Elasto-plastic Materials

Johnson-Cook (/MAT/LAW2)

In LAW2 there are three parts to the stress calculation.
  • Influence of plastic strain
  • Influence of strain rate
  • Influence of temperature change


Figure 1.

Material Parameters

There are two ways to input material parameter for LAW2.
  • Iflag=0: Classic input for Johnson-Cook parameter a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ , n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ is active
  • Iflag=1: New, simplified input with yield stress, UTS (engineering stress), or strain at UTS
Iflag = 0
(1) σ=a+b ε p n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyyaiabgUcaRiaadkgacqGHflY1cqaH1oqzdaWgaaWcbaGa amiCaaqabaGcdaahaaWcbeqaaiaad6gaaaaaaa@41AB@
Where,
a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
The yield stress which could be read from material test and converted to true stress.
b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
The material parameters. Fitting the material stress-strain curve (for example, Altair Compose script) can result in these two parameter.
Iflag = 1
With this new input, you will need yield stress ( σ y ), Ultimate tensile engineer stress (UTS) and engineer strain ( ε U T S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadwfacaWGubGaam4uaaqabaaaaa@3A55@ ) at necking point. With this new input, Radioss automatically calculates the equivalent value for a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ .


Figure 2. Tension Test

Strain Rate

Strain rate has a major effect of material character on crash performance in tensile or in fracture. In Johnson-Cook theory, the yield stress is affected directly by the strain rate and is described as:(2) σ=( a+b ε p n )( 1+cln ε ˙ ε ˙ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaeWaaeaacaWGHbGaey4kaSIaamOyaiabgwSixlabew7aLnaa BaaaleaacaWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkai aawMcaamaabmaabaGaaGymaiabgUcaRiaadogaciGGSbGaaiOBamaa laaabaGafqyTduMbaiaaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWa aabeaaaaaakiaawIcacaGLPaaaaaa@4D90@
Generally, yield stress increases with increasing the test strain rate. With the strain rate coefficient, c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DF@ , you can scale the factor of yield stress increase. No effect of strain rate could also be defined, if c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DF@ =0; or with ε ˙ 0 = 10 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaaGimamaaCaaa leqabaGaaG4maiaaicdaaaaaaa@3CB6@ or ε ˙ ε ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aacqGHKjYOcuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaa@3BF2@ .


Figure 3.

Temperature Change

Yield stress decreases with increasing temperature. In LAW2 influence is considered with ( 1 T * m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaeyOeI0IaamivamaaCaaaleqabaGaaiOkaiaad2gaaaaakiaa wIcacaGLPaaaaaa@3BD8@ .(3) σ=( a+b ε p n )( 1+cln ε ˙ ε ˙ 0 )( 1 T *m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaeWaaeaacaWGHbGaey4kaSIaamOyaiabgwSixlabew7aLnaa BaaaleaacaWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkai aawMcaamaabmaabaGaaGymaiabgUcaRiaadogaciGGSbGaaiOBamaa laaabaGafqyTduMbaiaaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWa aabeaaaaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaWG ubWaaWbaaSqabeaacaGGQaGaamyBaaaaaOGaayjkaiaawMcaaaaa@5371@ with(4) T * = T T r T m e l t T r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCa aaleqabaGaaiOkaaaakiabg2da9maalaaabaGaamivaiabgkHiTiaa dsfadaWgaaWcbaGaamOCaaqabaaakeaacaWGubWaaSbaaSqaaiaad2 gacaWGLbGaamiBaiaadshaaeqaaOGaeyOeI0IaamivamaaBaaaleaa caWGYbaabeaaaaaaaa@4455@
Where,
T m e l t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGTbGaamyzaiaadYgacaWG0baabeaaaaa@3AC2@
Melt temperature in unit Kelvin.
T r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGYbaabeaaaaa@37F3@
Room temperature in unit Kelvin.
With T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGubaaaa@3990@ computed as:(5) T = T i + E int ρ C p ( V o l u m e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2 da9iaadsfadaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaWcaaqaaiaa dweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaOqaaiabeg8aYj aadoeadaWgaaWcbaGaamiCaaqabaGcdaqadaqaaiaadAfacaWGVbGa amiBaiaadwhacaWGTbGaamyzaaGaayjkaiaawMcaaaaaaaa@4970@
Where,
E int MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaaciGGPbGaaiOBaiaacshaaeqaaaaa@39C6@
Internal energy.
Change of internal energy will affect the yield stress in Johnson-Cook law.

