/MAT/LAW112 (PAPER or XIA)

Block Format Keyword The Paperboard law models an orthotropic and dissymmetric elasto-plastic material from proposed by Xia, 2002.

The basic principle is to fully uncouple the behavior in the plane of the paper sheet and the behavior out of the plane. A yield stress is defined for each directions of loading, in tension and compression.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW112/mat_ID/unit_ID or /MAT/PAPER/mat_ID/unit_ID or /MAT/XIA/mat_ID/unit_ID
mat_title
ρ i                
E1 E2 E3 Ires Itab Ismooth  
ν 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGOmaiaaigdaaeqaaaaa@3ACA@ G12 G23 G13  
K E3C CC    
ν 1p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ ν 2 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ ν 4 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ ν 5 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@  
If Itab = 0, insert continuous yield stresses
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
S01 A01 B01 C01    
S02 A02 B02 C02    
S03 A03 B03 C03    
S04 A04 B04 C04    
S05 A05 B05 C05    
ASIG BSIG CSIG        
TAU0 ATAU BTAU        
If Itab = 1, insert tabulated yield stresses
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
TAB_YLD1 MAT_Xscale1 MAT_Yscale1        
TAB_YLD2 MAT_Xscale2 MAT_Yscale2        
TAB_YLD3 MAT_Xscale3 MAT_Yscale3        
TAB_YLD4 MAT_Xscale4 MAT_Yscale4        
TAB_YLD5 MAT_Xscale5 MAT_Yscale5        
TAB_YLDC MAT_XscaleC MAT_YscaleC        
TAB_YLDS MAT_XscaleS MAT_YscaleS        

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Optional unit identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

ρ i Initial density.

(Real)

[ kg m 3 ]
Ei Young’s modulus in the ith orthotropic direction.

(Real)

[ Pa ]
ν i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaamyAaiaadQgaaeqaaaaa@3B30@ Poisson's ratio related to the ith and jth orthotropic direction.

(Real)

G i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@3A44@ Shear modulus related to the ith and jth orthotropic direction.

(Real)

[ Pa ]
Ires Resolution method for plasticity.
= 0
Set to 2
= 1
NICE (Next Increment Correct Error) explicit method.
= 2 (Default)
Newton iterative method - cutting plane.

(Integer)

Itab Yield stresses computation.
= 0
Continuous yield stresses.
= 1
Tabulated yield stresses.

(Integer)

Ismooth Interpolation type (in case of tabulated yield function).
= 1 (Default)
Linear interpolation.
= 2
Logarithmic interpolation base 10.
=3
Logarithmic interpolation base n.

(Integer)

K In-plane yield surface exponent.

Default = 1.0 (Real)

 
E3C First elastic compression parameter.

Default = E3 (Real)

[ Pa ]
CC Second elastic compression parameter.

Default = 1.0 (Real)

 
ν 1p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ Tensile plastic Poisson’s ratio in direction 1.

(Real)

 
ν 2 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ Tensile plastic Poisson’s ratio in direction 2.

(Real)

 
ν 4 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ Compressive plastic Poisson’s ratio in direction 1.

(Real)

 
ν 5 p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBlm aaBaaabaGaaGymaiaadchaaeqaaaaa@3B03@ Compressive plastic Poisson’s ratio in direction 2.

(Real)

 
S0i Initial yield stress in the ith direction of loading.
Each direction is associated to a given loading direction following the order:
i=1
Tension in orthotropic direction 1.
i=2
Tension in orthotropic direction 2.
i=3
In-plane shear.
i=4
Compression in orthotropic direction 1.
i=5
Compression in orthotropic direction 2.
i=C
Compression in out-of-plane direction 3.
i=S
Transverse shear direction.

Default = 1.0e20 (Real)

[ Pa ]
A0i First hardening parameter in the ith direction of loading.

(Real)

[ Pa ]
B0i Second hardening parameter in the ith direction of loading.

