/MAT/LAW93 (ORTH_HILL) or (CONVERSE)

Block Format Keyword This law describes the orthotropic elastic behavior material with Hill plasticity and is applicable to shell and solid elements (/BRICK, /TETRA4 and /TETRA10).

It could be used with property set /PROP/TYPE11, /PROP/TYPE17, /PROP/TYPE51, /PROP/PCOMPP for shell and /PROP/TYPE6 for solid.

Format


(1)	(2)	(3)	(5)	(7)	(9)
/MAT/LAW93/`mat_ID`/`unit_ID` or /MAT/ORTH_HILL/`mat_ID`/`unit_ID` or /MAT/CONVERSE/`mat_ID`/`unit_ID`/
`mat_title`
$ρ_{i}$
`E`₁₁		`E`₂₂	`E`₃₃	`G`₁₂	$ν_{12}$
`G`₁₃		`G`₂₃	$ν_{13}$	$ν_{23}$
`N`_rate	`VP`	`F`_cut

Curve input for yield if

$N_{r a t e} > 0$ , define N_rate plasticity function per line:


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fct_ID`_i		`Fscale`_i		${\dot{ε}}_{i}$

Parameter input for yield:


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$σ_{y}$		`QR`₁		`CR`₁		`QR`₂		`CR`₂

Yield stress ratio for HILL criteria:


(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`R`₁₁		`R`₂₂		`R`₁₂
`R`₃₃		`R`₁₃		`R`₂₃

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`₁₁	Young’s modulus in direction 11. (Real)	$[Pa]$
`E`₂₂	Young’s modulus in direction 22. (Real)	$[Pa]$
`E`₃₃	Young’s modulus in direction 33. (Real)	$[Pa]$
`G`₁₂	Shear modulus in direction 12. (Real)	$[Pa]$
`G`₁₃	Shear modulus in direction 13. (Real)	$[Pa]$
`G`₂₃	Shear modulus in direction 23. (Real)	$[Pa]$
$ν_{12}$	Poisson's ratio 12. (Real)
$ν_{13}$	Poisson's ratio 13. (Real)
$ν_{23}$	Poisson's ratio 23 (Real)
`N`_rate	Number of yield function.
`VP`	Strain rate choice flag. = 1 Strain rate effect on yield stress depends on the plastic strain rate. = 2 (Default) Strain rate effect on yield depends on the total strain rate. = 3 Strain rate effect on yield depends on the deviatoric strain rate. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. Default = 1.0 x 10⁴ (Real)	$[Hz]$
`fct_ID`_i	Plasticity curves i^th function identifier (i=1, `N`_rate). (Integer)
`Fscale`_i	Scale factor for i^th function (i=1, `N`_rate). Default = 1.0 (Real)	$[Pa]$
${\dot{ε}}_{i}$	Strain rate for i^th function (i=1, `N`_rate). (Real)	$[\frac{1}{s}]$
$σ_{y}$	Initial yield stress. Default = 1E30 (Real)	$[Pa]$
`QR`₁	Parameter of hardening. Default = 0.0 (Real)	$[Pa]$
`CR`₁	Parameter of hardening. Default = 0.0 (Real)
`QR`₂	Parameter of hardening. Default = 0.0 (Real)	$[Pa]$
`CR`₂	Parameter of hardening. Default = 0.0 (Real)
`R`₁₁	Yield stress ratio in direction 11. Default = 1.0 (Real)
`R`₂₂	Yield stress ratio in direction 22. Default = 1.0 (Real)
`R`₃₃	Yield stress ratio in direction 33. Default = 1.0 (Real)
`R`₁₂	Yield stress ratio in direction 12. Default = 1.0 (Real)
`R`₁₃	Yield stress ratio in direction 13. Default = 1.0 (Real)
`R`₂₃	Yield stress ratio in direction 23. Default = 1.0 (Real)

