Tsai-Wu Formulation (Iform =0)

Block Format Keyword This law describes the composite shell and solid material using the Tsai-Wu formulation.

The material is assumed to be orthotropic-elastic before the Tsai-Wu criterion is reached. The material becomes nonlinear afterwards. For solid elements, the material is assumed to be linearly elastic in the transverse direction. The Tsai-Wu criterion limit can be set dependent on the plastic work and strain rate to model material hardening. Strain and plastic energy criterion for brittle damage and failure are available. A simplified delamination criterion based on out-of-plane shear angle can be used.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW25/mat_ID/unit_ID or /MAT/COMPSH/mat_ID/unit_ID
mat_title
ρ i                
E11 E22 υ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8YjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqyXdu3aaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3CE0@ Iform   E33
G12 G23 G31 ε f 1 ε f 2
ε t 1 ε m 1 ε t 2 ε m 2 dmax
Composite Plasticity Hardening
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
W p max W p ref Ioff   Ratio    
b n fmax        
Composite Yield Stress in Tension Compression
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
σ 1 y t σ 2 y t σ 1 y c σ 2 y c α
Yield Stress in Shear and Strain Rate
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
σ 12 y c σ 12 y t c ɛ ˙ ICC  
Delamination
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
γ i n i γ max d3max        
Strain Rate Filtering
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Fsmooth Fcut              

Definitions

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 ]
E11 Young's modulus in direction 1.

(Real)

[ Pa ]
E22 Young's modulus in direction 2.

(Real)

[ Pa ]
υ 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8YjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqyXdu3aaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3CE0@ Poisson's ratio.

(Real)

 
Iform Formulation flag. 1
= 0
Tsai-wu formulation

(Integer)

 
E33 Young's modulus in direction 33. 2

(Real)

[ Pa ]
G12 Shear modulus in direction 12.

(Real)

[ Pa ]
G23 Shear modulus in direction 23.

(Real)

[ Pa ]
G31 Shear modulus in direction 31.

(Real)

[ Pa ]
ε f 1 Maximum tensile strain for element deletion in material direction 1.

Default = 1.2 x 1020 (Real)

 
ε f 2 Maximum tensile strain for element deletion in material direction 2.

Default = 1.2 x 1020 (Real)

 
ε t 1 Tensile failure strain in the material direction 1 at which stress starts to reduce. 4

Default = 1.0 x 1020 (Real)

 
ε m 1 Maximum tensile strain in material direction 1 at which the stress at the element is set to zero, if dmax = 1. 4

Default = 1.1 x 1020 (Real)

 
ε t 2 Tensile failure strain in the material direction 2 at which the stress starts to reduce.

Default = 1.0 x 1020 (Real)

 
ε m 2 Maximum tensile strain in material direction 2 at which the stress in the element is set to zero, if dmax=1.

Default = 1.1 x 1020 (Real)

 
dmax Maximum damage factor dmax ≤ 1). 4

Default = 0.999 (Real)

 
W p max Maximum plastic work per unit shell volume.

Default = 1020 (Real)

[ J m 3 ]
W p ref Reference plastic work per unit shell volume. 4

Default = 1.0 (Real)

