/MAT/LAW78

Block Format Keyword This law is the Yoshida-Uemori model for describing the large-strain cyclic plasticity of metals. The law is based on the framework of two surfaces theory: the yielding surface and the bounding surface.

During the plastic deformation, a yield surface will move within the bounding surface and will never change its size, and the bounding surface can change both in size and location. The plastic-strain dependency of the Young's modulus and the work-hardening stagnation effect are also taken into account. Concerning SPH, it is compatible with solid only, this can be verified with the /SPH/WavesCompression test. The solid version is only isotropic. The shell version is anisotropic based on Hill criterion.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW78/mat_ID/unit_ID
mat_title
ρ i
E ν
Y b C 2 KH MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa WcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@393A@ h B0
m Rsat OptR C1 C2
r00 r45 r90 Mexp Icrit
fct_IDE Einf CE C 1 K H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa WcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@393A@

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 Young's modulus.

(Real)

[ Pa ]
ν Poisson's ratio.

(Real)

Y Yield stress.

(Real)

[ Pa ]
b Center of the bounding surface.

(Real)

[ Pa ]
C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa WcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@393A@ Parameter for kinematic hardening rule of yield surface (or used in calculation of C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeaaaa@36B4@ if C 1 K H > C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaa0 baaSqaaiaaigdaaeaacaWGlbGaamisaaaakiabg6da+iaadoeadaqh aaWcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@3E0B@ ). 5

(Real)

h Material parameter for controlling work hardening stagnation.

(Real)

B0 Initial size of the bounding surface.

(Real)

[ Pa ]
m Parameter for isotropic and kinematic hardening of the bounding surface.

(Real)

Rsat Saturated value of the isotropic hardening stress.

(Real)

[ Pa ]
OptR Modified isotropic hardening rule flag (available for shells only):
=0 (Default)
Yoshida formulation.
=1
Modified formulation (define C1 and C2).

(Integer)

C1, C2 Constant used in the modified formulation of the isotropic hardening of bounding surface (available for shells only).

(Real)

r00 Lankford parameter (0 degree) used for shell elements.

Default = 1.0 (Real)

r45 Lankford parameter (45 degree) used for shell elements.

Default = 1.0 (Real)

r90 Lankford parameter (90 degree) used for shell elements.

Default = 1.0 (Real)

Mexp Exponent M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaSqaaabaaaaaaaaape Gaamytaaaa@36DF@ in Barlat's 1989 Yield Criterion for shell elements. 8.
> 2.0
Any value greater than 2.0 is valid.
= 6.0 (Default)
For Body Centered Cubic (BCC) material.
= 8.0
For Face Centered Cubic (FCC) material.

(Real)

Icrit Plastic criterion selection flag.
= 0
Set to 1
= 1 (Default)
Hill 1948
= 2
Barlat 1989, only available for shell elements.
(Integer)
fct_IDE ID of the function defining the scale factor of Young's modulus evolution versus effective plastic strain. 9

(Integer)

Einf Asymptotic value of Young's modulus.

(Real)

[ Pa ]
CE Parameter controlling the dependency of Young's modulus on the effective plastic strain.

(Real)

C 1 K H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa WcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@393A@ Parameter C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeaaaa@36B4@ for kinematic hardening rule of yield surface. 5

Ignore if C 1 K H C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaa0 baaSqaaiaaigdaaeaacaWGlbGaamisaaaakiabgsMiJkaadoeadaqh aaWcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@3EB8@ .

