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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
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.
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
| (I5) | |
|---|---|
| NINTV | Number of sub-steps used to integrate numerically the constitutive equation in a time step |
| NPOINT | = -1 Law described by parameters (LUDWIK) |
| = 0 Law described by parameters (VOCE) | |
| > 0 Law described by points | |
| MININTV | Maximum number of sub-steps (0 –> 100) |
| MITERA | Number of sub-iteration (0 –> 10) |
| MUTIP | Number of multiplicator for sub-steps (0 –> 2) |
| IVISC | = 0 (EP LAW) |
| = 1 (EVP LAW) | |
| ICBIF | Bifurcation indice |
| ITRAC | = Number of groups of B(d) |
| = (1 For 2 groups (Traction, Compression)) | |
Real parameters
| Line 1 (6G10.0) | |
|---|---|
| ANU | POISSON's ratio |
| DNMAX | = 0 For EP without damage |
| = (0,1) –> Max. damage value at initial fracture | |
| otherwise –> 0.95 Limit damage value | |
| TAU1 | Ratio of volumetric damage to deviatoric damage in tensile state |
| TAU2 | Ratio of volumetric damage to deviatoric damage in compression state |
| ECROU | = 0 For isotropic hardening |
| ECROU | = 1 For kinematic hardening |
| ECROU | = [0,1] For mixed hardening |
| PROC | = Precision of iteration |
| (=0 –> 1.D-3) | |
| VISCO | = Viscosity parameter (unit: time) |
| THICK | = Thickness for plane state |
| Line 2 (6G10.0) | |
| POND | = Weight of volumetric energy |
| (= 0 by default) | |
| DLIM | = Coalescence limit |
| (= 1 by default) | |
| FMULP | = Slope multiplicator |
| (= 1 by default) | |
| Line 3 (6G10.0) (If NPOINT= -1) | |
| E | = YOUNG's elastic modulus |
| SIGY | = Lower yield limit |
| AKP | = LUDWIK hardening coefficient |
| ANP | = LUDWIK hardening exponent |
| LUDWIK law: SIG = SIGY + AKP*(EPSP**ANP) | |
| B0 | = Initial damage limit |
| DTG | = Damage tangent modulus |
| Line 3 (6G10.0) (If NPOINT= 0) | |
| E | = YOUNG's elastic modulus |
| SIGY | = Lower yield limit |
| AKP | = VOCE hardening coefficient |
| ANP | = VOCE hardening exponent |
| VOCE law: SIG = SIGY + AKP*(1-EXP(-ANP*EPSP)) | |
| B0 | = Initial damage limit |
| DTG | = Damage tangent modulus |
Reaped NPOINT times
Reaped NPOINT times
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) | $\sigma_{yy}$ |
| Q(3) | $\sigma_{zz}$ |
| Q(4) | $\sigma_{xy}$ |
| Q(5) | $\sigma_{xz}$ |
| Q(6) | $\sigma_{yz}$ |
| Q(7) | $\sigma_{xx}$ |
| Q(8) | $\sigma_{yy}$ |
| Q(9) | $\sigma_{zz}$ |
| Q(N) | $\sigma_{xy}$ |
| Q(N+1) | $\sigma_{xz}$ |
| Q(N+2) | $\sigma_{yz}$ |
| Q(N+3) | $\sigma_{xx}$ |
| Q(N+4) | $\sigma_{yy}$ |
| 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 |