[ J m 3 ]
Ioff Flag that controls shell and thick shell element deletion depending on failure modes in the element layers. 4
= 0
Shell is deleted if max. plastic work reached for one element layer.
= 1
Shell is deleted if max. plastic work reached for all element layers.
= 2
Shell is deleted if for each element layer,(1)
C o n d i t i o n 1 : { e i t h e r max . p l a s t i c w o r k r e a c h e d o r ε 1 > ε m 1 i n d i r e c t i o n 1 o r d 1 > d max i n d i r e c t i o n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4qaiaad+gacaWGUbGaamizaiaadMgacaWG0bGaamyAaiaad+ga caWGUbqbaeqabeqaaaqaaaaacaaIXaGaaiOoamaaceaabaqbaeqabm qaaaqaaiaadwgacaWGPbGaamiDaiaadIgacaWGLbGaamOCauaabeqa beaaaeaaaaGaciyBaiaacggacaGG4bGaaiOlauaabeqabeaaaeaaaa GaamiCaiaadYgacaWGHbGaam4CaiaadshacaWGPbGaam4yauaabeqa beaaaeaaaaGaam4Daiaad+gacaWGYbGaam4Aauaabeqabeaaaeaaaa GaamOCaiaadwgacaWGHbGaam4yaiaadIgacaWGLbGaamizaaqaauaa beqabeaaaeaaaaqbaeqabeqaaaqaaaaacaWGVbGaamOCauaabeqabe aaaeaaaaqbaeqabeqaaaqaaaaacqaH1oqzdaWgaaWcbaGaaGymaaqa baGccqGH+aGpfaqabeqabaaabaGaeqyTdu2aaSbaaSqaaiGac2gaca aIXaaabeaakuaabeqabeaaaeaaaaGaamyAaiaad6gafaqabeqabaaa baaaaiaadsgacaWGPbGaamOCaiaadwgacaWGJbGaamiDaiaadMgaca WGVbGaamOBauaabeqabeaaaeaaaaGaaGymaaaaaeaafaqabeqabaaa baaaauaabeqabeaaaeaaaaGaam4Baiaadkhafaqabeqabaaabaaaau aabeqabeaaaeaaaaGaamizamaaBaaaleaacaaIXaaabeaakiabg6da +uaabeqabeaaaeaacaWGKbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaa qabaGcfaqabeqabaaabaaaaiaadMgacaWGUbqbaeqabeqaaaqaaaaa caWGKbGaamyAaiaadkhacaWGLbGaam4yaiaadshacaWGPbGaam4Bai aad6gafaqabeqabaaabaaaaiaaigdaaaaaaaGaay5Eaaaaaa@871B@
= 3
Shell is deleted if for each element layer,(2)
C o n d i t i o n 2 : { e i t h e r max . p l a s t i c w o r k r e a c h e d o r ε 2 > ε m 2 i n d i r e c t i o n 2 o r d 2 > d max i n d i r e c t i o n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4qaiaad+gacaWGUbGaamizaiaadMgacaWG0bGaamyAaiaad+ga caWGUbqbaeqabeqaaaqaaaaacaaIYaGaaiOoamaaceaabaqbaeqabm qaaaqaaiaadwgacaWGPbGaamiDaiaadIgacaWGLbGaamOCauaabeqa beaaaeaaaaGaciyBaiaacggacaGG4bGaaiOlauaabeqabeaaaeaaaa GaamiCaiaadYgacaWGHbGaam4CaiaadshacaWGPbGaam4yauaabeqa beaaaeaaaaGaam4Daiaad+gacaWGYbGaam4Aauaabeqabeaaaeaaaa GaamOCaiaadwgacaWGHbGaam4yaiaadIgacaWGLbGaamizaaqaauaa beqabeaaaeaaaaqbaeqabeqaaaqaaaaacaWGVbGaamOCauaabeqabe aaaeaaaaqbaeqabeqaaaqaaaaacqaH1oqzdaWgaaWcbaGaaGOmaaqa baGccqGH+aGpfaqabeqabaaabaGaeqyTdu2aaSbaaSqaaiaad2gaca aIYaaabeaakuaabeqabeaaaeaaaaGaamyAaiaad6gafaqabeqabaaa baaaaiaadsgacaWGPbGaamOCaiaadwgacaWGJbGaamiDaiaadMgaca WGVbGaamOBauaabeqabeaaaeaaaaGaaGOmaaaaaeaafaqabeqabaaa baaaauaabeqabeaaaeaaaaGaam4Baiaadkhafaqabeqabaaabaaaau aabeqabeaaaeaaaaGaamizamaaBaaaleaacaaIYaaabeaakiabg6da +uaabeqabeaaaeaacaWGKbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaa qabaGcfaqabeqabaaabaaaaiaadMgacaWGUbqbaeqabeqaaaqaaaaa caWGKbGaamyAaiaadkhacaWGLbGaam4yaiaadshacaWGPbGaam4Bai aad6gafaqabeqabaaabaaaaiaaikdaaaaaaaGaay5Eaaaaaa@8720@
= 4
Shell is deleted if for each element layer, condition 1 and condition 2 are satisfied.
= 5
Shell is deleted if for all element layers, condition 1 or condition 2 is satisfied.
= 6
Shell is deleted if for each element layer condition 1 or condition 2 is satisfied.

