/MAT/LAW72 (HILL_MMC)

Block Format Keyword Describes the anisotropic Hill material with a modified Mohr fracture criteria. This law is available for shell and solid.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW72/mat_ID/unit_ID or /MAT/HILL_MMC/mat_ID/unit_ID
mat_title
ρ i
E v
σy0 εp0 n F G
H N L M
C1 C2 C3 m Dc

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 ]
E Initial Young's modulus.

(Real)

[ Pa ]
v Poisson's ratio.

(Real)

σ y 0 Initial yield stress.

Default = 1020 (Real)

[ Pa ]
ε p 0 Initial plastic strain.

Default = 10-20 (Real)

n Exponent for the isotropic function for the swift hardening:

σ y = σ y 0 ( ε p + ε p 0 ) n

It is also used as an exponent in the MMC failure equations. 2

Default = 1.0 (Real)

F, G, H, L, M, N Six HILL Materials anisotropic parameters (> 0).

(float)

C1 First parameter for MMC fracture model.

(Real)

C2 Second parameter for MMC fracture model.

Default = σ y 0 (Real)

[ Pa ]
C3 Third parameter for MMC fracture model.

(Real)

m Exponent for the softening function. 3

Default = 1.0 (Real)

Dc Critical damage.
= 1 (Default)
The element is deleted when damage reaches one.
> 1
The yield stress is modified by using a softening function. 3
If damage reaches the critical damage value, the element is deleted.

(Real)

Example (Metal)

Material softening and failure are considered in the material example. Using the MMC parameters C1, C2, and C3 the failure strain in a uniaxial tension test is calculated to be 0.98. In a uniaxial tension test with m=0.5, the material starts to soften at 0.98 until 0.98 x Dc is reached.
Figure 1.


#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW72/1/1
Metal
#              RHO_I
              0.0028
#                  E                  nu
              200E+3                 0.3
#               Sig0                Eps0                   n                   F                   G
                1276             1.63E-3               0.265                 0.5                 0.5
#                  H                   N                   L                   M
                 0.5                 1.5                   0                   0
#                 C1                  C2                  C3                   m                  Dc
                0.12                 720               1.095                 0.5                 1.1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. 3D equivalent Hill stress:
    f = F ( σ yy σ zz ) 2 + G ( σ zz σ xx ) 2 + H ( σ xx σ yy ) 2 + 2 L σ yz 2 + 2 M σ zx 2 + 2 N σ xy 2

    For shell element, take:

    f = F σ yy 2 + G σ xx 2 + H ( σ xx σ yy ) 2 + 2 N σ xy 2

  2. MMC fracture criteria:
    D = 0 ε p d ε p ε f ( θ , η )
    With
    ε f ( θ ¯ , η ) = { σ y 0 C 2 [ C 3 + 3 2 3 ( 1 C 3 ) ( sec ( θ ¯ π 6 ) 1 ) ] [ 1 + C 1 2 3 cos ( θ ¯ π 6 ) + C 1 ( η + 1 3 sin ( θ ¯ π 6 ) ) ] } 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaGcdaqadaqaaiqbeI7aXzaaraGaaiilaiab eE7aObGaayjkaiaawMcaaiabg2da9maacmGabaWaaSaaaeaacqaHdp WCdaqhaaWcbaGaamyEaaqaaiaaicdaaaaakeaacaWGdbWaaSbaaSqa aiaaikdaaeqaaaaakmaadmGabaGaam4qamaaBaaaleaacaaIZaaabe aakiabgUcaRmaalaaabaWaaOaaaeaacaaIZaaaleqaaaGcbaGaaGOm aiabgkHiTmaakaaabaGaaG4maaWcbeaaaaGcdaqadiqaaiaaigdacq GHsislcaWGdbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaWa aeWaceaaciGGZbGaaiyzaiaacogadaqadiqaamaalaaabaGafqiUde NbaebacqaHapaCaeaacaaI2aaaaaGaayjkaiaawMcaaiabgkHiTiaa igdaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaWadiqaamaakaaaba WaaSaaaeaacaaIXaGaey4kaSIaam4qamaaDaaaleaacaaIXaaabaGa aGOmaaaaaOqaaiaaiodaaaaaleqaaOGaci4yaiaac+gacaGGZbWaae WaceaadaWcaaqaaiqbeI7aXzaaraGaeqiWdahabaGaaGOnaaaaaiaa wIcacaGLPaaacqGHRaWkcaWGdbWaaSbaaSqaaiaaigdaaeqaaOWaae WaceaacqaH3oaAcqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZaaaaiGa cohacaGGPbGaaiOBamaabmGabaWaaSaaaeaacuaH4oqCgaqeaiabec 8aWbqaaiaaiAdaaaaacaGLOaGaayzkaaaacaGLOaGaayzkaaaacaGL BbGaayzxaaaacaGL7bGaayzFaaWaaWbaaSqabeaacqGHsisldaWcaa qaaiaaigdaaeaacaWGUbaaaaaaaaa@84F9@
    • For 3D solid elements

