Ogden Materials

Hyperelastic materials can be used to model the isotropic, nonlinear elastic behavior of rubber, polymers, and similar materials. These materials are nearly incompressible in their behavior and can be stretched to very large strains.

In Radioss, material laws LAW42, LAW62, LAW69, LAW82, and LAW88 utilize different strain energy density functions of the Ogden material model 1 to model hyperelastic materials. 2

Material Definition

Stretch (also called stretch ratio) λ is the ratio of final length and initial length. It is used for materials with large deformations. For a cube in tension:
ε 1 = Δ l l 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaqGuoGaamiBaaqa aiaadYgadaWgaaWcbaGaaGimaiaaigdaaeqaaaaaaaa@3E41@
Engineering strain (also called nominal strain) in direction 1
λ 1 = l 1 l 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWGSbWaaSbaaSqa aiaaigdaaeqaaaGcbaGaamiBamaaBaaaleaacaaIWaGaaGymaaqaba aaaaaa@3E25@
Stretch in direction 1


Figure 1.
Thus, strain and stretch are related as:(1) λ = 1 + ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey ypa0JaaGymaiabgUcaRiabew7aLbaa@3BF4@
Principal stretch λ i can be used to describe the volumetric deformation by calculating the relative volume, J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaaaa@36C5@ , computed as: (2) J = V V 0 = l 1 l 2 l 3 l 01 l 02 l 03 = λ 1 λ 2 λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maalaaabaGaamOvaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaa aOGaeyypa0ZaaSaaaeaacaWGSbWaaSbaaSqaaiaaigdaaeqaaOGaey yXICTaamiBamaaBaaaleaacaaIYaaabeaakiabgwSixlaadYgadaWg aaWcbaGaaG4maaqabaaakeaacaWGSbWaaSbaaSqaaiaaicdacaaIXa aabeaakiabgwSixlaadYgadaWgaaWcbaGaaGimaiaaikdaaeqaaOGa eyyXICTaamiBamaaBaaaleaacaaIWaGaaG4maaqabaaaaOGaeyypa0 Jaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaeq4UdW2aaSba aSqaaiaaikdaaeqaaOGaeyyXICTaeq4UdW2aaSbaaSqaaiaaiodaae qaaaaa@5FC1@
For an incompressible material the volume should not change and thus J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaaaa@36C5@ =1 and thus, the stretch can be calculated for the following material tests.
  • Uniaxial test:

    λ 1 = λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBcaaMe8oaaa@3D4B@ and λ 2 2 = λ 3 2 = 1 λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGH9aqp cqaH7oaBdaWgaaWcbaGaaG4maaqabaGcdaahaaWcbeqaaiaaikdaaa GccqGH9aqpdaWcaaqaaiaaigdaaeaacqaH7oaBaaaaaa@421D@

  • Biaxial test:

    λ 1 = λ 2 = λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBdaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcqaH7oaBcaaMe8oaaa@40F7@ and λ 3 = λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaG4maaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiab gkHiTiaaikdaaaaaaa@3D96@

  • Planar (shear) test:

    λ 1 = λ ; λ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBcaaMe8Uaai4oaiaa ysW7cqaH7oaBdaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIXaaaaa@43FF@ and λ 2 = λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiab gkHiTiaaigdaaaGccaaMb8oaaa@3F28@

/MAT/LAW42 (Ogden)

This material model defines a hyperelastic, viscous, and incompressible material specified using the Ogden, Neo-Hookean, or Mooney-Rivlin material models. This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. This material can be used with shell and solid elements.

LAW42 uses the following strain energy density representation of the Ogden material model.(3) W ( λ 1 , λ 2 , λ 3 ) = p = 1 5 μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) + K 2 ( J 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaae WaaeaacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZa aabeaaaOGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaaaeaacqaH 8oqBdaWgaaWcbaGaamiCaaqabaaakeaacqaHXoqydaWgaaWcbaGaam iCaaqabaaaaOWaaeWaaeaadaqdaaqaaiabeU7aSbaadaWgaaWcbaGa aGymaaqabaGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabe aaaaGccqGHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaGOmaaqa baGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccq GHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaG4maaqabaGcdaah aaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccqGHsislca aIZaaacaGLOaGaayzkaaaaleaacaWGWbGaeyypa0JaaGymaaqaaiaa iwdaa0GaeyyeIuoakiabgUcaRmaalaaabaGaam4saaqaaiaaikdaaa WaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaaa@6C02@
Where,
W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbaaaa@373A@
Strain energy density
λ i
ith principal engineering stretch
J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@
Relative volume defined as: J = λ 1 λ 2 λ 3 = ρ 0 ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbGaey ypa0Jaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaeyyXICTaeq4UdW2aaSbaaSqaaiaaio daaeqaaOGaeyypa0ZaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqa baaakeaacqaHbpGCaaaaaa@4A3F@
λ ¯ i = J 1 3 λ i
Deviatoric stretch
α p and μ p
Material constants coefficient pairs.
Up to 5 material constant pairs can be defined.
The initial shear modulus μ and bulk modulus ( K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@372E@ ) are given by:(4) μ = p = 1 5 μ p α p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaSaaaeaadaaeWbqaaiabeY7aTnaaBaaaleaacaWGWbaabeaa kiabgwSixlabeg7aHnaaBaaaleaacaWGWbaabeaaaeaacaWGWbGaey ypa0JaaGymaaqaaiaaiwdaa0GaeyyeIuoaaOqaaiaaikdaaaaaaa@4720@
and(5) K = μ 2 ( 1 + ν ) 3 ( 1 2 ν )

Where, ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4gaaa@37AF@ is the Poisson's ratio and is only used for computing the bulk modulus.

