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====== LHILCQ ======
===== Description =====
Hill's type constitutive law for 3D shell ([[elements:coqj4|COQJ4]]) element, numerically integrated on the thickness.\\
This element uses the Green-Lagrance strain tensor and the Piola-Kirkhoff stress tensor for the plastic deformation but it uses the true strain and Cauchy stress tensor for the elastic deformation. \\
You should then NOT USE THIS ELEMENT when the ELASTIC DEFORMATION is LARGE

==== The model ====
This law is only used for mechanical analysis of elasto-anisotropic plastic with anisotropic or isotropic hardening. Numerical integration on the thickness is used.
==== Files ====
Prepro: LLHILCQ.F \\
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| = 32 |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (7I5) ^^
|NPI|number of integration points across the thickness ( $\leq$ 10 )|
|IND|index of the thickness variable|
|:::|= 0: thickness updated by elastoplastic strain|
|:::|= 1: thickness updated by elastic strain|
|:::|= 2: thickness constant|
|:::|= 3: thickness updated by elastoplastic strain (green Lagrange)|
|:::|= -1 thickness updated by global approach|
|NPO|= number of points to define the uniaxial constitutive law ($\geq$ 0)|
|:::|= **0**: bilinear law|
|:::|$\geq$ **2**: multi-linear law (NPO segments, elastic part included)|
|:::|= **-1**: for Swift law: $\sigma = K ( \varepsilon_0 + \varepsilon^{pl})^n$ with $K$ and $n$ defined by the user ($\varepsilon_0$ is computed  by Prepro with: $\varepsilon_0 = \left( SY11/K \right)^{1/n}$)|
|:::|= **-2**: for Swift law: $\sigma = K ( \varepsilon_0 + \varepsilon^{pl})^n$ with $K$, $\varepsilon_0$ and $n$ defined by the user ( this law is adapted to inverse method: OPTIM)|
|:::|= **-3**: for Voce law: $\sigma = \sigma_0 + K (1- \exp(-n \times \varepsilon^{pl})) + \varepsilon^{pl})^n$ with $\sigma_0$, $K$ and $n$ defined by the user ( this law is adapted to inverse method: OPTIM)|
|METH|the method used to calculate the IP on thickness|
|:::|= 0 → GAUSS|
|:::|= 1 → LOBATTO|
|:::|= 2 → NEWTON-COTE|
|NINTV|number of sub-steps used to integrate numerically the constitutive equation in a time step (default = 1)|
|IND_FULL_SAVING|$\neq$ 0 if the user wants to save extra variables (see internal variables)|
|IND_KH|index for kinematic hardening|
|:::|= 0: no kinematic hardening|
|:::|= 10: Armstrong-Frederick kinematic hardening – No initial backstress|
|:::|= 11: Armstrong-Frederick kinematic hardening and initial backstress|
|:::|= 20: Ziegler  kinematic hardening – No initial backstress|
|:::|= 21: Ziegler  kinematic hardening and initial backstress|

Amstrong – Frederick hardening
\[
\dot{\vec{X}} = C_X\left( X_{sat}\dot{\vec{\varepsilon}}^{plastic} \vec{X}\right)
\]

Ziegler hardening
\[
\dot{\vec{X}} = C_A \frac{1}{\sigma^0} \left( \vec{\sigma} - \vec{X}\right) \dot{\vec{\varepsilon}}^{plastic} - G_A \vec{X}\dot{\vec{\varepsilon}}^{plastic}
\]
==== Real parameters ====
**__Global Data__**
^Line 1 (2G10.0)^^
|E|Young Modulus = // param(1, ilaw) //|
|ANU|Poisson's coefficient = // param(2, ilaw) //|

**__Uniaxial Constitutive Law__** \\
**For NPO = 0** 
^Line 1 (2G10.0)^^
|SY11|Initial yield stress of tension or compression in 1 – direction (RD) =// param (3, ilaw)//|
|ET11| The tangent modulus of tension or compression in 1 – direction (RD) =// param (4, ilaw)//|
^Line 2 (2G10.0)^^
|SY22| Initial yield stress of tension or compression in 2 – direction =// param (5, ilaw)//|
|ET22| The tangent modulus of tension or compression in 2 – direction =// param (6, ilaw)//|
^Line 3 (2G10.0)^^
|SY33| Initial yield stress of tension or compression in 3 – direction =// param (7, ilaw)//|
|ET33| The tangent modulus of tension or compression in 3 – direction =// param (8, ilaw)//|
^Line 4 (2G10.0)^^
|SY12| Initial yield shear stress  in 1 - 2 plan =// param (9, ilaw)//|
|ET12| The tangent modulus in 1 - 2 plan =// param (10, ilaw)//|

__Remark:__ \\
* the initial yield stress parameters ($SY_{ij}$) can be computed with $\sigma_F^{initial}$ and the Hill parameters (for further details, see section 4.3 in HILL3D law)
* for NPO = 0, it is possible to use the anisotropic or isotropic hardening law. This depends on whether these tangent modulus are equal or not, respectively.

