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ZDMG
Description
Elastic(-visco)-plastic constitutive law fully coupled with damage for solid elements at constatnt temperature.
Implemented by: Zhu Yongui, 1992
Improved by: Sylvie Castagne, 1997
Ehssen Betaieb, 2019
The model
The Lemaitre model is a fully coupled elastoplastic damage model based on energy equivalence. In this approach, damage is defined phenomenologically or experimentally instead of analytically or microscopically. The constitutive equations of the damaged material follow directly from thermodynamic considerations with two internal variables $d$ and $δ$.
Files
Prepro: LZDMG.F
Lagamine: ZDMG2A.F, ZDMG2E.F, ZDMG2S.F, ZDMG3D.F
Availability
| Plane stress state | YES |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
Input file
Parameters defining the type of constitutive law
| (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 225 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (8I5) | |
|---|---|
| NINTV | Number of sub-steps used to integrate numerically the constitutive equation in a time step |
| NPOINT | |
| Line 3 (6G10.0) | |
| E | = YOUNG's elastic modulus |
| SIGY | = Lower yield limit |
| AKP | = VOCE hardening coefficient |
| ANP | = VOCE hardening exponent |
| B0 | = Initial damage limit |
| DTG | = Damage tangent modulus |
Stresses
Number of stresses
6 for 3D state
4 for the other cases
Meaning
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
For the 3-D state:
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
| SIG(5) | $\sigma_{xz}$ |
| SIG(6) | $\sigma_{yz}$ |
For the other cases:
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{xy}$ |
| SIG(4) | $\sigma_{zz}$ |
State variables
Number of state variables
27 for 3D state
25 for the other cases
Meaning
| Q(1) | = Element thickness (t) in plane stress state |
| = 1 in plane strain state | |
| = Circumfrential strain rate ($\dot{\epsilon}_{θ}$) in axisymmetrical state | |
| = 0 in 3-D state | |
| = Element thickness (t) in generalized plane state | |
| Q(2) | = 0 If the current state is elastic |
| = 1 If the current state is elasto-plastic | |
| Q(3) | = 0 If the current state is not damage |
| = 1 If the current state is damage | |
| Q(4) | = Generalized plastic strain ($α$) |
| Q(5) | = Amount of current deviatoric damage ($d$) |
| Q(6) | = Amount of current volumetric damage ($δ$) |
| Q(7) | = Plastic hardening level ($R$) |
| Q(8) | = Damage hardening level ($B$) |
| Q(9) | = Back stresses for kinematic and mixed hardening |
| Q(N) | (N=14 for 3-D state, N=12 for other cases) |
| Q(N+1) | = Equivalent plastic strain |
| Q(N+2) | = Equivalent stress |
| Q(N+3) | = Thermodynamic reaction conjugated to deviatoric damage ($Y_{d}$) |
| Q(N+4) | = Thermodynamic reaction conjugated to volumetric damage ($Y_{δ}$) |
| Q(N+5) | = Plastic work per unit volume |
| Q(N+6) | = Damage work per unit volume |
| Q(N+7) | = Total strain energy per unit volume (elastic + plastic + damage) |
| Q(N+8) | = Fracture criteria |
