TsaiWu Formulation (I_{form} =0)
Block Format Keyword This law describes the composite shell and solid material using the TsaiWu formulation.
The material is assumed to be orthotropicelastic before the TsaiWu criterion is reached. The material becomes nonlinear afterwards. For solid elements, the material is assumed to be linearly elastic in the transverse direction. The TsaiWu 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 outofplane 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  
${\rho}_{i}$  
E_{11}  E_{22}  ${\upsilon}_{12}$  I_{form}  E_{33}  
G_{12}  G_{23}  G_{31}  ${\epsilon}_{f1}$  ${\epsilon}_{f2}$  
${\epsilon}_{t1}$  ${\epsilon}_{m1}$  ${\epsilon}_{t2}$  ${\epsilon}_{m2}$  d_{max} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\text{W}}_{p}^{\text{max}}$  ${W}_{p}^{\mathit{ref}}$  I_{off}  Ratio  
b  n  f_{max} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\sigma}_{1y}^{t}$  ${\sigma}_{2y}^{t}$  ${\sigma}_{1y}^{c}$  ${\sigma}_{2y}^{c}$  α 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\sigma}_{12y}^{c}$  ${\sigma}_{12y}^{t}$  c  $\dot{\varepsilon}$  ICC 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\gamma}_{ini}$  ${\gamma}_{\mathrm{max}}$  d_{3max} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

F_{smooth}  F_{cut} 
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) 

${\rho}_{i}$  Initial
density (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E_{11}  Young's modulus in
direction 1. (Real) 
$\left[\text{Pa}\right]$ 
E_{22}  Young's modulus in
direction 2. (Real) 
$\left[\text{Pa}\right]$ 
${\upsilon}_{12}$  Poisson's
ratio. (Real) 

I_{form}  Formulation flag. 1
(Integer) 

E_{33}  Young's modulus in
direction 33. 2 (Real) 
$\left[\text{Pa}\right]$ 
G_{12}  Shear modulus in direction
12. (Real) 
$\left[\text{Pa}\right]$ 
G_{23}  Shear modulus in direction
23. (Real) 
$\left[\text{Pa}\right]$ 
G_{31}  Shear modulus in direction
31. (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{f1}$  Maximum tensile strain for
element deletion in material direction 1. Default = 1.2 x 10^{20} (Real) 

${\epsilon}_{f2}$  Maximum tensile strain for
element deletion in material direction 2. Default = 1.2 x 10^{20} (Real) 

${\epsilon}_{t1}$  Tensile failure strain in
the material direction 1 at which stress starts to reduce. 4 Default = 1.0 x 10^{20} (Real) 

${\epsilon}_{m1}$  Maximum tensile strain in
material direction 1 at which the stress at the element is set to
zero, if d_{max} = 1. 4 Default = 1.1 x 10^{20} (Real) 

${\epsilon}_{t2}$  Tensile failure strain in
the material direction 2 at which the stress starts to
reduce. Default = 1.0 x 10^{20} (Real) 

${\epsilon}_{m2}$  Maximum tensile strain in
material direction 2 at which the stress in the element is set to
zero, if d_{max}=1. Default = 1.1 x 10^{20} (Real) 

d_{max}  Maximum damage factor d_{max} ≤ 1). 4 Default = 0.999 (Real) 

${\text{W}}_{p}^{\text{max}}$  Maximum plastic work per
unit shell volume. Default = 10^{20} (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
${W}_{p}^{\mathrm{ref}}$  Reference plastic work per
unit shell volume. 4 Default = 1.0 (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}}\right]$ 
I_{off}  Flag that controls shell
and thick shell element deletion depending on failure modes in the
element layers. 4
(Integer) 