Material Parameters

Parameters α p and μ p must be chosen so that initial shear modulus is:(6) μ= p=1 5 μ p α p 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaSaaaeaadaaeWbqaaiabeY7aTnaaBaaaleaacaWGWbaabeaa kiabgwSixlabeg7aHnaaBaaaleaacaWGWbaabeaaaeaacaWGWbGaey ypa0JaaGymaaqaaiaaiwdaa0GaeyyeIuoaaOqaaiaaikdaaaGaeyOp a4JaaGimaaaa@48E2@
For material stability, it is required that each material constant pair (7) μ p α p > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaamiCaaqabaGccqGHflY1cqaHXoqydaWgaaWcbaGaamiC aaqabaGccqGH+aGpcaaIWaaaaa@4015@

In general, the Ogden model can be used for strains up to 700%. The number of terms material pairs, α p and μ p , needed depends on the range of experimental data that is fit and curve fitting accuracy desired. In practice, 3 material pairs fit most data. If the material pairs are not known for a particular material, then a curve fit of uniaxial test data can be done in Radioss using LAW69 or via separate fitting software.

Neo-Hookean Model

A simple case of the Ogden material model is the Neo-Hooken model represented using the following equation for the strain energy density function:(8) W= C 10 ( I 1 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9iaadoeadaWgaaWcbaGaaGymaiaaicdaaeqaaOWaaeWaaeaacaWG jbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawM caaaaa@3F3E@
Where,
I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIXaaabeaaaaa@37AC@
The first invariants of the right Cauchy-Green Tensor
C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@3860@
Material constant

This representation can be derived from the LAW42 Ogden strain energy density function when:

μ 1 =2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ ; α 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ , and μ 2 = α 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeqySde2aaSbaaSqaaiaaikda aeqaaOGaeyypa0JaaGimaaaa@3DF6@

The Neo-Hookean model is a simple model that is typically only accurate for strains less than 20%.

Mooney-Rivlin Model

A slightly more complex case of the LAW42 Ogden material model is the Mooney-Rivlin model, which can be represented using the following equation for the strain energy density function:(9) W= C 10 ( I 1 3 )+ C 01 ( I 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9iaadoeadaWgaaWcbaGaaGymaiaaicdaaeqaaOWaaeWaaeaacaWG jbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maaGaayjkaiaawM caaiabgUcaRiaadoeadaWgaaWcbaGaaGimaiaaigdaaeqaaOWaaeWa aeaacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaaG4maaGaay jkaiaawMcaaaaa@4786@
Where,
I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIXaaabeaaaaa@37AC@ and I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIXaaabeaaaaa@37AC@
The first and second invariants of the right Cauchy-Green Tensor
C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@3860@ and C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaaaaa@3860@
Material constants

This representation can be derived from the LAW42 Ogden strain energy density function when:

μ 1 = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , μ 2 = 2 C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGOmaiabgwSixlaa doeadaWgaaWcbaGaaGimaiaaigdaaeqaaaaa@4001@ , α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ , and α 2 = -2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@

Mooney-Rivlin constants are available from a material supplier or testing company. If they are not available, then a curve fit of uniaxial test data can be done in Radioss using LAW69 or via separate fitting software. The Mooney-Rivlin material law is accurate for strains up to 100%.

Poisson's Ratio and Material Incompressibility

If a material is truly incompressible, then ν = 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaiaac6cacaaI1aaaaa@3AE0@ . However, in practice is not possible to use because that would result in an infinite bulk modulus, an infinite speed of sound, and thus an infinitely small solid element Time Step. (10) K = μ 2 ( 1 + ν ) 3 ( 1 2 ν ) = μ 2 ( 1 + ν ) 3 ( 1 2 * 0.5 ) = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9iabeY7aTjabgwSixpaalaaabaGaaGOmamaabmaabaGaaGymaiab gUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaaiodadaqadaqaaiaaig dacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaaiabg2da9iab eY7aTjabgwSixpaalaaabaGaaGOmamaabmaabaGaaGymaiabgUcaRi abe27aUbGaayjkaiaawMcaaaqaaiaaiodadaqadaqaaiaaigdacqGH sislcaaIYaGaaiOkaiaaicdacaGGUaGaaGynaaGaayjkaiaawMcaaa aacqGH9aqpcqGHEisPaaa@5C83@

The effect of different Poisson’s ratio input can be seen in Figure 2. The largest difference in the results is at higher amounts of strain. The results will match the test data better when ν = 0.4997 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@ but this results in a time step that is 4 times lower than ν = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@ . Thus, to balance the computation time and accuracy it is recommended to use ν = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jh9qqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaIWaGaaiOlaiaaisdacaaI5aGaaGyoaiaaiEdaaaa@3E7A@ for incompressible rubber material.

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


Figure 2.

Higher values of the Poisson’s ratio may lead to a very small time step or divergence for explicit simulations.

