/MAT/LAW15 (CHANG)

Block Format Keyword This law is used to model composite shell elements, similar to LAW25. The plastic behavior is based on the Tsai-Wu criteria (/MAT/LAW25 (COMPSH) for Tsai-Wu description) and failure is based on the Chang-Chang failure criterion is used.

It is recommended to use material LAW25 in combination with a separate Chang-Chang failure criteria (/MAT/LAW25 with /FAIL/CHANG keywords), instead of material LAW15.

Format

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW15/`mat_ID`/`unit_ID` or /MAT/CHANG/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$
`E`₁₁		`E`₂₂		$ν_{12}$
`G`₁₂		`G`₂₃		`G`₃₁
`b`		`n`		`f`_max
$W_{p}^{max}$		$W_{p}^{ref}$		`Ioff`
$σ_{1 y}^{t}$		$σ_{2 y}^{t}$		$σ_{1 y}^{c}$		$σ_{2 y}^{c}$	α
$σ_{12 y}^{c}$		$σ_{12 y}^{t}$		`c`		${\dot{ε}}_{0}$	`ICC`
$β$		$τ_{\max}$		`S`₁		`S`₂	`S`₁₂
`F`_smooth	`F`_cut		`C`₁		`C`₂

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)	$[\frac{kg}{m^{3}}]$
`E`₁₁	Young's modulus in direction 1. (Real)	$[Pa]$
`E`₂₂	Young's modulus in direction 2. (Real)	$[Pa]$
$ν_{12}$	Poisson's ratio. (Real)
`G`₁₂	Shear modulus. (Real)	$[Pa]$
`G`₂₃	Shear modulus. (Real)	$[Pa]$
`G`₃₁₁	Shear modulus. (Real)	$[Pa]$
`b`	Hardening parameter. (Real)
`n`	Hardening exponent. Default = 1.0 (Real)
`fmax`	Maximum value of yield function. 2 Default = 10³⁰ (Real)	$[Pa]$
$W_{p}^{max}$	Maximum plastic energy per volume unit. Default = 10³⁰ (Real)	$[\frac{J}{m^{3}}]$
$W_{p}^{ref}$	Reference plastic energy per volume unit. Default = 1.0 (Real)	$[\frac{J}{m^{3}}]$
`Ioff`	Total element failure criteria. 4 = 0 Shell is deleted if $W_{p}^{} > W_{p}^{max }$ for one layer. = 1 Shell is deleted if $W_{p}^{} > W_{p}^{max }$ for all layers. = 2 If for each layer, $W_{p}^{} > W_{p}^{max }$ or tensile failure in direction 1 = 3 If for each layer, $W_{p}^{} > W_{p}^{max }$ or tensile failure in direction 2. = 4 If for each layer, $W_{p}^{} > W_{p}^{max }$ or tensile failure in directions 1 and 2. = 5 If for all layers: $W_{p}^{} > W_{p}^{max }$ or tensile failure in direction 1 or if for all layers: $W_{p}^{} > W_{p}^{max }$ or tensile failure in direction 2. = 6 If for each layer, $W_{p}^{} > W_{p}^{max }$ or tensile failure in direction 1 or 2. (Integer)
$σ_{1 y}^{t}$	Composite yield stress in tension in direction 1. 2 (Real)	$[Pa]$
$σ_{2 y}^{t}$	Composite yield stress in tension in direction 2. (Real)	$[Pa]$
$σ_{1 y}^{c}$	Composite yield stress in compression in direction 1. (Real)	$[Pa]$
$σ_{2 y}^{c}$	Composite yield stress in compression in direction 2. (Real)	$[Pa]$
α	`F12` reduction factor. 2 Default set to 1.0 (Real)
$σ_{12 y}^{c}$	Yield stress in shear and strain rate compression in direction 12. (Real)	$[Pa]$
$σ_{12 y}^{t}$	Yield stress in shear and strain rate tension in direction 12. (Real)	$[Pa]$
`c`	Yield stress in shear and strain rate coefficient. 2 = 0 No strain rate dependency. (Real)
${\dot{ε}}_{0}$	Yield stress in shear and strain rate reference. (Real)	$[\frac{1}{s}]$
`ICC`	Strain rate computation flag. 2 = 1 (Default) Strain rate effect on `f`_max no effect on $W_{p}^{max}$ . = 2 No strain rate effect on `f`_max and $W_{p}^{max}$ = 3 Strain rate effect on `f`_max and $W_{p}^{max}$ . = 4 No strain rate effect on `f`_max effect on $W_{p}^{max}$ . (Integer)
$β$	Shear scaling factor. 1 (Real)
$τ_{\max}$	Time relaxation. 3 Default = 10³⁰ (Real)	$[s]$
`S`₁	Longitudinal tensile strength. 1 Default = 10³⁰ (Real)	$[Pa]$
`S`₂	Transverse tensile strength. Default = 10³⁰ (Real)	$[Pa]$
`S`₁₂	Shear strength. Default = 10³⁰ (Real)	$[Pa]$
`F`_smooth	Smooth strain rate option flag. = 0 (Default) No strain rate smoothing = 1 Strain rate smoothing active (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. Default = 10³⁰ (Real)	$[Hz]$
`C`₁	Longitudinal compressive strength. 1 Default = 10³⁰ (Real)	$[Pa]$
`C`₂	Transverse compressive strength. Default = 10³⁰ (Real)	$[Pa]$

