Bergstrom-Boyce (/MAT/LAW95)

この材料則は、エラストマー状の材料の非線形時間依存を予測するための構成モデルです。非線形粘弾性時間依存材料の応答を表すために、超弾性材料応答およびBergstrom-Boyce材料モデルに多項式材料モデルを使用します。

この材料則はソリッド要素とのみ適合性があります。

材料の応答は、2つの並列ネットワークAとBを用いて表すことができます。ネットワークAは、非線形超弾性コンポーネントを伴う均衡ネットワークです。ネットワークBでは、非線形超弾性コンポーネントは非線形粘弾性流れ要素と直列であり、したがって、時間依存のネットワークです。


図 1.

材料パラメータ

両方のネットワークで、超弾性コンポーネントに同じ多項式ひずみエネルギー密度定式化が用いられます。ネットワークBでは、係数 S b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaadkgaaeqaaaaa@384A@ によってスケーリングされます。ひずみエネルギー密度はその後、ネットワークの超弾性コンポーネント用に書き出されます:(1)
W A = i + j = 1 3 C i j ( I ¯ 1 3 ) i ( I ¯ 2 3 ) j + i = 1 3 1 D i ( J 1 ) 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadgeaaeqaaOGaeyypa0ZaaabCaeaacaWGdbWaaSbaaSqa aiaadMgacaWGQbaabeaakmaabmaabaGabmysayaaraWaaSbaaSqaai aaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqa baGaamyAaaaakiabgwSixpaabmaabaGabmysayaaraWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaamaaCaaaleqa baGaamOAaaaaaeaacaWGPbGaey4kaSIaamOAaiabg2da9iaaigdaae aacaaIZaaaniabggHiLdGccqGHRaWkdaaeWbqaamaalaaabaGaaGym aaqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaOWaaeWaaeaacaWGkb GaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaa dMgaaaaabaGaamyAaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLd aaaa@614A@
および(2)
W B = S b W A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadkeaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGIbaa beaakiabgwSixlaadEfadaWgaaWcbaGaamyqaaqabaaaaa@3F4B@
ここで、
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@
I ¯ 2 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcuaH7oaBgaqeamaaDaaa leaacaaIXaaabaGaeyOeI0IaaGOmaaaakiabgUcaRiqbeU7aSzaara Waa0baaSqaaiaaikdaaeaacqGHsislcaaIYaaaaOGaey4kaSIafq4U dWMbaebadaqhaaWcbaGaaG4maaqaaiabgkHiTiaaikdaaaaaaa@482F@
λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@
C i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3930@ および D i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadMgaaeqaaaaa@3842@
材料パラメータ
超弾性コンポーネントCauchy応力は次のように計算されます:(3)
σ i = λ i J W λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaamyAaaqabaaakeaacaWGkbaaamaalaaabaGaeyOaIyRaam4vaa qaaiabgkGi2kabeU7aSnaaBaaaleaacaWGPbaabeaaaaaaaa@4421@
全応力は、ネットワークAとネットワークBの応力の和です。


図 2.

σ= σ ¯ A + σ ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jafq4WdmNbaebadaWgaaWcbaGaamyqaaqabaGccqGHRaWkcuaH dpWCgaqeamaaBaaaleaacaWGcbaabeaaaaa@3F47@

W B = S b W A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadkeaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGIbaa beaakiabgwSixlaadEfadaWgaaWcbaGaamyqaaqabaaaaa@3F4B@ なので、 σ ¯ B = S b σ ¯ A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae badaWgaaWcbaGaamOqaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaa dkgaaeqaaOGaeyyXICTafq4WdmNbaebadaWgaaWcbaGaamyqaaqaba aaaa@40E1@ で、全応力は σ = ( 1 + S b ) σ ¯ A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaaiikaiaaigdacqGHRaWkcaWGtbWaaSbaaSqaaiaadkgaaeqa aOGaaiykaiabgwSixlqbeo8aZzaaraWaaSbaaSqaaiaadgeaaeqaaa aa@42C2@ となります。

たとえば1つの引張試験で、 S b = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGIbaabeaakiabg2da9iaaikdaaaa@39AE@ を使用すると、応力は、粘度を考慮しない(すなわち超弾性のみを考慮する)ケースの3倍となります。


図 3.
C i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3930@ の特別な値について、多項式モデルは下記の材料モデルに縮小することが可能です。
  • Yeoh:

    j = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2 da9iaaicdaaaa@38A5@

    ここで、 C 10 , C 20 , C 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EA8@ は、非ゼロ。