(Integer)

 
Ratio Ratio parameter which controls the deletion of shell elements based on the number of failed layers.
< 0.0
The element will be deleted if, all but one of the layers fails (that is, the number of layers that did not fail is equal to 1). 4
> 0.0
The element will be deleted if:(3)
n u m b e r o f f a i l e d l a y e r s n u m b e r o f t o t a l l a y e r s r a t i o

Default = 1.0 (Real)

 
b Plastic hardening parameter.

Default = 0.0 (Real)

 
n Plastic hardening exponent.

Default = 1.0 (Real)

 
fmax Maximum value of the Tsai-Wu criterion limit.

Default = 1020 (Real)

[ Pa ]
σ 1 y t Yield stress in tension in direction 1.

Default = 0.0 (Real)

[ Pa ]
σ 2 y t Yield stress in tension in direction 2.

Default = 0.0 (Real)

[ Pa ]
σ 1 y c Yield stress in compression in direction 1.

Default = 0.0 (Real)

[ Pa ]
σ 2 y c Yield stress in compression in direction 2.

Default = 0.0 (Real)

[ Pa ]
α Reduction factor for F12 coefficient calculation in Tsai-Wu criterion.

Default set to 1.0 (Real)

 
σ 12 y c Yield stress in compression in 45 degree of fiber direction.

Default = 0.0 (Real)

[ Pa ]
σ 12 y t Yield stress in tension in 45 degree of fiber direction.

Default = 0.0 (Real)

[ Pa ]
c Strain rate coefficient for plastic work criteria.
= 0
No strain rate dependency.

(Real)

 
ε ˙ 0 Reference strain rate.

If ε ˙ ε ˙ 0 , no strain rate effect.

(Real)

[ 1 s ]
ICC Strain rate effect flag. 4
= 1 (Default)
Strain rate effect on fmax is taken into account, no effect of strain rate on W p max .
= 2
There is no strain rate effect on both fmax and W p max .
= 3
There is strain rate effect on both fmax and W p max .
= 4
Strain rate effect on W p max is taken into account, but there is no effect of strain rate on fmax.

(Integer)

 
γ ini Out-of-plane shear strain when delamination begins. 4

Default = 1020 (Real)

 
γ max Out-of-plane shear strain when delamination ends and the element is deleted. 4

Default = 1.1 * 1020 (Real)

 
d3max Maximum delamination damage factor (d3max < 1). 4

Default = 1.0 (Real)

 
Fsmooth Strain rate filtering flag.
= 0 (Default)
Strain rate smoothing is inactive.
= 1
Strain rate smoothing is active.

(Integer)

 
Fcut Cutoff frequency for strain rate filtering.

Default = 1020 (Real)

[Hz]

Example (Composite)

#RADIOSS STARTER
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/COMPSH/1/1
composite example
#              RHO_I
             .001506
#                E11                 E22                NU12     Iform                           E33
              144000               10000                 .25         0                         20000
#                G12                 G23                 G31              EPS_f1              EPS_f2
                4200                4200                4200                   0                   0
#             EPS_t1              EPS_m1              EPS_t2              EPS_m2                dmax
                   0                   0                   0                   0                   0
#              Wpmax               Wpref      Ioff                         ratio
             1000000                   0         0                             0
#                  b                   n                fmax
                   0                   0             1000000
#            sig_1yt             sig_2yt             sig_1yc             sig_2yc               alpha
               10100               10100               10100               10100                   0
#           sig_12yc            sig_12yt                   c          Eps_rate_0       ICC
               10068               10068                   0                   0         0
#          GAMMA_ini           GAMMA_max               d3max
                   0                   0                   0
#  Fsmooth                Fcut
         0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