In LAW42, material incompressibility is provided by using a penalty approach, which calculates the pressure proportional to a change in density.(11) P = K F s c a l e b l k f b l k ( J ) ( J 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4saiabgwSixlaadAeacaWGZbGaam4yaiaadggacaWGSbGa amyzamaaBaaaleaacaWGIbGaamiBaiaadUgaaeqaaOGaeyyXICTaci OzamaaBaaaleaacaWGIbGaamiBaiaadUgaaeqaaOWaaeWaaeaacaWG kbaacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWGkbGaeyOeI0IaaG ymaaGaayjkaiaawMcaaaaa@5293@
Where,
K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@372E@
Bulk modulus
J = V V 0 = m ρ 0 m ρ = ρ 0 ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9maalaaabaGaamOvaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaa aOGaeyypa0ZaaSaaaeaacaWGTbGaeqyWdi3aaSbaaSqaaiaaicdaae qaaaGcbaGaamyBaiabeg8aYbaacqGH9aqpdaWcaaqaaiabeg8aYnaa BaaaleaacaaIWaaabeaaaOqaaiabeg8aYbaaaaa@4771@
Relative volume which simplifies to relative density if mass is constant
f b l k ( J ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadkgacaWGSbGaam4AaaqabaGcdaqadaqaaiaadQeaaiaa wIcacaGLPaaaaaa@3CA0@
Bulk coefficient scale factor versus relative volume function
F s c a l e b l k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaam 4CaiaadogacaWGHbGaamiBaiaadwgadaWgaaWcbaGaamOyaiaadYga caWGRbaabeaaaaa@3EBE@
Abscissa scale factor for function f b l k ( J ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadkgacaWGSbGaam4AaaqabaGcdaqadaqaaiaadQeaaiaa wIcacaGLPaaaaaa@3CA0@
The bulk modulus ( K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@372E@ ) of hyperelastic materials is generally a very high value which provides the needed pressure-resistance to maintain the incompressibility condition ( J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@ =1). But if a material starts to compress ( J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@ < 1) then the bulk modulus can be increased by including the fct_IDblk input function which allows the scaling of the bulk coefficient value as a function of J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@ . By default, there is no scaling and; thus, if the function identifier is zero and the value of the bulk scaling function is equal to 1. It is advisable to output (/ANIM/BRICK/DENS) and review the material density of LAW42 components to make sure that the density variation is small, that is the value of J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@ is close to 1 and the material is incompressible.


Figure 3. Bulk Modulus Scale Factor Function fct_IDblk

Viscous (Rate) Effects

Viscous (rate) effects are modeled in LAW42 using a Maxwell model, which can be described in a simplified manner as a system of η springs with stiffness' G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaadMgaaeqaaaaa@3844@ and dampers η i :

law82_maxwell_model
Figure 4. Maxwell Model
The Maxwell model is represented using Prony series inputs ( G i , τ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakiaacYcacqaHepaDdaWgaaWcbaGaamyAaaqa baaaaa@3B76@ ). The hyperelastic initial shear modulus μ is the same as the long-term shear modulus G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiabg6HiLcqabaaaaa@38C7@ in the Maxwell model, and τ i is the relaxation time:(12) τ i = η i G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcaaqaaiabeE7aOnaaBaaa leaacaWGPbaabeaaaOqaaiaadEeadaWgaaWcbaGaamyAaaqabaaaaa aa@3F13@

The G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaadMgaaeqaaaaa@3844@ and τ i values must be positive.

/MAT/LAW62 (VISC_HYP)

A hyper visco-elastic material law in Radioss that can be used to model polymers and elastomers.

The hyperelastic behavior in this material law is defined using the following strain energy density function:(13) W( λ 1 , λ 2 , λ 3 )= i=1 N 2 μ i α i 2 ( λ 1 α i + λ 2 α i + λ 3 α i 3+ 1 β ( J α i β 1) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiaacI cacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2aaSba aSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZaaabe aakiaacMcacqGH9aqpdaaeWbqaamaalaaabaGaaGOmaiabeY7aTnaa BaaaleaacaWGPbaabeaaaOqaaiabeg7aHnaaBaaaleaacaWGPbaabe aakmaaCaaaleqabaGaaGOmaaaaaaGcdaqadaqaaiabeU7aSnaaBaaa leaacaaIXaaabeaakmaaCaaaleqabaGaeqySde2aaSbaaWqaaiaadM gaaeqaaaaakiabgUcaRiabeU7aSnaaBaaaleaacaaIYaaabeaakmaa CaaaleqabaGaeqySde2aaSbaaWqaaiaadMgaaeqaaaaakiabgUcaRi abeU7aSnaaBaaaleaacaaIZaaabeaakmaaCaaaleqabaGaeqySde2a aSbaaWqaaiaadMgaaeqaaaaakiabgkHiTiaaiodacqGHRaWkdaWcaa qaaiaaigdaaeaacqaHYoGyaaGaaiikaiaadQeadaahaaWcbeqaaiab gkHiTiabeg7aHnaaBaaameaacaWGPbaabeaaliabek7aIbaakiabgk HiTiaaigdacaGGPaaacaGLOaGaayzkaaaaleaacaWGPbGaeyypa0Ja aGymaaqaaiaad6eaa0GaeyyeIuoaaaa@7228@
Where,
W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36F3@
Strain energy density
λ i
i th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaWG0bGaamiAaaqabaaaaa@38F6@ principal stretch
J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36F3@
Relative volume defined in Equation 13
β= ν ( 12ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGycq GH9aqpdaWcaaqaaiabe27aUbqaamaabmaabaGaaGymaiabgkHiTiaa ikdacqaH9oGBaiaawIcacaGLPaaaaaaaaa@4072@
α i and μ i
Material constants coefficient pairs.
Up to 5 material constant pairs can be defined.
Poisson’s ratio must be 0 < ν < 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iabe27aUjabgYda8iaaicdacaGGUaGaaGynaaaa@3C9C@ . This law can be used to model compressible or sometimes called hyperfoam materials by defining a low Poisson’s ratio value.
Note: The μ i material coefficients are different, but can be converted using:(14) μ i LAW62 = μ i LAW42 α i LAW42 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda qhaaWcbaGaamyAaaqaaiaadYeacaWGbbGaam4vaiaaiAdacaaIYaaa aOGaeyypa0ZaaSaaaeaacqaH8oqBdaqhaaWcbaGaamyAaaqaaiaadY eacaWGbbGaam4vaiaaisdacaaIYaaaaOGaeyyXICTaeqySde2aa0ba aSqaaiaadMgaaeaacaWGmbGaamyqaiaadEfacaaI0aGaaGOmaaaaaO qaaiaaikdaaaaaaa@4EBD@

