In Radioss, /FAIL/BIQUAD is the most user-friendly failure model for ductile materials. It uses a simplified, nonlinear strain-based failure criteria with linear damage accumulation.

The failure strain is described by two parabolic functions calculated using curve fitting from up to 5 user input failure strains.

By default, /FAIL/BIQUAD (S-Flag=1) uses two parabolic curves to describe the plastic failure strain ${\epsilon }_{f}$ , as a function of stress triaxiality ${\sigma }^{*}$ . The two parabolic curves use:(1)
${f}_{1}\left(x\right)=a{x}^{2}+bx+c$
(2)
${f}_{2}\left(x\right)=d{x}^{2}+ex+f$
Where,
$a$ , $b$ , $c$ , $d$ , $e$ , and $f$
Parabolic coefficients
$x$
Stress triaxiality
${f}_{1}\left(x\right)$ and ${f}_{2}\left(x\right)$
Plastic failure strain
The parabolic coefficients $a$ , $b$ , $c$ , $d$ , $e$ , and $f$ are computed by Radioss using a curve fit based on the plastic failure strain c1-c5 input values. If the calculated parabolic failure strain curves have negative failure strain values, these negative values will be replaced by a failure strain of 1E-6 which results in a very high damage accumulation and brittle behavior. The results of the curve fit are in the Starter *0000.out file.
      Bi-Quadratic FAILURE
--------------------
c1. . . . . . . . . . . . . . . . . . .=  0.2419E+00
c2. . . . . . . . . . . . . . . . . . .=  0.1900E+00
c3. . . . . . . . . . . . . . . . . . .=  0.1585E+00
c4. . . . . . . . . . . . . . . . . . .=  0.1437E+00
c5. . . . . . . . . . . . . . . . . . .=  0.1394E+00

COEFFICIENTS OF FIRST PARABOLA
-----------------------------
a . . . . . . . . . . . . . . . . . . .=  0.9180E-01
b . . . . . . . . . . . . . . . . . . .= -0.1251E+00
c . . . . . . . . . . . . . . . . . . .=  0.1900E+00

COEFFICIENTS OF SECOND PARABOLA
-----------------------------
d . . . . . . . . . . . . . . . . . . .=  0.3753E-01
e . . . . . . . . . . . . . . . . . . .= -0.9483E-01
f . . . . . . . . . . . . . . . . . . .=  0.1859E+00
The c1c5 plastic failure strains definitions are:
c1
Plastic failure strain in uniaxial compression
c2
Plastic failure strain in shear
c3
Plastic failure strain in uniaxial tension
c4
Plastic failure strain in plane strain tension
c5
Plastic failure strain in biaxial tension

## M-Flag Input Options

Depending on the M-Flag input option, there are three different ways to define the c1-c5 values.
• M-Flag=0, User-defined Test Data

For this case, you must enter c1-c5 which represents the plastic failure strain for the 5 different stress states. Ideally this data would be obtained from test or the material supplier.

• M-Flag=1-7, Predefined Material Data
If failure strain data is not available, you can pick from 7 predefined materials. Figure Figure 2 shows the plastic strain at failure curves for the 7 materials.
Note: The predefined values are supplied for early design exploration and it is your responsibility to verify that their material has the same properties.
• M-Flag=99, Plastic Failure Strain Ratio Input, r1-r5
The last input method is to enter the plastic failure strain in uniaxial tension, c3, and plastic failure strain ratios for the other four stress states. These ratios are defined as:
r1
Failure plastic strain ratio, Uniaxial Compression (c1) to Uniaxial Tension (c3), so $c1=r1\cdot c3$
r2
Failure plastic strain ratio, Pure Shear (c2) to Uniaxial Tension (c3), so $c2=r2\cdot c3$
r4
Failure plastic strain ratio, Plane Strain Tension (c4) to Uniaxial Tension (c3), so $c4=r4\cdot c3$
r5
Failure plastic strain ratio, Biaxial Tension (c5) to Uniaxial Tension (c3), so $c5=r5\cdot c3$
Using this method, it is easy to change the failure curve by adjusting the single plastic failure strain in uniaxial tension value, c3.

## Default Behavior

By default, the values different than 0 for c1 to c5 need to be entered. 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 (M-Flag=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 user- defined c3 value.
• In case the material behavior is known, M-Flag is defined and c3 can be used to adjust the failure model according the expected tensile failure. The selected material behavior is scaled by the user-defined c3 value.
• For all other cases, all c1 to c5 are intended to be defined and default value of 0.0 is used.

## Element Failure Treatment

A cumulative damage method is used to sum the amount of plastic strain that has occurred at each integration point in the element using:(3)
$D=\sum \frac{\text{Δ}{\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$
Where,
$D$
Damage
$\text{Δ}{\epsilon }_{p}$
The change in plastic strain of the integration point
${\epsilon }_{f}$
Plastic failure strain for the current stress triaxiality

In shell elements after an integration point reaches $D=1$ , the integration point’s stress tensor is set to zero. The element fails and is deleted when the percentage of through thickness failed integration points equals P_thickfail. In solid elements, the element is deleted when any integration point reaches $D=1$ .

## Plane Strain as Global Minimum

The S-Flag=2 option can be used to force the global minimum of the plastic failure strain curve to occur at the plane strain stress triaxiality location, c4. This is accomplished by splitting the second equation into 2 separate quadratic sub-functions.

## Modeling Material Instability (Localized Necking)

In materials such as sheet metal, thickness thinning and diffuse necking may appear during tensile loading of the material. This is called localized necking and normally occurs in the stress triaxiality range of $\frac{1}{3}\le {\sigma }^{*}\le \frac{2}{3}$ :
In /FAIL/BIQUAD it is possible to simulate this localized necking using the option, S-Flag=3 and Inst_start. This option uses the same plastic failure strain curve as S-Flag=2 and adds 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 a user-defined value in the Inst_start field and occurs at plane strain tension ${\sigma }^{*}=\frac{1}{\sqrt{3}}$ . Using this localized necking curve, a second localized necking damage value is calculated and failure due to necking only occurs when all integration points reach $D=1$ . The localized necking criteria is based on the Marciniak-Kuczynski analysis. 1

When using S-Flag=1 or 2, the damage accumulation begins once the plastic strain reaches in failure curve (red in Figure 6).

If S-Flag=3is used to describe the localized necking, damage accumulation begins once the plastic strain reaches in the localized necking curve (blue curve in Figure 6). For localized necking, the element is deleted when all integration points reach damage, $D=1$ , whereas element deletion not due to localized necking is defined by P_thickfail.

## Perturbation of the Failure Limit

Due to a materials imperfection or production process, a material’s failure strain may not be exactly the same everywhere and thus very small perturbations of failure limit may exist. When using M-Flag>0 with /PERTURB/FAIL/BIQUAD, a statistical distribution of the failure limit is applied to each element assigned the failure model. This is accomplished by calculating the normal or random distribution of a failure scale factor which is applied to /FAIL/BIQUAD, c3. The two different distributions methods are shown in Figure 7 and Figure 8.
/FAIL/BQUAD uses small perturbations of c3 generated by /PERTURB/FAIL/BIQUAD to scale the entire failure curve using the strain ratio values for the predefined materials or user-defined ratios, r1, r2, r4 and r5, depending on the value of M-Flag.

## Reference

1 Pack, Keunhwan, and Dirk Mohr. "Combined necking & fracture model to predict ductile failure with shell finite elements." Engineering Fracture Mechanics 182 (2017): 32-51