Elasto-plastic constitutive law for porous ductile metals at constant temperature, using the GURSON model.
This law is used for mechanical analysis of elasto-plastic isotropic porous ductile solids undergoing large strains. Combined isotropic and kinematic hardening is assumed.
Prepro: LGUR.F
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 80 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (2I5) | |
|---|---|
| NINTV | = 0 : NINTV will be calculated in the law |
| > 0 : Number of sub-steps used to integrate numerically the constitutive equation in a time step | |
| ICBIF | = 0 : Nothing |
| = 1 : RICE bifurcation criterion | |
| Line 1 (6G10.0) | |
|---|---|
| E | YOUNG's elastic modulus |
| ANU | POISSON's ratio |
| SIGY1 | Initial yield limit ($\sigma_{y1}$) |
| BB | Parameter of hardening (0$\leq$BB$\leq$1) |
| = 0 : Pure kinematic hardening | |
| = 1 : Pure isotropic hardening | |
| DN | > 0 : Strain hardening exponent ($n$) for piecewise power law |
| $\leq$ 0 : Plastic tangent modulus ($-E_t$) for bi-linear law | |
| FO | Initial void volume fraction |
| Line 2 (7G10.0) | |
| FC | Parameter of material ($f_C$) |
| FF | Parameter of material ($f_F$) |
| RFS | Parameter of material ($f_F^*/f_F$) |
| $Q_{UN}$ | Parameter of material ($Q_1$) |
| $Q_{DEUX}$ | Parameter of material ($Q_2$) |
| $Q_{TR}$ | Parameter of material ($Q_3$) |
| $\alpha_n$ | Parameter for calculation of NINTV (if NINTV=0 and $\alpha_n$=0 : its value will be $1.0\times10^{-4}$) |
| Line 3 (7G10.0) | |
| EPN | Mean strain for nucleation ($\varepsilon_N$) |
| SPN | Mean stress for nucleation ($\sigma_N$) |
| FN | Volume fraction of void nucleating particles ($f_N$) |
| SEP | Corresponding standard deviation for strain ($S_{\varepsilon}$) |
| SSIG | Corresponding standard deviation for stress ($S_{\sigma}$) |
| DETEP | Failure parameter of material ($\Delta_{\varepsilon}$) |
| DNCEPS | Central parameter of material for nucleation and failure |
| = 0 : Nucleation of new voids and failure of material are not considered | |
| = $\pm$1 : Nucleation is controlled by the plastic strain | |
| = $\pm$2 : Nucleation is controlled by the maximum normal stress | |
| = $\pm$3 : Both strain controlled and stress controlled nucleation take place | |
| = -4 : Nucleation of new voids is not considered | |
| > 0 : Failure of material is not considered | |
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{xy}$ |
| SIG(4) | $\sigma_{zz}$ |
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| Q(1) | Element thickness (t) in plane stress state |
| = 1 : Plane strain state | |
| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state | |
| = 0 : 3-D state | |
| Element thickness (t) in generalized plane state | |
| Q(2) | Current yield limit in tension (its initial value is $\sigma_{y1}$) |
| Q(3) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(4) | = $\left(\varepsilon_M^p\right)_{max}$ : Current maximum value of equivalent plastic strain of matrix material (its initial value is $\varepsilon_N$) |
| Q(5) | Current maximum value of $\sigma_m+\sigma_K^K/3$ (its initial value is $f_N$) |
| Q(6) | Current void volume fraction (f) (its initial value is $f_o$) |
| Q(7) | Equivalent plastic strain of matrix material ($\varepsilon_M^P$) |
| Q(8) | A |
| Q(9) | B |
| Q(10) | C |
| Q(11) | Yield surface centre for $\sigma_{xx}$ (its initial value is 0) |
| Q(12) | Idem for $\sigma_{yy}$ |
| Q(13) | Idem for $\sigma_{zz}$ |
| Q(14) | Idem for $\sigma_{xy}$ |
| Q(15)$\rightarrow$Q(26) | Modules for the analysis of bifurcation |