Viscous (Rate) Effects

Viscous (rate) effects are modeled in LAW62 using a Maxwell model which can be described in a simplified manner as a system of n springs with stiffness’ G i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaaaaa@37DC@ and dampers η i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaS baaSqaaiaadMgaaeqaaaaa@38BD@ .

law82_maxwell_model
Figure 5. Maxwell Model
The Maxwell model is represented using a Prony series with inputs. The initial shear modulus is:(15) G 0 = i=1 N μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0ZaaabCaeaacqaH8oqBdaWgaaWc baGaamyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGobaani abggHiLdaaaa@41A9@
The sum of μ i should be greater than 0.(16) G 0 = G + i G i
The stiffness ratio is:(17) γ = G G 0 = 1 i γ i (18) γ i = G i G 0
With, (19) γ i [ 0 , 1 ] , i γ i < 1
and the ground shear modulus(20) G 0 = G + i G i
The relative time, τ i must be positive:(21) τ i = η i G i
Note: When viscosity is included, the shear modulus in LAW62 is the initial shear modulus G 0 = i = 1 N μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0ZaaabCaeaacqaH8oqBdaWgaaWc baGaamyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGobaani abggHiLdaaaa@41A9@ which includes viscosity, but in LAW42 the shear modulus is the long-term shear modulus, which does not include viscosity G = p = 1 5 μ p α p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacqGHEisPaeqaaOGaeyypa0ZaaSaaaeaadaaeWbqaaiabeY7a TnaaBaaaleaacaWGWbaabeaakiabgwSixlabeg7aHnaaBaaaleaaca WGWbaabeaaaeaacaWGWbGaeyypa0JaaGymaaqaaiaaiwdaa0Gaeyye IuoaaOqaaiaaikdaaaaaaa@47DD@ .

/MAT/LAW69

This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and incompressible material specified using the Ogden or Mooney-Rivlin material models. Unlike LAW42, where the material parameters are input, this law computes the material parameters using test data from a uniaxial engineering stress-strain curve.

This material can be used with shell and solid elements.

The strain energy density formulation used depends on the law_ID:
  • law_ID = 1 (Ogden law):(22) W ( λ 1 , λ 2 , λ 3 ) = p = 1 5 μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) + K 2 ( J 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaae WaaeaacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZa aabeaaaOGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaaaeaacqaH 8oqBdaWgaaWcbaGaamiCaaqabaaakeaacqaHXoqydaWgaaWcbaGaam iCaaqabaaaaOWaaeWaaeaadaqdaaqaaiabeU7aSbaadaWgaaWcbaGa aGymaaqabaGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabe aaaaGccqGHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaGOmaaqa baGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccq GHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaG4maaqabaGcdaah aaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccqGHsislca aIZaaacaGLOaGaayzkaaaaleaacaWGWbGaeyypa0JaaGymaaqaaiaa iwdaa0GaeyyeIuoakiabgUcaRmaalaaabaGaam4saaqaaiaaikdaaa WaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaaa@6C02@
  • law_ID = 2 (Mooney-Rivlin law):(23) W = C 10 ( I 1 3 ) + C 01 ( I 2 3 )

Material Parameters

After reading the stress-strain curve (fct_ID1), Radioss calculates the corresponding material parameter pairs using a nonlinear least-square fitting algorithm. For the classic Ogden law, (law_ID=1), the calculated material parameter pairs are μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CE@ and α p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadchaaeqaaaaa@38B7@ where the value of p is defined via the N_pair input. The maximum value is N_pair=5 with a default value of 2. Usually no more than N_pair=3 is needed for a good fit.

For the Mooney-Rivlin law (law_ID =2), the material parameter C 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaaigdacaaIWaaabeaaaaa@38C7@ and C 01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaaigdacaaIWaaabeaaaaa@38C7@ are calculated remembering that μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CE@ and α p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadchaaeqaaaaa@38B7@ for the LAW42 Ogden law can be calculated using this conversion:

μ 1 = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , μ 2 = 2 C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGOmaiabgwSixlaa doeadaWgaaWcbaGaaGimaiaaigdaaeqaaaaa@4001@ , α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ , and α 2 = -2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@

The minimum test data input should be a uniaxial tension engineering stress strain curve. If uniaxial compression data is available, the engineering strain should increase monotonically from a negative value in compression to a positive value in tension. In compression, the engineering strain should not be less than -1.0 since -100% strain is physically not possible.

