/MAT/LAW32 (HILL)

Block Format Keyword This law describes the Hill orthotropic plastic material. It is applicable only to shell elements. This law differs from LAW43 (HILL_TAB) only in the input of yield stress.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW32/mat_ID/unit_ID or /MAT/HILL/mat_ID/unit_ID
mat_title
${\rho }_{i}$
E $\nu$
a ${\epsilon }_{0}$ n ${\epsilon }_{p}^{max}$ ${\sigma }_{\mathrm{max}}$
${\stackrel{˙}{\epsilon }}_{0}$ m
r00 r45 r90     Iyield0

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)

a Yield parameter.

(Real)

$\left[\text{Pa}\right]$
${\epsilon }_{0}$ Hardening parameter.

(Real)

n Hardening exponent.

(Real)

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

Default = 1030 (Real)

${\sigma }_{\mathrm{max}}$ Maximum stress.

Default = 1030 (Real)

$\left[\text{Pa}\right]$
${\stackrel{˙}{\epsilon }}_{0}$ Minimum strain rate.

Default = 1.0 (Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
m Strain rate exponent.

Default = 0.0 (Real)

r00 Lankford parameter 0 degree. 5

Default = 1.0 (Real)

r45 Lankford parameter 45 degrees.

Default = 1.0 (Real)

r90 Lankford parameter 90 degrees.

Default = 1.0 (Real)

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

(Integer)

Example (Steel)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
kg                  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/1/1
void_steel
#              RHO_I
7.8E-7
#                  E                  NU
210                  .3
#                  A           EPSILON_0                   n             EPS_max           SIGMA_max
.17                  .2                 .45                   0                   0
#          EPS_DOT_0                   m
0                   0
#                r00                 r45                 r90                       Iyield0
.75                   1                1.25                             0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|


1. The yield stress is defined as follows:(1)
${\sigma }_{y}=a{\left({\epsilon }_{0}+{\epsilon }_{p}\right)}^{n}\mathrm{max}{\left(\stackrel{˙}{\epsilon },{\stackrel{˙}{\epsilon }}_{0}\right)}^{m}$
The elastic limit is given by:(2)
${\sigma }_{0}=a{\left({\epsilon }_{0}\right)}^{n}{\left({\stackrel{˙}{\epsilon }}_{0}\right)}^{m}$
Where,
${\epsilon }_{p}$
Plastic strain
$\stackrel{˙}{\epsilon }$
Strain rate
2. The yield stress is compared to the equivalent stress:(3)
${\sigma }_{eq}=\sqrt{{A}_{1}{\sigma }_{1}^{2}+{A}_{2}{\sigma }_{2}^{2}-{A}_{3}{\sigma }_{1}{\sigma }_{2}+{A}_{12}{\sigma }_{12}^{2}}$
3. This material law must be used with property set type /PROP/TYPE10 (SH_COMP) or /PROP/TYPE9 (SH_ORTH).
4. Iterative projection (Iplas =1) and radial return (Iplas =2) for shell plane stress plasticity are available.
5. Angles for Lankford parameters are defined with respect to orthotropic direction 1.
(4)
$\begin{array}{ll}R=\frac{{r}_{00}+2{r}_{45}+{r}_{90}}{4}& H=\frac{R}{1+R}\\ {A}_{1}=H\left(1+\frac{1}{{r}_{00}}\right)& {A}_{2}=H\left(1+\frac{1}{{r}_{90}}\right)\\ {A}_{3}=2H& {A}_{12}=2H\left({r}_{45}+0.5\right)\left(\frac{1}{{r}_{00}}+\frac{1}{{r}_{90}}\right)\\ {r}_{00}=\frac{{A}_{3}}{2{A}_{1}-{A}_{3}}& {r}_{45}=\frac{1}{2}\left(\frac{{A}_{12}}{{A}_{1}+{A}_{2}-{A}_{3}}-1\right)\\ {r}_{90}=\frac{{A}_{3}}{2{A}_{2}-{A}_{3}}& \end{array}$
The Lankford parameters rα is the ratio of plastic strain in plane and plastic strain in thickness direction ${\epsilon }_{33}$ .(5)
${r}_{\alpha }=\frac{d{\epsilon }_{\alpha +\pi /2}}{d{\epsilon }_{33}}$

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

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

A higher value of R means better formability.

6. If the yield stresses have been obtained in the orthotropic direction 1, define Iyield0 =1; otherwise Iyield0 =0.
7. When ${\epsilon }_{p}$ reaches the value of ${\epsilon }_{p}^{max}$ , in one integration point, then the corresponding shell element is deleted.