/FAIL/BIQUAD
Block Format Keyword This failure model uses a simplified nonlinear, plastic strainbased, failure criteria with linear damage accumulation. The failure strain is described by two parabolic functions calculated using curve fitting from up to 5 userdefined failure strains.
A perturbation of the failure limit for each element can be activated using /PERTURB/FAIL/BIQUAD. This failure criteria is developed in a partnership with Christian Cremer from Ford.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/FAIL/BIQUAD/mat_ID/unit_ID  
c1  c2  c3  c4  c5 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

P_thick_{fail}  MFlag  SFlag  Inst_start  fct_ID_{el}  El_ref 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

r1  r2  r4  r5 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

ICOUP  DCRIT  EXP 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

fail_ID 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  Unit identifier. (Integer, maximum 10 digits) 

c1  Failure plastic strain at uniaxial
compression. Default = 0.0 (Real) 2 

c2  Failure plastic strain at shear.
Default = 0.0 (Real) 2 

c3  Failure plastic strain in uniaxial
tension. Default = 0.6 (Real) 2 

c4  Failure plastic strain at plain strain
tension. Default = 0.0 (Real) 2 

c5  Failure plastic strain at biaxial
tension. Default = 0.0 (Real) 2 

P_thick_{fail}  Ratio of through thickness integration
points that must fail before the element is deleted (shells only). Default = 0.0 (Real) 

MFlag  Material selector flag. 4


SFlag  Specific behavior flag. 7


Inst_start  Instability start value for localized
necking. Must be entered if SFlag=3. 7
(Real) 

fct_ID_{el}  Element size factor function
identifier. (Integer) 

El_ref  Reference element size. Default = 1.0 (Real) 
$\left[\text{m}\right]$ 
r1  Failure plastic strain ratio, Uniaxial
compression (c1) to Uniaxial Tension (c3) so
$c1=r1\cdot c3$
. Only used if MFlag=99. Default = 0.0 (Real) 

r2  Failure plastic strain ratio, Pure Shear
(c2) to Uniaxial Tension (c3)
$c2=r2\cdot c3$
. Only used if MFlag=99. Default = 0.0 (Real) 

r4  Failure plastic strain ratio, Plane
Strain Tension (c4) to Uniaxial Tension (c3)
so
$c4=r4\cdot c3$
. Only used if MFlag=99. Default = 0.0 (Real) 

r5  Failure plastic strain ratio, Biaxial
Tension (c5) to Uniaxial Tension (c3) so
$c5=r5\cdot c3$
. Only used if MFlag=99. Default = 0.0 (Real) 

ICOUP  Stress/damage coupling flag to create softening.


DCRIT  Critical damage for stress softening triggering. Default = 0.0 (Real) 

EXP  Stress softening exponent. Default = 1.0 (Real) 