To improve the quality of the nonlinear least square fit, it is recommended that:
  • The experimental data curve represents a smooth monotonically increasing function with uniform distribution of abscissa points. The number of data points in the experimental data curve should be greater than the number of parameter pairs (N_pair).
  • The engineering strain is negative in compression and positive in tension. For compression test data, the engineering strain should be greater than -1.0 (100% compression maximum) but tension only stress strain data can also be used.
  • If N_pair ≥ 3, then the test data should cover at least 100% of the tensile strain and/or 50% of the compressive strain.
  • N_pair should not be set to a very large value to avoid instabilities in the fitting procedure.

This material law is stable when μ p α p > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaamiCaaqabaGccqaHXoqydaWgaaWcbaGaamiCaaqabaGc cqGH+aGpcaaIWaaaaa@3DCB@ (with p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbaaaa@3753@ =1,…5) is satisfied for parameter pairs for all loading conditions. By default, Radioss tries to fit the curve by accounting for these conditions (Icheck=2). If a proper fit cannot be found, then Radioss uses a weaker condition (Icheck=1: ), which ensures that the initial shear hyperelastic modulus ( μ ) is positive.

To determine how well the calculated material parameters represent the input test data, the Radioss Starter outputs an “averaged error of fitting” value which is recommended to not exceed 10%. For visual comparison, the stress-strain curve calculated from the strain energy density and calculated material parameters is also output by the Radioss Starter.

Due to the friction involved in a uniaxial compression test, it is usually more accurate to take equal biaxial tension test data and convert it to uniaxial compressive data using these formulas 3 which are valid for incompressible materials.(24) ε c = 1 ( ε b + 1 ) 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaadaqa daqaaiabew7aLnaaBaaaleaacaWGIbaabeaakiabgUcaRiaaigdaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0IaaGym aaaa@4379@ (25) σ c = σ b ( 1 + ε b ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamOy aaqabaGcdaqadaqaaiaaigdacqGHRaWkcqaH1oqzdaWgaaWcbaGaam OyaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaaaa@43FA@
Where,
ε c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaaaaa@391A@
Uniaxial engineering compressive stain
ε b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaam4yaaqabaaaaa@391A@
Equal biaxial engineering tension strain
σ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaaaaa@3936@
Uniaxial engineering compressive stress
σ b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaam4yaaqabaaaaa@3936@
Equal biaxial engineering tension stress

Material Incompressibility

Material LAW69 uses the same method to maintain incompressibility as LAW42. For additional information, refer to Poisson's Ratio and Material Incompressibility in LAW42.

Viscous (Rate) Effects

/VISC/PRONY must be used with LAW69 to include viscous effects. Alternatively, LAW69 could be used to extract the Ogden or Mooney-Rivlin parameters and then those parameters can be used in LAW42 with viscosity added.

/MAT/LAW82

This material model defines a hyperelastic, and incompressible material specified using the Ogden, Neo-Hookean, or Mooney-Rivlin material models. This law is generally used to model incompressible rubbers, polymers, foams, and elastomers.

This material can be used with shell and solid elements. As compared to LAW42 or LAW62, this law uses a different Ogden strain energy density formulation given in Equation 26. The LAW82 strain energy density formulation matches what is used in some other finite elements solver’s hyperelastic model and; thus, the material parameters for this form of the Ogden strain energy density are sometimes available from material suppliers or other sources.(26) W = i = 1 N 2 μ i α i 2 ( λ ¯ 1 α i + λ ¯ 2 α i + λ ¯ 3 α i 3 ) + i = 1 N 1 D i ( J 1 ) 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaaSaaaeaacaaIYaGaeqiVd02aaSbaaSqaaiaadMga aeqaaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOWaaWbaaSqabe aacaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaa0Ga eyyeIuoakmaabmaabaGafq4UdWMbaebadaWgaaWcbaGaaGymaaqaba GcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGPbaabeaaaaGccqGH RaWkcuaH7oaBgaqeamaaBaaaleaacaaIYaaabeaakmaaCaaaleqaba GaeqySde2aaSbaaWqaaiaadMgaaeqaaaaakiabgUcaRiqbeU7aSzaa raWaaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacqaHXoqydaWgaa adbaGaamyAaaqabaaaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaaiab gUcaRmaaqahabaWaaSaaaeaacaaIXaaabaGaamiramaaBaaaleaaca WGPbaabeaaaaGcdaqadaqaaiaadQeacqGHsislcaaIXaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaGaamyAaaaaaeaacaWGPbGaeyypa0 JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa@6A1C@
Where,
W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36D3@
Strain energy density
N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36D3@
Number of material constants α i , μ i and D i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaWGPbaabeaaaaa@37DA@
λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@
Deviatoric stretch
J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36D3@
Relative volume as defined in Equation 2
The initial shear modulus:(27) μ= i=1 N μ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaabCaeaacqaH8oqBdaWgaaWcbaGaamyAaaqabaaabaGaamyA aiabg2da9iaaigdaaeaacaWGobaaniabggHiLdaaaa@413C@
The Bulk Modulus is calculated as K = 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A49@ based on these rules:
  • If ν = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaaaa@396F@ , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ should be entered.
  • If ν 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey iyIKRaaGimaaaa@3A30@ , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ input is ignored and will be recalculated and output in the Starter output using:(28) D 1 = 3 ( 1 2 v ) μ ( 1 + v ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaG4maiaacIcacaaI XaGaeyOeI0IaaGOmaiaadAhacaGGPaaabaGaeqiVd0Maaiikaiaaig dacqGHRaWkcaWG2bGaaiykaaaaaaa@43E2@
  • If ν = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaaaa@396F@ and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ =0, a default value of ν = 0.475 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaiaac6cacaaI0aGaaG4naiaaiwdaaaa@3C5F@ is used and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ is calculated using Equation 28

Neo-Hookean Model

Like LAW42, LAW82 can also be simplified to a Neo-Hooken model by using:

μ 1 =2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , α 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ and μ 2 = α 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeqySde2aaSbaaSqaaiaaikda aeqaaOGaeyypa0JaaGimaaaa@3DF6@

Mooney-Rivlin Model

Like LAW42, LAW82 can also be simplified to a Mooney-Rivlin model by using:

μ 1 = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , μ 2 = 2 C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ and α 2 = -2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@

Viscous (Rate) Effects

/VISC/PRONY must be used with LAW82 to include viscous effects.

