/FAIL/ORTHBIQUAD
Block Format Keyword This failure model uses an orthotropic simplified nonlinear, strainbased, failure criteria with linear damage accumulation.
For several loading directions, the failure strain is described by two parabolic functions calculated using curve fitting from up to 5 user input failure strains. For all loading directions that are not given in the input, an interpolation of the failure strain evolution with stress triaxiality will be done during the simulation. You can give up to 10 different set of parameters for 10 different directions equally distributed between 0 and 90 degrees.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/FAIL/ORTHBIQUAD/mat_ID/unit_ID 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

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

c5  ${\dot{\epsilon}}_{0}$  C_{JC}  fct_ID_{rate}  Rate_Scale 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

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

c1  c2  c3  c4  Inst_start 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

fail_ID 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  (Optional) Unit Identifier. (Integer, maximum 10 digits) 

P_thick_{fail}  Ratio of through thickness
integration points that must fail before the element is deleted.
(shells only). Default = 1.0 (Real) 

MFlag  Material selector flag.
(Integer) 

SFlag  Specific behavior flag.


N_{angle}  Number of experimental
angles. (Integer) 

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

El_ref  Reference element size. Default = 1.0 (Real) 
$\left[\text{m}\right]$ 
c5  Failure plastic strain in biaxial
tension (same for all directions). Default = 0.0 (Real) 

${\dot{\epsilon}}_{0}$  Inviscid limit for the strain
rate. Default = 0.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
C_{JC}  JohnsonCook strain rate
coefficient. Default = 0.0 (Real) 

fct_ID_{rate}  Strain rate dependency factor
tabulated function identifier. (Integer) 

Rate_Scale  Abscissa scale factor for strain
rate dependency tabulated function. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
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), so
$c2=r2\cdot c3$
. 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) 

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

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

c3  Failure plastic strain in uniaxial
tension. Default = 0.0 (Real) 

c4  Failure plastic strain in plane
strain tension. Default = 0.0 (Real) 

