Law name : GTNB
3D elasto-plastic constitutive combining isotropic and kinematic hardening, anisotropic yield locus and nucleation, growth and coalescence of voids. Extended to shear loads. Applied on porous ductile materials.
Prepro: LGUR3.F
Lagamine: GUR2DEXT.F, GUR3DEXT.F
| Plane stress state | NO |
| Plane strain state | NO |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 364 (3D) and 365 (axi) |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (7I5) | |
|---|---|
| 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) | |
| = 2 : Analytical tangent matrix | |
| NTYPHP | Type of hardening |
| = 1 : Swift law : $\sigma_Y = K(\varepsilon_0+\varepsilon_M^P)^n$ | |
| = 2 : Voce law : $\sigma_Y = \sigma_0 + K[1-\exp(-n.\varepsilon_M^P)]$ | |
| = 3 : Ludwik : $\sigma_Y = \sigma_0 + K(\varepsilon_M^P)^n$ (not working) | |
| NTYPEX | = 0 : Classic Gurson (only void growth) |
| = 1 : GTN model (Nucleation+growth+coalescence) | |
| = 2 : Shear (Nahshon and Hutchinson, 2008) | |
| = 3 : GTN+Shear (Nahshon and Hutchinson, 2008) | |
| = 4 : GTN+Shear (Xue, 2008) | |
| NTYCOA | = 1 : Coalescence criterion (Tvergaard & Needemen) |
| = 2 : Coalescence criterion Thomason (Zhang et al. 2000) | |
| Line 1 (2G10.0) | ||
|---|---|---|
| E | YOUNG's elastic modulus | = param(1,ilaw) |
| ANU | POISSON's ratio | = param(2,ilaw) |
| Line 2 (3G10.0) | ||
| CK | Coefficient of the hardening law ($K$) | = param(3,ilaw) |
| CW | Hardening coefficient ($\varepsilon_0$ or $\sigma_0$) | = param(5,ilaw) |
| CN | Strain hardening exponent ($n$) | = param(4,ilaw) |
| Line 3 (2G10.0) | ||
| CX | Parameter of the kinematic hardening ($C_X$) | = param(6,ilaw) |
| XSAT | Parameter of the kinematic hardening ($X_{sat}) \[\dot{\underline{X}} = C_X\left(X_{sat}\;\dot{\underline{\varepsilon}}^p-\underline{X}\;\dot{\bar{\varepsilon}}^p\right)\] | = param(7,ilaw) |
| Line 4 (6G10.0) | ||
| F | Hill's coefficients | = param(8,ilaw) |
| G | = param(9,ilaw) | |
| H | = param(10,ilaw) | |
| L | = param(11,ilaw) | |
| M | = param(12,ilaw) | |
| N | = param(13,ilaw) | |
| Line 5 (7G10.0) | ||
| QUN | Damage parameter ($q_1$) | = param(15,ilaw) |
| QDEUX | Damage parameter ($q_2$) | = param(16,ilaw) |
| QTR | Damage parameter ($q_3$) | = param(17,ilaw) |
| F0 | Initial porosity ($f_0$) | = param(18,ilaw) |
| If NTYPEX=2,3 (Nahshon and Hutchinson, 2008) | ||
| KOMEGA | Shear parameter ($k_{\omega}$) | = param(19,ilaw) |
| TR1 | Shear parameter ($T_1$) | = param(26,ilaw) |
| TR2 | Shear parameter ($T_2$) | = param(27,ilaw) |
| If NTYPEX=4 (Xue, 2008) | ||
| QQ3 | Shear parameter ($k_g$) | = param(19,ilaw) |
| QQ4 | Shear parameter ($q_4$) \[\dot{D}_{shear} = k_g\;f^{q_4}\;g_{\theta}\;\varepsilon_{eq}\;\dot{\varepsilon}_{eq}\] | = param(20,ilaw) |
| If NTYPEX=1,3,4 | ||
| Line 6 (5G10.0) | ||
| FNUC | Nucleation parameter ($f_N$) | = param(21,ilaw) |
| SNUC | Nucleation parameter ($S_N$) | = param(22,ilaw) |
| ENUC | Nucleation parameter ($\varepsilon_N$) | = param(23,ilaw) |
| FCRIT | Coalescence parameter ($f_c$) | = param(24,ilaw) |
| FFAIL | Coalescence parameter ($f_U$) | = param(25,ilaw) |
Note : If FCRIT=FFAIL=0, there is no coalescence.
