaaO qaaiaaikdaaaaaaa@4EBD@
LAW82
α i L A W 82 = α i L A W 42 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaaiaadYeacaWGbbGaam4vaiaaiIdacaaIYaaa aOGaeyypa0JaeqySde2aa0baaSqaaiaadMgaaeaacaWGmbGaamyqai aadEfacaaI0aGaaGOmaaaaaaa@44C0@
μ i L A W 82 = μ i L A W 42 α i L A W 42 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda qhaaWcbaGaamyAaaqaaiaadYeacaWGbbGaam4vaiaaiIdacaaIYaaa aOGaeyypa0ZaaSaaaeaacqaH8oqBdaqhaaWcbaGaamyAaaqaaiaadY eacaWGbbGaam4vaiaaisdacaaIYaaaaOGaeyyXICTaeqySde2aa0ba aSqaaiaadMgaaeaacaWGmbGaamyqaiaadEfacaaI0aGaaGOmaaaaaO qaaiaaikdaaaaaaa@4EBF@

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


Figure 9. Results of LAW69, LAW42 and LAW82 Compared with Test Data

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.


Figure 10. Itype=1: Uniaxial test


Figure 11. Itype=2: Equibiaxial test


Figure 12. Itype=3: Planar test

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBca GGSaGaaGjcVlaaysW7caWGebGaaiilaiaaysW7cqaH7oaBdaWgaaWc baGaamyBaaqabaaaaa@42A7@ 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. Equivalent LAW92 Used with Curve Input and Parameter Input
Figure 13 shows the same results using curve input and parameter input in LAW92.
Note: For positive values of shear modulus, μ , and Limit of stretch, λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaaaaa@3931@ , 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebaaaa@3814@ parameter. The D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebaaaa@3814@ material parameter is used to calculate the bulk modulus, K = 2 D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaey ypa0ZaaSaaaeaacaaIYaaabaGaamiraaaaaaa@3AB6@ .
Fitted D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebaaaa@3814@ in LAW92 Curve Input
ν = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGynaaaa@3DB5@
6.4695283364675E-02
ν = 0.4997 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@
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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBca GGSaGaaGjcVlaaysW7caWGebGaaiilaiaaysW7cqaH7oaBdaWgaaWc baGaamyBaaqabaaaaa@42A7@ from three different test types.(6) μ = 1 3 μ u n i a x i a l + 1 3 μ e q u i b i a x i a l + 1 3 μ p l a n a r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpdaWcaaqaaiaaigdaaeaacaaIZaaaaiabeY7aTnaaCaaaleqa baGaamyDaiaad6gacaWGPbGaamyyaiaadIhacaWGPbGaamyyaiaadY gaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZaaaaiabeY7aTnaa CaaaleqabaGaamyzaiaadghacaWG1bGaamyAaiaadkgacaWGPbGaam yyaiaadIhacaWGPbGaamyyaiaadYgaaaGccqGHRaWkdaWcaaqaaiaa igdaaeaacaaIZaaaaiabeY7aTnaaCaaaleqabaGaamiCaiaadYgaca WGHbGaamOBaiaadggacaWGYbaaaaaa@5D84@ (7) D = 1 3 D u n i a x i a l + 1 3 D e q u i b i a x i a l + 1 3 D p l a n a r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaSaaaeaacaaIXaaabaGaaG4maaaacaWGebWaaWbaaSqabeaa caWG1bGaamOBaiaadMgacaWGHbGaamiEaiaadMgacaWGHbGaamiBaa aakiabgUcaRmaalaaabaGaaGymaaqaaiaaiodaaaGaamiramaaCaaa leqabaGaamyzaiaadghacaWG1bGaamyAaiaadkgacaWGPbGaamyyai aadIhacaWGPbGaamyyaiaadYgaaaGccqGHRaWkdaWcaaqaaiaaigda aeaacaaIZaaaaiaadseadaahaaWcbeqaaiaadchacaWGSbGaamyyai aad6gacaWGHbGaamOCaaaaaaa@59D0@ (8) λ m = 1 3 λ m u n i a x i a l + 1 3 λ m e q u i b i a x i a l + 1 3 λ m p l a n a r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaI ZaaaaiabeU7aSnaaDaaaleaacaWGTbaabaGaamyDaiaad6gacaWGPb GaamyyaiaadIhacaWGPbGaamyyaiaadYgaaaGccqGHRaWkdaWcaaqa aiaaigdaaeaacaaIZaaaaiabeU7aSnaaDaaaleaacaWGTbaabaGaam yzaiaadghacaWG1bGaamyAaiaadkgacaWGPbGaamyyaiaadIhacaWG PbGaamyyaiaadYgaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZa aaaiabeU7aSnaaDaaaleaacaWGTbaabaGaamiCaiaadYgacaWGHbGa amOBaiaadggacaWGYbaaaaaa@617A@

