/FAIL/TAB2
Block Format Keyword This advanced failure model allows the plastic strain at failure to be defined as a function of stress triaxiality, strain rate, Lode angle, element size, and temperature.
A coupling with stress computation generating a stress softening is also available through different features. It can be fully coupled or triggered by other phenomena like instability strain to control necking. Damage is accumulated based using an exponent evolution. This criterion is compatible with both solids and shells and can be used with nonlocal regularization.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/FAIL/TAB2/mat_ID/unit_ID  
EPSF_ID  FCRIT  FAILIP  PTHICKFAIL 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

N  DCRIT  INST_ID  ECRIT 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

FCT_EXP  EXP_REF  EXP 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

TAB_EL  IREG  EL_REF  SR_REF1  FSCALE_EL 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

SHRF  BIAXF 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

FCT_SR  SR_REF2  FSCALE_SR  C_JCOOK 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

FCT_DLIM  FSCALE_DLIM 
(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  (Optional) Unit Identifier. (Integer, maximum 10 digits) 

EPSF_ID  Plastic strain at failure table
identifier. (Integer) 

FCRIT  Scale factor for failure plastic
strain table. Default = 1.0 (Real) 

FAILIP  Number of failed integration point
prior to solid element deletion. Defaut = 1 (Integer) 

PTHICKFAIL  Percentage of failed layers prior
to shell element deletion. Default = 0.0 (Real) 

N  Damage accumulation
exponent. Default = 1.0 (Real) 

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

INST_ID  Instability (necking) plastic
strain table identifier. (Integer) 

ECRIT  Scale factor for necking plastic
strain table identifier. (Real) 

FCT_EXP  Stress softening exponent function
identifier. (Integer) 

EXP_REF  Reference element size for stress
softening exponent function. Default = 1.0 (Real) 
$\left[\text{m}\right]$ 
EXP  Scale factor for stress softening
exponent function. Default = 1.0 (Real) 

TAB_EL  Element size scaling table
identifier. (Integer) 

IREG  Element size regularization flag.
(Integer) 

EL_REF  Reference element size for size
scaling table. Default = 1.0 (Real) 
$\left[\text{m}\right]$ 
SR_REF1  Reference strain rate for size
scaling table. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
FSCALE_EL  Scale factor for element size
scaling function. Default = 1.0 (Real) 

SHRF  Lower stress triaxiality boundary
for element size scaling. Default = 1.0 (Real) 

BIAXF  Upper stress triaxiality boundary
for element size scaling. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
FCT_SR  Strain rate dependency function
identifier. (Integer) 

SR_REF2  Reference strain rate for strain
rate dependency function. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
FSCALE_SR  Scale factor for strain rate
dependency function. Default = 1.0 (Real) 

C_JCOOK  JohnsonCook strain rate dependency
factor. Default = 0.0 (Real) 

FCT_DLIM  Damage limit function
identifier. (Integer) 

FSCALE_DLIM  Damage limit function scale
factor. Default = 1.0 (Real) 

