/MAT/LAW104 (JOHNS_VOCE_DRUCKER)
Block Format Keyword An elastoplastic constitutive material law using the 6^{th} order Drucker model with a mixed Voce and linear hardening.
Dependence on the JohnsonCook strain rate and thermal softening effects due to selfheating can also be modeled. The law is available for isotropic shell and solid elements.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW104/mat_ID/unit_ID or /MAT/JOHNS_VOCE_DRUCKER/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $v$  I_{res}  
${\sigma}_{yld}^{0}$  H  Q  B  C_{DR}  
C_{JC}  ${\dot{\epsilon}}_{0}$  Fcut  
$\mu $  T_{ref}  T_{ini}  
$\eta $  C_{p}  ${\dot{\epsilon}}_{iso}$  ${\dot{\epsilon}}_{ad}$ 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  (Optional) 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) 
$\left[\text{Pa}\right]$ 
$v$  Poisson’s
ratio. (Real) 

I_{res}  Resolution method for plasticity.
(Integer) 

${\sigma}_{yld}^{0}$  Initial yield
stress. (Real) 
$\left[\text{Pa}\right]$ 
H  Linear hardening
module. (Real) 
$\left[\text{Pa}\right]$ 
Q  Voce hardening
coefficient. (Real) 
$\left[\text{Pa}\right]$ 
B  Voce hardening
exponent. (Real) 

C_{DR}  Drucker
coefficient. (Real) 

C_{JC}  JohnsonCook strain rate
coefficient. (Real) 

${\dot{\epsilon}}_{0}$  Inviscid limit for the plastic
strain rate. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Fcut  Plastic strain rate filtering
frequency. Default = 10 kHz (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
$\mu $  Temperature softening
slope. (Real) 
$\left[\frac{\text{1}}{\text{K}}\right]$ 
T_{ref}  Reference temperature at which the
hardening law was identified in experiment. (Real) 
$\left[\text{K}\right]$ 
T_{ini}  Initial temperature of material in
simulation. (Real) 
$\left[\text{K}\right]$ 
$\eta $  TaylorQuinney
coefficient. (Real) 

C_{p}  Specific heat. (Real) 
$\left[\frac{\text{J}}{\text{kg}\cdot \text{K}}\right]$ . 
${\dot{\epsilon}}_{iso}$  Plastic strain rate at isothermic
conditions. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
${\dot{\epsilon}}_{ad}$  Plastic strain rate at adiabatic
conditions. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Comments
 The law uses 6^{th}
order Drucker equivalent stress definition: $${\sigma}_{eq}=k{\left({J}_{2}^{3}{C}_{DR}{J}_{3}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$6$}\right.}$$
Where, ${J}_{2}$ , ${J}_{3}$ are respectively the second and third invariant of the deviatoric stress tensor $k={\left(\frac{1}{27}{C}_{DR}\frac{4}{{27}^{2}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$6$}\right.}$ .
The parameter is userdefined and allows to define several yield surfaces (Figure 1). To respect the convexity, its value must respect 27/8 ≤ C_{DR} ≤ 2.25.  The yield function is
defined as:$$\varphi =\frac{{\sigma}_{eq}^{2}}{{\sigma}_{yld}^{2}}1=0$$
and
$${\sigma}_{yld}=\left({\sigma}_{yld}^{0}+H{\epsilon}_{p}+Q\left(1{e}^{B{\epsilon}_{p}}\right)\right)\left(1+{C}_{JC}\text{ln}\left(\frac{{\dot{\epsilon}}_{f}}{{\dot{\epsilon}}_{0}}\right)\right)\left(1\mu \left(\text{T}{T}_{ref}\right)\right)$$Where, ${\sigma}_{yld}^{0}$
 Initial yield stress.
 H
 Linear hardening.
 $Q,B$
 Voce hardening parameters.
 ${C}_{JC}$
 JohnsonCook strain rate coefficient.
 ${\dot{\epsilon}}_{f}$
 Filtered plastic strainrate.
 ${\dot{\epsilon}}_{0}$
 Inviscid limit plastic strain rate.
 $\mu $
 Thermal softening slope.
The evolution of this flow stress equation with plasticity.  If
/HEAT/MAT is not used for this material, the
temperature is calculated internally using the incremental
formula:$$dT=\omega \left({\dot{\epsilon}}_{p}\right)\frac{\eta}{{C}_{p}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}d{W}_{p}$$Where,
 $d{W}_{p}$
 Plastic work increment.
 $\eta $
 TaylorQuinney coefficient that must respect $0\le \eta \le 1$ .
 $\omega \left({\dot{\epsilon}}_{p}\right)$
 Coefficient that defines the transition between isothermal and adiabatic conditions (Figure 3).
$$\omega \left(\dot{{\epsilon}_{p}}\right)=\frac{{\left({\dot{\epsilon}}_{p}{\dot{\epsilon}}_{iso}\right)}^{2}\left(3{\dot{\epsilon}}_{ad}2{\dot{\epsilon}}_{p}{\dot{\epsilon}}_{iso}\right)}{{\left({\dot{\epsilon}}_{ad}{\dot{\epsilon}}_{iso}\right)}^{3}}$$