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CAZACUTN
Description
3D Coupled damage law for porous hexagonal closed packed (HCP) materials exhibiting orthotropy and strength differential effect.
The model
The mathematical model was developed by (J. Stewart & O. Cazacu, 2011),following a Gurson-type approach where the material yield stress is determined by the CPB06 yield criterion. The damage is modeled in the form of porosity ratio, and its evolution is ruled by phenomenological models of growth, nucleation and coalescence of voids. This constitutive law also integrates an automatic definition of coalescence onset, throughout the implementation of the Thomason-Zhang coalescence extension.
The inverse of the orthotropic elastic matrix is defined:
\[\begin{pmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \varepsilon_{12} \\ \varepsilon_{13} \\ \varepsilon_{23} \end{pmatrix} = \begin{pmatrix} \frac{1}{E_{1}} & \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{13}}{E_{1}} & 0 & 0 & 0\\ \frac{-\nu_{12}}{E_{1}} & \frac{-\nu_{12}}{E_{2}} & \frac{1}{E_{2}} & 0 & 0 & 0\\ \frac{-\nu_{13}}{E_{1}} & \frac{-\nu_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{2G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2G_{23}} \end{pmatrix} \begin{pmatrix} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\\ \end{pmatrix}\]
The yield locus of this damage law is defined as:
\[\Phi= \bar{\Sigma}_{CPB06} - \sigma_{y}\cdot{STF} = 0 \]
Where:
- $\bar{\Sigma}_{CPB06}$ is the CPB06 yield stress:
\[\bar{\Sigma}_{CPB06}=\tilde{m}\Big[ \overset{3}{\underset{J=1}{\Sigma}} \big(|\Sigma_{J}| - k\Sigma_{J}\big)^a \Big] ^{\frac{1}{a}}\]
- $\sigma_{y}$ is the current yield stress of the material:
\[\sigma_{y} = \sigma_{0} + S_{R}\big[1-exp\big(-C_{R}\bar{\epsilon}^{p}\big)\big]\]
- STF is the stress transformation function, containing all the damage-related variables:
\[STF=2fq_{1}cosh\Big[\frac{3q_{2}\big(\sigma_{m} - X_{m}\big)}{h\sigma_{y}}\Big] - q_{3}f^{2} - 1\]
Files
Prepro: LCAZACUTN.F
Lagamine: CAZACUTN.F
Thomason-Zhang coalescence onset criterion: THZCOAL.F
Subroutines
| Files | Contained subroutines | Description |
|---|---|---|
| LCAZACUTN.F | LCAZACUTN | Main Prepro LCAZACUTN subroutine |
| CAZACUTN.F | CAZACUTN | Main Lagamine CAZACUTN subroutine |
| CAZACUTNFUN | Calculation of CAZACUTN yield locus | |
| THZCOAL.F | THOMASON_ZHANG | Calculation of coalescence criterion |
Availability
| Plane stress state | NO |
| Plane strain state | NO |
| Axisymmetric state | NO |
| 3D state | YES |
| Generalized plane state | NO |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 337 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (1I5) | |
|---|---|
| MAXIT | Maximal number of iterations during stress integration |
Real parameters
| Line 1 (6G10.0) | |
|---|---|
| $E_{1}$ | YOUNG's orthotropic elastic moduli |
| $E_{2}$ | |
| $E_{3}$ | |
| $\nu_{12}$ | Orthotropic POISSON's ratios |
| $\nu_{13}$ | |
| $\nu_{23}$ | |
| Line 2 (1G10.0) | |
| A | degree of homogeneity, param(16, ilaw) |
| Line 3 (1G10.0) | |
| k | Asymmetry parameter, param(17, ilaw) |
| Line 4 (3G10.0) Components of orthotropic constants tensor | |
| $C_{11}$ | param(18,ilaw) |
| $C_{12}$ | param(19,ilaw) |
| $C_{13}$ | param(20,ilaw) |