  1. The formulation flag Iform should be set to 0, for the TSAI-WU. Compared with Iform=1, in this formulation:
    • The TSAI-WU criterion limit F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOraiGacI cacaWGxbWaa0baaSqaaiaadchaaeaacaGGQaaaaOGaaiilaiqbew7a LzaacaGaaiykaaaa@3D33@ is function of plastic work and strain rate
    • It allows the simulation of the brittle failure by formation of crack
    • Considering different plastic and failure behaviors in tension, in compression and in shear
  2. Usage with property and element type.
    • This material requires orthotropic shell properties (/PROP/TYPE9 (SH_ORTH), /PROP/TYPE10 (SH_COMP) or /PROP/TYPE11 (SH_SANDW)) and composite shell properties (/PROP/TYPE17 (STACK), /PROP/TYPE51, /STACK). These properties prescribe the orthotropic directions; therefore, it is not compatible with the isotropic shell property (/PROP/TYPE1 (SHELL))
    • This material is available with under-integrated Q4 (Ishell= 1,2,3,4) and fully integrated BATOZ (Ishell=12) shell formulations.
    • This material is compatible with orthotropic solid property (/PROP/TYPE6 (SOL_ORTH)), the orthotropic thick shell property (/PROP/TYPE21 (TSH_ORTH)) and the composite thick shell property (/PROP/TYPE22 (TSH_COMP)). These properties specify the orthotropic directions. It is assumed that for solids and thick shells, the material is elastic in transverse direction and the E33 value must be specified in such cases.
    • For shell and thick shell composite parts, with /PROP/SH_COMP, /PROP/SH_SANDW, /PROP/TSH_ORTH or /PROP/TSH_COMP, material is defined directly in the property card. The failure criteria defined within this material (for example, LAW25) are accounted for. Material referred to in the corresponding /PART card is only used for time step and interface stiffness calculation.
    • From version 14.0 global material properties (membrane stiffness, bending stiffness, mass, and inertia) are calculated based on the material properties and layup (thicknesses) given in composite properties TYPE11, TYPE16, TYPE19 and PLY card. They are used for stability, mass and interface stiffness. A material is still required at part definition level but is only used for pre- and post- (visualization “by material”) and its physical characteristics are ignored. The previous formulation where stiffness and mass were calculated from the material associated to the part is still used, if the version number of the input file is 13.0 or earlier.
    • Failure criterion in LAW25 is not applicable to solid elements. To determine failure for solid elements /FAIL card should be used.
  3. The Tsai-Wu criterion:
    The material is assumed to be elastic until the Tsai-Wu criterion is fulfilled. After exceeding the Tsai-Wu criterion limit F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOraiGacI cacaWGxbWaa0baaSqaaiaadchaaeaacaGGQaaaaOGaaiilaiqbew7a LzaacaGaaiykaaaa@3D33@ the material becomes nonlinear:
    • If F ( σ ) < F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOraiGacI cacqaHdpWCcaGGPaGaeyipaWJaciOraiGacIcacaWGxbWaa0baaSqa aiaadchaaeaacaGGQaaaaOGaaiilaiqbew7aLzaacaGaaiykaaaa@4221@ : Elastic
    • If F ( σ ) > F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOraiGacI cacqaHdpWCcaGGPaGaeyOpa4JaciOraiGacIcacaWGxbWaa0baaSqa aiaadchaaeaacaGGQaaaaOGaaiilaiqbew7aLzaacaGaaiykaaaa@4225@ : Nonlinear
    Where, Stress F ( σ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOraiGacI cacqaHdpWCcaGGPaaaaa@39E0@ in element for Tsai-Wu criterion computed as:(4)
    F ( σ ) = F 1 σ 1 + F 2 σ 2 + F 11 σ 1 2 + F 22 σ 2 2 + F 44 σ 12 2 + 2 F 12 σ 1 σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaHdpWCaiaawIcacaGLPaaacqGH9aqpcaWGgbWaaSbaaSqa aiaaigdaaeqaaOGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaOGaey4kaS IaamOramaaBaaaleaacaaIYaaabeaakiabeo8aZnaaBaaaleaacaaI YaaabeaakiabgUcaRiaadAeadaWgaaWcbaGaaGymaiaaigdaaeqaaO Gaeq4Wdm3aa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamOr amaaBaaaleaacaaIYaGaaGOmaaqabaGccqaHdpWCdaqhaaWcbaGaaG OmaaqaaiaaikdaaaGccqGHRaWkcaWGgbWaaSbaaSqaaiaaisdacaaI 0aaabeaakiabeo8aZnaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaa GccqGHRaWkcaaIYaGaamOramaaBaaaleaacaaIXaGaaGOmaaqabaGc cqaHdpWCdaWgaaWcbaGaaGymaaqabaGccqaHdpWCdaWgaaWcbaGaaG Omaaqabaaaaa@63D8@

