# Tabulated Failure Model /FAIL/TAB1

In Radioss, /FAIL/TAB1 is the most sophisticated failure model for ductile material. The plastic failure strain can be defined as a function of: stress triaxiality, strain rate, Lode angle, element size, temperature, and instability strain.

Damage is accumulated based on user-defined functions. The functionality of this failure model will be described starting with the most basic input to the most complex options.

## Plastic Failure Strain

`table_ID`

_{1}. This method can be used for shell and solid elements.

`dimension`=1

`table_ID`

_{1}.

```
/TABLE/1/4711
failure plastic-strain vs triaxiality
#dimension
1
# Triaxiality Failure_Strain
-0.7000 0.3386
-0.6000 0.3068
-0.5000 0.2794
-0.4000 0.2558
-0.3333 0.2419
-0.3000 0.2355
-0.2000 0.2180
-0.1000 0.2029
0.0000 0.1900
0.1000 0.1789
0.2000 0.1693
0.3000 0.1610
0.3333 0.1585
0.4000 0.1539
0.5000 0.1478
0.6000 0.1425
0.7000 0.1380
```

## Strain Rate Dependency

/FAIL/TAB1 can also include the influence of strain rate on material failure. For this case the /TABLE must be defined such that, the first dimension is the function ID for the failure curve and the second dimension is the strain rate where that failure curve is applied.

`dimension`=2

```
/TABLE/1/4711
failure plastic-strain vs triaxiality and strain rate
#dimension
2
# FCT_ID strain_rate
3000 1E-4
3001 0.1
3002 1.0
/FUNCT/3000
failure plastic-strain vs triaxiality
# Triaxiality Failure_Strain
-0.7000 0.3386
-0.6000 0.3068
-0.5000 0.2794
-0.4000 0.2558
-0.3333 0.2419
-0.3000 0.2355
-0.2000 0.2180
-0.1000 0.2029
0.0000 0.1900
0.1000 0.1789
0.2000 0.1693
0.3000 0.1610
0.3333 0.1585
0.4000 0.1539
0.5000 0.1478
0.6000 0.1425
0.7000 0.1380
/FUNCT/3001
failure plastic-strain vs triaxiality
# Triaxiality Failure_Strain
-0.7 0.27088
-0.6 0.24544
-0.5 0.22352
-0.4 0.20464
-0.3333 0.19352
-0.3 0.1884
-0.2 0.1744
-0.1 0.16232
0 0.152
0.1 0.14312
0.2 0.13544
0.3 0.1288
0.3333 0.1268
0.4 0.12312
0.5 0.11824
0.6 0.114
0.7 0.1104
```

## Lode Angle with Solid Elements

For solid elements, the failure strain can also depend on the 3D stress state defined using the Lode angle.

This can be included by adding failure strain as a function of Lode angle parameter
in the /TABLE entity referenced by `table_ID1`.
For shells elements, it is only necessary to define the failure strain as a function
of stress triaxiality. But for solid elements, it is more accurate to include the
failure strain as a function of stress triaxiality and Lode angle.

In Radioss, the Lode angle is entered using a normalized and dimensionless Lode Angle parameter $\xi $ and is defined here.

`P`, could be expressed with principle stresses (${\sigma}_{1},{\sigma}_{2},{\sigma}_{3}$) or could also be expressed using stress invariants (${I}_{1},{J}_{2},{J}_{3}$). The advantage of using stress invariants is that they are constant and do not depend on the orientation of the coordinate system. In Figure 2, to correctly describe stress state of point

`P`${\sigma}_{1},{\sigma}_{2},{\sigma}_{3}$ using stress invariants, the magnitude of $OO\text{'}$ as:

- ${\sigma}_{m}$
- Mean stress
- ${I}_{1}$
- First stress invariant ${I}_{1}={\text{\sigma}}_{1}+{\text{\sigma}}_{2}+{\text{\sigma}}_{3}$

Here, $${J}_{2}$$ is the second invariant of deviatoric stress $$s\left(s=\sigma -p\right)$$ with $${J}_{2}=\frac{1}{2}({S}_{1}{}^{2}+{S}_{2}{}^{2}+{S}_{3}{}^{2})=\frac{1}{2}\left[{\left({\sigma}_{1}-{\sigma}_{2}\right)}^{2}+{\left({\sigma}_{2}-{\sigma}_{3}\right)}^{2}+{\left({\sigma}_{3}-{\sigma}_{1}\right)}^{2}\right]$$.

