# /FAIL/BIQUAD

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:

- $a$ , $b$ , $c$ , $d$ , $e$ , and $f$
- Parabolic coefficients
- $x$
- Stress triaxiality
- ${f}_{1}\left(x\right)$ and ${f}_{2}\left(x\right)$
- Plastic failure strain

`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
```

`c1`–

`c5`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

`M-Flag`input option, there are three different ways to define the

`c1`-

`c5`values.

`M-Flag`=0, User-defined Test DataFor 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 DataIf failure strain data is not available, you can pick from 7 predefined materials. 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

`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 (`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 to 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:

- $D$
- Damage
- $\text{\Delta}{\epsilon}_{p}$
- 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 ratio 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)

`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).

`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 is not due to localized necking is defined by

`P_thickfail`.

## Perturbation of the Failure Limit

`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 distribution methods are shown in Figure 7 and Figure 8.

`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