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

fail_ID  (Optional) Failure criteria
identifier. (Integer, maximum 10 digits) 
Example 1
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
Mg mm s
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/LAW93/2/1
Steel
# RHO_I
7.8E09
# E11 E22 E33 G12 Nu12
190000 190000 190000 70000 0.3
# G13 G23 Nu13 Nu23
70000 70000 0.3 0.3
# NL VP Fcut
0 1 0
# SigY QR1 CR1 QR2 CR2
290 580 1 200 25
# R11 R22 R12
1 1 1
# R33 R13 R23
1 1 1
#12345678910
/FAIL/ORTHBIQUAD/2/1
# PTHK MFLAG SFLAG NANGLE FCT_IDEL EL_REF
1 2 1 2 101 .3
# C5 DEPS0 C_JCOOK FCT_ID_RATE RATE_SCALE
0 0 0 0 0
# C1 C2 C3 C4 INST
0 0 .7 0 .3
0 0 .35 0 .15
#12345678910
/FUNCT/101
Regularization DP450_ODG3_MED5
0 1
1 1
6 .35
10 .35
#12345678910
#enddata
#12345678910
Example 2
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
Mg mm s
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/LAW93/2/1
Steel
# RHO_I
7.8E09
# E11 E22 E33 G12 Nu12
190000 190000 190000 70000 0.3
# G13 G23 Nu13 Nu23
70000 70000 0.3 0.3
# NL VP Fcut
0 1 0
# SigY QR1 CR1 QR2 CR2
290 580 1 200 25
# R11 R22 R12
1 1 1
# R33 R13 R23
1 1 1
#12345678910
/FAIL/ORTHBIQUAD/2/1
# PTHK MFLAG SFLAG NANGLE FCT_IDEL EL_REF
1 0 1 2 101 .3
# C5 DEPS0 C_JCOOK FCT_ID_RATE RATE_SCALE
.56 0 0 0 0
# C1 C2 C3 C4 INST
3.01 .98 .7 .42 .3
1.505 .49 .35 .21 .15
#12345678910
/FUNCT/101
Regularization DP450_ODG3_MED5
0 1
1 1
6 .35
10 .35
#12345678910
#enddata
#12345678910
Example 3
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
Mg mm s
#12345678910
# 1. MATERIALS:
#12345678910
/MAT/LAW93/2/1
Steel
# RHO_I
7.8E09
# E11 E22 E33 G12 Nu12
190000 190000 190000 70000 0.3
# G13 G23 Nu13 Nu23
70000 70000 0.3 0.3
# NL VP Fcut
0 1 0
# SigY QR1 CR1 QR2 CR2
290 580 1 200 25
# R11 R22 R12
1 1 1
# R33 R13 R23
1 1 1
#12345678910
/FAIL/ORTHBIQUAD/2/1
# PTHK MFLAG SFLAG NANGLE FCT_IDEL EL_REF
1 99 1 2 101 .3
# C5 DEPS0 C_JCOOK FCT_ID_RATE RATE_SCALE
0 0 0 0 0
# r1 r2 r4 r5
4.3 1.4 .6 1.6
# C1 C2 C3 C4 INST
0 0 .7 0 .3
0 0 .35 0 .15
#12345678910
/FUNCT/101
Regularization DP450_ODG3_MED5
0 1
1 1
6 .35
10 .35
#12345678910
#enddata
#12345678910
Comments
 For each input
direction, the failure criteria is defined using failure plastic strain
versus stress triaxiality (state of stress) as it is the case for the
/FAIL/BIQUAD criterion. 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.
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 criteria is usable for all elastoplastic material models with shells and solids.
 To consider the failure orthotropy, several set of c1, c2, c3, c4 and Inst_start parameters can be given. You can input up to 10 different set of parameters for 10 different experimental loading direction (marked by an angle denoted $\theta $ ). The number of input tested loading direction is set by the parameter N_{angle}. The directions must be equally distributed 0 and 90 degrees following the table.
 For each input
direction, the 2 parabolic curves parameters (
$a$
,
$b$
,
$c$
,
$d$
,
$e$
,
$f$
) are computed in the Radioss Starter. During the simulation, for all
loading directions located between the input directions, a Fourier series
interpolation is used to determine the corresponding curves parameters (
$a$
,
$b$
,
$c$
,
$d$
,
$e$
,
$f$
) and, if defined, the necking instability
strain (Inst_start):
(1) $$\begin{array}{ccc}a={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{a}\mathrm{cos}(2m\theta )}& d={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{d}\mathrm{cos}(2m\theta )}& \\ b={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{b}\mathrm{cos}(2m\theta )}& e={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{e}\mathrm{cos}(2m\theta )}& Inst={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{Inst}\mathrm{cos}(2m\theta )}\\ c={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{c}\mathrm{cos}(2m\theta )}& f={\displaystyle \sum _{m=0}^{{N}_{angle}}{Q}_{m}^{f}\mathrm{cos}(2m\theta )}& \end{array}$$Where, ${Q}_{m}^{i}$ are the interpolation factors for parameter $i$ . These interpolation factors are automatically computed in the Radioss Starter.
For instance, if two sets of parameters are given for the direction 0 and 90 degrees, the Fourier series interpolation enables to determine the 45 degrees curve as:Loading directions not given in the input are interpolated using Fourier series. All directions have the same plastic strain at failure in biaxial tension.
 It must be noted that plastic strain at failure in biaxial tension c5 is common to all directions (Figure 3). Indeed, for this loading condition, the directions have no meaning as all directions are loaded in the same way. Thus, the failure behavior is the same.
 By default, values
different than 0 for c1 to c5 need to
be entered. However, specific default behaviors exist 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. Note: If c5 = 0.0, it is automatically computed depending on the value of MFlag. In this case, the minimum value computed among all input directions will be retained.
 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 materials.
 If
MFlag > 0, the entered
c1, c2,
c4 and c5 values will not
be used, but will be calculated as follows, using the predefined
ratio values from the table.
$c1=r1\cdot c3$
$c2=r2\cdot c3$
$c3=c3$
$c4=r4\cdot c3$
$c5=r5\cdot c3$
 If
c3=0, the default value
shown in the table is 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) 0.30 Optional input Optional input Optional input Optional input Important: 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 c5 = 0.0, the minimum value of among all input directions will be retained.
 If the MFlag = 99, failure strain ratios r1, r2, r4 and r5 must be input in a following additional line.
 If
MFlag > 0, the entered
c1, c2,
c4 and c5 values will not
be used, but will be calculated as follows, using the predefined
ratio values from the table.
 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.
(2) $$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{f}^{\theta}(\eta )}}$$For shell elements when an integration point reaches $D=1$ , the integration points stress tensor is set to 0. Shell elements are deleted based on the value of P_thick_{fail}.
If P_thick_{fail} is set blank or 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 is ignored and the value entered in this failure model is used.
For values of P_thick_{fail} set > 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 $D=1$ .
 Special features are
activated by this flag:
 SFlag = 1: the failure curves are created, as shown in Comment 1. In this case, the curves may not reach their minimum value for the same stress triaxiality (Figure 4).
 SFlag = 2: is set by default. It ensures value c4 as global minimum for all directions. To achieve this for all curves, the second equation is split into 2 separate quadratic subfunctions where the minimum value of the curves is at c4; where, ${\sigma}^{*}=\frac{1}{\sqrt{3}}$ (Figure 5).
 SFlag = 3: same as
SFlag = 2, plus a
simplified localized necking criterion (only for shells). The
localized necking criteria 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
userdefined in the Inst_start field and occurs
at plane strain tension,
${\sigma}^{*}=\frac{1}{\sqrt{3}}$
(Figure 6).
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.
 A strain rate
dependency can be applied on the failure criterion, if:

