/MAT/LAW73

This law differs from /MAT/LAW43 (HILL_TAB) by the fact that yield stress not only depends on strain rate and plastic strain, but also on temperature (it is defined by a user table).

Format

Definition

Comments

1. This material law must be used with property set /PROP/TYPE9 (SH_ORTH) or /PROP/TYPE10 (SH_COMP).
2. The yield stress is defined by a user function and the yield stress is compared to equivalent stress:
   $${\sigma }_{eq}=\sqrt{A_1{{\sigma }_1}^2+A_2{{\sigma }_2}^2-A_3{\sigma }_1{\sigma }_2+A_{\mathrm{12}}{{\sigma }_{\mathrm{12}}}^2}$$

3. Angles for Lankford parameters are defined with respect to orthotropic direction 1.

   $R=\frac{r_{00}+2r_{45}+r_{90}}{4}$	$H=\frac{R}{1+R}$
   $A_1=H\left(1+\frac{1}{r_{00}}\right)$	$A_2=H\left(1+\frac{1}{r_{90}}\right)$
   $A_3=2H$	$A_{12}=2H(r_{45}+0.5)\left(\frac{1}{r_{00}}+\frac{1}{r_{90}}\right)$
   $r_{00}=\frac{A_3}{2A_1-A_3}$	$r_{45}=\frac{1}{2}\left(\frac{A_{12}}{A_1+A_2-A_3}-1\right)$
   $r_{90}=\frac{A_3}{2A_2-A_3}$

   The Lankford parameters $r_{\alpha }$ is ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon }_{\mathrm{33}}$ .

   $$r_{\alpha }=\frac{d{\epsilon }_{\alpha +\pi /2}}{d{\epsilon }_{33}}$$

   Where, $\alpha$ is the angle to the orthotropic direction 1.

   This Lankford parameters $r_{\alpha }$ could be determined from a simple tensile test at an angle $\alpha$ .

   A higher value of R means better formability.

4. If ${\epsilon }_p$ (plastic strain) reaches ${\epsilon }_p^{max}$ , in one integration point, the corresponding shell element is deleted.
5. If largest principal strain ${\epsilon }_1>{\epsilon }_t$ , stress is reduced using the following relation:
   $$\sigma =\sigma \left(\frac{{\epsilon }_m-{\epsilon }_1}{{\epsilon }_m-{\epsilon }_t}\right)$$
6. If ${\epsilon }_1>{\epsilon }_m$ , the stress is reduced to 0 (but the element is not deleted).
7. This law always uses iterative projection for plasticity (I_plas from the property set is ignored).
8. This law is not available with global formulation for plasticity (N=0 in the property shell is not available).
9. The table for yield stress definition must be a 3-dimensional table whose parameters respectively represent plastic strain, strain rate, and temperature $\left({\epsilon }^p,\dot{\epsilon },T\right)$ . Values of the table are yield stress values.
10. If ${\epsilon }_{m-1}^p\le {\epsilon }^p\le {\epsilon }_m^p$ and ${\dot{\epsilon }}_{n-1}\le \dot{\epsilon }\le {\dot{\epsilon }}_n$ and $T_{q-1}\le T\le T_q$ yield is linearly interpolated between the eight values of the table corresponding to $\left({\epsilon }_i^p,{\dot{\epsilon }}_j,T_k\right),i=m-1,m;j=n-1,n;k=q-1,q$ .
11. If $\left({\epsilon }^p,\dot{\epsilon },T\right)$ falls out of the range of the table, yield stress is obtained by linear extrapolation. Thus it is necessary to input into the table the static curves corresponding to zero strain rate (entry $\dot{\epsilon }=0$ should belong to the table definition).

    If the /HEAT/MAT option is not associated to the material identifier, adiabatic conditions are assumed and temperature is computed as:

    $${\rm T}=T_i+\frac{E_{int}}{\rho C_p\left(Volume\right)}$$
    Where,

    $E_{\mathrm{int}}$

    Internal energy computed by Radioss,

    $\sigma$

    Volume are the current density, and volume

    $C_p$

    Heat capacity per mass unit

    Otherwise, the finite element formulation for heat transfer must be asked for (I_form =1 in option /HEAT/MAT); initial temperature and specific heat input in the option /HEAT/MAT will then be used.

12. If the yield stresses have been obtained in the orthotropic direction 1, define I_yield0 =1; otherwise I_yield0=0.