# /MAT/LAW2 (PLAS_JOHNS)

Block Format Keyword This law represents an isotropic elasto-plastic material using the Johnson-Cook material model.

This model expresses material stress as a function of strain, strain rate and temperature. A built-in 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 $\nu$ Iflag
If Iflag=0, insert the next line with classic input:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
a b n ${\epsilon }_{p}^{max}$ ${\sigma }_{\mathrm{max}\text{​}0}$
If Iflag=1, insert the next line with simplified input:
(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 ${\stackrel{˙}{\epsilon }}_{0}$ ICC Fsmooth Fcut Chard
m Tmelt $\rho {C}_{p}$ Tr

## 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)

Iflag Input type flag. 3
=0 (Default)
Classic input for Johnson-Cook using parameters a, b, and n.
=1
Simplified input type using ${\sigma }_{y}$ , UTS, and ${\epsilon }_{UTS}$ .

(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 = 1030 (Real)

${\sigma }_{\mathrm{max}\text{​}0}$ Maximum stress.

Default = 1030 (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)

$\left[\text{Pa}\right]$
c Strain rate coefficient $c\ge 0$ .
=0
No strain rate effect.

Default = 0.00 (Real)

${\stackrel{˙}{\epsilon }}_{0}$ Reference strain rate.

If $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{0}$ , no strain rate effect.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
ICC Strain rate computation flag. 9
=0
Set to 1.
=1 (Default)
Strain rate effect on ${\sigma }_{\mathrm{max}}$
=2
No strain rate effect on ${\sigma }_{\mathrm{max}}$

(Integer)

Fsmooth Strain rate smoothing flag.
=0
Set to 1.
=1 (Default)
Strain rate smoothing is active.

(Integer)

Fcut Cutoff frequency for strain rate smoothing. Only available for shell and solid elements, Appendix: Filtering.

Default = 1030 (Real)

$\text{[Hz]}$
Chard Hardening coefficient (unloading).
=0 (Default)
Isotropic model.
=1
Kinematic Prager-Ziegler model.
= value between 0 and 1
Hardening behavior is interpolated between the two models. 16

(Real)

m Temperature exponent. 13

Default = 1.00 (Real)

Tmelt Melting temperature.
=0
No temperature effect

Default = 1030 (Real)

$\left[\text{K}\right]$
$\rho {C}_{p}$ Specific heat per unit volume. 11

(Real)

$\left[\frac{\text{J}}{{\text{m}}^{\text{3}}\text{K}}\right]$
Tr Reference temperature. 11

Default = 298 K (Real)

$\left[\text{K}\right]$

## Example (Classic Parameter 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/PLAS_JOHNS/1/1
Steel
#              RHO_I
7.8E-9
#                  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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

## Example (Simplified Input - Experimental Data)

#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/PLAS_JOHNS/1/1
Steel (use ultimate tensile stress(UTS) and engineering strain )
#              RHO_I
7.8E-9
#                  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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

## Comments

1. This is an elasto-plastic material model that includes strain rate and temperature effects with true stress and strain output.
2. In this model the material behaves as a linear-elastic 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)
$\sigma =\left(a+b{{\epsilon }_{p}}^{n}\right)\left(1+c\mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\left(1-{\left({T}^{\ast }\right)}^{m}\right)$
Where,(2)
${T}^{*}=\frac{T-{T}_{r}}{{T}_{melt}-{T}_{r}}$
Where,
${\epsilon }_{p}$
Plastic strain
$\stackrel{˙}{\epsilon }$
Strain rate
$T$
Temperature
Tr
Ambient temperature
Tmelt
Melting temperature
3. If Iflag=0, the Johnson-Cook equation parameters a, b, and n values are entered.

If Iflag=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.

4. The plastic yield stress should always be greater than zero. To model pure elastic behavior, the plastic yield stress will be set to 1030.
5. 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.
6. The plastic hardening exponent, n must be less than or equal to 1.
7. The strain rate has no effect on truss elements.
8. To eliminate the effect of the strain rate, you can either set the value of c equal to 0 or the reference strain rate ( ${\stackrel{˙}{\epsilon }}_{0}$ ) can be set equal to 1030. There is no effect of strain rate when $\stackrel{˙}{\epsilon }$ is less than ${\stackrel{˙}{\epsilon }}_{0}$ .
9. 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.
10. There is no effect of temperature on trusses and beams.
11. The temperature is constant ( $T={T}_{r}$ ), if $\rho {C}_{p}=0$ .
12. Adiabatic conditions are assumed for thermal simulations with initial temperature equal to reference temperature (Tr) and:(3)
$T={T}_{r}+\frac{{E}_{\mathrm{int}}}{\rho {C}_{p}\left(Volume\right)}$

Where, Eint is the internal deformation energy.

13. The strain rate coefficient, c and reference strain rate ${\stackrel{˙}{\epsilon }}_{0}$ must be defined to include thermal effects.
14. When /HEAT/MAT (with Iform=1) references this material model, the values of Tr and $\rho {C}_{p}$ defined in this card will be overwritten by the corresponding ${T}_{0}$ and $\rho {\text{ }}_{0}{C}_{p}$ defined in /HEAT/MAT.
15. When the temperature is not initialized using /HEAT/MAT or /INITEMP, the reference temperature (Tr) is also the initial temperature.
16. The hardening coefficient is used to describe the hardening model (during unloading). The values of the hardening coefficient should be between 0 and 1.