(Real)

 
C0i Third hardening parameter in the ith direction of loading.

(Real)

[ Pa ]
ASIG Initial out-of-plane yield stress in compression.

Default = 1.0e20 (Real)

[ Pa ]
BSIG First out-of-plane hardening parameter in compression.

(Real)

[ Pa ]
CSIG Second out-of-plane hardening parameter in compression.

(Real)

 
TAU0 Initial transverse shear yield stress.

Default = 1.0e20 (Real)

[ Pa ]
ATAU First transverse shear hardening parameter.

(Real)

[ Pa ]
BTAU Second transverse shear hardening parameter.

(Real)

 
TAB_YLDi Tabulated yield stress – plastic strain - strain rate function identifier in the ith direction of loading.

(Integer)

MAT_Xscalei X scale factor of the tabulated yield – plastic strain - strain rate function in the ith direction of loading.

Default = 1.0 (Real)

[Hz]
MAT_Yscalei Y scale factor of the tabulated yield – plastic strain - strain rate function in the ith direction of loading.

Default = 1.0 (Real)

[ Pa ]

Example (Paper)

#RADIOSS STARTER
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW112/1/1
Xia
#              RHO_I
            7.83E-10
#                 E1                  E2                  E3      Ires      Itab   Ismooth       
                4193                1554                1554         2         0         0
#               nu21                 G12                 G23                 G13  
              0.1011                 988                  76                  76
#                  K                 E3C                  CC                 
                 2.0                47.2               24.46                 
#               nu1p                nu2p                nu4p                nu5p
               0.555              0.1537                0.18               0.145
#                S01                 A01                 B01                 C01
                12.0                19.0               260.0               800.0
#                S02                 A02                 B02                 C02
                 6.5                40.0               160.0               250.0
#                S03                 A03                 B03                 C03       
                 6.0                11.0               100.0               125.0
#                S04                 A04                 B04                 C04
                 7.3                 6.0               160.0               300.0
#                S05                 A05                 B05                 C05
                 6.3                 9.0               310.0               225.0
#               ASIG                BSIG                CSIG
               16.55               16.55                3.16
#               TAU0                ATAU                BTAU
                 2.1                 9.0                 2.0    
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA

Example (Tabulated)

#RADIOSS STARTER
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|          
/MAT/LAW112/1/1
Xia_tab
#              RHO_I
            7.83E-10
#                 E1                  E2                  E3      Ires      Itab   Ismooth       
                4193                1554                1554         1         1         1
#               nu21                 G12                 G23                 G13 
              0.1011                 988                  76                  76
#                  K                 E3C                  CC                 
                 2.0                47.2               24.46                 
#               nu1p                nu2p                nu4p                nu5p
               0.555              0.1537                0.18               0.145
#           TAB_YLD1         MAT_Xscale1         MAT_Yscale1
                  25                 1.0                 1.0
#           TAB_YLD2         MAT_Xscale2         MAT_Yscale2
                  25                 1.0                0.35
#           TAB_YLD3         MAT_Xscale3         MAT_Yscale3
                  25                 1.0                0.75
#           TAB_YLD4         MAT_Xscale4         MAT_Yscale4
                  25                 1.0              0.6341
#           TAB_YLD5         MAT_Xscale5         MAT_Yscale5
                  25                 1.0                 0.5
#           TAB_YLDC         MAT_XscaleC         MAT_YscaleC
                  25                 1.0                 0.5
#           TAB_YLDS         MAT_XscaleS         MAT_YscaleS
                  25                 1.0                 0.5 
/FUNCT/46
ecoulement2   
#     plastic strain              stress                                                                                
                 0.0	           12.00
               0.012	 32.979020979021
               0.025	50.4615384615385
                0.05	            74.5
               0.075	90.9473684210526
                 0.1	102.909090909091
               0.125	          112.00
                0.15	119.142857142857
               0.175	124.903225806452
                 0.2	129.647058823529
                0.25	          137.00
                 0.3	142.434782608696
                 0.4	149.931034482759
                 0.5	154.857142857143
                 1.0	165.846153846154   
/TABLE/1/25
Yld Functions : plastic strain + strain rate dependency
#DIMENSION
         2
#   FCT_ID                   strain rate                                                     Scale_y
        46                           0.0                                                        1.00
        46                           1.0                                                        1.10
        46                           5.0                                                        1.15
        46                          10.0                                                        1.20
        46                         100.0                                                        1.25
        46                      100000.0                                                        1.35  
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