Hardening Coefficient

Metal deformed up to yield and then generally hardened (yield stress increased). Different materials show different ways of hardening (isotropic hardening, kinematic hardening, etc.). This is also a very important material character (for spring-back).

In LAW2, use option Chard (hardening coefficient) to describe which hardening model is used for the material. This feature is also available in material LAW36, 43, 44, 57, 60, 66, 73 and 74.

The value of Chard is from 1 to 0. Chard =0 for isotropic model, Chard=1 for kinematic Prager-Ziegler model, or between 1 and 0 for hardening between the above two models.
Chard = 0: Isotropic Model
In a one dimension case, material strengthens after yield stress. The maximum stress of the last tension is the yield in the subsequent loading, and this new yield stress is the same in subsequent tension and compression.


Figure 4.
Chard = 1: Kinematic Prager-Ziegler Model
To model the Bauschinger effect (after hardening in tension, there is softening in a subsequent compression which mean yield in compression is decreased), use kinematic hardening.


Figure 5.

Elastic Plastic Piecewise Linear Material (/MAT/LAW36)

In LAW36, the numbers of plastic stress-strain curves can be directly defined for different strain rates.

Plastic stress-strain of high strain rate should always be above the lower plastic stress-stain curve.


Figure 6.

Young's Modulus

Young's modulus can be updated (decreased) in unloading with options fct_IDE, Einf and CE. Using this feature improves the accuracy of spring-back (in unloading phase) for high strength steel. This feature is also available in material LAW43, LAW57, LAW60, LAW74 and LAW78.
  • Use fct_IDE to update the Young's modulus (fct_IDE ≠ 0):


    Figure 7.
  • Use Einf and CE to update the Young's modulus (fct_IDE = 0):


    Figure 8.

Material Behavior

fct_IDp is used to distinguish the behavior in tension and compression for certain materials (pressure dependent yield). The effective yield stress is then obtained by multiplying the nominal yield stress by the yield factor corresponding to the actual pressure.


Figure 9.

HILL Materials

In Radioss material laws LAW32, LAW43, LAW72, LAW73, LAW74, LAW78 and LAW93 use HILL criteria.