fail_ID  (Optional) Failure criteria
identifier. 9 (Integer, maximum 10 digits) 
Example (Steel)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
# MUNIT LUNIT TUNIT
kg mm ms
#12345678910
/MAT/PLAS_JOHNS/1/1
Steel
# RHO_I
7.8E6 0
# E Nu Iflag
210 .3 0
# a b n EPS_p_max SIG_max0
.4 .5 .5 0 0
# c EPS_DOT_0 ICC Fsmooth F_cut Chard
0 0 0 0 0 0
# m T_melt rhoC_p T_r
0 0 0 0
#12345678910
/FAIL/TAB2/1/1
# EPSF_ID FCRIT FAILIP PTHICKFAIL
52 0.9 0 1.0
# N DCRIT INST_ID ECRIT
2.0 0 53 0.5
# FCT_EXP EXP_REF EXP
0 0 2.5
# TAB_EL IREG EL_REF SR_REF1 FSCALE_EL
0 0 0 0 0
# SHRF BIAXF
0 0
# FCT_SR SR_REF2 FSCALE_SR C_JCOOK
0 0 0 0
# FCT_DLIM FSCALE_DLIM
0 0
#12345678910
/FUNCT/52
epsf vs triax
0.333 3.009955556
0.3 2.728211
0.25 2.33840625
0.2 1.987976
0.15 1.67692025
0.1 1.405239
0.05 1.17293225
0 0.98
0.05 0.82644225
0.1 0.712259
0.15 0.63745025
0.2 0.602016
0.25 0.60595625
0.3 0.649271
0.333 0.700985898
0.35 0.663237826
0.4 0.567983816
0.45 0.496266718
0.5 0.448086532
0.55 0.423443259
0.577 0.419921961
0.6 0.428924499
0.625 0.459765672
0.65 0.512542413
0.666 0.559913333
#12345678910
/FUNCT/53
0.333 0.700985898
0.35 0.647131801
0.4 0.511235623
0.45 0.408918886
0.5 0.34018159
0.55 0.305023734
0.577350269 0.3
0.6 0.316714456
0.625 0.373975356
0.65 0.471962671
0.666666667 0.559913333
#12345678910
#enddata
/END
#12345678910
Comments
 The
/FAIL/TAB2 failure criterion is a tabulated criterion
that offers you the freedom to define your own map of plastic strain at
failure with stress triaxiality, Lode parameter and other dependency. This
plastic strain at failure is used to compute a damage variable evolution
described below. This criterion also offers the possibility to generate a
stress softening effect with the damage computation as:$$\sigma ={\sigma}_{eff}\left(1{\left(\frac{D{D}_{crit}}{1{D}_{crit}}\right)}^{EXP}\right)$$Where,
 $\sigma $
 Damaged stress tensor.
 ${\sigma}_{eff}$
 Undamaged effective stress tensor.
 ${D}_{crit}$
 Critical damage value that triggers stress softening.
 $EXP$
 Exponent parameter.
 To use /FAIL/TAB2, it is required to define a
plastic strain at failure used to compute the damage accumulation presented
below. It can be either constant using FCRIT parameter
alone or tabulated if EPSF_ID table identifier is
specified. The tabulated plastic strain at failure is defined with respect
to stress triaxiality, Lode parameter and temperature
${\epsilon}_{p}^{f}(\eta ,\xi ,T)$
(Figure 1). If EPSF_ID is defined, FCRIT becomes a scale factor to quickly increase or decrease the entire plastic strain at failure map.
 FAILIP is an integer value that is used only with higher order or fullyintegrated solid elements. It defines the number of failed integration points prior to solid element deletion.
 PTHICKFAIL parameter is a real parameter used for
shell elements. If PTHICKFAIL is blank or set to 0.0, the
value of PTHICKFAIL from the shell property is used. If
PTHICKFAIL > 0.0, any PTHICKFAIL
value defined in the shell properties are ignored and the value entered in
this failure model is used.
For values of PTHICKFAIL > 0.0, shell elements fail and are deleted when the ratio of through thickness failed integration points equals or exceeds PTHICKFAIL.
 The damage variable evolution is given by:$$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{p}^{f}(\eta ,\xi ,T)\cdot {\text{f}}_{crit}}}\cdot n\cdot {D}^{\left(1\frac{1}{n}\right)}$$
The parameter N then allows you to change the shape of the damage evolution with plastic strain shape from linear (N = 1, set by default) to nonlinear (N ≠ 1) (Figure 2(a)). An increase of N also creates a delay of the stress softening effect (Figure 2(b)).
 The DCRIT parameter allows you to define a damage variable trigger value for stress softening (Figure 3). By default DCRIT = 0.0, which means that damage variable always has an effect on stress computation, generating a softening effect from the beginning of the plasticity. However, you may want to delay this stress softening effect to a higher value of damage variable (0 < D < 1) or cut the effect of stress softening to obtain a fully failure criterion approach where elements lose their load carrying capacity when damage reaches the value 1.
 Instead of using a constant value of DCRIT to
trigger stress softening, a necking control process can be used. This is
defined using the parameter INST_ID and/or
ECRIT. The definition of those parameters will imply
the computation of another variable evolution called instability variable
and denoted
$f$
: $$f={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{p}^{inst}(\eta ,\xi ,T)\cdot {E}_{crit}}}\cdot n\cdot {f}^{\left(1\frac{1}{n}\right)}$$
The evolution of this instability variable is similar to the damage variable but represents the criterion to reach to trigger stress softening, implying the start of the strain localization and then necking especially observed at high stress triaxiality. When the criterion is reached ( $f$ = 1), the instant value taken by 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.
$\begin{array}{l}D={\displaystyle \int \text{\Delta}D}\\ f={\displaystyle \int \text{\Delta}f}\\ {D}_{crit}=\left\{\begin{array}{c}\begin{array}{ccc}1& \text{while}& f<1\end{array}\\ \begin{array}{ccc}D& \text{when}& f\ge 1\end{array}\end{array}\right.\end{array}$
