/MAT/LAW2 (PLAS_JOHNS)
Block Format Keyword This law represents an isotropic elastoplastic material using the JohnsonCook material model.
This model expresses material stress as a function of strain, strain rate and temperature. A builtin failure criterion based on the maximum plastic strain is available.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW2/mat_ID/unit_ID or /MAT/PLAS_JOHNS/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  I_{flag} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

a  b  n  ${\epsilon}_{p}^{max}$  ${\sigma}_{\mathrm{max}\text{}0}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

${\sigma}_{y}$  UTS  ${\epsilon}_{UTS}$  ${\epsilon}_{p}^{max}$  ${\sigma}_{\mathrm{max}\text{}0}$ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

c  ${\dot{\epsilon}}_{0}$  ICC  F_{smooth}  F_{cut}  C_{hard}  
m  T_{melt}  $\rho {C}_{p}$  T_{r} 
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) 

I_{flag}  Input type flag. 3
(Integer) 

a  Yield stress. 2 (Real) 
$\left[\text{Pa}\right]$ 
b  Plastic hardening parameter b. (Real) 
$\left[\text{Pa}\right]$ 
n  Plastic hardening exponent n. 6
Default = 1.0 (Real) 

${\epsilon}_{p}^{max}$  Failure plastic strain. Default = 10^{30} (Real) 

${\sigma}_{\mathrm{max}\text{}0}$  Maximum stress. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
${\sigma}_{y}$  Yield stress. (Real) 
$\left[\text{Pa}\right]$ 
UTS  Ultimate tensile stress (engineering
stress). Input
$UTS>{\sigma}_{y}$
. (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{UTS}$  Engineering strain at
UTS. Default = 1.0 (Real) 

c  Strain rate coefficient
$c\ge 0$
.
Default = 0.00 (Real) 

${\dot{\epsilon}}_{0}$  Reference strain rate. If $\dot{\epsilon}\le {\dot{\epsilon}}_{0}$ , no strain rate effect. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
ICC  Strain rate computation flag. 9
(Integer) 

F_{smooth}  Strain rate smoothing flag.
(Integer) 

F_{cut}  Cutoff frequency for strain rate smoothing. Only available for shell and solid
elements, Appendix: Filtering. Default = 10^{30} (Real) 
$\text{[Hz]}$ 
C_{hard}  Hardening coefficient (unloading).
(Real) 

m  Temperature exponent. 13 Default = 1.00 (Real) 

T_{melt}  Melting temperature.
Default = 10^{30} (Real) 
$\left[\text{K}\right]$ 
$\rho {C}_{p}$  Specific heat per unit volume. 11 (Real) 
$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}\text{K}}\right]$ 
T_{r}  Reference temperature. 11 Default = 298 K (Real) 
$\left[\text{K}\right]$ 
Example (Classic Parameter Input)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/PLAS_JOHNS/1/1
Steel
# RHO_I
7.8E9
# E Nu Iflag
210000 .3 0
# a b n EPS_max SIG_max0
270 450.0 0.6 0 0
# c EPS_DOT_0 ICC Fsmooth F_cut Chard
0 0 0 0 0 0
# m T_melt rhoC_p T_r
0 0 0 0
#12345678910
#ENDDATA
/END
#12345678910
Example (Simplified Input  Experimental Data)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/PLAS_JOHNS/1/1
Steel (use ultimate tensile stress(UTS) and engineering strain )
# RHO_I
7.8E9
# E Nu Iflag
210000 .3 1
# SIG_y UTS EPS_UTS EPS_max SIG_max0
270 362.8 0.2885 0 0
# c EPS_DOT_0 ICC Fsmooth F_cut Chard
0 0 0 0 0 0
# m T_melt rhoC_p T_r
0 0 0 0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 This is an elastoplastic material model that includes strain rate and temperature effects with true stress and strain output.
 In this model the material behaves as a
linearelastic material when the equivalent stress is lower than the plastic yield stress.
For higher stress values, the material behavior is plastic, and the true stress is
calculated as:
(1) $$\sigma =\left(a+b{{\epsilon}_{p}}^{n}\right)\left(1+c\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\left(1{({T}^{\ast})}^{m}\right)$$Where,(2) $${T}^{*}=\frac{T{T}_{r}}{{T}_{melt}{T}_{r}}$$Where, ${\epsilon}_{p}$
 Plastic strain
 $\dot{\epsilon}$
 Strain rate
 $T$
 Temperature
 T_{r}
 Ambient temperature
 T_{melt}
 Melting temperature
 If I_{flag}=0, the JohnsonCook equation parameters a, b, and n values are entered.
If I_{flag}=1, experimental engineering stress and stain data can be entered for ${\sigma}_{y}$ , UTS and ${\epsilon}_{UTS}$ and the parameters a, b and n are calculated and printed in the Starter output file. If the a, b and n parameters cannot be automatically fit, then a Starter warning message will contain important information about changes to the material input.
 The plastic yield stress should always be greater than zero. To model pure elastic behavior, the plastic yield stress will be set to 10^{30}.
 When
${\epsilon}_{p}$
reaches the value of
${\epsilon}_{p}^{max}$
in one integration point, then based on the element type:
 Shell elements: The corresponding shell element is deleted.
 Solid elements: The deviatoric stress of the corresponding integral point is permanently set to 0; however, the solid element is not deleted.
 The plastic hardening exponent, n must be less than or equal to 1.
 The strain rate has no effect on truss elements.
 To eliminate the effect of the strain rate, you can either set the value of c equal to 0 or the reference strain rate ( ${\dot{\epsilon}}_{0}$ ) can be set equal to 10^{30}. There is no effect of strain rate when $\dot{\epsilon}$ is less than ${\dot{\epsilon}}_{0}$ .
 The ICC flag defines the effect of strain
rate on the maximum material stress
${\sigma}_{\mathrm{max}}$
. Figure 1 shows the value of for
${\sigma}_{\mathrm{max}}$
the corresponding ICC flag.
$\sigma =\left(a+b{{\epsilon}_{p}}^{n}\right)\left(1+c\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$ $\sigma =\left(a+b{{\epsilon}_{p}}^{n}\right)\left(1+c\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}0}\left(1+c\mathrm{ln}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{o}}\right)\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}0}$  There is no effect of temperature on trusses. Beam element (with /PROP/TYPE3 or /PROP/TYPE18) together with /HEAT/MAT or /THERM_STRESS/MAT could consider thermal effect.
 The temperature is constant ( $T={T}_{r}$ ), if $\rho {C}_{p}=0$ .
 Adiabatic conditions are assumed for
thermal simulations with initial temperature equal to reference temperature (T_{r}) and:
(3) $$T={T}_{r}+\frac{{E}_{\mathrm{int}}}{\rho {C}_{p}(Volume)}$$Where, E_{int} is the internal deformation energy.
 The strain rate coefficient, c and reference strain rate ${\dot{\epsilon}}_{0}$ must be defined to include thermal effects.
 When /HEAT/MAT (with I_{form}=1) references this material model, the values of T_{r} and $\rho {C}_{p}$ defined in this card will be overwritten by the corresponding ${T}_{0}$ and $\rho {\text{\hspace{0.05em}}}_{0}{C}_{p}$ defined in /HEAT/MAT.
 When the temperature is not initialized using /HEAT/MAT or /INITEMP, the reference temperature (T_{r}) is also the initial temperature.
 The hardening coefficient is used to describe the hardening model (during unloading). The values of the hardening coefficient should be between 0 and 1.