HILL Criteria

The typical HILL criteria is:
  • 3D equivalent HILL stress:(6) f= F ( σ yy σ zz ) 2 +G ( σ zz σ xx ) 2 +H ( σ xx σ yy ) 2 +2L σ yz 2 +2M σ zx 2 +2N σ xy 2    = ( G+H ) σ xx 2 + ( F+H ) σ yy 2 + ( F+G ) σ zz 2 2H σ xx σ yy 2F σ yy σ zz 2G σ zz σ xx + 2L σ yz 2 + 2M σ zx 2 + 2N σ xy 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadA gacqGH9aqpdaGcaaqaaiaadAeadaqadiqaaiabeo8aZnaaBaaaleaa caWG5bGaamyEaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOEai aadQhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSIaam4ramaabmGabaGaeq4Wdm3aaSbaaSqaaiaadQhacaWG6b aabeaakiabgkHiTiabeo8aZnaaBaaaleaacaWG4bGaamiEaaqabaaa kiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGib WaaeWaceaacqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaeyOe I0Iaeq4Wdm3aaSbaaSqaaiaadMhacaWG5baabeaaaOGaayjkaiaawM caamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaygW7caaIYaGaamit aiabeo8aZnaaDaaaleaacaWG5bGaamOEaaqaaiaaikdaaaGccqGHRa WkcaaIYaGaamytaiabeo8aZnaaDaaaleaacaWG6bGaamiEaaqaaiaa ikdaaaGccqGHRaWkcaaIYaGaamOtaiabeo8aZnaaDaaaleaacaWG4b GaamyEaaqaaiaaikdaaaaabeaaaOqaaiaabccacaqGGaGaaeiiaiaa b2dadaGcaaqaamaayaaabaWaaeWaaeaacaWGhbGaey4kaSIaamisaa GaayjkaiaawMcaaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG 4bGaamiEaaqaaiaaikdaaaGccqGHRaWkdaagaaqaamaabmaabaGaam OraiabgUcaRiaadIeaaiaawIcacaGLPaaaaSqaaaGccaGL44pacqaH dpWCdaqhaaWcbaGaamyEaiaadMhaaeaacaaIYaaaaOGaey4kaSYaaG baaeaadaqadaqaaiaadAeacqGHRaWkcaWGhbaacaGLOaGaayzkaaaa leaaaOGaayjo+dGaeq4Wdm3aa0baaSqaaiaadQhacaWG6baabaGaaG OmaaaakiabgkHiTmaayaaabaGaaGOmaiaadIeaaSqaaaGccaGL44pa cqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaeq4Wdm3aaSbaaS qaaiaadMhacaWG5baabeaakiabgkHiTmaayaaabaGaaGOmaiaadAea aSqaaaGccaGL44pacqaHdpWCdaWgaaWcbaGaamyEaiaadMhaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaakiabgkHiTmaayaaa baGaaGOmaiaadEeaaSqaaaGccaGL44pacqaHdpWCdaWgaaWcbaGaam OEaiaadQhaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaa kiabgUcaRmaayaaabaGaaGzaVlaaikdacaWGmbaaleaaaOGaayjo+d Gaeq4Wdm3aa0baaSqaaiaadMhacaWG6baabaGaaGOmaaaakiabgUca RmaayaaabaGaaGOmaiaad2eaaSqaaaGccaGL44pacqaHdpWCdaqhaa WcbaGaamOEaiaadIhaaeaacaaIYaaaaOGaey4kaSYaaGbaaeaacaaI YaGaamOtaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG4bGaam yEaaqaaiaaikdaaaaabeaaaaaa@DD25@
  • Shell element:(7) f = F σ y y 2 + G σ x x 2 + H ( σ x x σ y y ) 2 + 2 N σ x y 2 = ( G + H ) σ x x 2 + ( F + H ) σ y y 2 2 H σ x x σ y y + 2 N σ x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaey ypa0ZaaOaaaeaacaWGgbGaeq4Wdm3aa0baaSqaaiaadMhacaWG5baa baGaaGOmaaaakiabgUcaRiaadEeacqaHdpWCdaqhaaWcbaGaamiEai aadIhaaeaacaaIYaaaaOGaey4kaSIaamisamaabmGabaGaeq4Wdm3a aSbaaSqaaiaadIhacaWG4baabeaakiabgkHiTiabeo8aZnaaBaaale aacaWG5bGaamyEaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIYaGaamOtaiabeo8aZnaaDaaaleaacaWG4b GaamyEaaqaaiaaikdaaaaabeaakiabg2da9maakaaabaWaaGbaaeaa daqadaqaaiaadEeacqGHRaWkcaWGibaacaGLOaGaayzkaaaaleaaaO Gaayjo+dGaeq4Wdm3aa0baaSqaaiaadIhacaWG4baabaGaaGOmaaaa kiabgUcaRmaayaaabaWaaeWaaeaacaWGgbGaey4kaSIaamisaaGaay jkaiaawMcaaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG5bGa amyEaaqaaiaaikdaaaGccqGHsisldaagaaqaaiaaikdacaWGibaale aaaOGaayjo+dGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiab eo8aZnaaBaaaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaai aaikdacaWGobaaleaaaOGaayjo+dGaeq4Wdm3aa0baaSqaaiaadIha caWG5baabaGaaGOmaaaaaeqaaaaa@863E@

    Where, F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ and N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ are six HILL anisotropic parameters. For shell elements, only F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ and N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ are the four HILL parameters needed.

