Arruda-Boyce (/MAT/LAW92)

LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions.

It assumes that the chain molecules are located on the average along the diagonals of the cubic in principal stretch space.

Material Parameters

The strain energy density function is:(1)
W = μ i = 1 5 c i ( λ m ) 2 i 2 ( I ¯ 1 i 3 i ) W ( I ¯ 1 ) + 1 D ( J 2 1 2 + ln ( J ) ) U ( J ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0ZaaGbaaeaacqaH8oqBdaaeWbqaamaalaaabaGaam4yamaaBaaa leaacaWGPbaabeaaaOqaaiaacIcacqaH7oaBdaWgaaWcbaGaamyBaa qabaGccaGGPaWaaWbaaSqabeaacaaIYaGaamyAaiabgkHiTiaaikda aaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaaGynaaqdcqGHris5aO WaaeWaaeaaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaadMgaaaGc cqGHsislcaaIZaWaaWbaaSqabeaacaWGPbaaaaGccaGLOaGaayzkaa aaleaacaWGxbWaaeWaaeaaceWGjbGbaebadaWgaaadbaGaaGymaaqa baaaliaawIcacaGLPaaaaOGaayjo+dGaey4kaSYaaGbaaeaadaWcaa qaaiaaigdaaeaacaWGebaaamaabmaabaWaaSaaaeaacaWGkbWaaWba aSqabeaacaaIYaaaaOGaeyOeI0IaaGymaaqaaiaaikdaaaGaey4kaS IaciiBaiaac6gaciGGOaGaamOsaiaacMcaaiaawIcacaGLPaaaaSqa aiaadwfadaqadaqaaiaadQeaaiaawIcacaGLPaaaaOGaayjo+daaaa@692A@

The material constant, c i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaadMgaaeqaaaaa@3861@ are:

c 1 = 1 2 , c 2 = 1 20 , c 3 = 11 1050 , c 4 = 19 7000 , c 5 = 519 673750 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaGaaiilaiaaysW7caWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGOmaiaaicdaaaGaaiilaiaaysW7caWG JbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaaG ymaaqaaiaaigdacaaIWaGaaGynaiaaicdaaaGaaiilaiaaysW7caWG JbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaaG yoaaqaaiaaiEdacaaIWaGaaGimaiaaicdaaaGaaGjbVlaacYcacaaM e8Uaam4yamaaBaaaleaacaaI1aaabeaakiabg2da9maalaaabaGaaG ynaiaaigdacaaI5aaabaGaaGOnaiaaiEdacaaIZaGaaG4naiaaiwda caaIWaaaaaaa@625B@
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@
First strain invariant
λ i
i t h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaWG0bGaamiAaaqabaaaaa@38F6@ principal engineering stretch
A material with LAW92 can be defined in two different ways:
  • Parameter Input

    Shear modulus, bulk modulus and strain stretch ( μ , D , λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBca GGSaGaamiraiaacYcacqaH7oaBdaWgaaWcbaGaamyBaaqabaaaaa@3D10@ )

    Where, only the above 3 parameters with clear physical meaning are necessary to define the material.

    μ is shear modulus at zero strain.(2)
    D = 2 K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaSaaaeaacaaIYaaabaGaam4saaaaaaa@39CA@
    Where,
    K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C6@
    Bulk coefficient at zero strain
    λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaaaaa@3931@
    Defines the limit of stretch
    Also called locking stretch. It specifies the beginning of the hardening phase in tension (locking strain in tension). Default = 7.0.


    Figure 1. Locking Stretch λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaaaaa@3931@
    In parametric input, Poisson’s ratio is computed as:(3)
    ν = 3 K 2 μ 6 K + 2 μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpdaWcaaqaaiaaiodacaWGlbGaeyOeI0IaaGOmaiabeY7aTbqa aiaaiAdacaWGlbGaey4kaSIaaGOmaiabeY7aTbaaaaa@42FD@
  • When using function input, Poisson ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBaa a@3817@ and Itype must be defined. Itype defines which type of engineering stress strain test data that is being used as input.


    • Figure 2. Itype = 1: Uniaxial data test


    • Figure 3. Itype = 2: Equibiaxial data test


    • Figure 4. Itype = 3: Planar data test

Poisson's Ratio and Material Incompressibility

If function input is defined, then parameters μ , D , λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBca GGSaGaamiraiaacYcacqaH7oaBdaWgaaWcbaGaamyBaaqabaaaaa@3D10@ are ignored and Radioss will calculate the material constant by fitting the input function. A nonlinear least squares algorithm is used to fit the Arruda-Boyce parameters by Radioss. The curve fitting is performed using the assumption that Poisson’s value is close to 0.5, which means the material is incompressible. Similar to the other hyperelastic material models, Poisson ratio values closer to 0.5 result in high bulk modulus and a lower timestep. For a good balance between incompressibility and a reasonable timestep, a Poisson’s ratio value of 0.495 is recommended.

The material fitting information can be found in the Starter output file (*0000.out).


Figure 5. LAW92 Function Example
The fitting error and fitted material parameters are printed in the Starter output file.


Figure 6.

Viscous (Rate) Effects

/VISC/PRONY must be used with LAW92 to include viscous effects.

References

1 Arruda, E. M. and Boyce, M. C., 1993, “A three-dimensional model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, 41(2), pp. 389–412