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaigdaaaa@3886@ ) 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9iaadoeadaWgaaWcbaGaaGymaiaaicdaaeqaaOWaaeWaaeaaceWG jbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaaacaGLOa GaayzkaaWaaWbaaSqabeaacaaIXaaaaOGaey4kaSIaam4qamaaBaaa leaacaaIYaGaaGimaaqabaGcdaqadaqaaiqadMeagaqeamaaBaaale aacaaIXaaabeaakiabgkHiTiaaiodaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaiodacaaIWa aabeaakmaabmaabaGabmysayaaraWaaSbaaSqaaiaaigdaaeqaaOGa eyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaaaa a@52E5@

Since the Yeoh model only depends on I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGymaaqabaGccqGH9aqpcuaH7oaBgaqeamaaDaaa leaacaaIXaaabaGaaGOmaaaakiabgUcaRiqbeU7aSzaaraWaa0baaS qaaiaaikdaaeaacaaIYaaaaOGaey4kaSIafq4UdWMbaebadaqhaaWc baGaaG4maaqaaiaaikdaaaaaaa@4567@ , only the uniaxial test is needed to calculate the Yeoh material parameters. Here λ 1 , λ 2 , λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2aaSbaaSqaaiaaikda aeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZaaabeaaaaa@3FA7@ are the stretch in principal direction 1, 2, 3 and λ i = ε i + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyAaaqabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaaIXaaaaa@3EA5@ . ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaaaaa@3920@ is the engineer strain in principal direction i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ .

Uniaxial Test

λ 1 = λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaaigdaaeqaaOGaeyypa0Jaeq4UdWgaaa@3B55@ and λ 2 = λ 3 = λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaaikdaaeqaaOGaeyypa0Jaeq4UdW2aaSbaaSqaaiaaioda aeqaaOGaeyypa0Jaeq4UdW2aaWbaaSqabeaacqGHsisldaWcaaqaai aaigdaaeaacaaIYaaaaaaaaaa@41A4@

Then with λ ¯ k = J 1 / 3 λ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga qeamaaBaaaleaacaWGRbaabeaakiabg2da9iaadQeadaahaaWcbeqa aiabgkHiTiaaigdacaGGVaGaaG4maaaakiabeU7aSnaaBaaaleaaca WGRbaabeaaaaa@4145@ and J = λ 1 λ 2 λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iabeU7aSnaaBaaaleaacaaIXaaabeaakiabeU7aSnaaBaaaleaa caaIYaaabeaakiabeU7aSnaaBaaaleaacaaIZaaabeaaaaa@3FB3@ you get I ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGymaaqabaaaaa@382C@ , as:(10) { λ ¯ 1 = J 1 / 3 λ 1 = λ 1 2 / 3 ( λ 2 λ 3 ) 1 / 3 = λ λ ¯ 2 = J 1 / 3 λ 2 = λ 2 2 / 3 ( λ 1 λ 3 ) 1 / 3 = λ 1 / 2 λ ¯ 3 = J 1 / 3 λ 3 = λ 3 2 / 3 ( λ 1 λ 2 ) 1 / 3 = λ 1 / 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGabaqaau aabeqadeaaaeaacuaH7oaBgaqeamaaBaaaleaacaaIXaaabeaakiab g2da9iaadQeadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaG4maa aakiabeU7aSnaaBaaaleaacaaIXaaabeaakiabg2da9iabeU7aSnaa BaaaleaacaaIXaaabeaakmaaCaaaleqabaGaaGOmaiaac+cacaaIZa aaaOWaaeWaaeaacqaH7oaBdaWgaaWcbaGaaGOmaaqabaGccqaH7oaB daWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaadaahaaWcbeqaai abgkHiTiaaigdacaGGVaGaaG4maaaakiabg2da9iabeU7aSvaabeqa beaaaeaaaaqbaeqabeqaaaqaaaaaaeaacuaH7oaBgaqeamaaBaaale aacaaIYaaabeaakiabg2da9iaadQeadaahaaWcbeqaaiabgkHiTiaa igdacaGGVaGaaG4maaaakiabeU7aSnaaBaaaleaacaaIYaaabeaaki abg2da9iabeU7aSnaaBaaaleaacaaIYaaabeaakmaaCaaaleqabaGa aGOmaiaac+cacaaIZaaaaOWaaeWaaeaacqaH7oaBdaWgaaWcbaGaaG ymaaqabaGccqaH7oaBdaWgaaWcbaGaaG4maaqabaaakiaawIcacaGL PaaadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaG4maaaakiabg2 da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIYaaa aaGcbaGafq4UdWMbaebadaWgaaWcbaGaaG4maaqabaGccqGH9aqpca WGkbWaaWbaaSqabeaacqGHsislcaaIXaGaai4laiaaiodaaaGccqaH 7oaBdaWgaaWcbaGaaG4maaqabaGccqGH9aqpcqaH7oaBdaWgaaWcba GaaG4maaqabaGcdaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaakmaa bmaabaGaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeq4UdW2aaSbaaS qaaiaaikdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsisl caaIXaGaai4laiaaiodaaaGccqGH9aqpcqaH7oaBdaahaaWcbeqaai abgkHiTiaaigdacaGGVaGaaGOmaaaaaaaakiaawUhaaaaa@9699@