Drücker Condition Stability Check

In LAW42 and LAW69, the Drücker stability is automatically calculated by the Radioss Starter.

The Drücker stability condition checks if the change in the Kirchhoff stress corresponding to the infinitesimal change in the logarithmic strain (true strain) satisfies the following inequality.(29) i=1 3 d τ i d ε i >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeWbqaai aadsgacqaHepaDdaWgaaWcbaGaamyAaaqabaGccaWGKbGaeqyTdu2a aSbaaSqaaiaadMgaaeqaaOGaeyOpa4JaaGimaaWcbaGaamyAaiabg2 da9iaaigdaaeaacaaIZaaaniabggHiLdaaaa@465E@

Where, i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaaleaacaWGPbaaaa@3857@ =1,2,3 principal direction.

With the change in logarithmic strain(30) d ε i = d λ i λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaeq yTdu2aaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGa eq4UdW2aaSbaaSqaaiaadMgaaeqaaaGcbaGaeq4UdW2aaSbaaSqaai aadMgaaeqaaaaaaaa@42C1@
d τ i =Jd σ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaeq iXdq3aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOsaiabgwSixlaa dsgacqaHdpWCdaWgaaWcbaGaamyAaaqabaaaaa@431F@
The change of Kirchhoff stress
dτ=D:dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaaC iXdiabg2da9iaahseacaGG6aGaamizaiaahw7aaaa@3E5C@
Relationship between Kirchhoff stress and logarithmic strain
The Drücker stability condition will be:(31) d ε : D : d ε > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeabqaai aadsgacaWH1oGaaiOoaiaahseacaGG6aGaamizaiaahw7acqGH+aGp caaIWaaaleqabeqdcqGHris5aaaa@41DB@
Here D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHebaaaa@3835@ is tangential material stiffness matrix and it is also the slope of stress-strain curve:(32) D=[ D 11 D 12 D 13 D 21 D 22 D 23 D 31 D 32 D 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHebGaey ypa0ZaamWaaeaafaqabeWadaaabaGaamiramaaBaaaleaacaaIXaGa aGymaaqabaaakeaacaWGebWaaSbaaSqaaiaaigdacaaIYaaabeaaaO qaaiaadseadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaamiramaa BaaaleaacaaIYaGaaGymaaqabaaakeaacaWGebWaaSbaaSqaaiaaik dacaaIYaaabeaaaOqaaiaadseadaWgaaWcbaGaaGOmaiaaiodaaeqa aaGcbaGaamiramaaBaaaleaacaaIZaGaaGymaaqabaaakeaacaWGeb WaaSbaaSqaaiaaiodacaaIYaaabeaaaOqaaiaadseadaWgaaWcbaGa aG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faaaaa@5173@
For a stable material, it requests tangential material stiffness D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHebaaaa@3835@ be positive (slope of stress-strain curve is positive). The tangential material matrix D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHebaaaa@3835@ is positive if following conditions satisfied:(33) I 1 = t r ( D ) = D 11 + D 22 + D 33 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaamiDaiaadkhadaqadaqaaiaa hseaaiaawIcacaGLPaaacqGH9aqpcaWGebWaaSbaaSqaaiaaigdaca aIXaaabeaakiabgUcaRiaadseadaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaey4kaSIaamiramaaBaaaleaacaaIZaGaaG4maaqabaGccqGH+a GpcaaIWaaaaa@4A64@ (34) I 2 = D 11 D 22 + D 22 D 33 + D 33 D 11 D 23 2 D 13 2 D 12 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaamiramaaBaaaleaacaaIXaGa aGymaaqabaGccaWGebWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgU caRiaadseadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaamiramaaBaaa leaacaaIZaGaaG4maaqabaGccqGHRaWkcaWGebWaaSbaaSqaaiaaio dacaaIZaaabeaakiaadseadaWgaaWcbaGaaGymaiaaigdaaeqaaOGa eyOeI0IaamiramaaBaaaleaacaaIYaGaaG4maaqabaGcdaahaaWcbe qaaiaaikdaaaGccqGHsislcaWGebWaaSbaaSqaaiaaigdacaaIZaaa beaakmaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadseadaWgaaWcba GaaGymaiaaikdaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeyOpa4Ja aGimaaaa@5983@ (35) I 3 =det( D )>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaiodaaeqaaOGaeyypa0JaciizaiaacwgacaGG0bWaaeWa aeaacaWHebaacaGLOaGaayzkaaGaeyOpa4JaaGimaaaa@4112@
The Kirchhoff stress for Ogden model is:(36) τ i = p μ p [ λ ¯ i α p 1 3 ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p ) ]+K( J 2 J ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaamyAaaqabaGccqGH9aqpdaaeqbqaaiabeY7aTnaaBaaa leaacaWGWbaabeaakmaadmaabaGafq4UdWMbaebadaWgaaWcbaGaam yAaaqabaGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaa aaGccqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaamaabmaabaGafq 4UdWMbaebadaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaiabeg7a HnaaBaaameaacaWGWbaabeaaaaGccqGHRaWkcuaH7oaBgaqeamaaBa aaleaacaaIYaaabeaakmaaCaaaleqabaGaeqySde2aaSbaaWqaaiaa dchaaeqaaaaakiabgUcaRiqbeU7aSzaaraWaaSbaaSqaaiaaiodaae qaaOWaaWbaaSqabeaacqaHXoqydaWgaaadbaGaamiCaaqabaaaaaGc caGLOaGaayzkaaaacaGLBbGaayzxaaGaey4kaSIaam4samaabmaaba GaamOsamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadQeaaiaawIca caGLPaaaaSqaaiaadchaaeqaniabggHiLdaaaa@66C7@
Since D i j = τ i λ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9maalaaabaGaeyOaIyRa eqiXdq3aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaeq4UdW2aaS baaSqaaiaadQgaaeqaaaaaaaa@43DE@ , then for a given Ogden parameter α p , μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda WgaaWcbaGaamiCaaqabaGccaGGSaGaeqiVd02aaSbaaSqaaiaadcha aeqaaaaa@3DB9@ with conditions I 1 > 0 , I 2 > 0  and  I 3 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaigdaaeqaaOGaeyOpa4JaaGimaiaacYcacaWGjbWaaSba aSqaaiaaikdaaeqaaOGaeyOpa4JaaGimaiaabccacaqGHbGaaeOBai aabsgacaqGGaGaamysamaaBaaaleaacaaIZaaabeaakiabg6da+iaa icdaaaa@46A0@ , the strain range of material in Drücker stability could then be calculated.


