# /MAT/LAW43 (HILL_TAB)

Block Format Keyword This law describes the Hill orthotropic material and is applicable only to shell elements. This law differs from LAW32 (HILL) only in the input of yield stress (here it is defined by a user function).

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW43/mat_ID/unit_ID or /MAT/HILL_TAB/mat_ID/unit_ID
mat_title
${\rho }_{i}$
E $\nu$
fct_IDE   Einf CE
r00 r45 r90 Chard Iyield0
${\epsilon }_{p}^{\mathrm{max}}$ ${\epsilon }_{t}$ ${\epsilon }_{m}$
fct_IDi   Fscalei ${\stackrel{˙}{\epsilon }}_{i}$

## Definitions

Field Contents SI Unit Example
mat_ID Material identifier

(Integer, maximum 10 digits)

unit_ID Unit Identifier

(Integer, maximum 10 digits)

mat_title Material title

(Character, maximum 100 characters)

${\rho }_{i}$ Initial density

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$
E Young's modulus

(Real)

$\left[\text{Pa}\right]$
$\nu$ Poisson's ratio

(Real)

fct_IDE Function identifier for the scale factor of Young's modulus, when Young's modulus is function of the plastic strain. 12

Default = 0: in this case the evolution of Young's modulus depends on Einf and CE.

(Integer)

Einf Saturated Young's modulus for infinitive plastic strain.

(Real)

$\left[\text{Pa}\right]$
CE Parameter for Young's modulus evolution. 12

(Real)

r00 Lankford parameter 0 degree. 3

Default = 1.0 (Real)

r45 Lankford parameter 45 degrees.

Default = 1.0 (Real)

r90 Lankford parameter 90 degrees.

Default = 1.0 (Real)

Chard Hardening coefficient.
= 0
Hardening is full isotropic model.
= 1
Hardening uses the kinematic Prager-Ziegler model.
= between 0 and 1
Hardening is interpolated between the two models.

(Real)

Iyield0 Yield stress flag.
= 0
Average yield stress input.
= 1
Yield stress in orthotropic direction 1.

(Integer)

${\epsilon }_{p}^{max}$ Failure plastic strain.

Default = 1.0 × 1030 (Real)

${\epsilon }_{t}$ Tensile failure strain at which stress starts to reduce.

Default = 1.0 × 1030 (Real)

${\epsilon }_{m}$ Maximum tensile failure strain at which the stress in element is set to zero.

Default = 2.0 × 1030 (Real)

fct_IDi Plasticity curves .ith function identifier.

(Integer)

Fscalei Scale factor for ith function.

Default set to 1.0 (Real)

${\stackrel{˙}{\epsilon }}_{i}$ Strain rate for ith function.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$

## Example (Metal)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
Mg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HILL_TAB/1/1
metal
#              RHO_I
80
#                  E                  NU
206000                  .3
#FUNCT_IDE                          EINF                  CE
0                             0                   0
#                r00                 r45                 r90              C_hard   Iyield0
1.73                1.34                2.24                   0         0
#           EPSP_max              EPS_t1               EPS_m
0                   0                   0
# func_IDi                      Fscale_i           EPS_dot_i
5                             0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  3. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/5
metal
#                  X                   Y
0                 260
.002                 270
.005                 280
.01                 297
.02                 322
.05                 370
.1                 422
.15                 457
.2                 485
.3                 528
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

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:(1)
${\sigma }_{eq}=\sqrt{{A}_{1}{\sigma }_{1}^{2}+{A}_{2}{\sigma }_{2}^{2}-{A}_{3}{\sigma }_{1}{\sigma }_{2}+{A}_{12}{\sigma }_{12}^{2}}$
3. Angles for Lankford parameters are defined with respect to orthotropic direction 1.(2)
$\begin{array}{ll}R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}& H=\frac{R}{1+R}\\ {A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)& {A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)\\ {A}_{3}=2H& {A}_{12}=2H\left({r}_{45}+0.5\right)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)\\ {r}_{00}=\frac{{A}_{3}}{2{A}_{1}-{A}_{3}}& {r}_{45}=\frac{1}{2}\left(\frac{{A}_{12}}{{A}_{1}+{A}_{2}-{A}_{3}}-1\right)\\ {r}_{90}=\frac{{A}_{3}}{2{A}_{2}-{A}_{3}}& \end{array}$

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

(3)
${r}_{\alpha }=\frac{d{\epsilon }_{\alpha +\pi /2}}{d{\epsilon }_{33}}$

Where, α is the angle to the orthotropic direction 1.

This Lankford parameters ra could be determined from a simple tensile test at an angle α.

A higher value of R means better formability.

4. 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 ${\epsilon }_{p}$ .
5. Element deletion:
• Once ${\epsilon }_{p}$ (plastic strain) reaches ${\epsilon }_{p}^{max}$ , in one integration point, the element is deleted.
• If ${\epsilon }_{1}$ reaches ${\epsilon }_{t}$ , the stress is reduced using the following relation: (4)
$\sigma =\sigma \left(\frac{{\epsilon }_{m}-{\epsilon }_{1}}{{\epsilon }_{m}-{\epsilon }_{t}}\right)$
• If ${\epsilon }_{1}$ (largest principal strain) reaches ${\epsilon }_{m}$ ( ${\epsilon }_{1}>{\epsilon }_{m}$ ), the stress in element is reduced to 0 (but the element is not deleted).
• Once ${\epsilon }_{1}$ (largest principal strain) reaches ${\epsilon }_{f}$ (maximum tensile failure strain), the element is deleted.
6. The maximum number of curves that can be input is 10.
7. If $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{n}$ , the yield is interpolated between ${f}_{n}$ and ${f}_{n-1}$ .
8. If $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{1}$ , function ${f}_{1}$ is used.
9. Above ${\stackrel{˙}{\epsilon }}_{\mathrm{max}}$ , yield is extrapolated.
10. Radial return is not available (only iterative plasticity).
11. If the yield stresses have been obtained in the orthotropic direction 1, define Iyield0 =1; otherwise Iyield0 =0.
12. The evolution of Young's modulus:
• If fct_IDE > 0, the curve defines a scale factor for Young's modulus evolution with equivalent plastic strain ${\overline{\epsilon }}_{p}$ , which means the Young's modulus is scaled by the function $\text{f}\left({\overline{\epsilon }}_{p}\right)$ :(5)
$\text{E}\left(t\right)=\text{f}\left({\overline{\epsilon }}_{p}\right)E$

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

• If fct_IDE = 0, the Young's modulus is calculated as:(6)
$\mathrm{E}\left(t\right)=E-\left(E-{E}_{\mathrm{inf}}\right)\left(1-\mathrm{exp}\left(-{C}_{E}{\overline{\epsilon }}_{p}\right)\right)$

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

Note:

If fct_IDE = 0 and CE = 0, Young's modulus E is kept constant.