  1. To describe the behavior of the paperboard material law, the following orthotropic direction is considered.


    Figure 1.
  2. The elastic behavior of this material law is orthotropic.
    The in-plane behavior should be fully uncoupled with the out-of-plane behavior, computed as:(1)
    { σ xx = C 11 ε xx + C 12 ε yy σ yy = C 21 ε xx + C 22 ε yy σ xy = G 12 γ xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe qaaiabeo8aZnaaBaaaleaacaWG4bGaamiEaaqabaGccqGH9aqpcaWG dbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabew7aLnaaBaaaleaaca WG4bGaamiEaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaigdacaaI Yaaabeaakiabew7aLnaaBaaaleaacaWG5bGaamyEaaqabaaakeaacq aHdpWCdaWgaaWcbaGaamyEaiaadMhaaeqaaOGaeyypa0Jaam4qamaa BaaaleaacaaIYaGaaGymaaqabaGccqaH1oqzdaWgaaWcbaGaamiEai aadIhaaeqaaOGaey4kaSIaam4qamaaBaaaleaacaaIYaGaaGOmaaqa baGccqaH1oqzdaWgaaWcbaGaamyEaiaadMhaaeqaaaGcbaGaeq4Wdm 3aaSbaaSqaaiaadIhacaWG5baabeaakiabg2da9iaadEeadaWgaaWc baGaaGymaiaaikdaaeqaaOGaeq4SdC2aaSbaaSqaaiaadIhacaWG5b aabeaaaaGccaGL7baaaaa@674D@

    With C = 1 1 ν 12 ν 21 [ E 1 ν 12 E 2 ν 21 E 1 E 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaC4qaiabg2 da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcqaH9oGBdaWgaaWc baGaaGymaiaaikdaaeqaaOGaeqyVd42aaSbaaSqaaiaaikdacaaIXa aabeaaaaGcdaWadaqaauaabeqaciaaaeaacaWGfbWaaSbaaSqaaiaa igdaaeqaaaGcbaGaeqyVd42aaSbaaSqaaiaaigdacaaIYaaabeaaki aadweadaWgaaWcbaGaaGOmaaqabaaakeaacqaH9oGBdaWgaaWcbaGa aGOmaiaaigdaaeqaaOGaamyramaaBaaaleaacaaIXaaabeaaaOqaai aadweadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLBbGaayzxaaaaaa@50BD@

    The transverse shear components are computed as:(2)
    { σ y z = G 23 ε y z σ z x = G 21 ε z x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe qaaiabeo8aZnaaBaaaleaacaWG5bGaamOEaaqabaGccqGH9aqpcaWG hbWaaSbaaSqaaiaaikdacaaIZaaabeaakiabew7aLnaaBaaaleaaca WG5bGaamOEaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamOEaiaadIha aeqaaOGaeyypa0Jaam4ramaaBaaaleaacaaIYaGaaGymaaqabaGccq aH1oqzdaWgaaWcbaGaamOEaiaadIhaaeqaaaaakiaawUhaaaaa@4DB3@
    The out-of-plane elastic behavior (for solids only) is treated as a uniaxial equivalent problem. However, the computation of the stress may differ between tension and compression. The elasticity becomes nonlinear for compressive loadings.(3)
    σ z z = E 3 ε z z e if ε z z e 0 σ z z = E 3 C ( 1 e C c ε z z e ) if ε z z e < 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaqbaeqabiWaaa qaaiabeo8aZnaaBaaaleaacaWG6bGaamOEaaqabaGccqGH9aqpcaWG fbWaaSbaaSqaaiaaiodaaeqaaOGaeqyTdu2aa0baaSqaaiaadQhaca WG6baabaGaamyzaaaaaOqaaiaabMgacaqGMbaabaGaeqyTdu2aa0ba aSqaaiaadQhacaWG6baabaGaamyzaaaakiabgwMiZkaaicdaaeaacq aHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaOGaeyypa0Jaamyramaa BaaaleaacaaIZaGaam4qaaqabaGccaGGOaGaaGymaiabgkHiTiGacw gadaahaaWcbeqaaiabgkHiTiaadoeadaWgaaadbaGaam4yaaqabaWc cqaH1oqzdaqhaaadbaGaamOEaiaadQhaaeaacaWGLbaaaaaakiaacM caaeaacaqGPbGaaeOzaaqaaiabew7aLnaaDaaaleaacaWG6bGaamOE aaqaaiaadwgaaaGccqGH8aapcaaIWaaaaaaa@6632@
  3. In the Xia 2002 formulation, the in-plane yield criterion, denoted as f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbaaaa@385A@ , is defined as:(4)
    f = I = 1 6 χ I ( σ : N I σ Y I ) 2 k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaey ypa0ZaaabCaeaacqaHhpWydaWgaaWcbaGaamysaaqabaGcdaqadaqa amaalaaabaGaaC4WdiaacQdacaWHobWaaSbaaSqaaiaahMeaaeqaaa GcbaGaeq4Wdm3aa0baaSqaaiaadMfaaeaacaWGjbaaaaaaaOGaayjk aiaawMcaaaWcbaGaamysaiabg2da9iaaigdaaeaacaaI2aaaniabgg HiLdGcdaahaaWcbeqaaiaaikdacaWGRbaaaOGaeyOeI0IaaGymaaaa @4E6C@

    Where,

    χ I = { 1 if σ : N I > 0 0 otherwise MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaadMeaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGadaaabaGa aGymaaqaaiaabMgacaqGMbaabaGaaC4WdiaahQdacaWHobWaaSbaaS qaaiaahMeaaeqaaOGaeyOpa4JaaGimaaqaaiaaicdaaeaacaqGVbGa aeiDaiaabIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4Caiaabwgaae aaaaaacaGL7baaaaa@4C54@
    χ I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHhpWyda WgaaWcbaGaamysaaqabaaaaa@3A20@
    Switching parameters.
    σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHdpaaaa@38BE@
    Cauchy stress tensor.
    N I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHobWaaS baaSqaaiaahMeaaeqaaaaa@3944@
    Normal direction of the yield planes.
    σ Y I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamywaaqaaiaadMeaaaaaaa@3B0B@
    Yield stresses.
    k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36C6@
    Positive integer.
    Each direction is associated to a given loading direction following the order defined below:
    1
    Tension in orthotropic direction 1.
    2
    Tension in orthotropic direction 2.
    3
    Positive in-plane shear.
    4
    Compression in orthotropic direction 1.
    5
    Compression in orthotropic direction 2.
    6
    Negative in-plane shear (same input as positive in-plane shear σ Y 6 = σ Y 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamywaaqaaiaaiAdaaaGccqGH9aqpcqaHdpWCdaqhaaWc baGaamywaaqaaiaaiodaaaaaaa@3F98@ ).

