====== CAZACUW ======
===== Description =====
3D constitutive law with an orthotropic yield criterion for hexagonal closed packed materials 
==== The model ====
Mechanical analysis of elasto-plastic HCP materials undergoing large strains. 
==== Files ====
Prepro: LCAZACW.F  \\
Lagamine: CAZACUW.F
===== 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| 524|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (3I5) ^^
|NWP| Number of plastic work levels|
|NTRAN| Number of linear transformations|
|MAXIT|Maximal number of iterations during stress integration|
==== Real parameters ====
^ Line 1 (6G10.0) ^^
|$E_{1}$|YOUNG's orthotropic elastic moduli|
|$E_{2}$|:::|
|$E_{3}$|:::|
|$\mbox{ANU}_{12}$|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:
  * $\Sigma_{J}^{(I)}$ is the Jth eigenvalue of $\underline{\Sigma}^{(I)}$ defined by the following relationship: \[\underline{\Sigma}^{(I)} = \underline{\underline{C}}^{(I)}:\underline{S}\] which represents the Ith linear transformation of the deviator __S__ of the Cauchy stress tensor. The tensor $\underline{\underline{C}}^{(I)}$ is defined: \[\underline{\underline{C}}^{(I)} = \begin{pmatrix}
      C_{11}^{I} & C_{12}^{I} & C_{13}^{I} & 0&0&0 \\
      C_{12}^{I} & C_{22}^{I} &C_{23}^{I}&0&0&0 \\
      C_{13}^{I}&C_{23}^{I}&C_{33}^{I}&0&0&0 \\
      0&0&0&C_{44}^{I}&0&0 \\
      0&0&0&0&C_{55}^{I}&0 \\
      0&0&0&0&0&C_{66}^{I} \\
      \end{pmatrix}\]
  * $k^{l}$ is linked to the differential effect (tension/compression asymmetry)
  * $a$ is the degree of homogeneity

^Line 1 (3G10.0) hardening parameters (Voce) ^^
|R0|initial yield stress (see [[laws:cazacuw#Hardening form|hardening form]]), //param(17+(10*NTRAN+1)*NWP,ilaw)//|
|RSAT|saturation value (see [[laws:cazacuw#Hardening form|hardening form]])// param(18+(10*NTRAN+1)*NWP,ilaw)//|
|CR|saturation rate (see [[laws:cazacuw#Hardening form|hardening form]])), //param(19+(10*NTRAN+1)*NWP,ilaw)//|
===== Stresses =====
==== Number of stresses ====
6
==== Meaning ====
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)|$\sigma_{YZ}$|


===== State variables =====
==== Number of state variables ====
9
==== List of state variables ====
|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|

==== Hardening form ====
$\sigma_{F} = R_{0} + R_{SAT}(1-exp(-C_{R}\overline{\varepsilon}^{p}))$

====**Important remark**====
It is required to use [[elements:blz3d|BWD3D]] elements and to work with local axes (for more details, see explanations of [[elements:blz3d|BLZ3D]] element).