`P`, the angle in the circular plane must be calculated. This angle is called the Lode Angle $\theta $:

Lode Angle Parameter $\xi $ | Lode Angle $\theta $ | Stress State |
---|---|---|

1 | 0 | Uniaxial tension + hydrostatic pressure (triaxial tension or axisymmetric tension) |

0 | 30 | Pure shear + hydrostatic pressure (plane strain) |

-1 | 60 | Uniaxial compression + hydrostatic pressure (axisymmetric compression) |

Example /TABLE, `dimension`=3

`table_ID`

_{1}

```
/TABLE/1/4711
failure plastic-strain vs triaxiality and strain rate
#dimension
3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
# FCT_ID strain_rate Lode_angle
3000 1E-4 -1
3001 0.1 0
3002 1.0 1
....
```

## Scaling Failure Strain

- $$Xscale1$$
- General scaling factor
- $$facto{r}_{el}$$
- Scale factor based on element size
- $$facto{r}_{T}$$
- Scale factor based on temperature

### Element Length Dependency

- $$Fscal{e}_{el}$$
- Element size function scale factor
- $${\mathrm{f}}_{el}\left(\frac{Siz{e}_{el}}{El\_ref}\right)$$
- failure strain scale factor as a function of the normalized element size
(defined via
`fct_ID`_{el}), and $$El\_ref$$ is the reference element size used to normalize the element size

### Temperature Dependency

- $$Fscal{e}_{T}$$
- Temperature function scale factor
- $${\mathrm{f}}_{T}\left({T}_{start}\right)$$
- Failure strain scale factor as a function temperature defined via
`fct_ID`_{T}.

Scaling the failure strain based on temperature works with any material that has /HEAT/MAT defined or materials that include thermos plasticity such as /MAT/LAW2 (PLAS_JOHNS).

`fct_ID`

_{T}, the temperature is defined relative to the melting and initial temperature.

## Element Failure Treatment

In /FAIL/TAB1 an accumulative damage model is used. The damage can be output for contour plotting using the Engine options, /ANIM/SHELL/DAMA or /ANIM/BRICK/DAMA.

- $$\text{\Delta}{\epsilon}_{p}$$
- The change in plastic strain of the integration point
- $${\epsilon}_{f}$$
- Plastic failure strain for the current stress triaxiality
- $${D}_{p}$$ and $$n$$
- Damage parameters

It is also interesting to understand the influence of the damage accumulation parameter, $$n$$ in Equation 10.

## Material Instability (Diffuse Necking)

In /FAIL/TAB1 it is possible to account for the
material instability (diffuse necking) with options
`table_ID`_{2}, or `Inst_start` and
`Fad_exp`.

with ${\epsilon}_{f}$ being the diffuse necking strain based on the current stress triaxiality.

The strain at which instability starts could be
either input with a curve (blue curve in Figure 13) using `table_ID`_{2} or input as a
constrain strain using the option `Inst_start`.

- If material instability is not included then damage is calculated using the red failure curve in Figure 13
- If material instability is included and modeled using the curve input
(
`table_ID`_{2}) then:- Damage due to diffuse necking starts when the plastic strain defined by the blue curve in Figure 13 is reached.
- Damage due to diffuse necking is linear if
`Fad_exp`=1 is used. Increasing the`Fad_exp`leads to more energy dissipated during damage. Figure 13 shows the influence of`Fad_exp`from 1 to 10 in stress-strain curve. It is recommend to use a`Fad_exp`value of 5 to 10. - Once the strain reaches the red curve the element fails.

- If only
`Inst_start`is used without curve input in`table_ID`_{2}, then the diffuse necking plastic strain is the constant value,`Inst_start`, for all stress triaxiality.

Currently diffuse necking (material instability) in /FAIL/TAB1 can only be used with material law numbers > 28.

^{1}Wierzbicki, Tomasz, "Addendum to the Research Proposal on Fracture of Advanced High Strength Steels", page 19, January 2007.

^{2}Wierzbicki, Tomasz. "Fracture of AHSS Sheets–Addendum to the Research Proposal on Fracture of Advanced High Strength Steels." Impact and Crashworthiness Laboratory, Massachusetts Institute of Technology (2007).