弾塑性材料

Johnson-Cook(/MAT/LAW2)

LAW2には、応力計算のための3つのパートがあります。


図 1.
  • 塑性ひずみの影響
  • ひずみ速度の影響
  • 温度変化の影響

材料パラメータ

LAW2で材料パラメータを入力する方法は2つあります。
  • Iflag=0: Johnson-Cookパラメータa MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@b MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@n MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@: アクティブ
  • Iflag=1: 降伏応力、UTS(公称応力)、またはUTSでのひずみによる、新しい簡素化された入力
Iflag= 0
(1) σ=a+bεpn MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyypa0JaamyyaiabgUcaRiaadkgacqGHflY1cqaH1oqzdaWgaaWcbaGaamiCaaqabaGcdaahaaWcbeqaaiaad6gaaaaaaa@41AB@
ここで、
a MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
材料試験で読み取られ、真応力に変換される可能性のある降伏応力です。
b MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@ および n MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@
材料パラメータ材料の応力-ひずみ曲線のフィッティングにより、これら2つのパラメータが求められます。試験から生成された応力-ひずみ曲線がない場合は、式 2 および式 3を使用して、b MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@n MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@を計算するためには2つの状態(2つの応力-ひずみポイント)が必要となります。最初の点はネッキングポイントで選ばれ(まず、 Rmの取得を試みる)、次にb MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@n MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@が、その結果は選ばれた点によって異なるために、曲線のそれぞれの他の点に対して計算され、平均化されます。(2) n=ln(σ1aσ2a)ln(ε1ε2) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9maalaaabaGaciiBaiaac6gadaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaamyyaaqaaiabeo8aZnaaBaaaleaacaaIYaaabeaakiabgkHiTiaadggaaaaacaGLOaGaayzkaaaabaGaciiBaiaac6gadaqadaqaamaalaaabaGaeqyTdu2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaaaaa@4D3A@ (3) b=σ1aε1n MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaamyyaaqaaiabew7aLnaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaamOBaaaaaaaaaa@4033@
このテストの目的は引張り試験結果を用いて材料則のパラメーターを引き出す方法を提案することです。
Iflag = 1
この新しい入力では、ネッキングポイントでの降伏応力(σy), 引張り強さ(UTS)および工学ひずみ(εUTS MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadwfacaWGubGaam4uaaqabaaaaa@3A55@)が必要です。この新しい入力により、Radiossは自動的にa MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@b MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@およびn MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DD@の等価値を計算します。


図 2. 引張試験

ひずみ速度

ひずみ速度は、引張または破壊における衝突パフォーマンスで、材料特性に大きな影響を与えます。Johnson-Cook理論では、降伏応力は直接ひずみ速度の影響を受け、次のように表されます:(4) σ=(a+bεpn)(1+clnε˙ε˙0) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyypa0ZaaeWaaeaacaWGHbGaey4kaSIaamOyaiabgwSixlabew7aLnaaBaaaleaacaWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMcaamaabmaabaGaaGymaiabgUcaRiaadogaciGGSbGaaiOBamaalaaabaGafqyTduMbaiaaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaaaaa@4D90@
一般に、試験ひずみ速度が増加すると降伏応力は増加します。ひずみ速度係数c MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DF@により、降伏応力の増加係数をスケーリングできます。c MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DF@=0の場合、またはε˙0=1030 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaaGimamaaCaaaleqabaGaaG4maiaaicdaaaaaaa@3CB6@ あるいはε˙ε˙0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaacqGHKjYOcuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaa@3BF2@の場合、ひずみ速度の影響もまた定義されません。


図 3.

