3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus, nucleation growth and coalescence of voids. Physically-based models. Applied on porous ductile materials.
Prepro: LGUR3.F
Lagamine: GTNB3DPHY.F, GUR3DBF.F
| Plane stress state | NO |
| Plane strain state | NO |
| Axisymmetric state | NO |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 363 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (8I5) | |
|---|---|
| NINTV | Number of sub-steps used to integrate numerically the constitutive equation in a time step |
| NINTEPS | Number of sub-intervals per unit of delta epsilon |
| Number of sub-steps = MAX(NINTV; NINTEPS*DELTA EPSILON) | |
| MAXIT | Maximum number of iterations in the N-R scheme |
| 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 : $\sigma_Y = \sigma_0 + K(\bar{\varepsilon}^p)^n$ | |
| NTYCOA | Void nucleation model |
| = 1 : Classic GTN | |
| = 2 : Bouaziz, Maire approach | |
| = 3 : Landron approach | |
| NTYGRO | Void growth model |
| = 1 : Classic GTN | |
| = 2 : Bouaziz, Maire approach | |
| NTYCOA | Void coalescence model |
| = 1 : Classic GTN | |
| = 2 : Brown and Embury (not ready) | |
| = 3 : Thomason | |
| Line 1 (2G10.0) | |
|---|---|
| E | YOUNG's elastic modulus |
| ANU | POISSON's ratio |
| Line 2 (3G10.0) | |
| CK | Coefficient of the hardening law ($K$) |
| CN | Strain hardening exponent ($n$) |
| CW | Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) |
| Line 3 (2G10.0) | |
| CX | First parameter of the kinematic hardening ($C_X.X_{sat}$) |
| XSAT | Second parameter of the kinematic hardening ($C_X$) : \[\dot{\underline{X}} = C_X\left(X_{sat}\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\bar{\dot{\varepsilon}}^p\right)\] |
| 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 (4G10.0) | |
| QUN | Initial damage parameter ($q_1$) |
| QDEUX | Initial damage parameter ($q_2$) |
| QTR | Initial damage parameter ($q_3$) |
| F0 | Initial porosity ($f_0$) |
| Void nucleation parameters (3G10.0) | |
| If NTYNUC = 1 | |
| FNUC | Nucleation parameter ($f_N$) |
| SNUC | Nucleation parameter ($S_N$) |
| ENUC | Nucleation parameter ($\varepsilon_N$) |
| If NTYNUC = 2 | |
| AA0 | $A$ : Number of nucleated voids per mm$^3$ |
| EPSN0 | $\varepsilon_{N0}$ : Critical strain for pure shear loading |
| If NTYNUC = 3 | |
| BB0 | Material parameter ($B$) |
| NN0 | Initial number of voids per unit volume ($N_0$) |
| SCRITM | Critical shear stress ($\sigma_C$) |
| Void growth parameters (3G10.0) | |
| If NTYGRO = 2 or NTYGRO = 3 | |
| RRi0 | = $R_0^i$ : Initial mean radius |
| Void coalescence parameters (3G10.0) | |
| If NTYCOA = 1 | |
| FCRIT | Coalescence parameter ($f_c$) |
| FFAIL | Coalescence parameter ($f_U$) |
6
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) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
| SIG(5) | $\sigma_{yz}$ |
| SIG(6) | $\sigma_{xz}$ |
25
| 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) | $f^*$ : (Effective) void volume fraction |
| Q(16) | $T$ : triaxiality (without backstress) |
| Q(17) | $\omega$ : Lode angle (Nahshon and Hutchinson, 2008) (not used) |
| Q(18) | $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
| Q(19) | $q$ : Effective eq. macroscopic stress |
| Q(20) | $p$ : Hydrostatic stress |
| Q(21) | $f$ : Void volume fraction |
| Q(22) | $N$ : Number of nucleated voids |
| Q(23) | $f_N$ : Porosity (Nucleation contribution) |
| Q(24) | $R$ : The updated void radius |
| Q(25) | $\ln(R_T/T_{T0})$ where $R_T$ and $R_{T0}$ are are the current and initial radius of the single equivalent porosity |
| Q(26) | $R_T$ : Radius of the single void cavity at the end of the time increment |
| Q(27) | $q_2$ |
| Q(28) | $q_1=1.5\; q_2$ |
| Q(29) | $q_3 = q_1^2$ |
Qtrial : anisotropic equivalent shifted stress with HILL criterion calculation.