(Real)

Example

#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/LAW78/1/1
DP600-HDG
#              RHO_I
              7.8E-9
#                  E                  NU
              206000                  .3
#                  Y                   B                C2KH                   H                  B0
                 420                 112                 200                   0                 555
#                  m                RSAT      OPTR                  C1                  C2
                  12                 190         0                   1                   1
#                 R0                 R45                 R90                Mexp     Icrit
                   1                   1                   1
#  Fct_IDE                          EINF                  CE                C1KH
         0                        163000                  50                   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. For solid elements, von Mises yield criterion is used, so the yield function is expressed as:
    f = 3 2 ( s α ) : ( s α ) Y 2
    Whereas for shell elements, Hill’s (1948) or Barlat’s (1989) yield criterion are used, which allows for the modeling of anisotropic materials:
    • The Hill’s criterion is expressed as:
      f = φ ( σ α ) Y 2
      Where,
      Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbaaaa@373C@
      Yield stress.
      α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHXoaaaa@379B@
      Total back stress. 4

      If A = σ α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHbbGaey ypa0JaaC4WdiabgkHiTiaahg7aaaa@3BA7@ , then φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAaa a@381B@ is expressed as:

      φ ( A ) = A x x 2 2 r 0 1 + r 0 A x x A y y + r 0 ( 1 + r 90 ) r 90 ( 1 + r 0 ) A y y 2 + r 0 + r 90 r 90 ( 1 + r 0 ) ( 2 r 45 + 1 ) A x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAca GGOaGaamyqaiaacMcacqGH9aqpcaWGbbWaa0baaSqaaiaadIhacaWG 4baabaGaaGOmaaaakiabgkHiTmaalaaabaGaaGOmaiaadkhadaWgaa WcbaGaaGimaaqabaaakeaacaaIXaGaey4kaSIaamOCamaaBaaaleaa caaIWaaabeaaaaGccaWGbbWaaSbaaSqaaiaadIhacaWG4baabeaaki aadgeadaWgaaWcbaGaamyEaiaadMhaaeqaaOGaey4kaSYaaSaaaeaa caWGYbWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaey4kaS IaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaakiaawIcacaGLPaaa aeaacaWGYbWaaSbaaSqaaiaaiMdacaaIWaaabeaakmaabmaabaGaaG ymaiabgUcaRiaadkhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGL PaaaaaGaamyqamaaDaaaleaacaWG5bGaamyEaaqaaiaaikdaaaGccq GHRaWkdaWcaaqaaiaadkhadaWgaaWcbaGaaGimaaqabaGccqGHRaWk caWGYbWaaSbaaSqaaiaaiMdacaaIWaaabeaaaOqaaiaadkhadaWgaa WcbaGaaGyoaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaey4kaSIaamOC amaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaaaadaqadaqaai aaikdacaWGYbWaaSbaaSqaaiaaisdacaaI1aaabeaakiabgUcaRiaa igdaaiaawIcacaGLPaaacaWGbbWaa0baaSqaaiaadIhacaWG5baaba GaaGOmaaaaaaa@786C@

    • The Barlat’s criterion is expressed as:
      f = ϕ ( σ α ) 2 Y M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiabg2da9iabew9aMnaabmaapaqaaGqad8qacaWFdpGaeyOe I0Iaa8xSdaGaayjkaiaawMcaaiabgkHiTiaaikdacaWGzbWdamaaCa aaleqabaWdbiaad2eaaaaaaa@4287@

      Where, M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaSqaaabaaaaaaaaape Gaamytaaaa@36DF@ is the exponent in Barlat's yield criterion.

      ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dygaaa@37D4@ is expressed as:

      ϕ ( A ) = a | K 1 + K 2 | M +   a | K 1 K 2 | M + c | 2 K 2 | M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2aaeWaa8aabaWdbiaadgeaaiaawIcacaGLPaaacqGH9aqp caWGHbWaaqWaa8aabaWdbiaadUeapaWaaSbaaSqaa8qacaaIXaaapa qabaGcpeGaey4kaSIaam4sa8aadaWgaaWcbaWdbiaaikdaa8aabeaa aOWdbiaawEa7caGLiWoapaWaaWbaaSqabeaapeGaamytaaaakiabgU caRiaacckacaWGHbWaaqWaa8aabaWdbiaadUeapaWaaSbaaSqaa8qa caaIXaaapaqabaGcpeGaeyOeI0Iaam4sa8aadaWgaaWcbaWdbiaaik daa8aabeaaaOWdbiaawEa7caGLiWoapaWaaWbaaSqabeaapeGaamyt aaaakiabgUcaRiaadogadaabdaWdaeaapeGaaGOmaiaadUeapaWaaS baaSqaa8qacaaIYaaapaqabaaak8qacaGLhWUaayjcSdWdamaaCaaa leqabaWdbiaad2eaaaaaaa@5A9E@

