/MAT/LAW104 (JOHNS_VOCE_DRUCKER)

Block Format Keyword An elasto-plastic constitutive material law using the 6th order Drucker model with a mixed Voce and linear hardening.

Dependence on the Johnson-Cook strain rate and thermal softening effects due to self-heating 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 $v$ Ires
${\sigma }_{yld}^{0}$ H Q B CDR
CJC ${\stackrel{˙}{\epsilon }}_{0}$ Fcut
$\mu$ Tref Tini
$\eta$ Cp ${\stackrel{˙}{\epsilon }}_{iso}$ ${\stackrel{˙}{\epsilon }}_{ad}$

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)

Ires Resolution method for plasticity.
= 0
Set to 1.
= 1 (Default)
NICE (Next Increment Correct Error) explicit method.
= 2
Newton iterative implicit method.

(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)

CDR Drucker coefficient.

(Real)

CJC Johnson-Cook strain rate coefficient.

(Real)

${\stackrel{˙}{\epsilon }}_{0}$ 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)

$\left[\frac{\text{1}}{\text{K}}\right]$
Tref Reference temperature at which the hardening law was identified in experiment.

(Real)

$\left[\text{K}\right]$
Tini Initial temperature of material in simulation.

(Real)

$\left[\text{K}\right]$
$\eta$ Taylor-Quinney coefficient.

(Real)

Cp Specific heat.

(Real)

$\left[\frac{\text{J}}{\text{kg}\cdot \text{K}}\right]$ .
${\stackrel{˙}{\epsilon }}_{iso}$ Plastic strain rate at isothermic conditions.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
${\stackrel{˙}{\epsilon }}_{ad}$ Plastic strain rate at adiabatic conditions.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$

1. The law uses 6th order Drucker equivalent stress definition:

Where, ${J}_{2}$ , ${J}_{3}$ are respectively the second and third invariant of the deviatoric stress tensor .

The parameter is user-defined and allows to define several yield surfaces (Figure 1). To respect the convexity, its value must respect -27/8 ≤ CDR ≤ 2.25.
2. The yield function is defined as:

and

Where,
${\sigma }_{yld}^{0}$
Initial yield stress.
H
Linear hardening.
$Q,B$
Voce hardening parameters.
${C}_{JC}$
Johnson-Cook strain rate coefficient.
${\stackrel{˙}{\epsilon }}_{f}$
Filtered plastic strain-rate.
Refer to Filtering in the User Guide.
${\stackrel{˙}{\epsilon }}_{0}$
Inviscid limit plastic strain rate.
$\mu$
Thermal softening slope.
The evolution of this flow stress equation with plasticity.
3. If /HEAT/MAT is not used for this material, the temperature is calculated internally using the incremental formula:
Where,
Plastic work increment.
$\eta$
Taylor-Quinney coefficient that must respect .
$\omega \left({\stackrel{˙}{\epsilon }}_{p}\right)$
Coefficient that defines the transition between isothermal and adiabatic conditions (Figure 3).