3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus, nucleation and growth of voids. Rupture criterion applied on porous ductile materials (GURSON model).
Prepro: LGUR3.F
Lagamine: GUR3DANI/GUR2DANI.F
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES (GUR2DANI) |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 358 (axi) and 361 (3D) |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (I5) | |
|---|---|
| NINTV | = 0 : Not used |
| IKAP | = 0 : Tangent matrix by perturbation (through LOAX3D) |
| = 1 : Tangent matrix by perturbation (calculated within the law) | |
| NTYPHP | Type of hardening |
| = 1 : Swift law : $\sigma_Y = K(\varepsilon_0+\bar{\varepsilon}^p)^n$ | |
| = 2 : Voce law : $\sigma_Y = \sigma_0 + K[1-\exp(-n.\bar{\varepsilon}^p)]$ | |
| = 3 : Ludwik : $\sigma_Y = \sigma_0 + K(\bar{\varepsilon}^p)^n$ | |
| Line 1 (2G10.0/3G10.0/2G10.0/3G10.0/7G10.0/2G10.0) | |
|---|---|
| E | YOUNG's elastic modulus |
| ANU | POISSON's ratio |
| Line 2 (3G10.0) | |
| SIGO | Coefficient of the hardening law ($K$) |
| DN | Strain hardening exponent ($n$) |
| EPS0 | Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) |
| Line 3 (2G10.0) | |
| HKIN | First parameter of the kinematic hardening ($C_X.X_{sat}$) |
| HNL | Second parameter of the kinematic hardening ($C_X$) : \[\dot{\underline{X}} = C_X\left(s\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\bar{\dot{\varepsilon}}^p\right)\] |
| Line 4 (3G10.0) | |
| R0 | Lankford coefficient in the direction 0° |
| R45 | Lankford coefficient in the direction 45° |
| R90 | Lankford coefficient in the direction 90° |
| Line 5 (7G10.0) | |
| QUN | Damage parameter ($q_1$) |
| QDEUX | Damage parameter ($q_2$) |
| QTR | Damage parameter ($q_3$) |
| F0 | Initial porosity |
| EPSN0 | Related to Bouaziz equation of void nucleation : $\varepsilon_n=\varepsilon_{n0}\;\exp(-T)$ |
| AA0 | Constant equal to 5000 voids/mm$^3$ for calculation of N (number of nucleated voids per mm$^3$) |
| RR0 | Initial radius of a singular void |
| Line 6 (2G10.0) | |
| AB | Thomason parameter (not used) |
| BETA | Thomason parameter (not used) |
4 (axi)
6 (3D)
The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.
GUR3DANI :
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
| SIG(5) | $\sigma_{yz}$ |
| SIG(6) | $\sigma_{xz}$ |
GUR2DANI :
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
22 (axi)
25 (3D)
GUR3DCLAS :
| Q(1) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(2) | Equivalent plastic strain in the matrix $\sigma_Y = K(\varepsilon_0+\varepsilon_m^p)^n$ |
| (not $\varepsilon_{eqa}^p$, which is the equivalent plastic strain of the macroscopic medium) | |
| Q(3)$\rightarrow$Q(8) | The six components of the macroscopic plastic strain : $\underline{\varepsilon}_{11}^p$, $\underline{\varepsilon}_{22}^p$, $\underline{\varepsilon}_{33}^p$, $\underline{\varepsilon}_{12}^p$, $\underline{\varepsilon}_{13}^p$, $\underline{\varepsilon}_{23}^p$ |
| Q(9)$\rightarrow$Q(14) | The six components of the macroscopic backstress : $X_{11}$, $X_{22}$, $X_{33}$, $X_{12}$, $X_{13}$, $X_{23}$ |
| Q(15) | f : the void porosity fraction |
| Q(16) | T: triaxiality |
| Q(17) | The equivalent macroscopic plastic strain |
| Q(18) | = N : Number of nucleated voids |
| Q(19) | The volume of the equivalent single void |
| Q(20) | The updated void radius computed by the integration equation |
| Q(21) | $q_2$ |
| Q(22) | $q_1=1.5\;q_2$ |
| Q(23) | $q_3=(1.5\;q_2)^2$ |
| Q(24) | ln=(RT/RT0) where RT and RT0 are are the current and initial radius of the single equivalent porosity |
| Q(25) | RT radius of the single void cavity at the end of the time increment |
Qtrial : anisotropic equivalent shifted stress with HILL criterion calculation.
GUR2ACLAS :
| Q(2) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(3) | Equivalent plastic strain in the matrix $\sigma_Y = K(\varepsilon_0+\varepsilon_m^p)^n$ |
| Q(4)$\rightarrow$Q(7) | The four components of the macroscopic plastic strain : $\underline{\varepsilon}_{11}^p$, $\underline{\varepsilon}_{22}^p$, $\underline{\varepsilon}_{33}^p$, $\underline{\varepsilon}_{12}^p$ |
| Q(8)$\rightarrow$Q(11) | The four components of the macroscopic backstress : $X_{11}$, $X_{22}$, $X_{33}$, $X_{12}$ |
| Q(12) | f : the void porosity fraction |
| Q(13) | T: triaxiality |
| Q(14) | The equivalent macroscopic plastic strain |
| Q(15) | N : Number of nucleated voids |
| Q(16) | The volume of the equivalent single void |
| Q(17) | The updated void radius computed by the integration equation |
| Q(18) | $q_2$ |
| Q(19) | $q_1=1.5\;q_2$ |
| Q(20) | $q_3=(1.5\;q_2)^2$ |
| Q(21) | ln=(RT/RT0). where RT and RT0 are are the current and initial radius of the single equivalent porosity |
| Q(22) | RT radius of the single void cavity at the end of the time increment |
Qtrial : anisotropic equivalent shifted stress with HILL criterion calculation.