Table of Contents
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 | = -2 Law described by parameters (SWIFT) SWIFT law: SIG = AKP*(EPSP + EPS0)∗∗ANP |
| = -1 Law described by parameters (LUDWIK) LUDWIK law: SIG = SIGY + AKP*(EPSP∗∗ANP) |
|
| = 0 Law described by parameters (VOCE) VOCE law : SIG = SIGY + AKP*(1-EXP(-ANP*EPSP)) |
|
| > 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 (8G10.0) | |
|---|---|
| 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 |
| = 1 For kinematic hardening | |
| = [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 (3G10.0) | |
| POND | = Weight of volumetric energy |
| (= 0 by default) | |
| DLIM | = Coalescence limit |
| (= 1 by default) | |
| FMULP | = Slope multiplicator |
| (= 1 by default) | |
| Line 3 (2G10.0) - Repeated NPOINT times (if NPOINT>0) | |
| EPS | = Strain for virgin by uniaxial testing |
| SIG | = Effective stress related to measured stress in tensile state |
 1) Effective stress-strain curve with hardening and softening phenomenon 2) The first point must be : $\sigma_{y}$ = The initial yiled limit $\epsilon_{e}$ = $\sigma_{y}$ / E |
|
| Line 3 + NPOINT (2G10.0) - Repeated NPOINT times (if NPOINT>0) | |
| DSHEAR | = Deviatoric damage variable |
| B | = Damage strenghthening force (Mpa) |
| 1) Possible for hardening and softening curve 2) The first point must be : $d$ = 0 $B_{0}$ = The initial damage limit |
|
If NPOINT= -2
| Line 3 (6G10.0) | |
|---|---|
| E | = YOUNG's elastic modulus |
| EPS0 | = EPS0 |
| AKP | = SWIFT hardening coefficient |
| ANP | = SWIFT hardening exponent |
| B0 | = Initial damage limit |
| DTG | = Damage tangent modulus |
If NPOINT= -1
| Line 3 (6G10.0) | |
|---|---|
| E | = YOUNG's elastic modulus |
| SIGY | = Lower yield limit |
| AKP | = LUDWIK hardening coefficient |
| ANP | = LUDWIK hardening exponent |
| B0 | = Initial damage limit |
| DTG | = Damage tangent modulus |
If NPOINT= 0
| 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 |