CAZACU

3D constitutive law with an orthotropic yield criterion for hexagonal closed packed materials

Mechanical analysis of elasto-plastic HCP materials undergoing large strains.

Prepro: LCAZAC2.F
Lagamine : CAZACU2.F

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

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

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

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

6

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.

9

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

It is required to use BWD3D elements and to work with local axes (for more details, see explanations of BLZ3D element).