/FAIL/HASHIN

Format

Definition

Comments

1. Example of ratio: if ratio=0.5, and I_{fail_sh}=2 (or I_{fail_so}=2), the element will be deleted, if more than half of the layers (or integration points) failed.
2. The 3D material failure model:
   • Uni-directional lamina model:
     Tensile/shear fiber mode:(1)

     $$F_1={\left(\frac{\langle {\sigma }_{11}\rangle }{{\sigma }_1^t}\right)}^2+\left(\frac{{\sigma }_{12}^2+{\sigma }_{13}^2}{{\sigma }_{12}^f{}^2}\right)$$
     Compression fiber mode: (2)

     $$F_2={\left(\frac{\langle {\sigma }_a\rangle }{{\sigma }_1^c}\right)}^2$$

     with, ${\sigma }_a=-{\sigma }_{11}+\text{}\langle -\frac{{\sigma }_{22}+{\sigma }_{33}}{2}\rangle$

     Crush mode:(3)

     $$F_3={\left(\frac{\langle p\rangle }{{\sigma }_c}\right)}^2$$

     with, $p=-\frac{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}{3}$

     Failure matrix mode:(4)

     $$F_4={\left(\frac{\langle {\sigma }_{22}\rangle }{{\sigma }_2^t}\right)}^2+{\left(\frac{{\sigma }_{23}}{S_{23}}\right)}^2+{\left(\frac{{\sigma }_{12}}{S_{12}}\right)}^2$$
     Delamination mode:(5)

     $$F_5=S_{del}^2\left[{\left(\frac{\langle {\sigma }_{33}\rangle }{{\sigma }_2^t}\right)}^2+{\left(\frac{{\sigma }_{23}}{{\tilde{S}}_{23}}\right)}^2+{\left(\frac{{\sigma }_{13}}{S_{13}}\right)}^2\right]$$

     Where,

     $\begin{array}{c}{S}_{12}={\sigma }_{12}^m+\langle -{\sigma }_{22}\rangle \tan \varphi \\ S_{23}={\sigma }_{23}^m+\langle -{\sigma }_{22}\rangle \tan \varphi \\ S_{13}={\sigma }_{13}^m+\langle -{\sigma }_{33}\rangle \tan \varphi \\ {\tilde{S}}_{23}={\sigma }_{23}^m+\langle -{\sigma }_{33}\rangle \tan \varphi \end{array}$

     Note: (6)

     $$\langle a\rangle =\{\begin{array}{c}aifa>0\\ 0ifa<0\end{array}$$
   • Fabric lamina model:
     Tensile/shear fiber mode:(7)

     $$F_1={\left(\frac{\langle {\sigma }_{11}\rangle }{{\sigma }_1^t}\right)}^2+\left(\frac{{\sigma }_{12}^2+{\sigma }_{13}^2}{{\sigma }_a^f{}^2}\right)$$

     (8)

     $$F_2={\left(\frac{\langle {\sigma }_{22}\rangle }{{\sigma }_2^t}\right)}^2+\left(\frac{{\sigma }_{12}^2+{\sigma }_{23}^2}{{\sigma }_b^f{}^2}\right)$$

     With ${\sigma }_a^f={\sigma }_{12}^f\text{},\text{}{\sigma }_b^f={\sigma }_{12}^f\frac{{\sigma }_2^t}{{\sigma }_1^t}$

     Compression fiber mode:(9)

     $$F_3={\left(\frac{\langle {\sigma }_a\rangle }{{\sigma }_1^c}\right)}^2$$
     with, ${\sigma }_a=-{\sigma }_{11}+\text{}\langle -{\sigma }_{33}\rangle$ (10)

     $$F_4={\left(\frac{\langle {\sigma }_b\rangle }{{\sigma }_2^c}\right)}^2$$

     with, ${\sigma }_b=-{\sigma }_{22}+\text{}\langle -{\sigma }_{33}\rangle$

     Crush mode:(11)

     $$F_5={\left(\frac{\langle p\rangle }{{\sigma }_c}\right)}^2$$

     with, $p=-\frac{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}{3}$

     Shear failure matrix mode:(12)

     $$F_6={\left(\frac{{\sigma }_{12}}{{\sigma }_{12}^m}\right)}^2$$
     Matrix failure mode:(13)

     $$F_7=S_{del}^2\left[{\left(\frac{\langle {\sigma }_{33}\rangle }{{\sigma }_3^t}\right)}^2+{\left(\frac{{\sigma }_{23}}{S_{23}}\right)}^2+{\left(\frac{{\sigma }_{13}}{S_{13}}\right)}^2\right]$$

     Where,

     $\begin{array}{c}{S}_{13}={\sigma }_{13}^m+\langle -{\sigma }_{33}\rangle \tan \varphi \\ S_{23}={\sigma }_{23}^m+\langle -{\sigma }_{33}\rangle \tan \varphi \end{array}$

     If the damage parameter is F_i ≥ 1.0, the stresses are decreased by using an exponential function to avoid numerical instabilities. A relaxation technique is used by decreasing the stress gradually:(14)

     $$\sigma (t)=\mathrm{f}(t)\cdot {\sigma }_d(t_r)$$
     With, (15)

     $$\mathrm{f}(t)=\exp \left(-\frac{t-t_r}{{\tau }_{\max }}\right)$$

     and $t\ge t_r$

     Where,

     $t$

     Time

     $t_r$

     Start time of relaxation when the damage criteria is assumed

     ${\tau }_{\text{max}}$

     Time of dynamic relaxation

     ${\sigma }_d\left(t_r\right)$

     Stress at the beginning of damage

3. The damage value, D is 0 ≤ D ≤ 1. The status for fracture is:
   • Free, if 0 ≤ D > 1
   • Failure, if D=1

   with $D=Max\left(F_1,F_2,F_3,F_4,F_5\right)$ for uni-directional lamina model and $D=Max\left(F_1,F_2,F_3,F_4,F_5,F_6,F_7\right)$ for fabric lamina model. This damage value shows with /ANIM/BRICK/DAMA or /ANIM/SHELL/DAMA.

4. The fail_ID is used with /STATE/BRICK/FAIL and /INIBRI/FAIL. There is no default value. If the line is blank, no value will be output for failure model variables in the /INIBRI/FAIL (written in .sta file with /STATE/BRICK/FAIL option).
5. After the failure criterion is reached, the ${\tau }_{\max }$ value determines a period of time when the stress in the failed element is gradually reduced to zero. When the stress reaches 1% of the stress value at the start of failure, the element is deleted. This is necessary to avoid instabilities coming from a sudden element deletion and a failure “chain reaction” in the neighboring elements. Even if the failure criterion is reached, the default value of ${\tau }_{\max }=1.0E30$ results in no element deletion. Therefore, it is recommended to define ${\tau }_{\max }$ 10 times larger than the simulation time step.
6. The I_Dam option improves damage calculation and stability calculating damage.