RD-V: 0210 Yeoh Hyperelastic Material

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

Figure 1.


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

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

Properties: /PROP/SOLID with, Isolid=24, Ismstr=10, and Icpre=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, Isolid=10, Icpre=1, with Isolid=24. If hourglassing occurs, then Isolid=18 can be used.

Simulation Iterations

  • Yeoh theory:

    The Yeoh energy density is:

    W = i = 1 3 [ C i 0 ( I ¯ 1 3 ) i + 1 D i ( J 1 ) 2 i ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaamWaaeaacaWGdbWaaSbaaSqaaiaadMgacaaIWaaa beaakmaabmaabaGabmysayaaraWaaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqabaGaamyAaaaakiab gUcaRmaalaaabaGaaGymaaqaaiaadseadaWgaaWcbaGaamyAaaqaba aaaOWaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaiaadMgaaaaakiaawUfacaGLDbaaaSqaaiaadM gacqGH9aqpcaaIXaaabaGaaG4maaqdcqGHris5aaaa@51C8@

    This example assumes an incompressible ( J = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaigdaaaa@3886@ ) hyperelastic material, then the strain energy density of Yeoh simplified, as:

    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@

    With

    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@

    Where,
    λ ¯ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga qeamaaBaaaleaacaWGPbaabeaaaaa@3945@
    Deviatoric stretch with λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga qeamaaBaaaleaacaWGPbaabeaakiabg2da9iaadQeadaahaaWcbeqa aiabgkHiTmaalaaabaGaaGymaaqaaiaaiodaaaaaaOGaeq4UdW2aaS baaSqaaiaadMgaaeqaaaaa@409E@ and λ i = ε i + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyAaaqabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaaIXaaaaa@3EA5@ .
    λ 1 , λ 2 , λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2aaSbaaSqaaiaaikda aeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZaaabeaaaaa@3FA7@
    The stretch in principal direction 1, 2, 3.
    ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaaaaa@3920@
    Engineer strain in principal direction I.

    It shows only three parameters C 10 , C 20 , C 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaaJSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaaJSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EBC@ need to be defined for the (incompressible) Yeoh model.

    Uniaxial test is used in this example, then:

    λ 1 = λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBaaa@3BBE@ and λ 2 = λ 3 = λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH7oaBdaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiabgkHiTmaalaaaba GaaGymaaqaaiaaikdaaaaaaaaa@420C@

    The Cauchy stress of Yeoh model is computed as:

    σ i = λ 1 J W λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaaGymaaqabaaakeaacaWGkbaaamaalaaabaGaeyOaIyRaam4vaa qaaiabgkGi2kabeU7aSnaaBaaaleaacaaIXaaabeaaaaaaaa@43BB@

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

    σ 1 = λ W I ¯ 1 d I ¯ 1 d λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaaeqaaOGaeyypa0Jaeq4UdW2aaSaaaeaacqGHciIT caWGxbaabaGaeyOaIyRabmysayaaraWaaSbaaSqaaiaaigdaaeqaaa aakmaalaaabaGaamizaiqadMeagaqeamaaBaaaleaacaaIXaaabeaa aOqaaiaadsgacqaH7oaBaaaaaa@4661@

    The engineer stress is:

    σ 1 _ e n g = σ 1 ( ε 1 _ e n g + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaGGFbGaamyzaiaad6gacaWGNbaabeaakiabg2da 9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGcbaGaaiikai abew7aLnaaBaaaleaacaaIXaGaai4xaiaadwgacaWGUbGaam4zaaqa baGccqGHRaWkcaaIXaGaaiykaaaaaaa@495B@

    In this example, three material parameters are defined below:
    C 10 =0 .18 C 20 =-2e-3 C 30 = 5 e 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGdb WaaSbaaSqaaiaaigdacaaIWaaabeaakiaab2dacaqGWaGaaeOlaiaa bgdacaqG4aaabaGaam4qamaaBaaaleaacaaIYaGaaGimaaqabaGcca qG9aGaaeylaiaabkdacaqGLbGaaeylaiaabodaaeaacaWGdbWaaSba aSqaaiaaiodacaaIWaaabeaakiabg2da9iaaiwdacaWGLbGaeyOeI0 IaaGynaaaaaa@49BB@
    Figure 3. Engineer stress-strain curve


  • LAW94:

    The hyperelastic model uses polynomial model and the Yeoh model is one of the polynomial model with only three parameters. In this example, viscous was not considered.

