====== CAZACU ======
===== 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: LCAZAC2.F  \\
Lagamine : CAZACU2.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| 521|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (3I5) ^^
|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, NTRAN**
^Line 1 (G10.0)^^
|ASYM| differential effect, //param(17+(I-1)*10,ilaw) //|
^Line 2 (3G10.0) material parameters (anisotropy)^^
|C11| //param(18+(I-1)*10,ilaw)//|
|C12|// param(19+(I-1)*10,ilaw)//|
|C13|//param(20+(I-1)*10,ilaw)//|
^Line 3 (3G10.0) material parameters (anisotropy)^^
|C22|//param(21+(I-1)*10,ilaw)//|
|C23|// param(22+(I-1)*10,ilaw)//|
|C33|//param(23+(I-1)*10,ilaw)//|
^Line 4 (3G10.0) material parameters (anisotropy)^^
|C44|// param(24+(I-1)*10,ilaw)//|
|C55|// param(25+(I-1)*10,ilaw)//|
|C66|// param(26+(I-1)*10,ilaw)//|

Yielding is described by:
\[f(\mathbf{\sigma}-\mathbf{X}, \overline{\varepsilon}_{p}) = \overline{\sigma}(\mathbf{\sigma}-\mathbf{X}, \overline{\varepsilon}_{p})-Y(\overline{\varepsilon}_{p})\]
Where $\overline{\sigma}$ is the effective stress while $\overline{\varepsilon}_{p}$ is the effective plastic strain associated to $\overline{\sigma}$  using the work-equivalence principle. The kinematic hardening is introduced by the back-stress tensor $\mathbf{X}$ and $Y$ denotes the yield stress whose evolution describes the size of the yield surface (isotropic hardening). \\

The effective stress $\overline{\sigma}$  is of the form:
\[\overline{\sigma} = B \big( \overset{NTRAN}{\underset{i=1}{\Sigma}}\big[ \big( |\Sigma_{1}^{(i)}| - k^{(i)}\Sigma_{1}^{(i)}\big)^{a} + \big( |\Sigma_{2}^{(i)}| - k^{(i)}\Sigma_{2}^{(i)}\big)^{a}+ \big( |\Sigma_{3}^{(i)}| - k^{(i)}\Sigma_{3}^{(i)}\big)^{a} \big]  \big) ^{\frac{1}{a}}\]
Where:\\

$\Sigma_{1}^{(i)}, \Sigma_{2}^{(i)}, \Sigma_{3}^{(i)}$ are the eigenvalue of  the tensor $\mathbf{\Sigma}^{(i)}$ defined as:\\
$$\mathbf{\Sigma}^{(i)} = \mathbf{C}^{(i)} : (\mathbf{s- X^{'}})$$
i.e. the i-th linear transformation of $\mathbf{s – X^{'}}$ where **s** and $\mathbf{X^{'}}$ are the deviator of the Cauchy stress tensor and the back-stress tensor, respectively. Each tensor $\mathbf{C}^{(i)}$ is represented in Voigt notations as follows:\\

\[ \mathbf{C}^{(i)} = \begin{pmatrix}
C_{11}^{i} & C_{12}^{i} & C_{13}^{i} & 0&0&0\\
C_{12}^{i} & C_{22}^{i} &C_{23}^{i}&0&0&0\\
C_{13}^{i}&C_{23}^{i}&C_{33}^{i}&0&0&0\\
0&0&0&C_{44}^{i}&0&0\\
0&0&0&0&C_{55}^{i}&0\\
0&0&0&0&0&C_{66}^{i}\\
\end{pmatrix}\]

  * the parameters $k^{i}$ allow for the description of strength differential effects;\\
  * $a$ is the degree of homogeneity;\\
  * $B$ is a constant defined such that $\overline{\sigma}$ reduces to the tensile yield stress in RD, i.e.:\\

\[B = \Big[ \overset{NTRAN}{\underset{i=1}{\Sigma}}\big[ \big( |\Phi_{1}^{(i)}| - k^{(i)}\Phi_{1}^{(i)}\big)^{a} + \big( |\Phi_{2}^{(i)}| - k^{(i)}\Phi_{2}^{(i)}\big)^{a}+ \big( |\Phi_{3}^{(i)}| - k^{(i)}\Phi_{3}^{(i)}\big)^{a} \big]  \Big] ^{\frac{-1}{a}}\]

With \\
$$\Phi_{1}^{(i)} = \frac{2}{3}C_{11}^{(i)}-\frac{1}{3}C_{12}^{(i)}-\frac{1}{3}C_{13}^{(i)}$$

$$\Phi_{2}^{(i)} = \frac{2}{3}C_{12}^{(i)}-\frac{1}{3}C_{22}^{(i)}-\frac{1}{3}C_{23}^{(i)}$$

$$\Phi_{3}^{(i)} = \frac{2}{3}C_{13}^{(i)}-\frac{1}{3}C_{23}^{(i)}-\frac{1}{3}C_{33}^{(i)}$$


^Line 1 (3G10.0) (see section [[laws:cazacu#Hardening form|hardening form]]) ^^
|R0|initial yield stress , // param(17+NTRAN*10,ilaw) //|
|SR|isotropic hardening saturation value, //param(18+NTRAN*10,ilaw) //|
|CR|isotropic hardening saturation rate, // param(19+NTRAN*10,ilaw)//|
^Line 2 (2G10.0) (see section [[laws:cazacu#Hardening form|hardening form]]) ^^
|SX|kinematic hardening saturation value, // param(20+NTRAN*10,ilaw)//|
|CX|kinematic hardening saturation rate, //param(21+NTRAN*10,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) $\rightarrow$ Q(8)|Back stress|
|Q(9)|Triaxiality|


==== Hardening form ====

Isotropic hardening (Voce):\\
\[Y(\overline{\varepsilon}_{p}) = R_{0} + R \\
dR = c_{R}(s_{R}-R)d\overline{\varepsilon}_{p}\]
Kinematic hardening (Armstrong-Frederick):\\
\[d\mathbf{X} = c_{X}(s_{X}d\mathbf{\varepsilon}_{p}-\textbf{X}d\overline{\varepsilon}_{p})\]

====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).