▸Example (Curve Input)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW93/1/1
plastic
#              RHO_I
           2.730E-09
#                E11                 E22                 E33                 G12                Nu12
              225654              195400              178526            75187.97                0.30
#                G13                 G23                Nu13                Nu23
            75187.97            75187.97                0.28                0.32
#     Nrate       VP                Fcut
         2         1                 0.0
#   Ifunct                        Yscale              Epsdot
         5                           1.0                0.01	     
         5                           1.5               100.0	       
#               SigY                 QR1                 CR1                 QR2                 CR2
                   0                   0                   0                 0.0                 0.0
#                R11                 R22                 R12
                 1.0             1.05626             0.96425
#                R33                 R13                 R23
              0.9337                 1.0                 1.0
/FUNCT/5
plastic
#                  X                   Y           
                   0         165.6362749
               0.002         173.8123558
               0.005         180.2967164
                0.01         186.5926709
                0.02         193.8182168
                0.05         204.4407991
                0.07         208.5903797
                 0.1         213.1182051
                0.12         215.4817557
                0.15         218.4183864
                0.17         220.0863912
                 0.2         222.2743041
                0.22         223.5689486
                0.25         225.3186882
                0.27         226.3794409
                 0.3         227.840544
                0.32         228.7406278
                0.35         229.996802
                0.37         230.7795124
                 0.4         231.8824363
                 0.5         235.0704031
                 0.6         237.7095003
                 0.7         239.9650034
                 0.8         241.9367878
                 0.9         243.689935
                   1         245.2692715
                 1.5         251.4456403
                   2         255.9237789
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The yield stress is compared to an equivalent stress in the orthotropic frame. For solid elements, this equivalent stress is defined as:
$σ_{e q} = \sqrt{F {(σ_{22} - σ_{33})}^{2} + G {(σ_{33} - σ_{11})}^{2} + H {(σ_{11} - σ_{22})}^{2} + 2 L σ_{23}^{2} + 2 M σ_{31}^{2} + 2 N σ_{12}^{2}}$
Where,

$F = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{33}^{2}} - \frac{1}{R_{11}^{2}})$

$G = \frac{1}{2} (\frac{1}{R_{33}^{2}} + \frac{1}{R_{11}^{2}} - \frac{1}{R_{22}^{2}})$

$H = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{11}^{2}} - \frac{1}{R_{33}^{2}})$

$L = \frac{3}{2 R_{23}^{2}}$

$M = \frac{3}{2 R_{31}^{2}}$

$N = \frac{3}{2 R_{12}^{2}}$

$R_{i i}^{} = \frac{σ_{F}^{i i}}{σ_{F}}$

Normal directions

$R_{i j}^{} = \frac{\sqrt{3} σ_{F}^{i j}}{σ_{F}}$

Shear directions

$σ_{F}^{i j}$

Yield stress in direction $i j$

$σ_{F}^{}$

Global flow stress that can either be defined with a sum of Voce hardening, or can be tabulated (see below).

Under plane-stress conditions, for shell elements, the equivalent yield stress becomes:
$σ_{e q} = \sqrt{(G + H) σ_{11}^{2} + (F + H) σ_{22}^{2} - 2 H σ_{11} σ_{22} + 2 N σ_{12}^{2}}$
The yield function $Φ$ will compare the Hill’s equivalent stress $σ_{e q}$ to the flow stress $σ_{F}^{}$ as:
$Φ = σ_{e q} - σ_{F}$
The two different ways to define the flow stress $σ_{F}^{}$ are: parameter input or curve input
- For parameter input, the flow stress is defined with an initial yield stress and a double Voce hardening as:
  $σ_{F} = σ_{Y}^{0} + R (ε_{p})$
  With $R (ε_{p}) = \sum_{i}^{2} Q R_{i} \cdot (1 - e^{- C R_{i} \cdot ε_{p}})$ .
- For curve input, the parameters input values will be ignored.
  The yield can be defined with using stress versus plastic strain curve taking in account the strain rate effect. When the stress versus strain curves are defined, this is the default method for defining the hardening.
  1. If $\dot{ε} \leq {\dot{ε}}_{n}$ , the yield is interpolated between $f_{n}$ and $f_{n - 1}$ .
  2. If $\dot{ε} \leq {\dot{ε}}_{1}$ function, $f_{1}$ is used.
  3. Above ${\dot{ε}}_{\max}$ , yield is extrapolated.
    
    Figure 1.
For tabulated flow stress, the strain rate $\dot{ε}$ computation depends on the value of the flag VP.
- If VP= 1, the plastic strain rate is used
- If VP= 2, the total strain rate is used
- If VP= 3, the total strain rate is used
In all cases the strain-rate computation includes a filtering. The cutoff frequency is automatically set for VP = 1. However, for VP = 1 or 3, you can input a cutoff frequency F_cut; otherwise, a default value will be set.