    図 4.
  • Mooney-Rivlin:

    i + j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgU caRiaadQgacqGH9aqpcaaIXaaaaa@3A76@

    ここで、 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EA8@ C 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EA8@ は非ゼロ、 D 2 = D 3 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIYaaabeaakiabg2da9iaadseadaWgaaWcbaGaaG4maaqa baGccqGH9aqpcaaIWaaaaa@3C33@

  • Neo-Hookean:

    C 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaI YaGaaGimaaqabaGccaGGSaGaam4qamaaBaaaleaacaaIZaGaaGimaa qabaaaaa@3EA8@ D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A6@ のみが非ゼロ。

ここで、
C i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3930@ および D i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadMgaaeqaaaaa@3842@
準-静的材料試験データ用にカーブフィッティングを行うことで計算することのできる材料パラメータ。

RD-E: 5600 超弾性材料と曲線入力には、Mooney-RivlinおよびYeoh材料モデル用のカーブフィッティングの例が含まれています。 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A6@ は、体積弾性係数から計算されるか、もしくは空白のままとされます。

初期せん断弾性係数と体積弾性係数は次のように計算されます:(4)
μ=2( S b +1 )( C 10 + C 01 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaaIYaWaaeWaaeaacaWGtbWaaSbaaSqaaiaadkgaaeqaaOGa ey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGaam4qamaaBaaale aacaaIXaGaaGimaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaicda caaIXaaabeaaaOGaayjkaiaawMcaaaaa@4643@
および(5)
K= 2 D 1 ( 1+ S b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbGaey ypa0ZaaSaaaeaacaaIYaaabaGaamiramaaBaaaleaacaaIXaaabeaa aaGcdaqadaqaaiaaigdacqGHRaWkcaWGtbWaaSbaaSqaaiaadkgaae qaaaGccaGLOaGaayzkaaaaaa@3FD6@

材料の体積弾性係数が既知である場合、 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A6@ は計算することが可能です。もしくは、 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A6@ =0である場合、非圧縮性材料が仮定されます。

粘性(速度)効果

ネットワークBでの有効クリープひずみ速度は、次のように与えられます:(6)
ε ˙ B v =A ( λ ˜ 1+ξ ) C σ ¯ B τ ref M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaDaaaleaacaWGcbaabaGaamODaaaakiabg2da9iaadgeadaqa daqaaiqbeU7aSzaaiaGaeyOeI0IaaGymaiabgUcaRiabe67a4bGaay jkaiaawMcaamaaCaaaleqabaGaam4qaaaakmaalaaabaGafq4WdmNb aebadaWgaaWcbaGaamOqaaqabaaakeaacqaHepaDdaWgaaWcbaGaam OCaiaadwgacaWGMbaabeaaaaGcdaahaaWcbeqaaiaad2eaaaaaaa@4D1A@
ここで、
λ ˜ = I ¯ 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga acaiabg2da9maakaaabaWaaSaaaeaaceWGjbGbaebadaWgaaWcbaGa aGymaaqabaaakeaacaaIZaaaaaWcbeaaaaa@3BE7@
σ ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qeamaaBaaaleaacaWGcbaabeaaaaa@392D@
ネットワークBでの有効応力。
A , ξ , M , C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaai ilaiaaysW7cqaH+oaEcaGGSaGaaGjbVlaad2eacaGGSaGaaGjbVlaa doeaaaa@4139@ そして τ r e f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaamOCaiaadwgacaWGMbaabeaaaaa@3B1C@
入力材料パラメータ

材料定数 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@ M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@ および C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@ は、リファレンスガイドで定義されている特定の実数範囲に制限されます。制限されたデータが入手可能な場合、試行錯誤法1を使ってこれらの定数を決定することができます。デフォルト値 ξ , M , C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaai ilaiaaysW7cqaH+oaEcaGGSaGaaGjbVlaad2eacaGGSaGaaGjbVlaa doeaaaa@4139@ S b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaaS baaSqaaiaadkgaaeqaaaaa@384A@ =1.6; および A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@ =5で開始します。続いて、少なくとも1つのひずみ速度についてモデルの予測と実験データとを比較し、 A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BC@ を調整して、ひずみ速度データにフィットするものを得ます。

参考文献

1 Bergström, J. S., and M. C. Boyce."Constitutive modeling of the large strain time-dependent behavior of elastomers."Journal of the Mechanics and Physics of Solids 46, no. 5 (1998): 931-954