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

(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)

[JkgK] .
˙εiso Plastic strain rate at isothermic conditions.

(Real)

[1s]
˙εad Plastic strain rate at adiabatic conditions.

(Real)

[1s]

Comments

  1. The law uses 6th order Drucker equivalent stress definition:
    σeq=k (J32CDR J23)16

    Where, J2 , J3 are respectively the second and third invariant of the deviatoric stress tensor k= (127CDR 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
  2. The yield function is defined as:
    ϕ= σ2eqσ2yld1=0

    and

    σyld=(σ0yld+Hεp+Q(1eB εp))(1+CJCln(˙εf˙ε0))(1μ(TTref))

    Where,
    σ0yld
    Initial yield stress.
    H
    Linear hardening.
    Q,B
    Voce hardening parameters.
    CJC
    Johnson-Cook strain rate coefficient.
    ˙εf
    Filtered plastic strain-rate.
    Refer to Filtering in the User Guide.
    ˙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
  3. If /HEAT/MAT is not used for this material, the temperature is calculated internally using the incremental formula:
    dT= ω(˙εp)η CpdWp
    Where,
    dWp 
    Plastic work increment.
    η
    Taylor-Quinney coefficient that must respect 0η 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)3


    Figure 3. Evolution of the temperature weight with the plastic strain rate