Ratio  Ratio parameter which
controls the deletion of shell elements based on the number of
failed layers.
Default = 1.0 (Real) 

b  Plastic hardening
parameter. Default = 0.0 (Real) 

n  Plastic hardening
exponent. Default = 1.0 (Real) 

f_{max}  Maximum value of the
TsaiWu criterion limit. Default = 10^{20} (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{1y}^{t}$  Yield stress in tension in
direction 1. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{2y}^{t}$  Yield stress in tension in
direction 2. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{1y}^{c}$  Yield stress in
compression in direction 1. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{2y}^{c}$  Yield stress in
compression in direction 2. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
α  Reduction factor for
F_{12} coefficient calculation in
TsaiWu criterion. Default set to 1.0 (Real) 

${\sigma}_{12y}^{c}$  Yield stress in
compression in 45 degree of fiber direction. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{12y}^{t}$  Yield stress in tension in
45 degree of fiber direction. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
c  Strain rate coefficient
for plastic work criteria.
(Real) 

${\dot{\epsilon}}_{0}$  Reference strain
rate. If $\dot{\epsilon}\le {\dot{\epsilon}}_{0}$ , no strain rate effect. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
ICC  Strain rate effect flag.
4
(Integer) 

${\gamma}_{\mathrm{ini}}$  Outofplane shear strain
when delamination begins. 4 Default = 10^{20} (Real) 

${\gamma}_{\mathrm{max}}$  Outofplane shear strain
when delamination ends and the element is deleted. 4 Default = 1.1 * 10^{20} (Real) 

d_{3max}  Maximum delamination
damage factor (d_{3max} < 1). 4 Default = 1.0 (Real) 

F_{smooth}  Strain rate filtering flag.
(Integer) 

F_{cut}  Cutoff frequency for
strain rate filtering. Default = 10^{20} (Real) 
$\text{[Hz]}$ 
Example (Composite)
#RADIOSS STARTER
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
g mm ms
#12345678910
/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
#12345678910
#enddata
#12345678910
Comments
 The formulation flag
I_{form} should be set to 0, for
the TSAIWU. Compared with I_{form}=1, in this
formulation:
 The TSAIWU 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
 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 underintegrated 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 E_{33} 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.
 The TsaiWu
criterion:The material is assumed to be elastic until the TsaiWu criterion is fulfilled. After exceeding the TsaiWu 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 TsaiWu criterion computed as:(4) $$\mathrm{F}\left(\sigma \right)={F}_{1}{\sigma}_{1}+{F}_{2}{\sigma}_{2}+{F}_{11}{\sigma}_{1}^{2}+{F}_{22}{\sigma}_{2}^{2}+{F}_{44}{\sigma}_{12}^{2}+2{F}_{12}{\sigma}_{1}{\sigma}_{2}$$Here, ${\sigma}_{1}$ , ${\sigma}_{2}$ and ${\sigma}_{12}$ are the stresses in the material coordinate system.
The F coefficients of the TsaiWu 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) $${F}_{1}=\frac{1}{{\sigma}_{1y}^{c}}+\frac{1}{{\sigma}_{1y}^{t}}$$(6) $${F}_{2}=\frac{1}{{\sigma}_{2y}^{c}}+\frac{1}{{\sigma}_{2y}^{t}}$$(7) $${F}_{11}=\frac{1}{{\sigma}_{1y}^{c}{\sigma}_{1y}^{t}}$$(8) $${F}_{12}=\frac{\alpha}{2}\sqrt{{F}_{11}{F}_{22}}$$(9) $${F}_{22}=\frac{1}{{\sigma}_{2y}^{c}{\sigma}_{2y}^{t}}$$(10) $${F}_{44}=\frac{1}{{\sigma}_{12y}^{c}{\sigma}_{12y}^{t}}$$The superscripts $c$ and $t$ represent compression and tension, respectively.
This criterion represents a second order closed threedimensional TsaiWu surface in ${\sigma}_{1}$ , ${\sigma}_{2}^{}$ and ${\sigma}_{12}$ space.
$\mathrm{F}\left({W}_{p}^{*},\dot{\epsilon}\right)$ is the variable TsaiWu criterion limit defined as a function of plastic work ( ${W}_{p}^{*}$ ) and the true strain rate ( $\dot{\epsilon}$ ).(11) $$\mathrm{F}\left({W}_{p}^{*},\dot{\epsilon}\right)=\left[1+b{\left({W}_{p}^{*}\right)}^{n}\right]\cdot \left[1+c\cdot \mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\right]$$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 TsaiWu surface is scaled outwards homothetically in all directions, due to increase in ${W}_{p}$ and $\dot{\epsilon}$ .
The max. of TsaiWu criterion limit $\mathrm{F}\left({W}_{p}^{*},\dot{\epsilon}\right)$ should be limited under: ${f}_{\mathrm{max}}$ , if ICC=2,4
 ${f}_{\mathrm{max}}\cdot \left(1+c\cdot \mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$ , if ICC=1,3
 Damage with tensile
strain and energy failure criterion.This material could describe in plane and outofplane 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:
${d}_{i}=\mathrm{min}\left(\frac{{\epsilon}_{i}{\epsilon}_{ti}}{{\epsilon}_{i}}\cdot \frac{{\epsilon}_{mi}}{{\epsilon}_{mi}{\epsilon}_{ti}},\text{}{d}_{\mathrm{max}}\right)$ in directions, $i$ = 1,2
 EmodulusEmodulus is reduced according to damage parameter if, ${\epsilon}_{ti}\le {\epsilon}_{i}\le {\epsilon}_{fi}$ :
(12) $${E}_{ii}^{reduced}={E}_{ii}(1{d}_{i})$$Emodulus is reduced according to damage paramter, if ${\epsilon}_{i}>{\epsilon}_{fi}$ :(13) $${E}_{ii}^{reduced}={E}_{ii}(1{d}_{\mathrm{max}})$$In this case, damage is set to ${d}_{\mathrm{max}}$ and it is not updated further.
 Outofplane damage (delamination) with
$\gamma $
.The simpliest delamination criterion is based on the evaluation of outofplane shear strains ( ${\gamma}_{31}$ and ${\gamma}_{23}$ ) with $\gamma =\sqrt{{({\gamma}_{13})}^{2}+{({\gamma}_{23})}^{2}}$ .
 Element stresses and are gradually reduced if, ${\gamma}_{\mathrm{max}}>\gamma >{\gamma}_{ini}$
 The element is completely removed (fails), if $\frac{\gamma {\gamma}_{ini}}{{\gamma}_{\mathrm{max}}{\gamma}_{ini}}>{d}_{3\mathrm{max}}$ 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:
 ${W}_{p}^{*}>{W}_{p}^{\mathrm{max}}{}^{*}$ if ICC = 1,2