4 (axi)
6 (3D)
GUR3DEXT :
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) | $\sigma_{xz}$ |
GUR2DEXT :
| SIG(1) | $\sigma_{xx}$ |
| SIG(2) | $\sigma_{yy}$ |
| SIG(3) | $\sigma_{zz}$ |
| SIG(4) | $\sigma_{xy}$ |
27 (axi)
30 (3D)
GUR3DEXT :
| Q(1) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(2) | Equivalent plastic strain in the matrix |
| (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 $f^*$) void volume fraction |
| Q(16) | $T$ : triaxiality (without backstress) |
| Q(17) | If NTYPEX=2,3 (Nahshon and Hutchinson, 2008) : = $\omega$ : Lode parameter corrected (Nielsen and Tvergaard, 2010) |
| If NTYPEX=4 (Xue, 2008) : = $g_{\theta}$ : Lode parameter |
|
| Q(18) | = $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
| Q(19) | $q$ : Effective eq. macroscopic stress |
| Q(20) | $p$ : Effective hydrostatic stress |
| Q(21) | $f$ : Void volume fraction |
| Q(22) | Porosity (Nucleation contribution) |
| Q(23) | Porosity (Growth contribution) |
| Q(24) | Porosity (Shear contribution) |
| Q(25) | $D$ : Damage variable |
| Q(26) | $\mu_{\sigma}$ : Lode parameter (Lode, 1926) |
| Q(27) | $X$ : Lode parameter (Wierzbicki et al., 2005) |
| Q(28) | $\bar{\theta}$ : Lode parameter (Bai and Wierzbicki, 2008) |
| Q(29) | $\omega$ : Lode parameter (Nahshon and Hutchinson, 2008) |
| Q(30) | $\theta_V$ : Lode parameter (Voyiadjis, 2012) |
| Q(31) | $\beta_{coal}$ if (.GE.0) then coalescence |
| Q(32) | Fcr computed by Thomason criterion |
GUR2DEXT :
| Q(2) | = 0 : Current state is elastic |
| = 1 : Current state is elasto-plastic | |
| Q(3) | Equivalent plastic strain in the matrix |
| (not $\varepsilon_{eqa}^p$, which is the equivalent plastic strain of the macroscopic medium) | |
| 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) | $f$ : (Effective $f^*$) void volume fraction |
| Q(13) | $T$ : triaxiality (without backstress) |
| Q(14) | If NTYPEX=2,3 (Nahshon and Hutchinson, 2008) : = $\omega$ : Lode parameter corrected (Nielsen and Tvergaard, 2010) |
| If NTYPEX=4 (Xue, 2008) : = $g_{\theta}$ : Lode parameter |
|
| Q(15) | $\varepsilon^p_{eqa}$ : Equivalent macroscopic plastic strain |
| Q(16) | $q$ : Effective eq. macroscopic stress |
| Q(17) | $p$ : Effective hydrostatic stress |
| Q(18) | $f$ : Void volume fraction |
| Q(19) | Porosity (Nucleation contribution) |
| Q(20) | Porosity (Growth contribution) |
| Q(21) | Porosity (Shear contribution) |
| Q(22) | $D$ : Damage variable |
| Q(23) | $\mu_{\sigma}$ : Lode parameter (Lode, 1926) |
| Q(24) | $X$ : Lode parameter (Wierzbicki et al., 2005) |
| Q(25) | $\bar{\theta}$ : Lode parameter (Bai and Wierzbicki, 2008) |
| Q(26) | $\omega$ : Lode parameter (Nahshon and Hutchinson, 2008) |
| Q(27) | $\theta_V$ : Lode parameter (Voyiadjis, 2012) |