Tsai-Wu Formulation (I_form =0)

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

Definition

Comments

1. The formulation flag I_form should be set to 0, for the TSAI-WU. Compared with I_form=1, in this formulation:
   • The TSAI-WU criterion limit $\mathrm{F}(W_p^{*},\dot{\epsilon })$ 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 (I_shell= 1,2,3,4) and fully integrated BATOZ (I_shell=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 $\mathrm{F}(W_p^{*},\dot{\epsilon })$ the material becomes nonlinear:

   • If $\mathrm{F}(\sigma )<\mathrm{F}(W_p^{*},\dot{\epsilon })$ : Elastic
   • If $\mathrm{F}(\sigma )>\mathrm{F}(W_p^{*},\dot{\epsilon })$ : Nonlinear

   Where, Stress $\mathrm{F}(\sigma )$ in element for Tsai-Wu criterion computed as:

   $$\mathrm{F}\left(\sigma \right)=F_1{\sigma }_1+F_2{\sigma }_2+F_{11}{\sigma }_1^2+F_{22}{\sigma }_2^2+F_{44}{\sigma }_{12}^2+2F_{12}{\sigma }_1{\sigma }_2$$

   Here, ${\sigma }_1$ , ${\sigma }_2$ and ${\sigma }_{12}$ 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:

   $$F_1=-\frac{1}{{\sigma }_{1y}^c}+\frac{1}{{\sigma }_{1y}^t}$$
   $$F_2=-\frac{1}{{\sigma }_{2y}^c}+\frac{1}{{\sigma }_{2y}^t}$$
   $$F_{11}=\frac{1}{{\sigma }_{1y}^c{\sigma }_{1y}^t}$$
   $$F_{12}=-\frac{\alpha }{2}\sqrt{F_{11}{F}_{22}}$$
   $$F_{22}=\frac{1}{{\sigma }_{2y}^c{\sigma }_{2y}^t}$$
   $$F_{44}=\frac{1}{{\sigma }_{12y}^c{\sigma }_{12y}^t}$$

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

   This criterion represents a second order closed three-dimensional Tsai-Wu surface in ${\sigma }_1$ , ${\sigma }_2^{}$ and ${\sigma }_{12}$ space.

   $\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)$ is the variable Tsai-Wu criterion limit defined as a function of plastic work ( $W_p^{*}$ ) and the true strain rate ( $\dot{\epsilon }$ ).

   $$\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)=\left[1+b{\left(W_p^{*}\right)}^n\right]\cdot \left[1+c\cdot \ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right]$$
   Where,

   $W_p^{ref}$

   Reference plastic work

   $W_p^{*}$

   Plastic work defined with $W_p^{*}=\frac{W_p}{W_p^{ref}}$

   b

   Plastic hardening parameter

   n

   Plastic hardening exponent

   ${\dot{\epsilon }}_0$

   Reference true strain rate

   c

   Strain rate coefficient

   This Tsai-Wu surface is scaled outwards homothetically in all directions, due to increase in $W_p$ and $\dot{\epsilon }$ .

   The max. of Tsai-Wu criterion limit $\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)$ should be limited under:

   • $f_{\max }$ , if ICC=2,4
   • $f_{\max }\cdot \left(1+c\cdot \ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_o}\right)\right)$ , 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 $d_i$

     Damage between ${\epsilon }_{ti}$ and ${\epsilon }_{fi}$ is controlled by the damage factor $d_i$ , which is given by:

     $d_i=\min \left(\frac{{\epsilon }_i-{\epsilon }_{ti}}{{\epsilon }_i}\cdot \frac{{\epsilon }_{mi}}{{\epsilon }_{mi}-{\epsilon }_{ti}},d_{\max }\right)$ in directions, $i$ = 1,2

   • E-modulus

     E-modulus is reduced according to damage parameter if, ${\epsilon }_{ti}\le {\epsilon }_i\le {\epsilon }_{fi}$ :

     $$E_{ii}^{reduced}=E_{ii}(1-d_i)$$

     E-modulus is reduced according to damage parameter, if ${\epsilon }_i>{\epsilon }_{fi}$ :

     $$E_{ii}^{reduced}=E_{ii}(1-d_{\max })$$

     In this case, damage is set to $d_{\max }$ and it is not updated further.

   • Out-of-plane damage (delamination) with $\gamma$ .
     The simplest delamination criterion is based on the evaluation of out-of-plane shear strains ( ${\gamma }_{31}$ and ${\gamma }_{23}$ ) with $\gamma =\sqrt{{({\gamma }_{13})}^2+{({\gamma }_{23})}^2}$ .

     • Element stresses and are gradually reduced if, ${\gamma }_{\max }>\gamma >{\gamma }_{ini}$
     • The element is completely removed (fails), if $\frac{\gamma -{\gamma }_{ini}}{{\gamma }_{\max }-{\gamma }_{ini}}>d_{3\max }$ 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:
     • $W_p^{*}>W_p^{\max }{}^{*}$ if ICC = 1,2
     • $W_p^{*}>W_p^{\max }{}^{*}\cdot \left(1+c\ln \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)$ if ICC = 3,4

       With $W_p^{*}=\frac{W_p}{W_p^{ref}}$ and $W_p^{\max }{}^{*}=\frac{W_p^{\max }}{W_p^{ref}}$ .

     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 I_off flag. The maximum plastic work criteria in option I_off is also depend on above ICC option.

     I_off = 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:

     • $W_p^{*}>W_p^{\max }{}^{*}$ if ICC = 1,2
     • $W_p^{*}>W_p^{\max }{}^{*}\cdot \left(1+c\ln \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)$ 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 $W_p^{ref}$ is energy per unit of volume. If set $W_p^{ref}$ 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 $W_p^{ref}=1\left[\frac{J}{m^3}\right]$
   • If unit system of Ton-mm-s used in model, then $W_p^{ref}=1\left[\frac{mJ}{m{m}^3}\right]$
   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 $W_p^{ref}$ will be automatically converted. Leaving the $W_p^{ref}$ 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
     $$d_1=\min \left(\frac{{\epsilon }_1-{\epsilon }_{t1}}{{\epsilon }_1}\cdot \frac{{\epsilon }_{m1}}{{\epsilon }_{m1}-{\epsilon }_{t1}},d_{\max }\right)$$
   • Mode 2: Tensile damage in direction 2
     $$d_2=\min \left(\frac{{\epsilon }_2-{\epsilon }_{t2}}{{\epsilon }_2}\cdot \frac{{\epsilon }_{m2}}{{\epsilon }_{m2}-{\epsilon }_{t2}},d_{\max }\right)$$
   • Mode 3: Global maximum plastic work
     $$d_3=\min \left(\frac{W_p}{W_p^{\max }},\text{ 1}\right)$$
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.