ECRIT allows you to define a constant necking plastic strain. However, the necking plastic strain can be defined with a table INST_ID depending on stress triaxiality, Lode parameter and temperature. In this case, ECRIT becomes a scale factor for the instability plastic strain table. The instability plastic strain must be lower than failure plastic strain to have a visible effect (Figure 4).  Even if the shape of the damage accumulation can be controlled with parameter N, another nonlinearity in stress softening can be defined with the exponent EXP (Figure 5). This exponent can be constant if EXP parameter is defined alone or can evolve with element size if FCT_EXP is specified. If the function is used, EXP_REF is the element reference size and EXP becomes a scale factor. By default, EXP is set to 1.0 leading to a linear decrease.
 Element size scaling can be
used to regularize the failure and ensure an almost constant fracture energy
dissipated with different mesh sizes is obtained. This element size
dependency is introduced by computing a size scale factor denoted
${\text{f}}_{size}$
defined by the table
TAB_EL. The dependencies of these tables depend on
the value of the IREG flag:
 IREG = 1: the table defines the evolution of the element size scaling factor with respect to initial element size and strain rate ${\text{f}}_{size}\left(\frac{{L}_{e}}{{L}_{ref}},\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)$ . In this case, EL_REF is the reference element size ${L}_{ref}$ , and SR_REF1 is the reference strain rate ${\dot{\epsilon}}_{0}$ .
 IREG = 2: the table defines the evolution of the element size scaling factor with respect to initial element size and stress triaxiality ${\text{f}}_{size}\left(\frac{{L}_{e}}{{L}_{ref}},\eta \right)$ . In this case, EL_REF is the reference element size ${L}_{ref}$ , and SR_REF1 is ignored.
In both cases, FSCALE_EL is a scale factor that quickly increases or decreases the values of the entire table. The element size scale factor thus computed is introduced in the damage variable evolution equation (and if defined, the instability variable evolution equation) as:$$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{p}^{f}(\eta ,\xi ,T)\cdot {\text{f}}_{crit}\cdot {\text{f}}_{size}}}\cdot n\cdot {D}^{\left(1\frac{1}{n}\right)}$$Note: If IREG = 1 is used, the element size scaling can be turned off, if the stress triaxiality is lower than the boundary SHRF, or is higher than the boundary BIAXF.  A strain rate dependency can be also applied to the failure
criterion. This dependency can be introduced in two different ways:
 If FCT_SR ≠ 0, a tabulated function of the strain
rate dependency factor
${\text{f}}_{rate}$
is used. In this case, you must
define a function to describe the evolution of the strain rate
factor (denoted
${\text{f}}_{rate}$
) with the strain rate. You can also
input a reference strain rate SR_REF2 denoted
${\dot{\epsilon}}_{0}$
in the equation, and a scale factor. Using
the tabulated strain rate dependency, the damage variable
computation (and if defined, the instability variable evolution
equation) becomes:$$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{p}^{f}(\eta ,\xi ,T)\cdot {\text{f}}_{crit}\cdot {\text{f}}_{rate}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)}}\cdot n\cdot {D}^{\left(1\frac{1}{n}\right)}$$
 If
${C}_{JC}\ne 0$
, the JohnsonCook strain rate
dependency is used and SR_REF2 becomes the
reference strain rate
${\dot{\epsilon}}_{0}$
. In this case, the plastic strain at
failure value is multiplied by the strain rate dependency factor
as:$$D={\displaystyle \sum _{t=0}^{\infty}\frac{\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{p}^{f}(\eta ,\xi ,T)\cdot {\text{f}}_{crit}\cdot \left(1+{C}_{JC}{\u2329\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\u232a}_{+}\right)}}\cdot n\cdot {D}^{\left(1\frac{1}{n}\right)}$$Where,
 ${\dot{\epsilon}}_{0}$
 Inviscid limit strain rate.
 ${C}_{JC}$
 Strain rate dependency parameter.
 ${\u2329\u232a}_{+}$
 Macaulay brackets, which consider only positive values.
If you are using a JohnsonCook material law coupled to the /FAIL/TAB2 criterion, the JohnsonCook parameters used for the constitutive law might not be the same for the failure criterion. The reference strain rate used in the presented equation is different from the one used for element size scaling when IREG = 1.
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.
 If FCT_SR ≠ 0, a tabulated function of the strain
rate dependency factor
${\text{f}}_{rate}$
is used. In this case, you must
define a function to describe the evolution of the strain rate
factor (denoted
${\text{f}}_{rate}$
) with the strain rate. You can also
input a reference strain rate SR_REF2 denoted
${\dot{\epsilon}}_{0}$
in the equation, and a scale factor. Using
the tabulated strain rate dependency, the damage variable
computation (and if defined, the instability variable evolution
equation) becomes:
 The stress softening can be restricted to a given range of stress triaxiality. To do so, a damage limit value (lower than 1 for which the element has lost its load carrying capacity) evolving with stress triaxiality may be defined using the function FCT_DLIM. Values taken by this function must be taken between 0 and 1. A scale factor can be used to quickly increase or decrease the entire function values.
 If the nonlocal regularization is used (/NONLOCAL/MAT), the nonlocal plastic strain is used to compute the damage evolution (and the instability variable if used). In that case, if an element size scaling is defined through TAB_EL or an element size dependent exponent parameter function FCT_EXP is used, the maximum nonlocal length parameter LE_MAX is used instead of the initial element size.