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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 [[https://www.sciencedirect.com/science/article/pii/S002076831000363X|(J. Stewart & O. Cazacu, 2011)]],following a Gurson-type approach where the material yield stress is determined by the [[https://www.sciencedirect.com/science/article/pii/S0749641905001257|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 [[https://www.sciencedirect.com/science/article/pii/S0013794400000552|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:
\[\bar{\Sigma}_{CPB06}=\tilde{m}\Big[ \overset{3}{\underset{i=1}{\Sigma}} \big(|\Sigma_{i}| - k\Sigma_{i}\big)^a \Big] ^{\frac{1}{a}}\]
  * $\sigma_{y}$ 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:
\[STF= 1 - 2fq_{1}cosh\Big[\frac{3q_{2}\big(\sigma_{m} - X_{m}\big)}{h\sigma_{y}}\Big] - q_{3}f^{2}\]


The corrected stress ($\hat{\sigma}$) and backstress ($\hat{X}$) are respectively calculated as:

\[\hat{\sigma} = L_{ijmn}T_{mnkl}\sigma_{kl}\]
\[\hat{X}= L_{ijmn}T_{mnkl}X_{kl}\]


The backstress tensor is calculated using the Armstrong-Frederick model:

\[dX = S_{x}[C_{x}d{\epsilon^p}-Xd\bar{\epsilon}^p]\]

==== Files ====
Prepro:  LCAZACUTN.F \\
Lagamine: CAZACUTN.F \\
Coalescence onset criterion: COALCITERIA.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 |
| COALCRITERIA.F | THOMASON_ZHANG       |Calculation of [[https://www.sciencedirect.com/science/article/pii/S0013794400000552|ThZ coalescence criterion]]|
| :::            | LSP_2006             |Calculation of [[https://www.sciencedirect.com/science/article/pii/S0013794405003048|LSP06 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 (3I5)                                                     ^^
|MAXIT  |Maximal number of iterations during stress integration     |
|IDAMAGE|Active damage mechanism(s) identifier, //param(35, ilaw)// |
|IDELEM |DELEM section identifier,                                  |
==== 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 (3G10.0) Isotropic hardening law parameters             ^^
|$\sigma_{0}$ | Initial Yield stress [MPa], //param(27, ilaw)//  |
|$S_{R}$ | Saturation rate [MPa], //param(28, ilaw)//            |
|$C_{R}$ | Saturation value [-], //param(29, ilaw)//             |
^Line 8 (2G10.0) Kinematic hardening parameters                 ^^
|$S_{X}$ | Saturation rate  [-], //param(30, ilaw)//             |
|$C_{X}$ | Saturation value [MPa], //param(31, ilaw)//           |
^Line 9 (4G10.0) Standard initial damage control parameters     ^^
|$f_{0}$| Initial porosity ratio, // VARIN(5,ilaw) //           |
|$q_{1}$| Tvergaard&Needleman parameter, // param(32,ilaw) //    |
|$q_{2}$| Tvergaard&Needleman parameter, // param(33,ilaw) //    |
|$q_{3}$| Tvergaard&Needleman parameter, // param(34,ilaw) //    |


|**SELECT CASE (IDAMAGE)**                                              ||||
^    |**CASE (0)**: //No damage increment is calculated//                |||
|                                                                       ||||
^    |**CASE (1)**: //Growth is the only active damage mechanism//       |||
|                                                                       ||||
^    |**CASE (2)**: //Growth and nucleation of voids are active//        |||
|        |^Line 10 (3G10.0) Nucleation model parameters                   ^^
|:::     ||$F_{N}$ | Total nucleated porosity ratio, //param(36, ilaw)//   |
|:::     ||$S_{N}$ | Standard deviation, //param(37, ilaw)//               |
|:::     ||$\epsilon_{N}$ | Standard mean, //param(38, ilaw)//             |
|                                                                       |||| 
^    |**CASE (3)**: //Growth, nucleation and coalescence are active//    |||
|        |^Line 10 (3G10.0) Nucleation model parameters                   ^^ 
|:::     ||$F_{N}$ | Total nucleated porosity ratio, //param(36, ilaw)//   |
|:::     ||$S_{N}$ | Standard deviation, //param(37, ilaw)//               |
|:::     ||$\epsilon_{N}$ | Standard mean, //param(38, ilaw)//             |
|:::     |^Line 11 (3G10.0) Coalescence model parameters                  ^^ 
|:::     ||$f_{U}$   | Ultimate porosity ratio, //param(39, ilaw)//        |
|:::     ||$f_{F}$   | Fracture porosity ratio, //param(40, ilaw)//        |
|:::     ||$f_{cr}$ | Critical porosity ratio for coalescence onset, //VARIN(19,ilaw) [If 0, Thomason criterion is applied]//  |

... If the previous was the $n^{th}$ line...
^    |IF **IDELEM = 1** THEN                                             |||
|        |^Line $(n+1)^{th}$ (2G10.0) DELEM control parameters        ^^ 
|        ||FDELEM | Porosity ratio at which element starts being deleted, //param(42, ilaw)//   |
|:::     ||TDELEM | Time for deleting the element (linear interpolation), //param(43, ilaw)//   |
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