Results

In LAW94 Yeoh model, three parameters C 10 , C 20 , C 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaaJSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaaJSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EBC@ defined same as analytical case.

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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9iaaikdacaWGdbWaaSbaaSqaaiaaigdacaaIWaaabeaakiaab2da caqGYaGaaeOkaiaabcdacaqGUaGaaeymaiaabIdacaqG9aGaaeimai aab6cacaqGZaGaaeOnaaaa@437F@
D 1 = 3 ( 1 2 ν ) G ( 1 + ν ) = 3 ( 1 2 × 0 .495 ) 0 .36 ( 1 + 0 .495 ) = 0.055741 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaaig dacqGH9aqpdaWcaaqaaiaabodadaqadaqaaiaaigdacqGHsislcaaI YaGaeqyVd4gacaGLOaGaayzkaaaabaGaam4ramaabmaabaGaaGymai abgUcaRiabe27aUbGaayjkaiaawMcaaaaacaqG9aWaaSaaaeaacaqG ZaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabgEna0kaabcdacaqGUa GaaeinaiaabMdacaqG1aaacaGLOaGaayzkaaaabaGaaeimaiaab6ca caqGZaGaaeOnamaabmaabaGaaGymaiabgUcaRiaabcdacaqGUaGaae inaiaabMdacaqG1aaacaGLOaGaayzkaaaaaiaab2daqaaaaaaaaaWd biaaicdacaGGUaGaaGimaiaaiwdacaaI1aGaaG4naiaaisdacaaIXa aaaa@5FF4@
1 D 1 = 17.94 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGaaeaaca qGXaaabaGaaeiraiaaigdaaaGaeyypa0deaaaaaaaaa8qacaaIXaGa aG4naiaac6cacaaI5aGaaGinaaaa@3D13@

C10 . . . . . . . . . . . . . . . . . . .= 0.1800000000000    
     C01 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     C20 . . . . . . . . . . . . . . . . . . .=-2.0000000000000E-03
     C11 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     C02 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     C30 . . . . . . . . . . . . . . . . . . .= 5.0000000000000E-05
     C21 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     C12 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     C03 . . . . . . . . . . . . . . . . . . .=  0.000000000000    
     1/D1  . . . . . . . . . . . . . . . . . .=  17.94001716615    
     1/D2  . . . . . . . . . . . . . . . . . .=  0.000000000000    
     1/D3  . . . . . . . . . . . . . . . . . .=  0.000000000000
Compare the above results, then show the difference with analytical results especially in high deformed area.
Figure 4.


In order to match the incompressible hyperelastic analytical results, the Poisson’s ratio needs to be defined as close to 0.5 as possible, but can not be simply defined as ν =0 .5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maae ypaiaabcdacaqGUaGaaeynaaaa@3A8A@ , which will lead to infinite small time step in numerical computation. In this example ν = 0.49999995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maae ypaabaaaaaaaaapeGaaGimaiaac6cacaaI0aGaaGyoaiaaiMdacaaI 5aGaaGyoaiaaiMdacaaI5aGaaGynaaaa@4009@ is used. It shows Radioss results are well matched with analytical results.
Figure 5.


Since material incompressible is assumed in analytical calculation., Poisson’s ratio closer to 0.5 is better, but the computation time will increase. Just one element tension model from ν =0 .495 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maae ypaiaabcdacaqGUaGaaeinaiaabMdacaqG1aaaaa@3BFD@ to ν = 0.49999995 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maae ypaabaaaaaaaaapeGaaGimaiaac6cacaaI0aGaaGyoaiaaiMdacaaI 5aGaaGyoaiaaiMdacaaI5aGaaGynaaaa@4009@ run time is more than 20 times increased. In this example, use ν =0 .4997 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maae ypaiaabcdacaqGUaGaaeinaiaabMdacaqG5aGaae4naaaa@3CBB@ ; therefore, set D1=0.003334.