    Here, σ 1 , σ 2 and σ 12 are the stresses in the material coordinate system.

    The F coefficients of the Tsai-Wu criterion are determined from the limiting stresses when the material becomes nonlinear in directions 1, 2 or 12 (shear) in compression or tension as:(5)
    F 1 = 1 σ 1 y c + 1 σ 1 y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqGHsisldaWc aaqaaiaaigdaaeaacqaHdpWCdaqhaaWcbaGaaGymaiaadMhaaeaaca WGJbaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiabeo8aZnaaDaaa leaacaaIXaGaamyEaaqaaiaadshaaaaaaaaa@48A3@
    (6)
    F 2 = 1 σ 2 y c + 1 σ 2 y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsisldaWc aaqaaiaaigdaaeaacqaHdpWCdaqhaaWcbaGaaGOmaiaadMhaaeaaca WGJbaaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiabeo8aZnaaDaaa leaacaaIYaGaamyEaaqaaiaadshaaaaaaaaa@48A6@
    (7)
    F 11 = 1 σ 1 y c σ 1 y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0ZaaSaa aeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaigdacaWG5baabaGaam 4yaaaakiabeo8aZnaaDaaaleaacaaIXaGaamyEaaqaaiaadshaaaaa aaaa@46C4@
    (8)
    F 12 = α 2 F 11 F 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeyOe I0YaaSaaaeaacqaHXoqyaeaacaaIYaaaamaakaaabaGaamOramaaBa aaleaacaaIXaGaaGymaaqabaGccaWGgbWaaSbaaSqaaiaaikdacaaI Yaaabeaaaeqaaaaa@450B@
    (9)
    F 22 = 1 σ 2 y c σ 2 y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0ZaaSaa aeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaikdacaWG5baabaGaam 4yaaaakiabeo8aZnaaDaaaleaacaaIYaGaamyEaaqaaiaadshaaaaa aaaa@46C8@
    (10)
    F 44 = 1 σ 12 y c σ 12 y t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAeadaWgaaWcbaGaaGinaiaaisdaaeqaaOGaeyypa0ZaaSaa aeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaGaamyEaa qaaiaadogaaaGccqaHdpWCdaqhaaWcbaGaaGymaiaaikdacaWG5baa baGaamiDaaaaaaaaaa@4842@

    The superscripts c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaS qaaiaadogaaaa@3A28@ and t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaS qaaiaadogaaaa@3A28@ represent compression and tension, respectively.

    This criterion represents a second order closed three-dimensional Tsai-Wu surface in σ 1 , σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4Wdm3aa0baaSqaaiaaikdaaeaaaaaaaa@3B58@ and σ 12 space.