Figure 6.

The Drücker stability criterion calculates the strain range where the material model will remain stable given a set of material parameters. This stability check cannot be made for every deformation but instead is commonly used to check material stability under uniaxial, biaxial and planar strain loading.

For example, using the following Ogden parameters:

μ 1 =13.99077258830 α 1 =3.788192935039 μ 2 =9.13454532223 α 2 =7.17617341059 μ 3 =8.904655103235 α 3 =7.27028137148 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqabeWaca aabaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaOGaeyypa0deaaaaaaaa a8qacaaIXaGaaG4maiaac6cacaaI5aGaaGyoaiaaicdacaaI3aGaaG 4naiaaikdacaaI1aGaaGioaiaaiIdacaaIZaGaaGimaaWdaeaacqaH XoqydaWgaaWcbaGaaGymaaqabaGccqGH9aqppeGaaG4maiaac6caca aI3aGaaGioaiaaiIdacaaIXaGaaGyoaiaaikdacaaI5aGaaG4maiaa iwdacaaIWaGaaG4maiaaiMdaa8aabaGaeqiVd02aaSbaaSqaaiaaik daaeqaaOGaeyypa0ZdbiabgkHiTiaaiMdacaGGUaGaaGymaiaaioda caaI0aGaaGynaiaaisdacaaI1aGaaG4maiaaikdacaaIYaGaaGOmai aaiodaa8aabaGaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaeyypa0Ja eyOeI0YdbiaaiEdacaGGUaGaaGymaiaaiEdacaaI2aGaaGymaiaaiE dacaaIZaGaaGinaiaaigdacaaIWaGaaGynaiaaiMdaa8aabaGaeqiV d02aaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZdbiaaiIdacaGGUaGaaG yoaiaaicdacaaI0aGaaGOnaiaaiwdacaaI1aGaaGymaiaaicdacaaI ZaGaaGOmaiaaiodacaaI1aaapaqaaiabeg7aHnaaBaaaleaacaaIZa aabeaakiabg2da98qacqGHsislcaaI3aGaaiOlaiaaikdacaaI3aGa aGimaiaaikdacaaI4aGaaGymaiaaiodacaaI3aGaaGymaiaaisdaca aI4aaaaaaa@8CBA@

Then the Drücker stability will be automatically checked in Radioss Starter, and results printed in Starter output file *0.out. This shows the strain at which instability can occur for the given Ogden parameters:
CHECK THE DRUCKER PRAGER STABILITY CONDITIONS   
      -----------------------------------------------
     MATERIAL LAW = OGDEN (LAW42) 
     MATERIAL NUMBER =         1
       TEST TYPE = UNIXIAL  
        COMPRESSION:    UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.3880000000000    
        TENSION:        UNSTABLE AT A NOMINAL STRAIN LARGER THAN  0.9709999999999    
       TEST TYPE = BIAXIAL  
        COMPRESSION:    UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.2880000000000    
        TENSION:        UNSTABLE AT A NOMINAL STRAIN LARGER THAN  0.2780000000000    
       TEST TYPE = PLANAR (SHEAR)
        COMPRESSION:    UNSTABLE AT A NOMINAL STRAIN LESS THAN -0.3680000000000    
        TENSION:        UNSTABLE AT A NOMINAL STRAIN LARGER THAN  0.5829999999999
Note: For a Neo-Hookean material with C 10 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGimaaqabaGccqGH+aGpcaaIWaaaaa@3A2C@ (or μ 1 > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyOpa4JaaGimaaaa@3A60@ ), the material is always stable and thus no critical value is found by the Drücker stability check.

