/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) |
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/MAT/LAW104/mat_ID/unit_ID or /MAT/JOHNS_VOCE_DRUCKER/mat_ID/unit_ID | |||||||||
mat_title | |||||||||
ρi | |||||||||
E | v | Ires | |||||||
σ0yld | H | Q | B | CDR | |||||
CJC | ˙ε0 | Fcut | |||||||
μ | Tref | Tini | |||||||
η | Cp | ˙εiso | ˙εad |
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) |
|
ρi | Initial
density. (Real) |
[kgm3] |
E | Young‘s
modulus. (Real) |
[Pa] |
v | Poisson’s
ratio. (Real) |
|
Ires | Resolution method for plasticity.
(Integer) |
|
σ0yld | Initial yield
stress. (Real) |
[Pa] |
H | Linear hardening
module. (Real) |
[Pa] |
Q | Voce hardening
coefficient. (Real) |
[Pa] |
B | Voce hardening
exponent. (Real) |
|
CDR | Drucker
coefficient. (Real) |
|
CJC | Johnson-Cook strain rate
coefficient. (Real) |
|
˙ε0 | Inviscid limit for the plastic
strain rate. (Real) |
[1s] |
Fcut | Plastic strain rate filtering
frequency. Default = 10 kHz (Real) |
[1s] |
μ | Temperature softening
slope. (Real) |
[1K] |
Tref | Reference temperature at which the
hardening law was identified in experiment. (Real) |
[K] |
Tini | Initial temperature of material in
simulation. (Real) |
[K] |
η | Taylor-Quinney
coefficient. (Real) |
|
Cp | Specific heat. (Real) |
[Jkg⋅K] . |
˙εiso | Plastic strain rate at isothermic
conditions. (Real) |
[1s] |
˙εad | Plastic strain rate at adiabatic
conditions. (Real) |
[1s] |
Comments
- The law uses 6th
order Drucker equivalent stress definition: σeq=k (J32−CDR J23)16
Where, J2 , J3 are respectively the second and third invariant of the deviatoric stress tensor k= (127−CDR 4272)−16 .
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.Figure 1. Drucker yield surfaces - The yield function is
defined as:ϕ= σ2eqσ2yld−1=0
and
σyld=(σ0yld+Hεp+Q(1−e−B εp))(1+CJCln(˙εf˙ε0))(1−μ(T−Tref))Where,- σ0yld
- Initial yield stress.
- H
- Linear hardening.
- Q,B
- Voce hardening parameters.
- CJC
- Johnson-Cook strain rate coefficient.
- ˙εf
- Filtered plastic strain-rate.
- ˙ε0
- Inviscid limit plastic strain rate.
- μ
- Thermal softening slope.
The evolution of this flow stress equation with plasticity.Figure 2. Flow stress evolution with plasticity - If
/HEAT/MAT is not used for this material, the
temperature is calculated internally using the incremental
formula:dT= ω(˙εp)η Cp dWpWhere,
- dWp
- Plastic work increment.
- η
- Taylor-Quinney coefficient that must respect 0≤η ≤1 .
- ω(˙εp)
- Coefficient that defines the transition between isothermal and adiabatic conditions (Figure 3).
ω(˙εp)= (˙εp− ˙εiso)2(3˙εad −2˙εp−˙εiso)(˙εad− ˙εiso)3Figure 3. Evolution of the temperature weight with the plastic strain rate