${\dot{\epsilon}}_{0}\ne 0$
and
${C}_{JC}\ne 0$
, the JohnsonCook’s strain rate
dependency is used. In this case, the plastic strain at failure
value is multiplied by the strain rate dependency factor
as:
(3) $$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{f}^{\theta}(\eta )\left(1+{C}_{JC}{\langle \mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\rangle}_{+}\right)}}$$Where, ${\dot{\epsilon}}_{0}$
 Inviscid limit strain rate
 ${C}_{JC}$
 Strain rate dependency parameter
 ${\langle \rangle}_{+}$
 Macaulay brackets, which considers only positive values
Note: If using a JohnsonCook material law coupled to the /FAIL/ORTHBIQUAD criterion, the JohnsonCook parameters used for the constitutive law might not be the same for the failure criterion.  fct_ID_{rate} ≠
0, a tabulated function of strain rate dependency factor is used. In
this case, you have to define a function (/FUNCT)
to describe the evolution of the strain rate factor (denoted
${f}_{rate}$
) with the strain rate. You can also
input a strain rate scale factor for the abscissa of the function:
Xscale_rate (by default, this scale factor is
set to 1.0). Using the tabulated strain rate dependency, the damage
variable computation becomes:
(4) $$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{f}^{\theta}(\eta ){f}_{rate}\left(\dot{\epsilon}\right)}}$$
Important: The strain rate dependency applied to the failure criterion can only be used with material laws that are strain rate dependent. The strain rate used for the constitutive law (total strain rate, deviatoric strain rate or plastic strain rate), will be the same used for the failure criterion. 
${\dot{\epsilon}}_{0}\ne 0$
and
${C}_{JC}\ne 0$
, the JohnsonCook’s strain rate
dependency is used. In this case, the plastic strain at failure
value is multiplied by the strain rate dependency factor
as:
 The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL. There is no default value. If the line is blank, a value will not be output for failure model variables in the /INIBRI/FAIL (written in .sta file with /STATE/BRICK/FAIL option).