    The normal direction vector to the yield planes are:

    N 1 = [ 1 1 + ν 1 p 2 ν 1 p 1 + ν 1 p 2 0 0 0 0 ] N 2 = [ ν 2 p 1 + ν 2 p 2 1 1 + ν 2 p 2 0 0 0 0 ] N 3 = [ 0 0 0 1 0 0 ] N 4 = [ 1 1 + ν 4 p 2 ν 4 p 1 + ν 4 p 2 0 0 0 0 ] N 5 = [ ν 5 p 1 + ν 5 p 2 1 1 + ν 5 p 2 0 0 0 0 ] N 6 = [ 0 0 0 1 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaah6 eadaWgaaWcbaGaaCymaaqabaGccqGH9aqpdaWadaqaauaabeqabyaa aaqaamaalaaabaGaaGymaaqaamaakaaabaGaaGymaiabgUcaRiabe2 7aUnaaDaaaleaacaaIXaGaamiCaaqaaiaaikdaaaaabeaaaaaakeaa cqGHsisldaWcaaqaaiabe27aUnaaBaaaleaacaaIXaGaamiCaaqaba aakeaadaGcaaqaaiaaigdacqGHRaWkcqaH9oGBdaqhaaWcbaGaaGym aiaadchaaeaacaaIYaaaaaqabaaaaaGcbaGaaGimaaqaaiaaicdaae aacaaIWaaabaGaaGimaaaaaiaawUfacaGLDbaaaeaacaWHobWaaSba aSqaaiaaikdaaeqaaOGaeyypa0ZaamWaaeaafaqabeqagaaaaeaacq GHsisldaWcaaqaaiabe27aUnaaBaaaleaacaaIYaGaamiCaaqabaaa keaadaGcaaqaaiaaigdacqGHRaWkcqaH9oGBdaqhaaWcbaGaaGOmai aadchaaeaacaaIYaaaaaqabaaaaaGcbaWaaSaaaeaacaaIXaaabaWa aOaaaeaacaaIXaGaey4kaSIaeqyVd42aa0baaSqaaiaaikdacaWGWb aabaGaaGOmaaaaaeqaaaaaaOqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiaaicdaaaaacaGLBbGaayzxaaaabaGaaCOtamaaBaaaleaaca aIZaaabeaakiabg2da9maadmaabaqbaeqabeGbaaaabaGaaGimaaqa aiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaaaa Gaay5waiaaw2faaaqaaiaah6eadaWgaaWcbaGaaGinaaqabaGccqGH 9aqpdaWadaqaauaabeqabyaaaaqaaiabgkHiTmaalaaabaGaaGymaa qaamaakaaabaGaaGymaiabgUcaRiabe27aUnaaDaaaleaacaaI0aGa amiCaaqaaiaaikdaaaaabeaaaaaakeaadaWcaaqaaiabe27aUnaaBa aaleaacaaI0aGaamiCaaqabaaakeaadaGcaaqaaiaaigdacqGHRaWk cqaH9oGBdaqhaaWcbaGaaGinaiaadchaaeaacaaIYaaaaaqabaaaaa GcbaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaaaaiaawUfa caGLDbaaaeaacaWHobWaaSbaaSqaaiaaiwdaaeqaaOGaeyypa0Zaam WaaeaafaqabeqagaaaaeaadaWcaaqaaiabe27aUnaaBaaaleaacaaI 1aGaamiCaaqabaaakeaadaGcaaqaaiaaigdacqGHRaWkcqaH9oGBda qhaaWcbaGaaGynaiaadchaaeaacaaIYaaaaaqabaaaaaGcbaGaeyOe I0YaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaey4kaSIaeqyVd4 2aa0baaSqaaiaaiwdacaWGWbaabaGaaGOmaaaaaeqaaaaaaOqaaiaa icdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaacaGLBbGaayzxaa aabaGaaCOtamaaBaaaleaacaaI2aaabeaakiabg2da9maadmaabaqb aeqabeGbaaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0 IaaGymaaqaaiaaicdaaeaacaaIWaaaaaGaay5waiaaw2faaaaaaa@AFBD@

