/MAT/LAW36 (PLAS_TAB)

For Solid and SPH elements, this material may account for the nonlinear dependence between pressure and density when corresponding Equation of State is prescribed.

Format

Definition

Comments

1. The first point of yield stress functions (plastic strain versus stress) should have a plastic strain value ${\epsilon }_p$ of zero. The yield stress functions must be strictly positive. The yield stress should be either constant or increasing (no softening to avoid numerical instabilities).
2. Element deletion:
   • Once ${\epsilon }_p$ (plastic strain) reaches ${\epsilon }_p^{max}$ , in one integration point, the element is deleted.
   • If ${\epsilon }_1$ (largest principal strain) reaches ${\epsilon }_t$ , the yield stress is reduced using the following relation:
     $${\sigma }_Y^d={\sigma }_Y\left(\frac{{\epsilon }_m-{\epsilon }_1}{{\epsilon }_m-{\epsilon }_t}\right)$$
   • If ${\epsilon }_1$ reaches ${\epsilon }_m$ ( ${\epsilon }_1>{\epsilon }_m$ ), the stress in element is reduced to 0 (but the element is not deleted).
   • Once ${\epsilon }_1$ reaches ${\epsilon }_f$ (maximum tensile failure strain), the element is deleted.
   • Once the damaged yield stress value ${\sigma }_Y^d$ , interpolated or extrapolated reaches 0, the element is deleted.

     
     
     

3. The kinematic hardening model is not available in global formulation (N=0 in shell property keyword) that is, hardening is fully isotropic.
4. In case of kinematic hardening and strain rate dependency, the yield stress depends on the strain rate.
5. Strain rate filtering is used to smooth strain rates.
6. The first function in fct_ID₁ is used for strain rate values from 0 to its corresponding strain rate, strain rate 1. However, the last function used in the model does not extend to the maximum strain rate; for higher strain rates, a linear extrapolation will be applied. This could lead to instability and/or unusual deformation. This instability can be overcome by repeating the stress strain curve, corresponding to the last strain rate, again with a much higher strain rate.

   
   
   

7. fct_ID_p is used to distinguish the behavior in tension and compression for certain materials (pressure dependent yield). This is available for both shell and solid elements. The effective yield stress is then obtained by multiplying the nominal yield stress by the yield factor corresponding to the actual pressure.
8. If $\dot{\epsilon }\le {\dot{\epsilon }}_n$ , the yield stress is interpolated between ${\mathrm{f}}_n$ and ${\mathrm{f}}_{n-1}$ .
9. If $\dot{\epsilon }\le {\dot{\epsilon }}_1$ , function ${\mathrm{f}}_1$ is used.
10. Beyond the last point of the plastic strain versus stress input curves, the yield stress is extrapolated.

    
    
    

    Where,

    $\dot{\epsilon }$

    Total strain rate for VP =0

    $\dot{\epsilon }$

    Plastic strain rate for VP =1

11. If the last point of the first (static) function equals 0 in stress, default value of ${\epsilon }_p^{max}$ is set to the corresponding value of ${\epsilon }_p$ .
12. Strain rate values must be given in strictly ascending order.
13. 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 $\text{f}\left({\overline{\epsilon }}_p\right)$ :
      $$\mathrm{E}\left(t\right)=f\left({\overline{\epsilon }}_p\right)E$$

      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:
      $$\mathrm{E}\left(t\right)=E-(E-E_{\inf })\left(1-\exp (-C_E{\overline{\epsilon }}_p)\right)$$
      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 $E$ remains constant.
14. When you specify ${\epsilon }_p^{max}$ or ${\epsilon }_t$ and ${\epsilon }_m$ , a damage output variable is available using /ANIM/ELEM/DAMG or /H3D/ELEM/DAMG. This damage variable is computed as:
    $$D=\max \left(\frac{{\epsilon }_p}{{\epsilon }_p^{\max }},1-\frac{{\epsilon }_m-{\epsilon }_1}{{\epsilon }_m-{\epsilon }_t}\right)$$
15. When ${\epsilon }_p^{max}$ is specified and the /NONLOCAL/MAT option is activated, non-local plastic strain is used to compute the damage variable.
16. When VP = 0 and F_cut is set to the default value, it is automatically updated according to the unit system used for the computation.
17. When VP = 1, the strain rate dependency is computed considering the plastic strain rate. This approach is more consistent with physics and is closer to a full visco-plastic approach. In that case, the plastic strain rate is always smoothed with a cut-off frequency of 10000Hz that cannot be changed by users.
18. By default, the hydrostatic pressure is linearly proportional to the density:
    $$P=K\mu$$

    Where,

    $K=\frac{E}{3\left(1-2\nu \right)}$

    is the bulk modulus and

    $\mu =\frac{V_0}{V}-1=\frac{\rho }{{\rho }_0}-1$ .

    An additional Equation of State (/EOS) card can refer to this material in order to define a nonlinear dependency between hydrostatic pressure and volumetric strain. This behavior is only available for solid and SPH elements.