3D constitutive law with an orthotropic yield criterion for hexagonal closed packed materials
Mechanical analysis of elasto-plastic HCP materials undergoing large strains.
Prepro: LCAZACW.F
Lagamine: CAZACUW.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 | 524 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (3I5) | |
|---|---|
| NWP | Number of plastic work levels |
| NTRAN | Number of linear transformations |
| MAXIT | Maximal number of iterations during stress integration |
| Line 1 (6G10.0) | |
|---|---|
| $E_{1}$ | YOUNG's orthotropic elastic moduli |
| $E_{2}$ | |
| $E_{3}$ | |
| $\mbox{ANU}_{12}$ | Orthotropic POISSON's ratios |
| $\mbox{ANU}_{13}$ | |
| $\mbox{ANU}_{23}$ | |
| Line 2 (3G10.0) | |
| $G_{12}$ | COULOMB's orthotropic elastic moduli |
| $G_{13}$ | |
| $G_{23}$ | |
| Line 3 (G10.0) | |
| A | degree of homogeneity ,param(16, ilaw) |
The inverse of the orthotropic elastic matrix is defined:
\[\begin{pmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \varepsilon_{12} \\ \varepsilon_{13} \\ \varepsilon_{23} \end{pmatrix} = \begin{pmatrix} \frac{1}{E_{1}} & \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{13}}{E_{1}} & 0 & 0 & 0\\ \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{12}}{E_{2}} & \frac{1}{E_{2}} & 0 & 0 & 0\\ \frac{-\nu_{13}}{E_{1}} & \frac{-\nu_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{2G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2G_{23}} \end{pmatrix} \begin{pmatrix} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\\ \end{pmatrix}\]
FOR I = 1, NWP
| Line 1 (G10.0) | |
|---|---|
| WP | plastic work level, param(17+(I-1)*(10*NTRAN + 1),ilaw) |
FOR J = 1, NTRAN
| Line 1 (G10.0) | |
|---|---|
| ASYM | differential effect, param(18+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| Line 2 (3G10.0) anisotropy coefficients | |
| C11 | param(19+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C12 | param(20+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C13 | param(21+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| Line 3 (3G10.0) anisotropy coefficients | |
| C22 | param(22+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C23 | param(23+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C33 | param(24+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| Line 4 (3G10.0) anisotropy coefficients | |
| C44 | param(25+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C55 | param(26+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
| C66 | param(27+(I-1)*(10*NTRAN + 1) + (J-1)*10,ilaw) |
The yiel locus is defined: \[F=\Big[ \overset{NTRAN}{\underset{I=1}{\Sigma}}\overset{3}{\underset{J=1}{\Sigma}} \big(|\Sigma_{J}^{(I)}| - k^{(I)}\Sigma_{J}^{(I)}\big)^a \Big] ^{\frac{1}{a}} - \sigma_{F} = 0 \] Where:
| Line 1 (3G10.0) hardening parameters (Voce) | |
|---|---|
| R0 | initial yield stress (see hardening form), param(17+(10*NTRAN+1)*NWP,ilaw) |
| RSAT | saturation value (see hardening form) param(18+(10*NTRAN+1)*NWP,ilaw) |
| CR | saturation rate (see hardening form)), param(19+(10*NTRAN+1)*NWP,ilaw) |
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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) | $\sigma_{XZ}$ |
| SIG(6) | $\sigma_{YZ}$ |
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| Q(1) | Yield criterion = 0 : the previous step was elastic = 1: the previous step was elasto-plastic |
| Q(2) | Accumulated equivalent plastic strain |
| Q(3) | Accumulated plastic work |
| Q(4) | Pointer for PARAM vector |
| Q(5) | Triaxiality |
| Q(6) | Equivalent stress |
$\sigma_{F} = R_{0} + R_{SAT}(1-exp(-C_{R}\overline{\varepsilon}^{p}))$