    In LAW78, the HILL criteria is:(8) φ ( A ) = 1 G + H A x x 2 2 r 0 1 + r 0 2 H A x x A y y + r 0 ( 1 + r 90 ) r 90 ( 1 + r 0 ) F + H A y y 2 + r 0 + r 90 r 90 ( 1 + r 0 ) ( 2 r 45 + 1 ) 2 N A x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAca GGOaGaamyqaiaacMcacqGH9aqpdaagaaqaaiaaigdaaSqaaiaadEea cqGHRaWkcaWGibaakiaawIJ=aiabgwSixlaadgeadaqhaaWcbaGaam iEaiaadIhaaeaacaaIYaaaaOGaeyOeI0YaaGbaaeaadaWcaaqaaiaa ikdacaWGYbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaaGymaiabgUcaRi aadkhadaWgaaWcbaGaaGimaaqabaaaaaqaaiaaikdacaWGibaakiaa wIJ=aiaadgeadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaamyqamaaBa aaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaamaalaaabaGa amOCamaaBaaaleaacaaIWaaabeaakmaabmaabaGaaGymaiabgUcaRi aadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGccaGLOaGaayzkaaaa baGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaGcdaqadaqaaiaaig dacqGHRaWkcaWGYbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzk aaaaaaWcbaGaamOraiabgUcaRiaadIeaaOGaayjo+dGaamyqamaaDa aaleaacaWG5bGaamyEaaqaaiaaikdaaaGccqGHRaWkdaagaaqaamaa laaabaGaamOCamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadkhada WgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaamOCamaaBaaaleaacaaI 5aGaaGimaaqabaGcdaqadaqaaiaaigdacqGHRaWkcaWGYbWaaSbaaS qaaiaaicdaaeqaaaGccaGLOaGaayzkaaaaamaabmaabaGaaGOmaiaa dkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGymaaGaay jkaiaawMcaaaWcbaGaaGOmaiaad6eaaOGaayjo+dGaamyqamaaDaaa leaacaWG4bGaamyEaaqaaiaaikdaaaaaaa@8B69@
    There are two ways to determine HILL parameters by using Lankford parameters.
    • Strain ratio r 00 , r 45 , r 90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI 0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaa qabaaaaa@3F42@ (LAW32, LAW43, LAW72, LAW73)
    • Yield stress ratio R 11 , R 22 , R 33 , R 12 , R 13 , R 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaGymaaqabaGccaGGSaGaamOuamaaBaaaleaacaaI YaGaaGOmaaqabaGccaGGSaGaamOuamaaBaaaleaacaaIZaGaaG4maa qabaGccaGGSaGaamOuamaaBaaaleaacaaIXaGaaGOmaaqabaGccaGG SaGaamOuamaaBaaaleaacaaIXaGaaG4maaqabaGccaGGSaGaamOuam aaBaaaleaacaaIYaGaaG4maaqabaaaaa@487B@ (LAW74, LAW93)

Strain Ratio

The Lankford parameters r α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@ is the ratio of plastic strain in plane and plastic strain in thickness direction ε 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaa@39FA@ .(9) r α = d ε α + π / 2 d ε 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaak8qacqGH9aqp daWcaaWdaeaapeGaamizaiabew7aL9aadaWgaaWcbaWdbiabeg7aHj abgUcaRiabec8aWjaac+cacaaIYaaapaqabaaakeaapeGaamizaiab ew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaaaa@47D3@

Where, α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaaleaaqaaaaaaaaa Wdbiabeg7aHbaa@381F@ is the angle to the orthotropic direction 1.

r α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@ could be measured with different samples which cut in different angle with orthotropic direction 1. Like r 00 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaIWaGaaGimaaqabaaaaa@388E@ measured from tensile test in which the loading direction is along the orthotropic direction 1. r 90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaI5aGaaGimaaqabaaaaa@3897@ measured from tensile test in which the loading is perpendicular to orthotropic direction 1.

The strain ratio is the strain in width direction of sample to strain in thickness direction of sample.


Figure 10.
In this case, the HILL parameters are:(10) F = r 00 r 90 ( r 00 + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqa aiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaOGaaiikaiaadkhada WgaaWcbaGaaGimaiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaa aa@4322@ (11) G = 1 ( r 00 + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey ypa0ZaaSaaaeaacaaIXaaabaGaaiikaiaadkhadaWgaaWcbaGaaGim aiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaaaa@3E93@ (12) H = r 00 ( r 00 + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqa aiaacIcacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiabgUcaRi aaigdacaGGPaaaaaaa@407A@ (13) N = ( 1 + 2 r 45 ) ( r 00 + r 90 ) 2 r 90 ( r 00 + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0ZaaSaaaeaacaGGOaGaaGymaiabgUcaRiaaikdacaWGYbWaaSba aSqaaiaaisdacaaI1aaabeaakiaacMcacaGGOaGaamOCamaaBaaale aacaaIWaGaaGimaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiMda caaIWaaabeaakiaacMcaaeaacaaIYaGaamOCamaaBaaaleaacaaI5a GaaGimaaqabaGccaGGOaGaamOCamaaBaaaleaacaaIWaGaaGimaaqa baGccqGHRaWkcaaIXaGaaiykaaaaaaa@4F27@

Here, G + H = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey 4kaSIaamisaiabg2da9iaaigdaaaa@3A9B@ .