I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 = λ 2 + λ 1 + λ 1 = λ 2 + 2 λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3ce WGjbGbaebadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcuaH7oaBgaqe amaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiqbeU7aSzaara Waa0baaSqaaiaaikdaaeaacaaIYaaaaOGaey4kaSIafq4UdWMbaeba daqhaaWcbaGaaG4maaqaaiaaikdaaaGccqGH9aqpcqaH7oaBdaahaa WcbeqaaiaaikdaaaGccqGHRaWkcqaH7oaBdaahaaWcbeqaaiabgkHi TiaaigdaaaGccqGHRaWkcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaig daaaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiaaikdaaaGccqGHRaWk caaIYaGaeq4UdW2aaWbaaSqabeaacqGHsislcaaIXaaaaaaa@5D39@

And the Cauchy stress of Yeoh model is computed as:(11) σ i = λ i J W λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaamyAaaqabaaakeaacaWGkbaaamaalaaabaGaeyOaIyRaam4vaa qaaiabgkGi2kabeU7aSnaaBaaaleaacaWGPbaabeaaaaaaaa@4421@
In uniaxial test the Cauchy stress in principal direction 1 is:(12) σ 1 = λ 1 λ 1 λ 2 λ 3 W λ 1 = 1 λ 1 / 2 λ 1 / 2 W λ = λ W λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaaGymaaqabaaakeaacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccq aH7oaBdaWgaaWcbaGaaGOmaaqabaGccqaH7oaBdaWgaaWcbaGaaG4m aaqabaaaaOWaaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaeq4UdW 2aaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaabaGaaGymaaqa aiabeU7aSnaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIYaaaaO Gaeq4UdW2aaWbaaSqabeaacqGHsislcaaIXaGaai4laiaaikdaaaaa aOWaaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaeq4UdWgaaiabg2 da9iabeU7aSnaalaaabaGaeyOaIyRaam4vaaqaaiabgkGi2kabeU7a Sbaaaaa@641C@
Run the Compose script, to get fitted Yeoh parameters C i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaaIWaaabeaaaaa@38FB@ in command window:


Figure 14. Fitted Yeoh Parameter with Compose Script

Next, the curve fit C i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaaIWaaabeaaaaa@38FB@ 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@ to get a good fit to the test data. D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaaigdaaeqaaaaa@380F@ can then be calculated from D 1 = 3 ( 1 2 v ) μ ( 1 + v ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaG4maiaacIcacaaI XaGaeyOeI0IaaGOmaiaadAhacaGGPaaabaGaeqiVd0Maaiikaiaaig dacqGHRaWkcaWG2bGaaiykaaaaaaa@43E2@ .

Where, μ = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0JaaGOmaiabgwSixlaadoeadaWgaaWcbaGaaGymaiaaicdaaeqa aaaa@3E22@ .

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----|


Figure 15. Results of LAW94 with Fitted Yeoh Parameter

References

1 Treloar, L. R. G. "Stress-strain data for vulcanised rubber under various types of deformation." Transactions of the Faraday Society 40 (1944): 59-70.
2 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

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

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