For a Mooney-Rivlin material, the Drücker stability should be checked since C 01  or  μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaake aacaWGdbWaaSbaaSqaaiaaicdacaaIXaaabeaakiaabccacaqGVbGa aeOCaiaabccacqaH8oqBdaWgaaWcbaGaaGOmaaqabaaaaa@4110@ could be negative, which leads material instability.

/MAT/LAW88

This law utilizes a tabulated uniaxial tension and compression engineering stress and strain test data at different strain rates to model incompressible materials. It is only compatible with solid elements.

The material is based on the following Ogden’s strain energy density function but does not require curve fitting to extract material constants like most other hyperelastic material models. 4(37) W = i = 1 3 j = 1 m μ j α j ( λ ¯ i α j 1 ) d e v i a t o r i c p a r t + K ( J 1 ln J ) s p h e r i c a l p a r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0ZaaGbaaeaadaaeWbqaamaaqahabaWaaSaaaeaacqaH8oqBdaWg aaWcbaGaamOAaaqabaaakeaacqaHXoqydaWgaaWcbaGaamOAaaqaba aaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aaWc baGaamyAaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLdGcdaqada qaaiqbeU7aSzaaraWaaSbaaSqaaiaadMgaaeqaaOWaaWbaaSqabeaa cqaHXoqydaWgaaadbaGaamOAaaqabaaaaOGaeyOeI0IaaGymaaGaay jkaiaawMcaaaWcbaGaaWizaiaamwgacaaJ2bGaaWyAaiaamggacaaJ 0bGaaW4BaiaamkhacaaJPbGaaW4yaiaamccacaaJWbGaaWyyaiaamk hacaaJ0baakiaawIJ=aiabgUcaRmaayaaabaGaam4samaabmaabaGa amOsaiabgkHiTiaaigdacqGHsislciGGSbGaaiOBaiaadQeaaiaawI cacaGLPaaaaSqaaiaamohacaaJWbGaaWiAaiaamwgacaaJYbGaaWyA aiaamogacaaJHbGaaWiBaiaamccacaaJWbGaaWyyaiaamkhacaaJ0b aakiaawIJ=aaaa@7B85@

Instead, this law determines the Ogden function directly from the uniaxial engineering stress strain curve tabulated data.

Unlike other Ogden material laws, the Bulk Modulus must be input from either test data or extracted from Starter output of the LAW69 Ogden curve fit. When comparing results between LAW42 or LAW69 to LAW88, the same bulk modulus must be used.

Unloading Behavior

Unloading can be represented using an unloading function or by providing hysteresis and shape factor inputs to a damage model based on energy.

If using the damage model, the loading curves are used for both loading and unloading and the unloading stress tensor is reduced by:(38) σ = ( 1 D ) σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHdpGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamiraaGaayjkaiaawMcaaiaa ho8aaaa@3DFC@
with(39) D = ( 1 H y s ) ( 1 ( W c u r W max ) S h a p e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamisaiaadMhacaWGZbaacaGL OaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaai aadEfadaWgaaWcbaGaam4yaiaadwhacaWGYbaabeaaaOqaaiaadEfa daWgaaWcbaGaciyBaiaacggacaGG4baabeaaaaaakiaawIcacaGLPa aadaahaaWcbeqaaiaadofacaWGObGaamyyaiaadchacaWGLbaaaaGc caGLOaGaayzkaaaaaa@4F7D@
Where,
W c u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiaadogacaWG1bGaamOCaaqabaaaaa@3A3F@
Current energy
W max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3A3A@
Maximum energy corresponding to the quasi-static behavior
H y s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaam yEaiaadohaaaa@3921@ and S h a p e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaaleaacaWGtbGaam iAaiaadggacaWGWbGaamyzaaaa@3AE9@
Input by user
If an unloading curve is provided, these options are available:
Tension
Loading and Unloading
= 0


Figure 7.
Loading use loading function fct_IDLi and ε ˙ L i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaBaaaleaacaWGmbGaamyAaaqabaaaaa@39FA@
Unloading use unloading function fct_IDunL
= 1


Figure 8.
Loading and unloading all use loading function fct_IDLi and ε ˙ L i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaBaaaleaacaWGmbGaamyAaaqabaaaaa@39FA@
= -1


Figure 9.
Loading use loading function fct_IDLi and ε ˙ L i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaBaaaleaacaWGmbGaamyAaaqabaaaaa@39FA@
  • Tension

    Unloading use unloading function fct_IDunL

  • Compression

    Unloading use loading fct_IDLi and ε ˙ L i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH1oqzga GaamaaBaaaleaacaWGmbGaamyAaaqabaaaaa@39FA@

Viscous (Rate) Effects

Strain rate effects can be modeled by including engineering stress strain test data at different strain rates. This can be easier than calculating viscous parameters for traditional hyperelastic material models.

Conclusion

Make sure to use the material law that best fits the test data available.

For example, if minimal test data is available and the strains are not too large than the LAW42 Neo-Hookean Model could be used. If the loading state is known, then it is important to have test data that represents that stress state and make sure the material model fits that test data.

References

1 Ogden, R. W., and Non-linear Elastic Deformations. "Ellis Horwood." New York (1984)
2 Miller, Kurt. "Testing Elastomers for Hyperelastic Material Models in Finite Element Analysis" Axel Products, Inc., Ann Arbor, MI (2017). Last modified April 5, 2017

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

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

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

4 Kolling, S., P. A. Du Bois, D. J. Benson, and W. W. Feng. "A tabulated formulation of hyperelasticity with rate effects and damage." Computational Mechanics 40, no. 5 (2007): 885-899