In LAW32, LAW43, and LAW73, the HILL criteria is:(14) σ e q = A 1 σ 1 2 + A 2 σ 2 2 A 3 σ 1 σ 2 + A 12 σ 12 2
R = r 00 + 2 r 45 + r 90 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaey ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiab gUcaRiaaikdacaWGYbWaaSbaaSqaaiaaisdacaaI1aaabeaakiabgU caRiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaaGinaaaa aaa@437F@ H = R 1 + R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey ypa0ZaaSaaaeaacaWGsbaabaGaaGymaiabgUcaRiaadkfaaaaaaa@3B8D@
A 1 = H ( 1 + 1 r 00 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiab gUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGimaiaaic daaeqaaaaaaOGaayjkaiaawMcaaaaa@407B@ A 2 = H ( 1 + 1 r 90 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiab gUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGyoaiaaic daaeqaaaaaaOGaayjkaiaawMcaaaaa@4085@
A 3 = 2 H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaaiodaaeqaaOGaeyypa0JaaGOmaiaadIeaaaa@3AA7@ A 12 = 2 H ( r 45 + 0.5 ) ( 1 r 00 + 1 r 90 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiaaigdacaaIYaaabeaakiabg2da9iaaikdacaWGibGaaiik aiaadkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGimai aac6cacaaI1aGaaiykamaabmaabaWaaSaaaeaacaaIXaaabaGaamOC amaaBaaaleaacaaIWaGaaGimaaqabaaaaOGaey4kaSYaaSaaaeaaca aIXaaabaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaaGccaGL OaGaayzkaaaaaa@4BBD@

They all request Lankford parameter (strain ratio) r 00 , r 45 , r 90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI 0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaa qabaaaaa@3F42@ and the HILL parameter A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaWGPbaabeaaaaa@37D6@ is automatically computed by Radioss.

Yield Stress Ratio

In LAW93, the yield stress ratio used is:(15) R i j = σ i j σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaa BaaaleaacaWGPbGaamOAaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaG imaaqabaaaaaaa@4076@
To get yield stress ratio R i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38D7@ , yield stress in two loading cases need to be measured.
  • Yield stress σ 11 , σ 22 , σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa aGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG 4maaqabaaaaa@41A0@ from tensile test
  • Yield shear stress σ 12 , σ 13 , σ 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa aGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG 4maaqabaaaaa@41A0@ from shear test

In LAW93, if parameter input is used, then take initial stress parameter σ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMhaaeqaaaaa@38E4@ as reference yield stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@38A0@ . If curve input is used, then take the yield stress from curve as reference yield stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@38A0@ .

Four HILL parameters for shell are automatically computed by Radioss.(16) F = 1 2 ( 1 R 22 2 + 1 R 33 2 1 R 11 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym aaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaG4maiaa iodaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk fadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaakiaacMcaaaa@489E@ (17)

G = 1 2 ( 1 R 33 2 + 1 R 11 2 1 R 22 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym aaqaaiaadkfadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaa igdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk fadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaakiaacMcaaaa@489F@

(18) H = 1 2 ( 1 R 22 2 + 1 R 11 2 1 R 33 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym aaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaa igdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk fadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaakiaacMcaaaa@48A0@ (19) N = 3 2 R 12 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maalaaabaGaaG4maaqaaiaaikdacaWGsbWaa0baaSqaaiaaigda caaIYaaabaGaaGOmaaaaaaaaaa@3C90@
In LAW74, yield stress ratio R i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38D7@ is used with yield stress σ 11 , σ 22 , σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa aGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG 4maaqabaaaaa@41A0@ and σ 12 , σ 13 , σ 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa aGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG 4maaqabaaaaa@41A0@ input directly, and then six HILL parameters for solid are automatically computed by Radioss.
F = 1 2 ( 1 σ 22 2 + 1 σ 33 2 1 σ 11 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaaaa@4BF9@ G = 1 2 ( 1 σ 22 2 + 1 σ 33 2 1 σ 11 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaaaa@4BFA@
H = 1 2 ( 1 σ 22 2 + 1 σ 33 2 1 σ 11 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaaaa@4BFB@ L = 1 2 σ 23 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI YaGaaG4maaqaaiaaikdaaaaaaaaa@3DE1@
M = 1 2 σ 31 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI ZaGaaGymaaqaaiaaikdaaaaaaaaa@3DE1@ N = 1 2 σ 12 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI XaGaaGOmaaqaaiaaikdaaaaaaaaa@3DE1@

For shell element, take M = N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey ypa0JaamOtaaaa@3909@ and L = N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey ypa0JaamOtaaaa@3909@ .