    F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiGacAeadaqadaqaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQca aaGccaGGSaGafqyTduMbaiaaaiaawIcacaGLPaaaaaa@40AA@ is the variable Tsai-Wu criterion limit defined as a function of plastic work ( W p * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaaaaa@3BEB@ ) and the true strain rate ( ε ˙ ).(11)
    F ( W p * , ε ˙ ) = [ 1 + b ( W p * ) n ] [ 1 + c ln ( ε ˙ ε ˙ 0 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiGacAeadaqadaqaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQca aaGccaGGSaGafqyTduMbaiaaaiaawIcacaGLPaaacqGH9aqpdaWada qaaiaaigdacqGHRaWkcaWGIbWaaeWaaeaacaWGxbWaa0baaSqaaiaa dchaaeaacaGGQaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGUb aaaaGccaGLBbGaayzxaaGaeyyXIC9aamWaaeaacaaIXaGaey4kaSIa am4yaiabgwSixlGacYgacaGGUbWaaeWaaeaadaWcaaqaaiqbew7aLz aacaaabaGafqyTduMbaiaadaWgaaWcbaGaaGimaaqabaaaaaGccaGL OaGaayzkaaaacaGLBbGaayzxaaaaaa@5C67@
    Where,
    W p r e f
    Reference plastic work
    W p * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaaaaa@3BEB@
    Plastic work defined with W p * = W p W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH9aqpdaWc aaqaaiaadEfadaWgaaWcbaGaamiCaaqabaaakeaacaWGxbWaa0baaS qaaiaadchaaeaacaWGYbGaamyzaiaadAgaaaaaaaaa@43DC@
    b
    Plastic hardening parameter
    n
    Plastic hardening exponent
    ε ˙ 0
    Reference true strain rate
    c
    Strain rate coefficient

    This Tsai-Wu surface is scaled outwards homothetically in all directions, due to increase in W p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaBaaaleaacaWGWbaabeaaaaa@3AA9@ and ε ˙ .

    The max. of Tsai-Wu criterion limit F ( W p * , ε ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiGacAeadaqadaqaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQca aaGccaGGSaGafqyTduMbaiaaaiaawIcacaGLPaaaaaa@40AA@ should be limited under:
    • f max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaaa@3D2A@ , if ICC=2,4
    • f max ( 1 + c ln ( ε ˙ ε ˙ o ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadAgadaWgaaWcbaGaciyBaiaacggacaGG4baabeaakiabgwSi xpaabmaabaGaaGymaiabgUcaRiaadogacqGHflY1ciGGSbGaaiOBam aabmaabaWaaSaaaeaacuaH1oqzgaGaaaqaaiqbew7aLzaacaWaaSba aSqaaiaad+gaaeqaaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaa aa@4DDD@ , if ICC=1,3
  4. Damage with tensile strain and energy failure criterion.
    This material could describe in plane and out-of-plane damage.
    • In plane damage with damage factor d i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgadaWgaa WcbaGaamyAaaqabaaaaa@37EF@

      Damage between ε t i and ε f i is controlled by the damage factor d i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgadaWgaa WcbaGaamyAaaqabaaaaa@37EF@ , which is given by:

      d i = min ( ε i ε t i ε i ε m i ε m i ε t i ,   d max ) in directions, i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ = 1,2

    • E-modulus
      E-modulus is reduced according to damage parameter if, ε t i ε i ε f i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8YjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqyTdu2aaSbaaSqaaiaadshacaWGPbaabeaakiabgsMiJkabew7a LnaaBaaaleaacaWGPbaabeaakiabgsMiJkabew7aLnaaBaaaleaaca WGMbGaamyAaaqabaaaaa@471B@ :(12)
      E i i r e d u c e d = E i i ( 1 d i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqhaa WcbaGaamyAaiaadMgaaeaacaWGYbGaamyzaiaadsgacaWG1bGaam4y aiaadwgacaWGKbaaaOGaeyypa0JaamyramaaBaaaleaacaWGPbGaam yAaaqabaGccaGGOaGaaGymaiabgkHiTiaadsgadaWgaaWcbaGaamyA aaqabaGccaGGPaaaaa@4837@
      E-modulus is reduced according to damage paramter, if ε i > ε f i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8YjY=LipeYth9vqqj=hEeei0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba GaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaeyOpa4JaeqyTdu2aaSba aSqaaiaadAgacaWGPbaabeaaaaa@40F5@ :(13)
      E i i r e d u c e d = E i i ( 1 d max ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqhaa WcbaGaamyAaiaadMgaaeaacaWGYbGaamyzaiaadsgacaWG1bGaam4y aiaadwgacaWGKbaaaOGaeyypa0JaamyramaaBaaaleaacaWGPbGaam yAaaqabaGccaGGOaGaaGymaiabgkHiTiaadsgadaWgaaWcbaGaciyB aiaacggacaGG4baabeaakiaacMcaaaa@4A1D@

      In this case, damage is set to d max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgadaWgaa WcbaGaciyBaiaacggacaGG4baabeaaaaa@39D4@ and it is not updated further.