fail_ID  Failure criteria
identifier. 8
(Integer, maximum 10 digits) 
Example 1 (Aluminum)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
kg mm ms
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/PLAS_JOHNS/2/1
Aluminium
# RHO_I
2.64E6 0
# E Nu Iflag
70 .3 0
# a b n EPS_max SIG_max0
.35 .45 .6 0 1000
# c EPS_DOT_0 ICC Fsmooth F_cut
0 1 1 0 0
# m T_melt rhoC_p T_r
0 0 0 298
#12345678910
/FAIL/BIQUAD/2/1
# c1 c2 c3 c4 c5
0 0 0 0 0
# P_thickfail MFlag SFlag Inst_start FCT_ID_EL EI_REF
1.0 4 3 0.1 0 0
# ICOUP DCRIT EXP
0
# Fail_ID
1
#12345678910
/PERTURB/FAIL/BIQUAD/2
test1
# Mean_value Deviation Min_cut Max_cut Seed Idistri
1.0 0.03 0.95 1.05 0 1
# Fail_ID parameter
1 c3
#12345678910
#enddata
#12345678910
Example 2 (Aluminum)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
kg mm ms
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/PLAS_JOHNS/2/1
Aluminium
# RHO_I
2.64E6 0
# E Nu Iflag
70 .3 0
# a b n EPS_max SIG_max0
.35 .45 .6 0 1000
# c EPS_DOT_0 ICC Fsmooth F_cut
0 1 1 0 0
# m T_melt rhoC_p T_r
0 0 0 298
#12345678910
/FAIL/BIQUAD/2/1
# c1 c2 c3 c4 c5
1.5 0.3 0.3 0.12 0.24
# P_thickfail MFlag SFlag Inst_start FCT_ID_EL EI_REF
1.0 0 3 0.1 0 0
# ICOUP DCRIT EXP
1 0.4 2.5
# Fail_ID
1
#12345678910
/PERTURB/FAIL/BIQUAD/2
test1
# Mean_value Deviation Min_cut Max_cut Seed Idistri
1.0 0.03 0.95 1.05 0 1
# Fail_ID parameter
1 c3
#12345678910
#enddata
#12345678910
Example 3 (Aluminum)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
kg mm ms
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/PLAS_JOHNS/2/1
Aluminium
# RHO_I
2.64E6 0
# E Nu Iflag
70 .3 0
# a b n EPS_max SIG_max0
.35 .45 .6 0 1000
# c EPS_DOT_0 ICC Fsmooth F_cut
0 1 1 0 0
# m T_melt rhoC_p T_r
0 0 0 298
#12345678910
/FAIL/BIQUAD/2/1
# c1 c2 c3 c4 c5
0 0 0.3 0 0
# P_thickfail MFlag SFlag Inst_start FCT_ID_EL EI_REF
1.0 99 3 0.1 0 0
# r1 r2 r4 r5
5.0 1.0 0.4 0.8
# ICOUP DCRIT EXP
0
# Fail_ID
1
#12345678910
/PERTURB/FAIL/BIQUAD/2
test1
# Mean_value Deviation Min_cut Max_cut Seed Idistri
1.0 0.03 0.95 1.05 0 1
# Fail_ID parameter
1 c3
#12345678910
#enddata
#12345678910
Comments
 This failure criterion is defined using
failure plastic strain versus stress triaxiality (state of stress). This allows for the
different plastic failure strains that materials exhibit depending on loading condition.
The curve is described by 2 parabolic functions that intersect at the triaxiality value of
$\frac{1}{3}$
which is uniaxial tension.
The parameters for the 2 parabolic failure strain curves versus the state of stress (stress triaxiality) are calculated iteratively by Radioss during the initialization phase using the input c1c5 values. The calculated parabolic failure curve parameters a, b, c, d, e and f, can be reviewed in the Starter output file.
If the calculated parabolic failure strain curves have negative failure strain values, these negative values will be replaced by a failure strain of 1E6 which results in a very high damage accumulation and brittle behavior.
This failure criterion is usable for all elastoplastic material models with shells and solids.