      With,

      K 1 =   A x x + h A y y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4sa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaGG GcWaaSaaa8aabaWdbiaadgeapaWaaSbaaSqaa8qacaWG4bGaamiEaa WdaeqaaOWdbiabgUcaRiaadIgacaWGbbWdamaaBaaaleaapeGaamyE aiaadMhaa8aabeaaaOqaa8qacaaIYaaaaaaa@4359@ and K 2 = ( A x x h A y y 2 ) 2 + p 2 A x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4sa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpdaGc aaWdaeaapeWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGbbWdamaaBa aaleaapeGaamiEaiaadIhaa8aabeaak8qacqGHsislcaWGObGaamyq a8aadaWgaaWcbaWdbiaadMhacaWG5baapaqabaaakeaapeGaaGOmaa aaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakiabgUca RiaadchapaWaaWbaaSqabeaapeGaaGOmaaaakiaadgeapaWaa0baaS qaa8qacaWG4bGaamyEaaWdaeaapeGaaGOmaaaaaeqaaaaa@4BFB@ .

      Parameters a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAaaaa@36F9@ , c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAaaaa@36F9@ , and h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAaaaa@36F9@ are computed from the Lankford coefficients.

      a = 2 2 r 00 1 + r 00   r 90 1 + r 90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyyaiabg2da9iaaikdacqGHsislcaaIYaWaaOaaa8aabaWdbmaa laaapaqaa8qacaWGYbWdamaaBaaaleaapeGaaGimaiaaicdaa8aabe aaaOqaa8qacaaIXaGaey4kaSIaamOCa8aadaWgaaWcbaWdbiaaicda caaIWaaapaqabaaaaOWdbiaacckadaWcaaWdaeaapeGaamOCa8aada WgaaWcbaWdbiaaiMdacaaIWaaapaqabaaakeaapeGaaGymaiabgUca RiaadkhapaWaaSbaaSqaa8qacaaI5aGaaGimaaWdaeqaaaaaa8qabe aaaaa@4ACC@ , c = 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4yaiabg2da9iaaikdacqGHsislcaWGHbaaaa@3A89@ , h =   r 00 1 + r 00   1 + r 90 r 90   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAaiabg2da9iaacckadaGcaaWdaeaapeWaaSaaa8aabaWdbiaa dkhapaWaaSbaaSqaa8qacaaIWaGaaGimaaWdaeqaaaGcbaWdbiaaig dacqGHRaWkcaWGYbWdamaaBaaaleaapeGaaGimaiaaicdaa8aabeaa aaGcpeGaaiiOamaalaaapaqaa8qacaaIXaGaey4kaSIaamOCa8aada WgaaWcbaWdbiaaiMdacaaIWaaapaqabaaakeaapeGaamOCa8aadaWg aaWcbaWdbiaaiMdacaaIWaaapaqabaaaaaWdbeqaaOGaaiiOaaaa@4AC0@

      Parameter p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiAaaaa@36F9@ is obtained by solving:

      2 M Y M ( f A x x +   f A y y ) σ 45 1 r 45 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaaikdacaWGnbGaamywa8aadaahaaWcbeqaa8qa caWGnbaaaaGcpaqaa8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgk Gi2kaadAgaa8aabaWdbiabgkGi2kaadgeapaWaaSbaaSqaa8qacaWG 4bGaamiEaaWdaeqaaaaak8qacqGHRaWkcaGGGcWaaSaaa8aabaWdbi abgkGi2kaadAgaa8aabaWdbiabgkGi2kaadgeapaWaaSbaaSqaa8qa caWG5bGaamyEaaWdaeqaaaaaaOWdbiaawIcacaGLPaaacqaHdpWCpa WaaSbaaSqaa8qacaaI0aGaaGynaaWdaeqaaaaak8qacqGHsislcaaI XaGaeyOeI0IaamOCa8aadaWgaaWcbaWdbiaaisdacaaI1aaapaqaba GcpeGaeyypa0JaaGimaaaa@5701@

