/FAIL/INIEVO

• 1^st step: the damage initiation phase, in which an internal variable evolution denoted is computed. This variable is a purely internal value that has no influence on the stress computation. Once this initiation variable reaches the value 1.0, the second stage of the failure is triggered by the computation of the damage variable evolution denoted $D$ .
• 2^nd step: The damage variable $D$ evolution is computed, generating a stress softening effect until the complete failure of the element and its deletion.

• If INITYPE = 1: A tabulated plastic strain at failure initiation map is given with respect to stress triaxiality and, optionally, with strain rate.

  $\begin{array}{ccc}{\epsilon }_p^{init}=f\left(\eta ,\dot{\epsilon }\right)& \text{with}& \eta =-\frac{p}{{\sigma }_{VM}}\end{array}$

  Where,

  $p$

  Hydrostatic pressure $-\text{Tr}\left(\sigma \right)/3$

  ${\sigma }_{VM}$

  von Mises stress

• If INITYPE = 2: A tabulated plastic strain at failure initiation map is given with respect to a shear influence variable denoted as and, optionally, with strain rate.

  $\begin{array}{ccc}{\epsilon }_p^{init}=f\left(\theta ,\dot{\epsilon }\right)& \text{with}& \theta =\frac{{\sigma }_{VM}+k_{s}p}{\tau }\end{array}$

  Where,

  $k_s$

  Pressure influence parameter

  $\tau$

  Maximum shear stress

  $\tau =\frac{{\sigma }_{\text{major}}-{\sigma }_{\text{minor}}}{2}$

• If INITYPE = 3: A tabulated modified plastic strain at failure initiation map is given with respect to principal strain rate ratio and, optionally, with strain rate. This initiation criterion is also denoted as MSFLD.

  $\begin{array}{ccc}{\epsilon }_p^{init}=f\left(\alpha ,\dot{\epsilon }\right)& \text{with}& \alpha =\frac{{\dot{\epsilon }}_{\text{minor}}^p}{{\dot{\epsilon }}_{\text{minor}}^p}\approx \frac{s_{\text{minor}}}{s_{\text{major}}}\end{array}$

  Where, $s_i$ are the principal deviatoric stress component.

• If INITYPE = 4: A tabulated plastic strain at failure initiation map is given with respect to principal strain rate ratio and, optionally, with strain rate. This initiation criterion is also denoted as FLD.

  $\begin{array}{ccc}{\epsilon }_p^{init}=f\left(\alpha ,\dot{\epsilon }\right)& \text{with}& \alpha =\frac{{\dot{\epsilon }}_{\text{minor}}^p}{{\dot{\epsilon }}_{\text{minor}}^p}\approx \frac{s_{\text{minor}}}{s_{\text{major}}}\end{array}$

• If INITYPE = 5: A tabulated plastic strain at failure initiation map is given with respect to a stress state parameter denoted as $\beta$ and, optionally, with strain rate. This initiation criterion is also denoted as FLD.
  $\begin{array}{ccc}{\epsilon }_p^{init}=f\left(\beta ,\dot{\epsilon }\right)& \text{with}& \beta =\end{array}\frac{{\sigma }_{VM}+k_{d}p}{{\sigma }_{\text{major}}}$

  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 INITYPE = 3 (MSFLD), a modified plastic strain is used, which only evolves when $\eta >0$ :
  $${\omega }_D={\displaystyle \sum _{t=0}^{\infty }\frac{\text{Δ}{\epsilon }_p}{{\epsilon }_p^{init}}}$$
• If INITYPE = 3 (MSFLD) and INITYPE = 4 (FLD) criteria, depending on PARAM value, a direct formulation can be used instead of the incremental one:
  $${\omega }_D=\frac{{\epsilon }_p}{{\epsilon }_p^{init}}$$

• Plastic displacement at failure (DISP), denoted $u_p^f$ , if EVOTYPE = 1.
• Dissipated fracture energy (ENER) denoted $G_f$ , if EVOTYPE = 2.

• If EVOSHAP = 1, the damage evolution is linear:
  • If EVOTYPE = 1: plastic displacement at failure is directly input with DISP:
    $$\text{Δ}D=\frac{L_e\text{Δ}{\epsilon }_p}{u_p^f}$$
    Where, $L_e$ is the initial element length.

    
    
    

  • If EVOTYPE = 2: plastic displacement at failure is deduced from the fracture energy input, which is calculated with ENER:

    $\begin{array}{ccc}\text{Δ}D=\frac{L_e\text{Δ}{\epsilon }_p}{u_p^f}& \text{with}& u_p^f=\frac{2G_f}{{\sigma }_Y^0}\end{array}$

    Where, ${\sigma }_Y^0$ is the yield stress at damage evolution triggering.

    Note: This expression considers that the plastic behavior is almost perfect at the onset of damage to ensure a dissipated energy equal to $G_f$ .

    
    
    

• If EVOSHAP = 2, the damage evolution is exponential.
  • If EVOTYPE = 1: plastic displacement at failure is directly input with DISP and shape of the exponential can be modified with ALPHA.
    $$D=\frac{1-e^{-\alpha \frac{L_e({\epsilon }_p-{\epsilon }_p^0)}{u_p^f}}}{1-e^{-\alpha }}$$

    
    
    

  • If EVOTYPE = 2: the area between the stress – plastic displacement curve of the exponential function corresponds to the failure energy ENER.

    $\begin{array}{ccc}D=1-e^{-\frac{E_{dis}}{G_f}}& \text{with}& E_{dis}={\displaystyle \underset{D=0}{\overset{D=1}{\int }}{\sigma }_Y\cdot L_e\text{Δ}{\epsilon }_p}\end{array}$

    
    
    

• If COMPTYP = 1: the damage variable of the criterion concerned, denoted as $D_i$ , is compared with the maximum value among all the criteria using COMPTYP = 1.

  $\begin{array}{ccc}{D}_{MAX}=\max \left(D_i,D_{MAX}\right)& with& i\in \end{array}{N}_{MAX}$

  Where, $N_{MAX}$ is the number of initiation/evolution card using COMPTYP = 1.

• If COMPTYP = 2: the damage variable of the criterion concerned, denoted as $D_i$ , is accumulated by a multiplicative formula among all the criteria using COMPTYP = 2.
  $$D_{MULT}=1-{\displaystyle \prod _{i\in N_{mult}}(1-D_i)}$$

  Where, $N_{MULT}$ is the number of initiation/evolution card using COMPTYP = 2.

Finally, the global damage variable used to calculate the damaged stress tensor is defined as the following maximum value:

ILEN=0

Initial geometrical formulation, where the characteristic length corresponds to the square root of the initial area for shells $L_e=\sqrt{A_0}$ and the cubic root of initial volume for solids $L_e=\sqrt[{}^3]{V_0}$ .

ILEN = 1

Initial critical timestep formulation where the characteristic length corresponds to the one used to compute the initial element critical timestep. The formula used may vary between the different formulations of shells or solids.

ILEN = 2

Current geometrical formulation, where the characteristic length corresponds to the square root of the current area for shells $L_e=\sqrt{A}$ .