${W}_{p}^{*}>{W}_{p}^{\mathrm{max}}{}^{*}\cdot \left(1+c\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)$
if ICC =
3,4
With ${W}_{p}^{*}=\frac{{W}_{p}}{{W}_{p}^{ref}}$ and ${W}_{p}^{\mathrm{max}}{}^{*}=\frac{{W}_{p}^{\mathrm{max}}}{{W}_{p}^{ref}}$ .
ICC flag defines the effect of strain rate on the maximum plastic work and on the TsaiWu criterion limit.
Element deletion is controlled by the I_{off} flag. The max. plastic work criteria in option I_{off} is also depend on above ICC option.
I_{off} = 0: Shell is deleted if max. plastic work is reached for one element layer.
In this case, shell element is deleted if plastic work ${W}_{p}^{*}$ and stress reaches the below criteria in one layer: ${W}_{p}^{*}>{W}_{p}^{\mathrm{max}}{}^{*}$ if ICC = 1,2
 ${W}_{p}^{*}>{W}_{p}^{\mathrm{max}}{}^{*}\cdot \left(1+c\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)$ 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.
 In plane damage with damage factor
${d}_{i}$
 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 kgms used in model, then ${W}_{p}^{ref}=1\left[\frac{J}{{m}^{3}}\right]$
 If unit system of Tonmms 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 postprocessor, 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.  Output for
postprocessing:
 To postprocess 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 postprocess this material in the timehistory 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.
 To postprocess this material in the animation file, the following
Engine cards should be used:
 /VISC/PRONY can be used with this material law to include viscous effects.