温度変化

温度が上昇すると降伏応力は低下します。LAW2では、影響は(1T*m) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIXaGaeyOeI0IaamivamaaCaaaleqabaGaaiOkaiaad2gaaaaakiaawIcacaGLPaaaaaa@3BD8@ により考慮されます。(5) σ=(a+bεpn)(1+clnε˙ε˙0)(1T*m) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaeyypa0ZaaeWaaeaacaWGHbGaey4kaSIaamOyaiabgwSixlabew7aLnaaBaaaleaacaWGWbaabeaakmaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMcaamaabmaabaGaaGymaiabgUcaRiaadogaciGGSbGaaiOBamaalaaabaGafqyTduMbaiaaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislcaWGubWaaWbaaSqabeaacaGGQaGaamyBaaaaaOGaayjkaiaawMcaaaaa@5371@ ここで、(6) T*=TTrTmeltTr MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaiOkaaaakiabg2da9maalaaabaGaamivaiabgkHiTiaadsfadaWgaaWcbaGaamOCaaqabaaakeaacaWGubWaaSbaaSqaaiaad2gacaWGLbGaamiBaiaadshaaeqaaOGaeyOeI0IaamivamaaBaaaleaacaWGYbaabeaaaaaaaa@4455@
ここで、
Tmelt MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWGTbGaamyzaiaadYgacaWG0baabeaaaaa@3AC2@
溶融温度(単位はケルビン)
Tr MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBaaaleaacaWGYbaabeaaaaa@37F3@
室温(単位はケルビン)
(7) T=Ti+EintρCp(Volume) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2da9iaadsfadaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaWcaaqaaiaadweadaWgaaWcbaGaciyAaiaac6gacaGG0baabeaaaOqaaiabeg8aYjaadoeadaWgaaWcbaGaamiCaaqabaGcdaqadaqaaiaadAfacaWGVbGaamiBaiaadwhacaWGTbGaamyzaaGaayjkaiaawMcaaaaaaaa@4970@
ここで、
Eint MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaaciGGPbGaaiOBaiaacshaaeqaaaaa@39C6@
内部エネルギー
内部エネルギーの変化は、Johnson-Cook則で降伏応力に影響を与えます。

硬化係数

金属は降伏するまで変形し、その後一般には硬化します(降伏応力は増加)。材料により硬化の様子は異なります(等方硬化、移動硬化など)。これは非常に重要な材料特性でもあります(スプリングバックの場合)。

LAW2では、オプションChard(硬化係数)を使用して、材料にどの硬化モデルを使用するかを記述します。この機能はLAW36、43、44、57、60、66、73、74でも使用できます。

Chardの値は1~0です。等方モデルの場合はChard=0、移動Prager-Zieglerモデルの場合はChard=1、これら2つのモデルの間の硬化の場合は1と0の間となります。
Chard= 0: 等方性モデル
1次元のケースでは、材料は降伏応力後に強化されます。前回の引張りの最大応力がそれに続く荷重での降伏となり、この新しい降伏応力はそれに続く引張りおよび圧縮での降伏応力と同じになります。


図 4.
Chard= 1: 運動学的Prager-Zieglerモデル
Bauschinger効果(引張りによる硬化の後、圧縮による軟化が発生し、圧縮での平均降伏が低下する)をモデル化するには、移動硬化を使用します。


図 5.

弾塑性区分線形材料(/MAT/LAW36)

LAW36では、さまざまなひずみ速度に対してさまざまな塑性応力-ひずみ曲線を直接定義できます。

大きなひずみ速度の塑性応力-ひずみ曲線は、必ず小さなひずみ速度の塑性応力-ひずみ曲線より上になります。


図 6.

ヤング率

ヤング率は、オプションfct_IDEEinf、およびCEを使用して、除荷時に更新(低減)できます。この機能の使用により、ハイテン鋼のスプリングバックの精度(除荷相時)が向上します。この機能は材料LAW43、LAW57、LAW60、LAW74およびLAW78でも使用できます。
  • fct_IDEを使用したヤング率の更新(fct_IDE ≠ 0):


    図 7.
  • EinfおよびCEを使用したヤング率の更新(fct_IDE = 0):


    図 8.

材料の挙動

fct_IDpは、特定の材料における引張と圧縮の挙動の区別(圧力依存降伏)に使用されます。したがって、有効降伏応力は公称降伏応力に実際の圧力に対応する降伏係数を乗じることによって得られます。


図 9.

HILL材料

Radioss材料則では、LAW32、LAW43、LAW72、LAW73、LAW74、LAW78およびLAW93はHILL基準を使用します。

HILL基準

代表的なHILL基準は:
  • 3D等価HILL応力:(8) f=F(σyyσzz)2+G(σzzσxx)2+H(σxxσyy)2+2Lσyz2+2Mσzx2+2Nσxy2   =(G+H)σxx2+(F+H)σyy2+(F+G)σzz22Hσxxσyy2Fσyyσzz2Gσzzσxx+2Lσyz2+2Mσzx2+2Nσxy2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=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aiabeo8aZnaaDaaaleaacaWG4bGaamyEaaqaaiaaikdaaaaabeaaaaaa@DD25@
  • シェル要素:(9) f=Fσyy2+Gσxx2+H(σxxσyy)2+2Nσxy2=(G+H)σxx2+(F+H)σyy22Hσxxσyy+2Nσxy2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr0=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aiabeo8aZnaaDaaaleaacaWG5bGaamyEaaqaaiaaikdaaaGccqGHsisldaagaaqaaiaaikdacaWGibaaleaaaOGaayjo+dGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiabeo8aZnaaBaaaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaaiaaikdacaWGobaaleaaaOGaayjo+dGaeq4Wdm3aa0baaSqaaiaadIhacaWG5baabaGaaGOmaaaaaeqaaaaa@863E@

