/MAT/LAW57 (BARLAT3)

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1. The anisotopic yield criteria F for plane stress is defined by:(1)
   $$F=a{\left|K_1+K_2\right|}^m+a{\left|K_1-K_2\right|}^m+c{\left|2K_2\right|}^m-2{{\sigma }_y}^m=0$$

   Where,

   ${\sigma }_y$ is the yield stress

   $K_1=\frac{{\sigma }_{xx}+h{\sigma }_{yy}}{2}$ and $K_2=\sqrt{{\left(\frac{{\sigma }_{xx}-h{\sigma }_{yy}}{2}\right)}^2+p^2{\sigma }_{xy}^2}$

2. Angles for Lankford parameters are defined with respect to orthotropic direction 1. The material constants a, c, h, and p are obtained from the three Lankford parameters:(2)
   $$\begin{array}{l}a=2-2\sqrt{\frac{r_{00}}{1+r_{00}}}\sqrt{\frac{r_{90}}{1+r_{90}}}\\ c=2-a\\ h=\sqrt{\frac{r_{00}}{1+r_{00}}}\sqrt{\frac{1+r_{90}}{r_{90}}}\end{array}$$
   Material constant p is calculated by solving:(3)

   $$\frac{2m\cdot {\sigma }_y{}^m}{\left(\frac{\partial F}{\partial {\sigma }_{xx}}+\frac{\partial F}{\partial {\sigma }_{yy}}\right){\sigma }_{45}}-1-r_{45}=0$$
3. If the last point of the first (static) function equals 0 in stress, the default value of ${\epsilon }_p^{max}$ is set to the corresponding value of ${\epsilon }_p$ .
4. If ${\epsilon }_p$ (plastic strain) reaches ${\epsilon }_p^{\max }$ , in one integration point, the corresponding shell element is deleted.
5. If the largest principal strain ${\epsilon }_1>{\epsilon }_t$ , the stress is reduced using the following relation:(4)
   $$\sigma =\sigma \left(\frac{{\epsilon }_m-{\epsilon }_1}{{\epsilon }_m-{\epsilon }_t}\right)$$
6. If ${\epsilon }_1>{\epsilon }_m$ , the stress is reduced to 0 (but the element is not deleted).
7. The maximum number of curves is 10.
8. If $\dot{\epsilon }\le {\dot{\epsilon }}_n$ , the yield is interpolated between f_n and f_n-1.
9. If $\dot{\epsilon }\le {\dot{\epsilon }}_1$ , function f₁ is used.
10. Above ${\dot{\epsilon }}_{\max }$ , yield is extrapolated.

    
    
    

    Figure 1.

11. The evolution of Young's modulus:
    • If fct_ID_E > 0, the curve defines a scale factor for Young's modulus evolution with equivalent plastic strain, which means the Young's modulus is scaled by the function $\mathrm{f}\left({\overline{\epsilon }}_p\right)$ :(5)
      $$E\left(t\right)=E\cdot \mathrm{f}\left({\overline{\epsilon }}_p\right)$$

      The initial value of the scale factor should be equal to 1 and it decreases.

    • If fct_ID_E = 0, the Young's modulus is calculated as:(6)
      $$E\left(t\right)=E-\left(E-E_{inf}\right)\left[1-\exp \left(-C_E{\overline{\epsilon }}_p\right)\right]$$

      Where, E and E_inf are respectively the initial and asymptotic value of Young's modulus, and ${\overline{\epsilon }}_p$ is the accumulated equivalent plastic strain.

      Note: If fct_ID_E = 0 and C_E = 0, Young's modulus, E is kept constant.
12. The parameters F_smooth and F_cut allows you to enable a strain-rate filtering. Three cases can be set:
    • If F_smooth = 0 and F_cut = 0.0, the strain-rate filtering is turned off.
    • If F_smooth = 1 and F_cut = 0.0, the strain-rate filtering uses a default cutoff frequency of 10 kHz.
    • If F_cut ≠ 0, F_smooth is automatically set to 1 and the strain-rate filtering uses the cutoff frequency given by you.