/MAT/LAW84
Block Format Keyword SwiftVoce elastoplastic law with JohnsonCook strain rate hardening and temperature softening. This law allows modeling a quadratic nonassociated flow rule.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW84/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  
P_{12}  P_{22}  P_{33}  Q  B  
G_{12}  G_{22}  G_{33}  K_{0}  $\alpha $  
A  ${\epsilon}_{0}$  n  C  ${\dot{\epsilon}}_{0}$  
$\eta $  C_{p}  T_{ini}  T_{ref}  T_{melt}  
m  ${\dot{\epsilon}}_{\alpha}$ 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E  Young's
modulus (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's
ratio (Real) 

P_{12}  Yield parameter Default = 0.5 (Real) 

P_{22}  Yield parameter Default = 1.0 (Real) 

P_{33}  Yield parameter Default = 3.0 (Real) 

G_{12}  Flow rule
parameter Default = P_{12} (Real) 

G_{22}  Flow rule
parameter Default = P_{22} (Real) 

G_{33}  Flow rule
parameter Default = P_{33} (Real) 

Q  Voce hardening
coefficient (Real) 
$\left[\text{Pa}\right]$ 
B  Voce plastic strain
coefficient Default = 0.0 (Real) 

K_{0}  Voce
parameter (Real) 

$\alpha $  Yield weighting
coefficient.
Default = 0.0 (Real) 

A  Swift hardening
coefficient. (Real) 
$\left[\text{Pa}\right]$ 
n  Swift hardening
exponent. Default = 1.0 (Real) 

${\epsilon}_{0}$  Swift hardening
parameter. Default = 0.00 (Real) 

C  Strain rate
coefficient.
Default = 0.00 (Real) 

${\dot{\epsilon}}_{0}$  Reference strain
rate. Default = 10^{30}, no strain rate effect (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
$\eta $  TaylorQuinney coefficient
quantifies the fraction of plastic work converted to
heat. (Real) 

C_{p}  Specific
heat. (Real) 
$\left[\frac{\text{J}}{\text{kg}\cdot \text{K}}\right]$ 
T_{ini}  Initial temperature used
in initialization when time=0. (Real) 
$\left[\text{K}\right]$ 
T_{ref}  Reference
temperature. (Real) 
$\left[\text{K}\right]$ 
T_{melt}  Melting
temperature. (Real) 
$\left[\text{K}\right]$ 
m  Temperature
exponent. (Real) 

${\dot{\epsilon}}_{\alpha}$  Strain rate optimization
parameter for temperature dependency. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Example (Metal)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
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# 2. MATERIALS:
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/MAT/LAW84/1/1
Swiftvoce (metal)
# Rho_i
8E9
# E Nu
206000 .3
# P12 P22 P33 Q B
.5 1 3 524 25
# G12 G22 G33 K0 ALPHA
.5 1 3 100 .5
# A EPS0 n C EPSDOT
1000 .00128 .2 .014 .0011
# ETA CP Tini Tref Tmelt
.9 42000000000 293 293 1700
# m EPSDOTA
.921 1.379
#12345678910
#ENDDATA
/END
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Comments
 Yield stress is computed using an analytic
expression with a combination of both Swift and Voce models, the strain rate
dependency and temperature dependency following JohnsonCook
law.
(1) $${\sigma}_{y}=\left\{\alpha \left[A{\left({\overline{\epsilon}}_{p}+{\epsilon}_{0}\right)}^{n}\right]+\left(1\alpha \right)\left[{K}_{0}+Q\left(1\mathrm{exp}\left(B{\overline{\epsilon}}_{p}\right)\right)\right]\right\}\left(1+C\mathrm{ln}\frac{{\dot{\overline{\epsilon}}}_{p}}{{\dot{\epsilon}}_{0}}\right)\left[1{\left(\frac{T{T}_{ref}}{{T}_{melt}{T}_{ref}}\right)}^{m}\right]$$  The effective stress is computed
as:
(2) $$\begin{array}{l}\overline{\sigma}=\mathrm{f}\left(\sigma \right)\\ \begin{array}{c}\end{array}=\sqrt{{\sigma}^{T}P\sigma}\\ \begin{array}{c}\end{array}={\sigma}_{11}^{2}+{P}_{22}{\sigma}_{22}^{2}+\left(1+{P}_{12}+{P}_{22}\right){\sigma}_{33}^{2}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}+2{P}_{12}{\sigma}_{11}{\sigma}_{22}2\left(1+{P}_{12}\right){\sigma}_{11}{\sigma}_{33}2\left({P}_{22}+{P}_{12}\right){\sigma}_{22}{\sigma}_{33}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}+\left({P}_{33}+3\right){\sigma}_{12}^{2}+3{\sigma}_{23}^{2}\end{array}$$  The plastic nonassociated flow rule is computed as:
(3) $$\text{\Delta}{\epsilon}_{p}=\text{\Delta}{\overline{\epsilon}}_{p}\frac{\partial \mathrm{g}\left(\sigma \right)}{\partial \sigma}$$Where,(4) $$\begin{array}{l}\mathrm{g}\left(\sigma \right)=\sqrt{{\sigma}^{T}G\sigma}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}={\sigma}_{11}^{2}+{G}_{22}{\sigma}_{22}^{2}+\left(1+{G}_{12}+{G}_{22}\right){\sigma}_{33}^{2}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+2{G}_{12}{\sigma}_{11}{\sigma}_{22}2\left(1+{G}_{12}\right){\sigma}_{11}{\sigma}_{33}2\left({G}_{22}+{G}_{12}\right){\sigma}_{22}{\sigma}_{33}\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+\left({G}_{33}+3\right){\sigma}_{12}^{2}+3{\sigma}_{23}^{2}\end{array}$$  Temperature is updated using:
(5) $$\text{\Delta}T=\omega \left({\dot{\overline{\epsilon}}}_{p}\right)\frac{\eta}{\rho {C}_{p}}\overline{\sigma}d{\overline{\epsilon}}_{p}$$Where, $\omega \left({\dot{\overline{\epsilon}}}_{p}\right)=\{\begin{array}{ccc}0& \text{if}& {\dot{\overline{\epsilon}}}_{p}<{\dot{\epsilon}}_{0}\\ 1& \text{if}& {\dot{\overline{\epsilon}}}_{p}>{\dot{\epsilon}}_{\alpha}\\ \frac{{\left({\dot{\overline{\epsilon}}}_{p}{\dot{\epsilon}}_{0}\right)}^{2}\left(3{\dot{\epsilon}}_{\alpha}2{\dot{\overline{\epsilon}}}_{p}{\dot{\epsilon}}_{0}\right)}{{\left({\dot{\epsilon}}_{\alpha}{\dot{\epsilon}}_{0}\right)}^{3}}& \text{if}& {\dot{\epsilon}}_{0}\le {\dot{\overline{\epsilon}}}_{p}\le {\dot{\epsilon}}_{\alpha}\end{array}$