  2. Yield stress, Poisson ratio and Young's modulus should be strictly positive. The other parameters should be non-negative value.
  3. The schematic illustration of the two-surface model is shown in Figure 1.
    Where, 0 is the original center of the yield surface, the yield surface with its center α and its radii Y, is moving kinematically, within a bounding surface that has a size indicated by B+R and tensor β indicating its center position.
    Figure 1. Schematic Drawing of the Two-surface Model

    law78_2-surface_model
  4. The yield surface is subjected to a kinematic hardening. The kinematic motion is described by α * that has the following evolution:
    • for shell elements
      α ˙ * = C [ ( a Y ) ( σ α ) a α * α * ] ε ¯ ˙ p
    • for solid elements
      α ˙ * = C [ ( 2 3 ) a ε ˙ p a α * α * ε ¯ ˙ P ]
    Where,
    ε ¯ ˙ p
    Equivalent plastic strain rate
    C and a
    Material parameters and a = B + R Y
    α = α * + β
    Total back stress
    Note: In case of a quadratic (using Hill or M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaSqaaabaaaaaaaaape Gaamytaaaa@36DF@ =2.0 with Barlat) and isotropic criterion ( r 00 =   r 45 = r 90 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaicdacaaIWaaapaqabaGcpeGaeyyp a0JaaiiOaiaadkhapaWaaSbaaSqaaiaaisdacaaI1aaabeaakiabg2 da98qacaWGYbWdamaaBaaaleaapeGaaGyoaiaaicdaa8aabeaakiab g2da9iaaigdaaaa@437D@ ), the shell kinematic motion equation equals the solid one.
  5. The parameter C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeaaaa@36B4@ for kinematic hardening rule can take two different values to describe a change of work hardening rate for forward and reverse deformations.
    • If C 1 K H > C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaa0 baaSqaaiaaigdaaeaacaWGlbGaamisaaaakiabg6da+iaadoeadaqh aaWcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@3E0B@ , { C = C 1 K H if M a x ( α * ) < B Y C = C 2 K H otherwise MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGabaqaau aabeqaceaaaeaafaqabeqadaaabaGaam4qaiabg2da9iaadoeadaqh aaWcbaGaaGymaaqaaiaadUeacaWGibaaaaGcbaGaaeyAaiaabAgaae aacaWGnbGaamyyaiaadIhadaqadaqaamaafmaabaGaaCySdmaaCaaa leqabaGaaiOkaaaaaOGaayzcSlaawQa7aaGaayjkaiaawMcaaiabgY da8iaadkeacqGHsislcaWGzbaaaaqaauaabeqabmaaaeaacaWGdbGa eyypa0Jaam4qamaaDaaaleaacaaIYaaabaGaam4saiaadIeaaaaake aacaqGVbGaaeiDaiaabIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4C aiaabwgaaeaafaqabeqacaaabaqbaeqabeGaaaqaauaabeqabiaaae aaaeaaaaaabaaaaaqaaaaaaaaaaaGaay5Eaaaaaa@597C@
    • If C 1 K H C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaa0 baaSqaaiaaigdaaeaacaWGlbGaamisaaaakiabgsMiJkaadoeadaqh aaWcbaGaaGOmaaqaaiaadUeacaWGibaaaaaa@3EB8@ , the change of work hardening rate is deactivated and C = C 2 K H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaey ypa0Jaam4qamaaDaaaleaacaaIYaaabaGaam4saiaadIeaaaaaaa@3B7A@ .
  6. The bounding surface is subjected to an isotropic-kinematic hardening. The evolution equation for isotropic hardening is:
