/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)	(3)	(5)	(7)	(9)
/MAT/LAW32/`mat_ID`/`unit_ID` or /MAT/HILL/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`	$ν$
`a`	$ε_{0}$	`n`	$ε_{p}^{m a x}$	$σ_{\max 0}$
${\dot{ε}}_{0}$	`m`
`r`₀₀	`r`₄₅	`r`₉₀		`I`_yield0

Definition

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
`a`	Yield parameter. (Real)	$[Pa]$
$ε_{0}$	Hardening parameter. (Real)
`n`	Hardening exponent. (Real)
$ε_{p}^{m a x}$	Failure plastic strain. Default = 10³⁰ (Real)
$σ_{\max 0}$	Maximum stress. Default = 10³⁰ (Real)	$[Pa]$
${\dot{ε}}_{0}$	Minimum strain rate. Default = 1.0 (Real)	$[\frac{1}{s}]$
`m`	Strain rate exponent. Default = 0.0 (Real)
`r`₀₀	Lankford parameter 0 degree. 5 Default = 1.0 (Real)
`r`₄₅	Lankford parameter 45 degrees. Default = 1.0 (Real)
`r`₉₀	Lankford parameter 90 degrees. Default = 1.0 (Real)
`I`_yield0	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-6
#                  E                  NU
                 210                  .3
#                  A           EPSILON_0                   n             EPS_max          SIGMA_max0
                 .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----|

Comments

The yield stress is defined as:(1)
$σ_{y} = a {(ε_{0} + ε_{p})}^{n} \max {(\dot{ε}, {\dot{ε}}_{0})}^{m}$

The elastic limit is given by:(2)
$σ_{0} = a {(ε_{0})}^{n} {({\dot{ε}}_{0})}^{m}$

Where,

$ε_{p}$

Plastic strain

$\dot{ε}$

Strain rate
The yield stress is compared to the equivalent stress:(3)
$σ_{e q} = \sqrt{A_{1} σ_{1}^{2} + A_{2} σ_{2}^{2} - A_{3} σ_{1} σ_{2} + A_{12} σ_{12}^{2}}$

Figure 1.
This material law must be used with property set type /PROP/TYPE10 (SH_COMP) or /PROP/TYPE9 (SH_ORTH).
Iterative projection (I_plas =1) and radial return (I_plas =2) for shell plane stress plasticity are available.
Angles for Lankford parameters are defined with respect to orthotropic direction 1.
(4)
$\begin{array}{l} R = \frac{r_{00} + 2 r_{45} + r_{90}}{4} & H = \frac{R}{1 + R} \\ A_{1} = H (1 + \frac{1}{r_{00}}) & A_{2} = H (1 + \frac{1}{r_{90}}) \\ A_{3} = 2 H & A_{12} = 2 H (r_{45} + 0.5) (\frac{1}{r_{00}} + \frac{1}{r_{90}}) \\ r_{00} = \frac{A_{3}}{2 A_{1} - A_{3}} & r_{45} = \frac{1}{2} (\frac{A_{12}}{A_{1} + A_{2} - A_{3}} - 1) \\ r_{90} = \frac{A_{3}}{2 A_{2} - A_{3}} \end{array}$

The Lankford parameters $r_{α}$ is the ratio of plastic strain in plane and plastic strain in thickness direction $ε_{33}$ .(5)
$r_{α} = \frac{d ε_{α + π / 2}}{d ε_{33}}$

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

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

A higher value of R means better formability.
If the yield stresses have been obtained in the orthotropic direction 1, define I_yield0 =1; otherwise I_yield0 =0.
When $ε_{p}$ reaches the value of $ε_{p}^{m a x}$ , in one integration point, then the corresponding shell element is deleted.