    ここで、F MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@G MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@H MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@L MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@M MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@およびN MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@は6つのHILL異方性パラメータ。シェル要素の場合、F MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@G MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@H MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@およびN MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@のみが、必要とされる4つのHILLパラメータです。

    LAW78ではHILL基準は:(10) φ(A)=1G+HAxx22r01+r02HAxxAyy+r0(1+r90)r90(1+r0)F+HAyy2+r0+r90r90(1+r0)(2r45+1)2NAxy2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAcaGGOaGaamyqaiaacMcacqGH9aqpdaagaaqaaiaaigdaaSqaaiaadEeacqGHRaWkcaWGibaakiaawIJ=aiabgwSixlaadgeadaqhaaWcbaGaamiEaiaadIhaaeaacaaIYaaaaOGaeyOeI0YaaGbaaeaadaWcaaqaaiaaikdacaWGYbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaaGymaiabgUcaRiaadkhadaWgaaWcbaGaaGimaaqabaaaaaqaaiaaikdacaWGibaakiaawIJ=aiaadgeadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaamyqamaaBaaaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaamaalaaabaGaamOCamaaBaaaleaacaaIWaaabeaakmaabmaabaGaaGymaiabgUcaRiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGccaGLOaGaayzkaaaabaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaGcdaqadaqaaiaaigdacqGHRaWkcaWGYbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaaaaaWcbaGaamOraiabgUcaRiaadIeaaOGaayjo+dGaamyqamaaDaaaleaacaWG5bGaamyEaaqaaiaaikdaaaGccqGHRaWkdaagaaqaamaalaaabaGaamOCamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaGcdaqadaqaaiaaigdacqGHRaWkcaWGYbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaaaamaabmaabaGaaGOmaiaadkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGymaaGaayjkaiaawMcaaaWcbaGaaGOmaiaad6eaaOGaayjo+dGaamyqamaaDaaaleaacaWG4bGaamyEaaqaaiaaikdaaaaaaa@8B69@
    ランクフォードのパラメータを使ってHILLパラメータを決定する方法が2つあります。
    • ひずみ比 r00,r45,r90 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaa@3F42@ (LAW32、LAW43、LAW72、LAW73)
    • 降伏応力比 R11,R22,R33,R12,R13,R23 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaaIXaGaaGymaaqabaGccaGGSaGaamOuamaaBaaaleaacaaIYaGaaGOmaaqabaGccaGGSaGaamOuamaaBaaaleaacaaIZaGaaG4maaqabaGccaGGSaGaamOuamaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGSaGaamOuamaaBaaaleaacaaIXaGaaG4maaqabaGccaGGSaGaamOuamaaBaaaleaacaaIYaGaaG4maaqabaaaaa@487B@ (LAW74、LAW93)

ひずみ比

ランクフォードパラメータrα MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@は、面内の塑性ひずみと厚み方向の塑性ひずみε33 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaa@39FA@との比率です。(11) rα=dεα+π/2dε33 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaamizaiabew7aL9aadaWgaaWcbaWdbiabeg7aHjabgUcaRiabec8aWjaac+cacaaIYaaapaqabaaakeaapeGaamizaiabew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaaaa@47D3@

ここで、α MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaaleaaqaaaaaaaaaWdbiabeg7aHbaa@381F@は、直交異方性方向1に対して成す角度です。

rα MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@ は、直交方向1の異なる角度で切断した異なる試料で測定することができます。r00 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaaaaa@388E@ は荷重方向が直交方向1に沿った引張試験から測定され、 r90 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaa@3897@ は荷重が直交方向1に直交する引張試験から測定されます。