    R ˙ = m ( R sat R ) ε ¯ ˙ p
    Default (if OptR = 0) Yoshida expression
    R = R sat [ ( C 1 + ε ¯ p ) C 2 C 1 C 2 ]
    Available for shell elements, if OptR = 1
    The evolution equation for kinematic hardening of bounding surface is:
    • for shell elements:
      β ˙ = m [ ( b Y ) ( σ α ) β ] ε ¯ ˙ p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaceWFYo GbaiaacqGH9aqpcaWGTbWaamWaaeaadaqadaqaamaalaaabaGaamOy aaqaaiaadMfaaaaacaGLOaGaayzkaaWaaeWaaeaacaWHdpGaeyOeI0 IaaCySdaGaayjkaiaawMcaaiabgkHiTiaa=j7aaiaawUfacaGLDbaa cuaH1oqzgaqegaGaamaaBaaaleaadaahaaadbeqaaiaadchaaaaale qaaaaa@4937@
    • for solid elements:
      β ˙ = m ( 2 3 b ε ˙ p β ε ¯ ˙ p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaceWFYo GbaiaacqGH9aqpcaWGTbWaaeWaaeaadaWcaaqaaiaaikdaaeaacaaI ZaaaaiaadkgaceWH1oGbaiaadaWgaaWcbaWaaWbaaWqabeaacaWHWb aaaaWcbeaakiabgkHiTiaa=j7acuaH1oqzgaqegaGaamaaBaaaleaa daahaaadbeqaaiaadchaaaaaleqaaaGccaGLOaGaayzkaaaaaa@459A@
    Note: In case of a quadratic (using Hill or Mexp=2 with Barlat) and isotropic criterion ( r 00 =   r 45 = r 90 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOCa8aadaWgaaWcbaWdbiaaicdacaaIWaaapaqabaGcpeGaeyyp a0JaaiiOaiaadkhapaWaaSbaaSqaaiaaisdacaaI1aaabeaakiabg2 da98qacaWGYbWdamaaBaaaleaapeGaaGyoaiaaicdaa8aabeaakiab g2da9iaaigdaaaa@437D@ ), the shell kinematic hardening equation equals the solid one.
  7. The work-hardening stagnation during unloading is described using a J2-type surface g σ with a radius r and a center q:
    g σ ( β , q , r ) = 3 2 ( β q ) : ( β q ) r 2 q ˙ = μ ( β q ) r = h Γ ˙ , Γ ˙ 3 ( β q ) : β ˙ 2 r
    Where,
    h ( 0 < h 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaays W7daqadaqaaiaaicdacqGH8aapcaWGObGaeyizImQaaGymaaGaayjk aiaawMcaaaaa@3F14@
    Denotes a material parameter that determines the rate of expansion of surface g σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacqaHdpWCaeqaaaaa@38D1@ .
    β
    Should be either inside or on the surface g σ .
  8. The exponent in Barlat’s (1989) yield criterion can be set by considering the microstructure of the material. Any value greater than 2.0 is valid, but typically:
    • Mexp = 6.0 (Default) for a Body Centered Cubic (BCC) material
    • Mexp = 8.0 for a Face Centered Cubic (FCC) material
  9. The evolution of Young's modulus:
    • If fct_IDE > 0, the curve defines a scale factor for Young's modulus evolution with equivalent plastic strain, which means the Young's modulus is scaled by the function f ( ε ¯ p ) :
      • E ( t ) = E f ( ε ¯ p )

      The initial value of the scale factor should be equal to 1 and it decreases.

    • If fct_IDE = 0, the Young's modulus is calculated as:
      E ( t ) = E ( E E inf ) [ 1 exp ( C E ε ¯ p ) ]
      Where,
      E and Einf
      Initial and asymptotic value of Young's modulus, respectively.
      ε ¯ p
      Accumulated equivalent plastic strain.
      Note: If fct_IDE = 0 and CE = 0, Young's modulus E is kept constant.