Example (Carbon)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW15/1/1
Carbon
#              RHO_I
              1.8E-6                   0
#                E11                 E22                nu12
                  41                 3.3                  .3
#                G12                 G23                 G31
                 5.2                 1.3                 1.3
#                  b                   n                fmax
                8E-6                   1              100000
#              Wpmax              Wpref       Ioff
              100000                   0         0
#          sigma_1yt           sigma_2yt           sigma_1yc          sigma_2yc               alpha
                .786               .1566                .786               .1566                   0
#         sigma_12yc          sigma_12yt                   c           Eps_dot_0       ICC
               .0655               .0655                   0                   0         0
#               beta                Tmax                  S1                  S2                 S12
                   1                 .01                   0                   0                   0
#  Fsmooth                Fcut                  C1                 C12
         0                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

Chang-Chang failure criteria

Six material parameters are used in the Chang-Chang failure criteria to describe the two different failure behaviors.

For fiber breakage, the failure criteria is:

Tensile fiber mode

$σ_{11} > 0$

(1)

$e_{f}^{2} = {(\frac{σ_{11}}{S_{1}})}^{2} + β {(\frac{σ_{12}}{S_{12}})}^{2} - 1.0$

$\begin{array}{l} \geq 0 failed \\ < 0 elastic - plastic \end{array}$

Compressive fiber mode

$σ_{11} < 0$

(2)

$e_{c}^{2} = {(\frac{σ_{11}}{C_{1}})}^{2} - 1.0$

$\begin{array}{l} \geq 0 failed \\ < 0 elastic - plastic \end{array}$

For matrix cracking, the failure criteria is:

Tensile fiber mode

$σ_{22} > 0$

(3)

$e_{m}^{2} = {(\frac{σ_{22}}{S_{2}})}^{2} + β {(\frac{σ_{12}}{S_{12}})}^{2} - 1.0$

$\begin{array}{l} \geq 0 failed \\ < 0 elastic - plastic \end{array}$

Compressive matrix mode $σ_{22} < 0$

(4)

$e_{d}^{2} = {(\frac{σ_{22}}{2 S_{12}})}^{2} + [{(\frac{C_{2}}{2 S_{12}})}^{2} - 1] \frac{σ_{22}}{C_{2}} + {(\frac{σ_{12}}{S_{12}})}^{2} - 1.0$

$\begin{array}{l} \geq 0 failed \\ < 0 elastic - plastic \end{array}$

Before failed (damage parameter $e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2}$ is less than 0), material is in elastic–plastic phase. The plastic behavior is based on the TSAI-WU criteria (see Tsai-Wu Formulation (Iform =0) for Tsai-Wu criterion description).
After failed (damage parameter $e_{f}^{2}, e_{c}^{2}, e_{m}^{2}, e_{d}^{2}$ is greater than or equal to 0), the stresses are decreased by using an exponential function to avoid numerical instabilities.
A relaxation technique is used by gradually decreasing the stress.(5)
$σ (t) = f (t) \cdot σ_{d} (t_{r})$

With function of relaxation:

$f (t) = \exp (- \frac{t - t_{r}}{τ_{\max}})$ and $t \geq t_{r}$

Where,

$t$

Time

$t_{r}$

Start time of relaxation when the damage criteria is assumed

$τ_{\max}$

Time of dynamic relaxation

$σ_{d} (t_{r})$

Stress components at the beginning of damage
If a shell has several layers with one material per layer (different materials, different I_off), the I_off used is the one which is associated to the shell in the shell element definition.