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
ρiρi
E11 E22 υ12υ12 Iform E33
G12 G23 G31 εf1εf1 εf2εf2
εt1εt1 εm1εm1 εt2εt2 εm2εm2 dmax
Composite Plasticity Hardening
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
WmaxpWmaxp WrefpWrefp Ioff Ratio
b n fmax
Composite Yield Stress in Tension Compression
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
σt1yσt1y σt2yσt2y σc1yσc1y σc2yσc2y αα
Yield Stress in Shear and Strain Rate
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
σc12yσc12y σt12yσt12y c ˙ɛ ICC
Delamination
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
γini γmax d3max
Strain Rate Filtering
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Fsmooth Fcut

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)

ρi Initial density.

(Real)

[kgm3]
E11 Young's modulus in direction 1.

(Real)

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

(Real)

[Pa]
υ12 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]
εf1 Maximum tensile strain for element deletion in material direction 1.

Default = 1.2 x 1020 (Real)

εf2 Maximum tensile strain for element deletion in material direction 2.

Default = 1.2 x 1020 (Real)

εt1 Tensile failure strain in the material direction 1 at which stress starts to reduce. 4

Default = 1.0 x 1020 (Real)

εm1 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)

εt2 Tensile failure strain in the material direction 2 at which the stress starts to reduce.

Default = 1.0 x 1020 (Real)

εm2 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)

Wmaxp Maximum plastic work per unit shell volume.

Default = 1020 (Real)

[Jm3]
Wrefp Reference plastic work per unit shell volume. 4

Default = 1.0 (Real)

[Jm3]
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,
Condition1:{eithermax.plasticworkreachedorε1>εm1indirection1ord1>dmaxindirection1
= 3
Shell is deleted if for each element layer,
Condition2:{eithermax.plasticworkreachedorε2>εm2indirection2ord2>dmaxindirection2
= 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:
numberoffailedlayersnumberoftotallayersratio

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)

σt1y Yield stress in tension in direction 1.

Default = 0.0 (Real)

[Pa]
σt2y Yield stress in tension in direction 2.

Default = 0.0 (Real)

[Pa]
σc1y Yield stress in compression in direction 1.

Default = 0.0 (Real)

[Pa]
σc2y 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)

σc12y Yield stress in compression in 45 degree of fiber direction.

Default = 0.0 (Real)

[Pa]
σt12y 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)

˙ε0 Reference strain rate.

If ˙ε˙ε0 , no strain rate effect.

(Real)

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

(Integer)

γini Out-of-plane shear strain when delamination begins. 4

Default = 1020 (Real)

γmax 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)

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) 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 specify 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) the material becomes nonlinear:
    • If F(σ)<F(W*p,˙ε) : Elastic
    • If F(σ)>F(W*p,˙ε) : Nonlinear

    Where, Stress F(σ) in element for Tsai-Wu criterion computed as:

    F(σ)=F1σ1+F2σ2+F11σ21+F22σ22+F44σ212+2F12σ1σ2

    Here, σ1 , σ2 and σ12 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:

    F1=1σc1y+1σt1y
    F2=1σc2y+1σt2y
    F11=1σc1yσt1y
    F12=α2F11F22
    F22=1σc2yσt2y
    F44=1σc12yσt12y

    The superscripts c and t represent compression and tension, respectively.

    This criterion represents a second order closed three-dimensional Tsai-Wu surface in σ1 , σ2 and σ12 space.

    F(W*p,˙ε) is the variable Tsai-Wu criterion limit defined as a function of plastic work ( W*p ) and the true strain rate ( ˙ε ).

    F(W*p,˙ε)=[1+b(W*p)n][1+cln(˙ε˙ε0)]

    Where,
    Wrefp
    Reference plastic work
    W*p
    Plastic work defined with W*p=WpWrefp
    b
    Plastic hardening parameter
    n
    Plastic hardening exponent
    ˙ε0
    Reference true strain rate
    c
    Strain rate coefficient

    This Tsai-Wu surface is scaled outwards homothetically in all directions, due to increase in Wp and ˙ε .

    The max. of Tsai-Wu criterion limit F(W*p,˙ε) should be limited under:
    • fmax , if ICC=2,4
    • fmax(1+cln(˙ε˙εo)) , 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 di

      Damage between εti and εfi is controlled by the damage factor di , which is given by:

      di=min(εiεtiεiεmiεmiεti, dmax) in directions, i = 1,2

    • E-modulus

      E-modulus is reduced according to damage parameter if, εtiεiεfi :

      Ereducedii=Eii(1di)

      E-modulus is reduced according to damage parameter, if εi>εfi :

      Ereducedii=Eii(1dmax)

      In this case, damage is set to dmax and it is not updated further.

    • Out-of-plane damage (delamination) with γ .
      The simplest delamination criterion is based on the evaluation of out-of-plane shear strains ( γ31 and γ23 ) with γ=(γ13)2+(γ23)2 .
      • Element stresses and are gradually reduced if, γmax>γ>γini
      • The element is completely removed (fails), if γγiniγmaxγini>d3max 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>Wmaxp* if ICC = 1,2
      • W*p>Wmaxp*(1+cln˙ε˙ε0) if ICC = 3,4

        With W*p=WpWrefp and Wmaxp*=WmaxpWrefp .

      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 maximum plastic work criteria in option Ioff is also depend on above ICC option.

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

      In this case, shell element is deleted if plastic work W*p and stress reaches the below criteria in one layer:
      • W*p>Wmaxp* if ICC = 1,2
      • W*p>Wmaxp*(1+cln˙ε˙ε0) 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 Wrefp is energy per unit of volume. If set Wrefp 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 Wrefp=1[Jm3]
    • If unit system of Ton-mm-s used in model, then Wrefp=1[mJmm3]
    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 Wrefp will be automatically converted. Leaving the Wrefp 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.
  8. The different modes of failure can be output using /H3D/ELEM/DAMG/ID=Mat_ID with the keyword MODE (=I or ALL). The correspondence between the modes and the damage variables are:
    • Mode 1: Tensile damage in direction 1
      d1=min(ε1εt1ε1εm1εm1εt1, dmax)
    • Mode 2: Tensile damage in direction 2
      d2=min(ε2εt2ε2εm2εm2εt2, dmax)
    • Mode 3: Global maximum plastic work
      d3=min(WpWmaxp, 1)
  9. A global failure index can be plotted using /H3D/ELEM/DAMG/(ID=Mat_ID) without MODE option. It corresponds to the maximum value between the 3 damage variables described above.