/FAIL/TAB2

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 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.

$\begin{array}{l}D={\displaystyle \int \text{Δ}D}\\ f={\displaystyle \int \text{Δ}f}\\ D_{crit}=\left\{\begin{array}{c}\begin{array}{ccc}1& \text{while}& f<1\end{array}\\ \begin{array}{ccc}D& \text{when}& f\ge 1\end{array}\end{array}\right.\end{array}$

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).

• The first way is by creating a 3-dimensions table in EPSF_ID, where the third dimension is the temperature dependency.
• The second way is to define the temperature dependency separately from the third dimension of tables defined with EPSF_ID and INST_ID. To do so, a specific temperature dependency function can be introduced with FCT_TEMP. In this case, the plastic strain at failure and the plastic strain at necking are both multiplied by the temperature dependent scale factor (denoted $f_{temp}$ ):
  $$D=\sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_p^f\left(\eta ,\xi \right)\cdot {\text{f}}_{crit}{\text{·f}}_{temp}\left({\displaystyle \frac{T}{T_{ref}}}\right)}\cdot n\cdot D^{\left(1-\frac{1}{n}\right)}$$
  $$f=\sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_p^{inst}\left(\eta ,\xi \right)· E_{crit}·f_{temp}\left({\displaystyle \frac{T}{T_{ref}}}\right)}\cdot n\cdot f^{\left(1-\frac{1}{n}\right)}$$

  In this case, the temperature dependency defined in EPSF and INST tables should be removed, else FCT_TEMP will be ignored.

• 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 ${\dot{\epsilon }}_0$ .
  Note: If the first strain rate value in the table is negative, the logarithmic scale is assumed.
• 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. A third dimension can be defined, introducing a dependency to Lode parameter ${\text{f}}_{size}\left(\frac{L_e}{L_{ref}},\eta ,\xi \right)$ .

• 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:
  $$D=\sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_p^f\left(\eta ,\xi ,{\displaystyle \frac{T}{T_{ref}}}\right)\cdot {\text{f}}_{crit}\cdot {\text{f}}_{rate}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)}\cdot n\cdot D^{\left(1-\frac{1}{n}\right)}$$
  Note: If the first strain rate value in the table is negative, the logarithmic scale is assumed.
• If $C_{JC}\ne 0$ , the Johnson-Cook 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:
  Where,

  ${\dot{\epsilon }}_0$

  Inviscid limit strain rate.

  $C_{JC}$

  Strain rate dependency parameter.

  ${〈〉}_{+}$

  Macaulay brackets, which consider only positive values.

  If you are using a Johnson-Cook material law coupled to the /FAIL/TAB2 criterion, the Johnson-Cook 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.