    • Out-of-plane damage (delamination) with γ .
      The simpliest delamination criterion is based on the evaluation of out-of-plane shear strains ( γ 31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeek0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4SdC2aaSbaaSqaaiaaiodacaaIXaaabeaaaaa@3CC8@ and γ 23 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=viVeYth9vqqj=hEeek0xXdbb a9frFf0=yqFf0dbba91qpepeI8k8fiI+fsY=rqaqpepae9pg0Firpe pesP0xe9Fve9Fve9qapdbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaeq4SdC2aaSbaaSqaaiaaiodacaaIXaaabeaaaaa@3CC8@ ) with γ = ( γ 13 ) 2 + ( γ 23 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=HhbHc9v8 qqaqFr0xb9pg0xb9qqaqFn0dXdHiVcFbIOFHK8Feea0dXdar=Jb9hs 0dXdHuk9fr=xfr=xfrpeWZqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiabeo7aNjabg2da9maakaaabaGaaiikaiabeo7aNnaaBaaaleaa caaIXaGaaG4maaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaaiikaiabeo7aNnaaBaaaleaacaaIYaGaaG4maaqabaGccaGG PaWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4928@ .
      • Element stresses and are gradually reduced if, γ max > γ > γ i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqGH+aGpcqaHZoWzcqGH +aGpcqaHZoWzdaWgaaWcbaGaamyAaiaad6gacaWGPbaabeaaaaa@4300@
      • The element is completely removed (fails), if γ γ i n i γ max γ i n i > d 3 max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaamaalaaabaGaeq4SdCMaeyOeI0Iaeq4SdC2aaSbaaSqaaiaadMga caWGUbGaamyAaaqabaaakeaacqaHZoWzdaWgaaWcbaGaciyBaiaacg gacaGG4baabeaakiabgkHiTiabeo7aNnaaBaaaleaacaWGPbGaamOB aiaadMgaaeqaaaaakiabg6da+iaadsgadaWgaaWcbaGaaG4maiGac2 gacaGGHbGaaiiEaaqabaaaaa@5087@ in one of the shell layers.
    • The element damage could also be controlled by plastic work (energy) failure criterion. Stress is set to zero in the layer, if:
      • W p * > W p max * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH+aGpcaWG xbWaa0baaSqaaiaadchaaeaaciGGTbGaaiyyaiaacIhaaaGcdaahaa WcbeqaaiaacQcaaaaaaa@42B4@ if ICC = 1,2
      • W p * > W p max * ( 1 + c ln ε ˙ ε ˙ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH+aGpcaWG xbWaa0baaSqaaiaadchaaeaaciGGTbGaaiyyaiaacIhaaaGcdaahaa WcbeqaaiaacQcaaaGccqGHflY1daqadaqaaiaaigdacqGHRaWkcaWG JbGaciiBaiaac6gadaWcaaqaaiqbew7aLzaacaaabaGafqyTduMbai aadaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaaaaa@4F5A@ if ICC = 3,4

        With W p * = W p W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH9aqpdaWc aaqaaiaadEfadaWgaaWcbaGaamiCaaqabaaakeaacaWGxbWaa0baaS qaaiaadchaaeaacaWGYbGaamyzaiaadAgaaaaaaaaa@43DC@ and W p max * = W p max W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiGac2gacaGGHbGaaiiEaaaa kmaaCaaaleqabaGaaiOkaaaakiabg2da9maalaaabaGaam4vamaaDa aaleaacaWGWbaabaGaciyBaiaacggacaGG4baaaaGcbaGaam4vamaa DaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaaaaaaa@49BC@ .