      η is stress triaxiality with η = 1 3 ( σ x x + σ y y + σ z z ) σ V M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH3oaAcq GH9aqpdaWcaaqaamaalaaabaGaaGymaaqaaiaaiodaaaWaaeWaceaa cqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaey4kaSIaeq4Wdm 3aaSbaaSqaaiaadMhacaWG5baabeaakiabgUcaRiabeo8aZnaaBaaa leaacaWG6bGaamOEaaqabaaakiaawIcacaGLPaaaaeaacqaHdpWCda WgaaWcbaGaamOvaiaad2eaaeqaaaaaaaa@4E7B@

      θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga qeaaaa@3935@ is shift Lode angle θ ¯ = 1 2 π a r cos ζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga qeaiabg2da9iaaigdacqGHsisldaWcaaqaaiaaikdaaeaacqaHapaC aaGaamyyaiaadkhaciGGJbGaai4BaiaacohacqaH2oGEaaa@44D9@

      with Lode angle ( θ ) parameter ζ = cos ( 3 θ ) = 27 2 J 3 σ V M 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH2oGEcq GH9aqpciGGJbGaai4BaiaacohadaqadaqaaiaaiodacqaH4oqCaiaa wIcacaGLPaaacqGH9aqpdaWcaaqaaiaaikdacaaI3aaabaGaaGOmaa aadaWcaaqaaiaadQeadaWgaaWcbaGaaG4maaqabaaakeaacqaHdpWC daqhaaWcbaGaamOvaiaad2eaaeaacaaIZaaaaaaaaaa@4A74@

      J 3 is the third invariant of the deviatoric stress.

    • For shell elements

      η is stress triaxiality with η = 1 3 ( σ x x + σ y y + σ z z ) σ V M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH3oaAcq GH9aqpdaWcaaqaamaalaaabaGaaGymaaqaaiaaiodaaaWaaeWaceaa cqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaey4kaSIaeq4Wdm 3aaSbaaSqaaiaadMhacaWG5baabeaakiabgUcaRiabeo8aZnaaBaaa leaacaWG6bGaamOEaaqabaaakiaawIcacaGLPaaaaeaacqaHdpWCda WgaaWcbaGaamOvaiaad2eaaeqaaaaaaaa@4E7B@

      θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga qeaaaa@3935@ is shift Lode angle θ ¯ = 1 2 π a r cos ζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH4oqCga qeaiabg2da9iaaigdacqGHsisldaWcaaqaaiaaikdaaeaacqaHapaC aaGaamyyaiaadkhaciGGJbGaai4BaiaacohacqaH2oGEaaa@44D9@

      with Lode angle ( θ ) parameter ζ = cos ( 3 θ ) = 27 2 η ( η 2 1 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH2oGEcq GH9aqpciGGJbGaai4BaiaacohadaqadaqaaiaaiodacqaH4oqCaiaa wIcacaGLPaaacqGH9aqpcqGHsisldaWcaaqaaiaaikdacaaI3aaaba GaaGOmaaaacqaH3oaAdaqadaqaaiabeE7aOnaaCaaaleqabaGaaGOm aaaakiabgkHiTmaalaaabaGaaGymaaqaaiaaiodaaaaacaGLOaGaay zkaaaaaa@4D7E@

  3. Fracture and damage with MMC fracture criteria:
    • When D = 1: fracture initiate
    • By 1 < D < Dc: the yield stress is multiplied by softening function β to reduce the deformation resistance.

      σ y = β σ y 0 ( ε p + ε p 0 ) n with β = ( D c D D c 1 ) m 0 < β < 1

    • If DDc, the element is deleted.
    • The exponent m is used to describe the softening behavior. It is recommended to use m > 0.

      If 0 < m < 1, then the softening curve is convex.

      If m > 1, then the softening curve is concave. The softening is between ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaaaaa@3A25@ and D c ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadogaaeqaaOGaeyyXICTaeqyTdu2aaSbaaSqaaiaadAga aeqaaaaa@3E56@ . Once the plastic strain is reached D c ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadogaaeqaaOGaeyyXICTaeqyTdu2aaSbaaSqaaiaadAga aeqaaaaa@3E56@ (in this case D > D c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey Opa4JaamiramaaBaaaleaacaWGJbaabeaaaaa@3B15@ ), then the element is deleted.
      Figure 2.


      Figure 3.


  4. It is possible to display user variables in animation files (with Engine /ANIM/Eltyp/Restype) and in Time history file (with Starter /TH/SHEL and /TH/BRIC):
    • USER1: Damage value
  5. It is also possible to display a normalized damage variable D n = D D c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWGebaabaGaamir amaaBaaaleaacaWGJbaabeaaaaaaaa@3D15@ in animation files with /ANIM/BRICK/DAMG, /ANIM/SHELL/DAMG, /H3D/SHELL/DAMG and /H3D/SOLID/DAMG.