    Each direction I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbaaaa@383D@ is then associated to a specific yield stress whose expression is:

    σ Y I = S I 0 + A I 0 tanh ( B I 0 ε p f ) + C I 0 ε p f with I [ 1 , 6 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqabeqaca aabaGaeq4Wdm3aa0baaSqaaiaadMfaaeaacaWGjbaaaOGaeyypa0Ja am4uamaaDaaaleaacaWGjbaabaGaaGimaaaakiabgUcaRiaadgeada qhaaWcbaGaamysaaqaaiaaicdaaaGcciGG0bGaaiyyaiaac6gacaGG ObWaaeWaaeaacaWGcbWaa0baaSqaaiaadMeaaeaacaaIWaaaaOGaeq yTdu2aa0baaSqaaiaadchaaeaacaWGMbaaaaGccaGLOaGaayzkaaGa ey4kaSIaam4qamaaDaaaleaacaWGjbaabaGaaGimaaaakiabew7aLn aaDaaaleaacaWGWbaabaGaamOzaaaaaOqaauaabeqabiaaaeaacaqG 3bGaaeyAaiaabshacaqGObaabaGaamysaiabgIGiopaadmaabaGaaG ymaiaacYcacaaI2aaacaGLBbGaayzxaaaaaaaaaaa@5F1B@

    Where, ε p f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda qhaaWcbaGaamiCaaqaaiaadAgaaaaaaa@3B23@ is the in-plane equivalent plastic strain (associated to the yield function f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@36C1@ ).

    The out-of-plane yield function is denoted as g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGNbaaaa@385B@ is defined as:

    g = σ z z σ Y C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGNbGaey ypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaakiab gkHiTiabeo8aZnaaDaaaleaacaWGzbaabaGaam4qaaaaaaa@42C8@ with σ Y C = A σ + B σ exp( C σ ε p g ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamywaaqaaiaadoeaaaGccqGH9aqpcaWGbbWaaSbaaSqa aiabeo8aZbqabaGccqGHRaWkcaWGcbWaaSbaaSqaaiabeo8aZbqaba GcciGGLbGaaiiEaiaacchacaGGOaGaam4qamaaBaaaleaacqaHdpWC aeqaaOGaeqyTdu2aa0baaSqaaiaadchaaeaacaWGNbaaaOGaaiykaa aa@4D2A@

    Where, ε p g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda qhaaWcbaGaamiCaaqaaiaadEgaaaaaaa@3B24@ is the out-of-plane equivalent plastic strain (associated to the yield function g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGNbaaaa@385B@ ).

    The transverse shear yield function is:(5)
    h = σ y z 2 + σ z x 2 σ Y S 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObGaey ypa0ZaaSaaaeaadaGcaaqaaiabeo8aZnaaDaaaleaacaWG5bGaamOE aaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaamOEaiaadI haaeaacaaIYaaaaaqabaaakeaacqaHdpWCdaqhaaWcbaGaamywaaqa aiaadofaaaaaaOGaeyOeI0IaaGymaaaa@4921@
    Where,
    σ Y S = τ 0 + [ A τ min ( 0 , σ z z ) B τ ] ε p h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamywaaqaaiaadofaaaGccqGH9aqpcqaHepaDdaWgaaWc baGaaGimaaqabaGccqGHRaWkdaWadaqaaiaadgeadaWgaaWcbaGaeq iXdqhabeaakiabgkHiTiGac2gacaGGPbGaaiOBaiaacIcacaaIWaGa aiilaiabeo8aZnaaBaaaleaacaWG6bGaamOEaaqabaGccaGGPaGaam OqamaaBaaaleaacqaHepaDaeqaaaGccaGLBbGaayzxaaGaeqyTdu2a a0baaSqaaiaadchaaeaacaWGObaaaaaa@5560@
    ε p h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda qhaaWcbaGaamiCaaqaaiaadIgaaaaaaa@3B25@
    Out-of-plane equivalent plastic strain (associated to the yield function h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGNbaaaa@385B@ ).

