/MAT/LAW112 (PAPER or XIA)

The in-plane behavior should be fully uncoupled with the out-of-plane behavior, computed as:

With $C=\frac{1}{1-{\nu }_{12}{\nu }_{21}}\left[\begin{array}{cc}{E}_1& {\nu }_{12}{E}_2\\ {\nu }_{21}{E}_1& E_2\end{array}\right]$

The transverse shear components are computed as:

The out-of-plane elastic behavior (for solids only) is treated as a uniaxial equivalent problem. However, the computation of the stress may differ between tension and compression. The elasticity becomes nonlinear for compressive loadings.

Where,

The normal direction vector to the yield planes are:

$\begin{array}{l}{N}_1=\left[\begin{array}{cccccc}\frac{1}{\sqrt{1+{\nu }_{1p}^2}}& -\frac{{\nu }_{1p}}{\sqrt{1+{\nu }_{1p}^2}}& 0& 0& 0& 0\end{array}\right]\\ N_2=\left[\begin{array}{cccccc}-\frac{{\nu }_{2p}}{\sqrt{1+{\nu }_{2p}^2}}& \frac{1}{\sqrt{1+{\nu }_{2p}^2}}& 0& 0& 0& 0\end{array}\right]\\ N_3=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\end{array}\right]\\ N_4=\left[\begin{array}{cccccc}-\frac{1}{\sqrt{1+{\nu }_{4p}^2}}& \frac{{\nu }_{4p}}{\sqrt{1+{\nu }_{4p}^2}}& 0& 0& 0& 0\end{array}\right]\\ N_5=\left[\begin{array}{cccccc}\frac{{\nu }_{5p}}{\sqrt{1+{\nu }_{5p}^2}}& -\frac{1}{\sqrt{1+{\nu }_{5p}^2}}& 0& 0& 0& 0\end{array}\right]\\ N_6=\left[\begin{array}{cccccc}0& 0& 0& -1& 0& 0\end{array}\right]\end{array}$

Each direction $I$ is then associated to a specific yield stress whose expression is:

$\begin{array}{cc}{\sigma }_Y^I=S_I^0+A_I^0\tanh \left(B_I^0{\epsilon }_p^f\right)+C_I^0{\epsilon }_p^f& \begin{array}{cc}\text{with}& I\in \left[1,6\right]\end{array}\end{array}$

Where, ${\epsilon }_p^f$ is the in-plane equivalent plastic strain (associated to the yield function $f$ ).

The out-of-plane yield function is denoted as $g$ is defined as:

$g=-{\sigma }_{zz}-{\sigma }_Y^C$ with ${\sigma }_Y^C=A_{\sigma }+B_{\sigma }\exp (C_{\sigma }{\epsilon }_p^g)$

Where, ${\epsilon }_p^g$ is the out-of-plane equivalent plastic strain (associated to the yield function $g$ ).

The transverse shear yield function is:

If the tabulated yield stress option is selected (Itab = 1), each yield stress is associated to a table (TAB_YLDi) to define the stress evolution with the plastic strain, at several plastic strain-rate. Two scale factors can be also defined in the X and Y direction for each table. In this case, the hardening parameters $S0i$ , $A0i$ , $B0i$ , $C0i$ , $A_{\sigma }$ , $B_{\sigma }$ , $C_{\sigma }$ , ${\tau }_0$ , $A_{\tau }$ , and $B_{\tau }$ are ignored, and the yield stress becomes:

$\begin{array}{l}\begin{array}{cc}{\sigma }_Y^I=f_Y^{tab\_YLDI}({\epsilon }_p^f,{\dot{\epsilon }}_p^f)& I\in \left[1,6\right]\end{array}\\ {\sigma }_Y^C=f_Y^{tab\_YLDC}({\epsilon }_p^g,{\dot{\epsilon }}_p^g)\\ {\sigma }_Y^S=f_Y^{tab\_YLDS}({\epsilon }_p^h,{\dot{\epsilon }}_p^h)\end{array}$

For output field, an equivalent “global” plastic strain is computed as: