====== ILYSH ======
===== Description =====
Anisotropic elastoplastic constitutive law, __analytically__ integrated on thickness \\
For:	
  * membrane elements ([[elements:mem2d|MEM2D]]) 
  * thin shell elements ([[elements:kirsh|KIRSH]])
  * thick shell elements ([[elements:minds|MINDS]])
in generalized plane state \\

==== The model ====
This law is only used for mechanical analysis of anisotropic elastoplastic thin bodies (membranes and shells) in generalized plane state.  This law deals with reduced stresses, analytically integrated on thickness.  The anisotropic ILYUSHIN criteria is assumed :
\[
\sqrt{Q_n + Q_m + \frac{3}{2}Q_{mn}^2} - 1 \leq 0
\]
Where 
\[
Q_n = \alpha_1 n_1^2 + \alpha_2 n_2^2 - \alpha_{12} n_1n_2 + 3\alpha_3 n_3^2\ n_i = \frac{N_i}{N_p}
\]
\[
Q_m = \alpha_1 m_1^2 + \alpha_2 m_2^2 - \alpha_{12} m_1m_2 \ m_i = \frac{M_i}{M_p}
\]
\[
Q_{mn} = \alpha_1 m_1n_1 + \alpha_2 m_2n_2 - \alpha_{12} \left(m_1n_2 + m_2n_1\right)/2
\]
And with 
  * $N_1 = N_x =$ normal effort in local x‑direction
  * $N_2 = N_z =$ normal effort in local z‑direction
  * $N_3 = T =$ shear effort
  * $M_1 = M_x =$ moment associated to the x‑direction
  * $M_2 = M_z =$ moment associated to the z‑direction

|$\alpha_1 = 1$|$\alpha_3 = 1$|$R_x =\frac{1 + r_x}{2 r_x}$|
|$\alpha_2 = R_z / R_X$ | $\alpha_{12} = 1/R_x$ | $R_x = \frac{1 + r_z}{2 r_z}$|

Coefficients $r_x$ and $r_z$ are classical parameters which can take into account the anisotropic behaviour, via $\alpha_i$’s coefficients. $r_x$ and $r_z$ have the following physical meaning :

$r_x = \frac{d\varepsilon_z^p}{d\varepsilon_y^p}$ when we do a tensile test in the local x‑direction\\
$r_z = \frac{d\varepsilon_x^p}{d\varepsilon_y^p}$ when we do a tensile test in the local z‑direction

The yield values $N_p$ and $M_P$ describe the material behaviour.\\
Hardening is isotropic and can be taken into account either by equivalent strains, or by work equivalence.

==== Files ====
Prepro: LILYSH.F \\
Lagamine: ILYSH.F
===== Availability =====
|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| NO |
|3D state| NO |
|Generalized plane state| YES |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| = 25 for a membrane behaviour ([[elements:mem2d|MEM2D]] elements) |
|:::| = 26 for a KIRCHOFF shell behaviour (KIRSH elements)|
|:::| = 27 for a MINDLIN shell behaviour (MINDS elements)|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (I5) ^^
|N|number of points defining the uniaxial constitutive law|

Remarks :\\

- The membrane constitutive law is defined by the constitutive law $\sigma_o(\varepsilon)$. The yield value $N_P$ is assumed equal to $N_P = e\times \sigma_o$ (e is the thickness of the shell).\\
The bending constitutive law is defined by the constitutive law $\sigma_x(\varepsilon)$, which is computed from the membrane constitutive law $\sigma_o(\varepsilon)$. The yield value $M_P$ is assumed equal to $ M_P = \frac{e^2}{4}\sigma_{\chi}$
- The constitutive laws are stored in central memory such that one must have :
\[
4 * N+7 \leq MPARA
\]
\[
2 * N+6 \leq MPARA\ \text{if}\ LTYPE = 25\ \text{(membrane law)}
\]
==== Real parameters ====
^Line 1 - Global Data (3G10.0)^^
|ANU|Poisson's coefficient|
|RT|$r_x$ anisotropic coefficient (default = 1)|
|RL|$r_z$ anisotropic coefficient (default = RT)|
^Line 2:N+1 - Uniaxial Constitutive Law (2G10.0) ^^
|SIG|$\sigma_0$ value at the considered point|
|EPS|$\varepsilon$  value at the considered point|modulus is kept:

{{ :laws:hilsh1.png?600 |}}
===== Results =====
==== Stresses ====
|SIG(1)|N1|
|SIG(2)|N2|
|SIG(3)|M1|
|SIG(4)|M2|
|SIG(5)|T|

These integrated stresses are given by unit of length.
{{ :laws:hilsh2.png?400 |}}
==== State variables ====
|Q(1)|$b$, out‑of‑plane thickness (along z‑direction), according to the generalized plane state|
|Q(2)|$e$, in‑plane thickness of the shell (updated at the end of each step)|
|Q(3)|$W_t$, total work associated to this integration point|
|Q(4)|$W_p$, plastic work associated to this integration point (by unit of shell area)|
|Q(5)|$\bar{\varepsilon}_0^p$, equivalent membrane plastic strain|
|Q(6)|$\bar{\chi}^p$, equivalent bending plastic strain (curvature)|
|Q(7)|$N_p$, membrane yield stress = $e\sigma_o$|
|Q(8)|$M_p$, bending yield stress = $\frac{e^2}{4}\sigma_{\chi}$|