    If the tabulated yield stress option is selected (Itab = 1), each yield stress is associated to a table (TAB_YLDi) to define the stress evolution with the plastic strain, at several plastic strain-rate. Two scale factors can be also defined in the X and Y direction for each table. In this case, the hardening parameters S 0 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaaic dacaWGPbaaaa@3856@ , A 0 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaaic dacaWGPbaaaa@3856@ , B 0 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaaic dacaWGPbaaaa@3856@ , C 0 i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaaic dacaWGPbaaaa@3856@ , A σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacqaHdpWCaeqaaaaa@388B@ , B σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacqaHdpWCaeqaaaaa@388B@ , C σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacqaHdpWCaeqaaaaa@388B@ , τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGimaaqabaaaaa@3A1A@ , A τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS baaSqaaiabes8a0bqabaaaaa@3A26@ , and B τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbWaaS baaSqaaiabes8a0bqabaaaaa@3A27@ are ignored, and the yield stress becomes:

    σ Y I = f Y t a b _ Y L D I ( ε p f , ε ˙ p f ) I [ 1 , 6 ] σ Y C = f Y t a b _ Y L D C ( ε p g , ε ˙ p g ) σ Y S = f Y t a b _ Y L D S ( ε p h , ε ˙ p h ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaauaabe qabiaaaeaacqaHdpWCdaqhaaWcbaGaamywaaqaaiaadMeaaaGccqGH 9aqpcaWGMbWaa0baaSqaaiaadMfaaeaacaWG0bGaamyyaiaadkgaca GGFbGaamywaiaadYeacaWGebGaamysaaaakiaacIcacqaH1oqzdaqh aaWcbaGaamiCaaqaaiaadAgaaaGccaGGSaGafqyTduMbaiaadaqhaa WcbaGaamiCaaqaaiaadAgaaaGccaGGPaaabaGaamysaiabgIGiopaa dmaabaGaaGymaiaacYcacaaI2aaacaGLBbGaayzxaaaaaaqaaiabeo 8aZnaaDaaaleaacaWGzbaabaGaam4qaaaakiabg2da9iaadAgadaqh aaWcbaGaamywaaqaaiaadshacaWGHbGaamOyaiaac+facaWGzbGaam itaiaadseacaWGdbaaaOGaaiikaiabew7aLnaaDaaaleaacaWGWbaa baGaam4zaaaakiaacYcacuaH1oqzgaGaamaaDaaaleaacaWGWbaaba Gaam4zaaaakiaacMcaaeaacqaHdpWCdaqhaaWcbaGaamywaaqaaiaa dofaaaGccqGH9aqpcaWGMbWaa0baaSqaaiaadMfaaeaacaWG0bGaam yyaiaadkgacaGGFbGaamywaiaadYeacaWGebGaam4uaaaakiaacIca cqaH1oqzdaqhaaWcbaGaamiCaaqaaiaadIgaaaGccaGGSaGafqyTdu MbaiaadaqhaaWcbaGaamiCaaqaaiaadIgaaaGccaGGPaaaaaa@837C@

    For output field, an equivalent “global” plastic strain is computed as:(6)
    ε p = ( ε p f ) 2 + ( ε p g ) 2 + ( ε p h ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamiCaaqabaGccqGH9aqpdaGcaaqaaiaacIcacqaH1oqz daqhaaWcbaGaamiCaaqaaiaadAgaaaGccaGGPaWaaWbaaSqabeaaca aIYaaaaOGaey4kaSIaaiikaiabew7aLnaaDaaaleaacaWGWbaabaGa am4zaaaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOa GaeqyTdu2aa0baaSqaaiaadchaaeaacaWGObaaaOGaaiykamaaCaaa leqabaGaaGOmaaaaaeqaaaaa@4F32@