/MAT/LAW92

Block Format Keyword This law describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior.

A stress versus strain curve can be defined as an input function in order to determine the material parameters by fitting this curve. This law is only compatible with solid elements.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW92/mat_ID/unit_ID
mat_title
ρ i
Parameter input
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
μ D λ m
Function input
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Itype fct_ID ν Fscale

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Unit Identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

ρ i Initial density.

(Real)

[ kg m 3 ]
μ Shear modulus.

Used only if fct_ID is not defined.

(Real)

[ Pa ]
D Material parameter.

If null, D is automatically computed from μ , λ m and ν =0.495. 2

Used only if fct_ID is not defined.

(Real)

[ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@
λ m The limit of stretch.

Used only if fct_ID is not defined.

Default = 7.0 (Real)

Itype Test data type (stress strain curve).
= 1 (Default)
Uniaxial data test
= 2
Equibiaxial data test
= 3
Planar data test

(Integer)

fct_ID Function identifier defining engineer stress versus engineer strain.

(Integer)

ν Poisson's ratio.

Used only if fct_ID is defined.

Default = 0.495 (Real)

Fscale Scale factor for ordinate (stress) in function fct_ID.

Default = 1.0 (Real)

[ Pa ]

Example (Rubber with Parameter Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
Generic RUBBER
#              RHO_I
               1E-09 
#                 mu                   D                 LAM           
                   5                 .05                 100                 
#    IType    fct_ID                  NU              Fscale
         0         0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example (Rubber with Function Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW92/1/1
rubber
#              RHO_I
            1.000E-9 
#                 mu                   D                 LAM           
                 
#    IType    fct_ID                  NU              Fscale
         1         2               0.495
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
LAW92  e.strain vs  e.stress from uniaxial test(IType=1) 
#                  X                   Y
                   0                   0                                                            
                 .03                 .30                                                           
                 .06                 .55                                                            
                 .10                 .80
                 .20                 1.4
                 .30                 2.0
                 .50                 2.7
                 .70                 3.4
                 1.0                 4.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. The Arruda-Boyce energy density.
    W=μ i=1 5 c i ( λ m ) 2i2 ( I ¯ 1 i 3 i )+ 1 D ( J 2 1 2 ln(J) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0JaeqiVd02aaabCaeaadaWcaaqaaiaadogadaWgaaWcbaGaamyA aaqabaaakeaacaGGOaGaeq4UdW2aaSbaaSqaaiaad2gaaeqaaOGaai ykamaaCaaaleqabaGaaGOmaiaadMgacqGHsislcaaIYaaaaaaaaeaa caWGPbGaeyypa0JaaGymaaqaaiaaiwdaa0GaeyyeIuoakmaabmaaba GabmysayaaraWaa0baaSqaaiaaigdaaeaacaWGPbaaaOGaeyOeI0Ia aG4mamaaCaaaleqabaGaamyAaaaaaOGaayjkaiaawMcaaiabgUcaRm aalaaabaGaaGymaaqaaiaadseaaaWaaeWaaeaadaWcaaqaaiaadQea daahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaaabaGaaGOmaaaacq GHsislciGGSbGaaiOBaiGacIcacaWGkbGaaiykaaGaayjkaiaawMca aaaa@5DC5@

    With

    c 1 = 1 2 , c 2 = 1 20 , c 3 = 11 1050 , c 4 = 19 7000 , c 5 = 519 673750

    and

    I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2

    with λ ¯ k = J 1 / 3 λ J = λ 1 λ 2 λ 3

    The Cauchy stress is computed as:

    σ i = λ i J W λ i

  2. If the stress strain curve, fct_ID is not defined, the material parameters in line 3, μ , D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@369F@ and λ m must be defined and the line 4 input is not used.

    Ground shear modulus is computed as:

    μ 0 = μ ( 1 + 3 5 ( 1 λ m 2 ) + 99 175 ( 1 λ m 4 ) + 513 875 ( 1 λ m 6 ) + 42039 67375 ( 1 λ m 8 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaicdaaeqaaOGaeyypa0JaeqiVd02aaeWaaeaacaaIXaGa ey4kaSYaaSaaaeaacaaIZaaabaGaaGynaaaadaqadaqaamaalaaaba GaaGymaaqaaiabeU7aSnaaDaaaleaacaWGTbaabaGaaGOmaaaaaaaa kiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaiMdacaaI5aaabaGaaG ymaiaaiEdacaaI1aaaamaabmaabaWaaSaaaeaacaaIXaaabaGaeq4U dW2aa0baaSqaaiaad2gaaeaacaaI0aaaaaaaaOGaayjkaiaawMcaai abgUcaRmaalaaabaGaaGynaiaaigdacaaIZaaabaGaaGioaiaaiEda caaI1aaaamaabmaabaWaaSaaaeaacaaIXaaabaGaeq4UdW2aa0baaS qaaiaad2gaaeaacaaI2aaaaaaaaOGaayjkaiaawMcaaiabgUcaRmaa laaabaGaaGinaiaaikdacaaIWaGaaG4maiaaiMdaaeaacaaI2aGaaG 4naiaaiodacaaI3aGaaGynaaaadaqadaqaamaalaaabaGaaGymaaqa aiabeU7aSnaaDaaaleaacaWGTbaabaGaaGioaaaaaaaakiaawIcaca GLPaaaaiaawIcacaGLPaaaaaa@6A4C@

    If D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@369F@ is not defined, the bulk modulus is calculated as:

    K = 2 ( 1 + υ ) μ 0 3 ( 1 2 υ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaiaacIcacaaIXaGaey4kaSIaeqyXduNaaiyk aiabeY7aTnaaBaaaleaacaaIWaaabeaaaOqaaiaaiodadaqadaqaai aaigdacqGHsislcaaIYaGaeqyXduhacaGLOaGaayzkaaaaaaaa@466D@

    Where, υ = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXduNaey ypa0JaaGimaiaac6cacaaI0aGaaGyoaiaaiwdaaaa@3C70@ and D = 2 K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maalaaabaGaaGOmaaqaaiaadUeaaaaaaa@3962@ .

    If D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@369F@ is defined, the formula should be K = 2 D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseaaaaaaa@3962@ , and the Poisson ratio is updated with υ = 3 K 2 μ 0 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXduNaey ypa0ZaaSaaaeaacaaIZaGaam4saiabgkHiTiaaikdacqaH8oqBdaWg aaWcbaGaaGimaaqabaaakeaacaaIZaaaaaaa@3F6C@ .
    Note: For positive values of shear modulus, μ , and Limit of stretch, λ m , this model is always stable.
  3. If the stress strain curve, fct_ID, is defined then the line 3 input parameters μ , D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@369F@ and λ m are ignored and are automatically identified by fitting of the provided stress versus strain curve.

    A nonlinear least squares algorithm is used to fit the Arruda-Boyce parameters. The model is fully incompressible in fitting the Arruda-Boyce constants to the test data, except in the volumetric test.

    E = k = 1 n d a t a ( N k t e s t N k t h N k t e s t ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaey ypa0ZaaabCaeaadaqadaqaamaalaaabaGaamOtamaaDaaaleaacaWG RbaabaGaamiDaiaadwgacaWGZbGaamiDaaaakiabgkHiTiaad6eada qhaaWcbaGaam4AaaqaaiaadshacaWGObaaaaGcbaGaamOtamaaDaaa leaacaWGRbaabaGaamiDaiaadwgacaWGZbGaamiDaaaaaaaakiaawI cacaGLPaaaaSqaaiaadUgacqGH9aqpcaaIXaaabaGaamOBaiaadsga caWGHbGaamiDaiaadggaa0GaeyyeIuoakmaaCaaaleqabaGaaGOmaa aaaaa@54B8@

    Where E is relative error. The material constants are obtained using a least-squares-fit procedure to minimize the relative error between the theoretical nominal stress and given experimental data.

    λ 1 = λ 2 = λ and λ 3 = λ 2 with λ = 1 + ε

    Where, N k test is a stress value from the test data and N i th is the theoretical nominal stress given by for each engineer strain i.

    N k t h = W λ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaa0 baaSqaaiaadUgaaeaacaWG0bGaamiAaaaakiabg2da9maalaaabaGa eyOaIyRaam4vaaqaaiabgkGi2kabeU7aSnaaBaaaleaacaWGRbaabe aaaaaaaa@41CD@

    The nominal stress is computed for each mode assuming the full incompressibility:
    J = λ 1 λ 2 λ 3 = 1
    • Uniaxial Mode:

      λ 1 = λ and λ 2 = λ 3 = λ 1 2 with λ = 1 + ε

      So

      N th = W λ = 2 μ ( λ λ 2 ) i = 1 5 ic i ( λ m ) 2 i 2 I ¯ 1 i 1 with I ¯ 1 = λ 2 + 2 λ

    • Equibiaxial Mode:

      λ 1 = λ 2 = λ and λ 3 = λ 2 with λ = 1 + ε

      So

      N th = W λ = 2 μ ( λ λ 5 ) i = 1 5 ic i ( λ m ) 2 i 2 I ¯ 1 i 1 with I ¯ 1 = 2 λ 2 + 1 λ 4

    • Planar (Shear Mode):

      λ 1 = λ , λ 3 = 1 and λ 2 = λ 1 with λ = 1 + ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBcaaMe8Uaaiilaiaa ysW7cqaH7oaBdaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIXaGaaG zbVlaabggacaqGUbGaaeizaiaaywW7cqaH7oaBdaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaigdaaa GccaaMb8UaaGzbVlaabEhacaqGPbGaaeiDaiaabIgacaaMf8Uaeq4U dWMaeyypa0JaaGymaiabgUcaRiabew7aLbaa@5F73@

      So

      N th = W λ = 2 μ ( λ λ 3 ) i = 1 5 ic i ( λ m ) 2 i 2 I ¯ 1 i 1 with I ¯ 1 = λ 2 + 1 + λ 2

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