By default, values different than 0 need to be entered for c1 to c5. However, specific default behaviors exists, in case failure information are missing.
 In case the material failure behavior is unknown, c1 to c5 are set to 0.0 and the mild steel behavior (MFlag=1) is used.
 If only the tensile failure value is known, c3 is defined ( $c1=c2=c4=c5=0.0$ ). The mild steel behavior is used and scaled by the userdefined c3 value.
 In case the material behavior is known, MFlag is defined and c3 can be used to adjust the failure model according to the expected tensile failure. The selected material behavior is scaled by the userdefined c3 value.
 For all other cases, all c1 to c5 are intended to be defined and default value of 0.0 is used.
 The plastic strain at failure from physical tests can be input as c1 – c5.
 If failure strain data is not available
then the material flag, MFlag, can be used to select predefined
failure values for some material.If MFlag > 0, the entered c1, c2, c4 and c5 values will not be used and instead will be calculated using the predefined ratio values from:
 $c1=r1\cdot c3$
 $c2=r2\cdot c3$
 $c3=c3$
 $c4=r4\cdot c3$
 $c5=r5\cdot c3$
If c3 =0 and MFlag ≠ 0, c3 values will also be used :MFlag Roughly Corresponds to Material c3 (Default)
r1 r2 r4 r5 1 Mild steel 0.60 3.5 1.6 0.6 1.5 2 HSS steel 0.50 4.3 1.4 0.6 1.6 3 UHSS steel 0.12 5.2 3.1 0.8 3.5 4 Aluminum AA5182 0.30 5.0 1.0 0.4 0.8 5 Aluminum AA6082T6 0.17 7.8 3.5 0.6 2.8 6 Plastic PA6GF30 0.10 3.6 0.6 0.5 0.6 7 Plastic PP T40 0.11 10.0 2.7 0.6 0.7 99 Selfdefined values (optional line) 0.60 Optional input Optional input Optional input Optional input *Neither Altair nor the authors assume any responsibility for the validity, accuracy or any results obtained from these values. You must verify your own values by reasonable test results. Usage is only recommended for early design exploration. If c3 > 0, the selected material behavior is scaled by c3 and the r1 to r5 predefined ratio values.
 If the MFlag=99, failure strain ratios r1, r2, r4 and r5 must be input in a following additional line.
 Damage is accumulated linearly and can
be postprocessed in the animation files using the output request
/ANIM/SHELL/DAMA/ALL or /ANIM/BRICK/DAMA/111. For
shells elements when an integration point reaches
$D=1$
, the integration point stress tensor is set to zero. Shell
elements are deleted based on the value of
P_thick_{fail}.
If P_thick_{fail} is blank or set to 0, the value of P_thick_{fail} from the shell property is used. If P_thick_{fail} > 0, any P_thick_{fail} value defined in the shell properties are ignored and the value entered in this failure model is used.
For values of P_thick_{fail} > 0, the element fails and is deleted when the ratio of through thickness failed integration points equals or exceeds P_thick_{fail}.
In solid elements, the element is deleted when any integration point reaches $D=1$ .
 If /PERTURB/FAIL/BIQUAD is used, MFlag > 0 must be used, and the scatter will be applied to the c3 value only. The resulting c1, c2, c4, and c5 values will be calculated using the failure strain ratios thus applying the perturbation noise to the entire failure strain curve.
 Special features are activated by this
flag:
SFlag= 1: failure curves created. Comment 1.
SFlag= 2: Ensures value c4 as global minimum. To achieve this, the second equation is split into 2 separate quadratic subfunctions where the minimum value of the curves is at c4.
Where, ${\sigma}^{*}=\frac{1}{\sqrt{3}}$ .
SFlag=3: Same as SFlag= 2 plus a simplified localized necking criterion. The localized necking criterion is based on the MarciniakKuczynski analysis. It uses two additional quadratic functions that define a curve that represents the start of localized necking between stress triaxiality $\frac{1}{3}$ and $\frac{2}{3}$ . The minimum value of this curve is defined by the user in the Inst_start field and occurs at plane strain tension ${\sigma}^{*}=\frac{1}{\sqrt{3}}$ . Using this curve, a second localized necking damage value is calculated, and failure only occurs when all integration points reach $D=1$ .
The Inst_start value can be estimated as the (true plastic) strain at maximum stress, from the uniaxial tension test.
 The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL and /PERTURB/FAIL/BIQUAD. There is no default value. If the line is blank, no value will be output for failure model variables in the /INIBRI/FAIL (written in .sta file with /STATE/BRICK/FAIL option).
 If nonlocal regularization is used (with /NONLOCAL/MAT), the element size scaling factor is not used. If a scaling function is still defined (fct_ID_{el} > 0), the parameters are scaled using LE_MAX parameter of the nonlocal card (either specified directly by you or computed from the Rlen parameter value).
 By default, a failure criterion approach with
stress computation is used for /FAIL/BIQUAD. This means that damage
evolution has no effect on the stress computation until the element deletion triggered by
D = 1 occurs. It is possible to soften the stress
during damage evolution. Therefore, the flag ICOUP must be set to a
value not equal to zero in order to activate the stress/damage coupling. For this purpose
the following stress softening equation (similarly to /FAIL/TAB2) is
introduced:$$\sigma ={\sigma}_{eff}\left(1{\left(\frac{D{D}_{crit}}{1{D}_{crit}}\right)}^{EXP}\right)$$Where,
 $\sigma $
 Damage stress tensor
 ${\sigma}_{eff}$
 Undamaged effective stress tensor
 ${D}_{crit}$
 Critical damage value that triggers stress softening
 $EXP$
 Exponent parameter
Two different approaches can be used: If ICOUP = 1 (for solids and shells), the critical damage parameter DCRIT is directly input by the user. By default, DCRIT = 0.0 which means that damage variable has an influence on stress computation from the beginning of plasticity. However, one may want to delay this stress softening effect to a higher value of damage variable (0 < D < 1). The exponent parameter EXP can be used to control the shape of the stress softening so that it is nonlinear when EXP is different from 1 (default value).
 If ICOUP = 2 (for shells with
SFlag = 3 only), the instability damage
variable computed from the instability curve (defined with
Inst_start) is used to deduce the DCRIT
value.Indeed, in this case the instability curve represents the criterion that must be reached to trigger stress softening, implying the beginning of the strain localization and then the necking observed especially at high stress triaxiality.Note: The instability curves only have an effect between simple tension and biaxial tension stress triaxiality. When the instability criterion is reached, the instantaneous value of the damage variable D is saved in the value DCRIT, which becomes an element history variable and not a constant value.
In this case, the DCRIT value defined in the input card is ignored. This allows the necking in shells to be triggered by generating a stress softening similar to /FAIL/TAB2. The exponent parameter still can be used with ICOUP = 2.