      ICC flag defines the effect of strain rate on the maximum plastic work and on the Tsai-Wu criterion limit.

      Element deletion is controlled by the Ioff flag. The max. plastic work criteria in option Ioff is also depend on above ICC option.

      Ioff = 0: Shell is deleted if max. plastic work is reached for one element layer.

      In this case, shell element is deleted if plastic work W p * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaaaaa@3BEB@ and stress reaches the below criteria in one layer:
      • W p * > W p max * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH+aGpcaWG xbWaa0baaSqaaiaadchaaeaaciGGTbGaaiyyaiaacIhaaaGcdaahaa WcbeqaaiaacQcaaaaaaa@42B4@ if ICC = 1,2
      • W p * > W p max * ( 1 + c ln ε ˙ ε ˙ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadEfadaqhaaWcbaGaamiCaaqaaiaacQcaaaGccqGH+aGpcaWG xbWaa0baaSqaaiaadchaaeaaciGGTbGaaiyyaiaacIhaaaGcdaahaa WcbeqaaiaacQcaaaGccqGHflY1daqadaqaaiaaigdacqGHRaWkcaWG JbGaciiBaiaac6gadaWcaaqaaiqbew7aLzaacaaabaGafqyTduMbai aadaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaaaaa@4F5A@ if ICC = 3,4

      The Ratio field can be used to provide stability to composite shell components. It allows you to delete unstable elements wherein. all but one layer has failed. This last layer may cause instability during simulation, due to a low stiffness value. This option is available for strain and plastic energy based brittle failure.

      Tensile strain and energy failure criterion of LAW25 is not available for orthotropic shells with /PROP/TYPE9.

  5. The unit of W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaaaa @3D76@ is energy per unit of volume. If set W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaaaa @3D76@ as default value (0) is encountered, the default value is 1 unit of the model.
    Example:
    • If unit system of kg-m-s used in model, then W p r e f = 1 [ J m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaOGa eyypa0JaaGymamaadmaabaWaaSaaaeaacaGGkbaabaGaaiyBamaaCa aaleqabaGaai4maaaaaaaakiaawUfacaGLDbaaaaa@43F0@
    • If unit system of Ton-mm-s used in model, then W p r e f = 1 [ m J m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaOGa eyypa0JaaGymamaadmaabaWaaSaaaeaacaGGTbGaaiOsaaqaaiaac2 gacaGGTbWaaWbaaSqabeaacaGGZaaaaaaaaOGaay5waiaaw2faaaaa @45D2@
    For proper conversion of this value if changing units in pre- and post-processor, it is advised to replace the default value by the true value “1”, so that the value of W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaaaa @3D76@ will be automatically converted. Leaving the W p r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8WjY=vipgYlh9vqqj=hEeeu0xXdbb a9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yq aiVgFr0xfr=xfr=xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGcba Gaam4vamaaDaaaleaacaWGWbaabaGaamOCaiaadwgacaWGMbaaaaaa @3D76@ field to “0” may result in errors in case of automatic conversion.
    Note: A local unit system can be created for the material to avoid conversion.
  6. Output for post-processing:
    • To post-process this material in the animation file, the following Engine cards should be used:
      • /ANIM/SHELL/WPLA/ALL for plastic work output
      • /ANIM/BRICK/WPLA for plastic work output
      • /ANIM/SHELL/TENS/STRAIN for strain tensor output in the elemental coordinate system
      • /ANIM/SHELL/TENS/STRESS for stress tensor output in the elemental coordinate system
      • /ANIM/SHELL/PHI angle between elemental and first material direction
      • /ANIM/SHELL/FAIL number of failed layers.
    • To post-process this material in the time-history file, the following definitions in /TH/SHEL or /TH/SH3N card should be used:
      • PLAS (or EMIN and EMAX) for minimum and maximum plastic work in the shell.
      • WPLAYJJ (JJ=0 to 99) for plastic work in a corresponding layer.
  7. /VISC/PRONY can be used with this material law to include viscous effects.