ひずみ比は、試料の幅方向のひずみと試料の厚さ方向のひずみとの比です。


図 10.
この場合、HILLパラメータは:(12) F=r00r90(r00+1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaeyypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqaaiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaOGaaiikaiaadkhadaWgaaWcbaGaaGimaiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaa aa@4322@ (13) G=1(r00+1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaiikaiaadkhadaWgaaWcbaGaaGimaiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaaaa@3E93@ (14) H=r00(r00+1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqaaiaacIcacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiabgUcaRiaaigdacaGGPaaaaaaa@407A@ (15) N=(1+2r45)(r00+r90)2r90(r00+1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaeyypa0ZaaSaaaeaacaGGOaGaaGymaiabgUcaRiaaikdacaWGYbWaaSbaaSqaaiaaisdacaaI1aaabeaakiaacMcacaGGOaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiMdacaaIWaaabeaakiaacMcaaeaacaaIYaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaGccaGGOaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaGccqGHRaWkcaaIXaGaaiykaaaaaaa@4F27@

ここで、G+H=1 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey4kaSIaamisaiabg2da9iaaigdaaaa@3A9B@

LAW32、LAW43およびLAW73では、HILL基準は:(16) σeq=A1σ12+A2σ22A3σ1σ2+A12σ122
R=r00+2r45+r904 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaeyypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiabgUcaRiaaikdacaWGYbWaaSbaaSqaaiaaisdacaaI1aaabeaakiabgUcaRiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaaGinaaaaaaa@437F@ H=R1+R MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0ZaaSaaaeaacaWGsbaabaGaaGymaiabgUcaRiaadkfaaaaaaa@3B8D@
A1=H(1+1r00) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGimaiaaicdaaeqaaaaaaOGaayjkaiaawMcaaaaa@407B@ A2=H(1+1r90) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaaaaOGaayjkaiaawMcaaaaa@4085@
A3=2H MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGOmaiaadIeaaaa@3AA7@ A12=2H(r45+0.5)(1r00+1r90) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaaikdacaWGibGaaiikaiaadkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGimaiaac6cacaaI1aGaaiykamaabmaabaWaaSaaaeaacaaIXaaabaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaaGccaGLOaGaayzkaaaaaa@4BBD@

これらはすべてランクフォードパラメータ(ひずみ比)r00,r45,r90 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBaaaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaa@3F42@ を要求し、HILLパラメータAi MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBaaaleaacaWGPbaabeaaaaa@37D6@ Radiossによって自動的に計算されます。

降伏応力比

LAW93では、使用される降伏応力比は:(17) Rij=σijσ0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaaa@4076@
降伏応力比 Rij MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@38D7@ を得るには2つの荷重ケースでの降伏応力を測定する必要があります。
  • 引張試験からの降伏応力 σ11,σ22,σ33 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaaaa@41A0@
  • せん断試験からの降伏せん断応力 σ12,σ13,σ23 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaaaaa@41A0@

LAW93では、パラメータ入力が使用されている場合は初期応力パラメータσy MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMhaaeqaaaaa@38E4@ を基準降伏応力 σ0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaaa@38A0@ とします。曲線入力を使用する場合は、曲線からの降伏応力を基準降伏応力 σ0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaaa@38A0@ とします。

シェル用の4つのHILLパラメータがRadiossによって自動的に計算されます。(18) F=12(1R222+1R3321R112) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaakiaacMcaaaa@489E@ (19) G=12(1R332+1R1121R222) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaakiaacMcaaaa@489F@ (20) H=12(1R222+1R1121R332) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaakiaacMcaaaa@48A0@ (21) N=32R122 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2da9maalaaabaGaaG4maaqaaiaaikdacaWGsbWaa0baaSqaaiaaigdacaaIYaaabaGaaGOmaaaaaaaaaa@3C90@
LAW74では、降伏応力比Rij MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@38D7@ は降伏応力σ11,σ22,σ33 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG4maaqabaaaaa@41A0@ およびσ12,σ13,σ23 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaaaaa@41A0@ 入力と直接使用され、ソリッド用の6つのHILLパラメータがRadiossによって自動的に計算されます。
F=12(1σ222+1σ3321σ112) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaaGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaiodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4BF9@ G=12(1σ222+1σ3321σ112) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaaGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaiodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4BFA@
H=12(1σ222+1σ3321σ112) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaaGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaiodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4BFB@ L=12σ232 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaIYaGaaG4maaqaaiaaikdaaaaaaaaa@3DE1@
M=12σ312 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaIZaGaaGymaaqaaiaaikdaaaaaaaaa@3DE1@ N=12σ122 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaaaaaaa@3DE1@

シェル要素の場合、M=N MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaeyypa0JaamOtaaaa@3909@ L=N MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaeyypa0JaamOtaaaa@3909@ とします。