====== HILSH ======
===== Description =====
Anisotropic elastoplastic constitutive law, __numerically__ 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 \\
\\
Implemented by: L. Grisard (1991)
==== The model ====
This law is only used for mechanical analysis of elastoplastic anisotropic thin bodies (membranes and shells) in generalized plane state.\\
Local stresses $\sigma_x$ and $\sigma_z$ are computed at some points across the thickness, and numerically integrated to obtain a law linking global stresses (shear and normal efforts, moments,...) to global strains (membrane shear and bending strains).  Locally, the anisotropic HILL criteria in plane stress state is assumed :
\[
\sqrt{\alpha_1 \sigma_x^2 + \alpha_2 \sigma_z^2 - \alpha_{12} \sigma_x \sigma_z + 3 \alpha_z \tau^2} \leq \sigma_0
\]
With 
|$\alpha_1 = 1$ | $\alpha_3 = 1$ | $R_x = \frac{1 + r_x}{2 r_x}$|
|$\alpha_2 = R_z / R_X$ | $\alpha_{12} = 1/R_x$ | $R_x = \frac{1 + r_z}{2 r_z}$|

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

$\varepsilon_y^p$ is, here, the plastic strain in the thickness direction. Isotropic hardening is assumed for $\sigma_0$, taken into account by an equivalent strain $\bar{\varepsilon}^p$.

==== Files ====
Prepro: LHILSH.F \\
Lagamine: HILSH.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| = 20 for a membrane behaviour ([[elements:mem2d|MEM2D]] elements) |
|:::| = 21 for a KIRCHOFF shell behaviour ([[elements:kirsh|KIRSH]] elements)|
|:::| = 22 for a MINDLIN shell behaviour ([[elements:minds|MINDS]] elements)|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (3I5) ^^
|NPI|number of integration points across the thickness (less than or equal to 10)|
|N|number of points defining the uniaxial constitutive law $\sigma_0(\varepsilon)$ (2 or greater)|
|ITAU|= 0 if shear stress is __not__ taken into account in plasticity|
|:::|= 1 if shear stress is taken into account in plasticity (this parameter has no meaning if LTYPE is not equal to 22).|
__Remarks:__\\

  * A trapezoidal rule is assumed for the integration across the thickness. This rule has been performed for moments.
  * The integration points are equally spaced across the thickness, with a point on both sides (upper and lower layers).
  * The constitutive law will be stored in central memory.  So, we must have the following condition : $2\times N + 6 \leq MPARA$
  * For a membrane law (ITYPE = 20), NPI is always equal to one.
==== 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|
Warning: The first point is the end of the elastic behaviour.\\
After the last point, tangent modulus is kept.

{{ :laws:hilsh1.png?600 |}}

===== Results =====
==== Stresses ====
|SIG(1)|N1, normal effort in x‑direction|
|SIG(2)|N2, normal effort in z‑direction|
|SIG(3)|M1, moment associated to the x‑direction|
|SIG(4)|M2, moment associated to the z‑direction|
|SIG(5)|T, shear effort|

{{ :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 (by unit of shell area)|
|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|
|Q(8)|$M_p$, bending yield stress|
__For i = 1 to NPI :__
|Q(9+5(i‑1))|$\sigma_x$, local stress in x‑direction at the $i^e$ integration point across the thickness e|
|Q(10+5(i‑1))|$\sigma_z$, local stress in z‑direction at the $i^e$ integration point across the thickness e|
|Q(11+5(i‑1))|$\tau$, local shear stress at the $i^e$ integration point across the thickness e|
|Q(12+5(i‑1))|$\sigma_0$, local yield stress at the $i^e$ integration point across the thickness e|
|Q(13+5(i‑1))|$\bar{\varepsilon^p}$, local equivalent plastic strain at the $i^e$ integration point across the thickness e.|

The total number of state variables is so : 8 + 5 * NPI.\\

Remarks : the shear stress $\tau$ is only stored in Q in the case of a MINDLIN shell behaviour.  In the other cases, only four parameters ($\sigma_x$, $\sigma_z$, $\sigma_0$, $\bar{\varepsilon}$) are stored in Q at each integration point, and the total number of state variables is so : 8 + 4 * NPI.