| Line 5 (3G10.0) Components of orthotropic constants tensor | |
| $C_{22}$ | param(21,ilaw) |
| $C_{23}$ | param(22,ilaw) |
| $C_{33}$ | param(23,ilaw) |
| Line 6 (3G10.0) Components of orthotropic constants tensor | |
| $C_{44}$ | param(24,ilaw) |
| $C_{55}$ | param(25,ilaw) |
| $C_{66}$ | param(26,ilaw) |
| Line 7 (4G10.0, 1I5) Damage control parameters | |
| $f_{0}$ | Initial porosity ratio, VARIN(21,ilaw) |
| $q_{1}$ | Tvergaard&Needleman parameter, param(27,ilaw) |
| $q_{2}$ | Tvergaard&Needleman parameter, param(28,ilaw) |
| $q_{3}$ | Tvergaard&Needleman parameter, param(29,ilaw) |
| IDAMAGE | Active damage mechanism ID, param(39,ilaw) |
| SELECT CASE (IDAMAGE) | |||
| CASE (0): No damage increment is calculated | |||
| Line 8 (3G10.0) Isotropic hardening law parameters | |||
| $\sigma_{0}$ | Initial Yield stress [MPa], param(30, ilaw) | ||
| $S_{R}$ | Saturation rate [MPa], param(31, ilaw) | ||
| $C_{R}$ | Saturation value [-], param(32, ilaw) | ||
| Line 9 (2G10.0) Kinematic hardening parameters | |||
| $S_{X}$ | Saturation rate [-], param(33, ilaw) | ||
| $C_{X}$ | Saturation value [MPa], param(34, ilaw) | ||
| CASE (1): Growth is the only active damage mechanism | |||
| Line 8 (3G10.0) Isotropic hardening law parameters | |||
| $\sigma_{0}$ | Initial Yield stress [MPa], param(30, ilaw) | ||
| $S_{R}$ | Saturation rate [MPa], param(31, ilaw) | ||
| $C_{R}$ | Saturation value [-], param(32, ilaw) | ||
| Line 9 (2G10.0) Kinematic hardening parameters | |||
| $S_{X}$ | Saturation rate [-], param(33, ilaw) | ||
| $C_{X}$ | Saturation value [MPa], param(34, ilaw) | ||
| CASE (2): Growth and nucleation of voids are considered | |||
| Line 8 (3G10.0) Nucleation model parameters | |||
| $F_{N}$ | Total nucleated porosity ratio, param(40, ilaw) | ||
| $S_{N}$ | Standard deviation, param(40, ilaw) | ||
| $\epsilon_{N}$ | Standard mean, param(40, ilaw) | ||
| Line 9 (3G10.0) Isotropic hardening law parameters | |||
| $\sigma_{0}$ | Initial Yield stress [MPa], param(30, ilaw) | ||
| $S_{R}$ | Saturation rate [MPa], param(31, ilaw) | ||
| $C_{R}$ | Saturation value [-], param(32, ilaw) | ||
| Line 10 (2G10.0) Kinematic hardening parameters | |||
| $S_{X}$ | Saturation rate [-], param(33, ilaw) | ||
| $C_{X}$ | Saturation value [MPa], param(34, ilaw) | ||
| CASE (3): Growth, nucleation and coalescence are active | |||
| Line 8 (3G10.0) Nucleation model parameters | |||
| $F_{N}$ | Total nucleated porosity ratio, param(40, ilaw) | ||
| $S_{N}$ | Standard deviation, param(41, ilaw) | ||
| $\epsilon_{N}$ | Standard mean, param(42, ilaw) | ||
| Line 9 (3G10.0) Coalescence model parameters | |||
| $f_{U}$ | Ultimate porosity ratio, param(43, ilaw) | ||
| $f_{F}$ | Fracture porosity ratio, param(44, ilaw) | ||
| $f_{cr}$ | Critical porosity ratio for onset of coalescence, param(45, ilaw) | ||
| Line 10 (3G10.0) Isotropic hardening law parameters | |||
| $\sigma_{0}$ | Initial Yield stress [MPa], param(30, ilaw) | ||
| $S_{R}$ | Saturation rate [MPa], param(31, ilaw) | ||
| $C_{R}$ | Saturation value [-], param(32, ilaw) | ||
| Line 11 (2G10.0) Kinematic hardening parameters | |||
| $C_{X}$ | Saturation rate [-], param(33, ilaw) | ||
| $S_{X}$ | Saturation value [MPa], param(34, ilaw) | ||