**For NPO > 0** \\
^Line 1 (4G10.0)^^
|SY11|These parameters can be computed with $\sigma_F^{initial}$ and the Hill parameters; for further details, see section 4.3 in HILL3D law \\ //= varin(1$\rightarrow$ 4, ilaw)//|
|SY22|:::|
|SY33|:::|
|SY12|:::|
^Line 2:NPO+1 (2G10.0)^^
|$\varepsilon_i$|the value of strain at the considered point in the referent direction|
|$\sigma_i$|the value of stree at the considered point in the referent direction. The first point defines the end of the elastic zone ($\sigma_1 = \sigma_{LE}$)|

**For NPO = -1** 
^Line 1 (4G10.0)^^
|SY11|These parameters can be computed with $\sigma_F^{initial}$ and the Hill parameters; for further details, see section 4.3 in HILL3D law //= varin(1$\rightarrow$ 4, ilaw)// \\ SY11 = param (3, ilaw)|
|SY22|:::|
|SY33|:::|
|SY12|:::|
^Line 2 (2G10.0)^^
|K|hardening parameters of the law: $\sigma = K(\varepsilon_0+ \varepsilon^{pl})^n$ \\ $\varepsilon_0$ is computed  by Prepro with: $\varepsilon_0 = \left( SY11/K \right)^{1/n}$ \\ K = param (4, ilaw) \\ n = param (5, ilaw) \\ $\varepsilon_0$ = param (6, ilaw)|
|n|:::|

**For NPO = -2 ** (adapted to OPTIM)\\

^Line 1 (4G10.0)^^
|F, G, H, N (N = L = M) Hill's coefficients defining the yield locus||
|F|= param (6, ilaw)|
|G|= param (7, ilaw)|
|H|= param (8, ilaw)|
|N = L = M|= param (9, ilaw)|
^Line 2 (3G10.0)^^
|Hardening parameters of the law: $\sigma = K(\varepsilon_0+ \varepsilon^{pl})^n$||
|K| = param (3, ilaw)|
|$\varepsilon_0$|= param (4, ilaw)|
|n| = param (5, ilaw)|

**For NPO = -3** (adapted to OPTIM)\\
^Line 1 (4G10.0)^^
|F, G, H, N (N = L = M) Hill's coefficients defining the yield locus||
|F|= param (6, ilaw)|
|G|= param (7, ilaw)|
|H|= param (8, ilaw)|
|N = L = M|= param (9, ilaw)|
^Line 2 (3G10.0)^^
|K, $\sigma_0$, n = hardening parameters of the Voce law||
|K|= param (3, ilaw)|
|$\sigma_0$|= param (4, ilaw)|
|n| = param (5, ilaw)|

__**Kinematic hardening parameters (only if IND_KH ≠ 0!) **__ \\
**For IND_KH = 10 or 11:  Armstrong-Frederick kinematic hardening parameters**      
\[
\dot{\vec{X}} = C_X\left( X_{sat}\dot{\vec{\varepsilon}}^{plastic} - \dot{\vec{\varepsilon}}^{plastic} \vec{X}\right)
\]
^Line 1 (2G10.0)^^			
|CX|kinematic hardening saturation rate = param (2*NPI+KPAR+1, ilaw)|
|Xsat|kinematic hardening saturation value = param (2*NPI+KPAR+2, ilaw) with KPAR= 6 (NPOINT= -1) or KPAR= 9 (NPOINT= -2)|

**For IND_KH = 20 or 21:  Ziegler hardening parameters**
\[
\dot{\vec{X}} = C_A \frac{1}{\sigma^0} \left( \bar{\sigma} - \vec{X}\right) \dot{\bar{\varepsilon}}^{plastic} - G_A \bar{X}\dot{\bar{\varepsilon}}^{plastic}
\]
^Line 1 (2G10.0)^^
|CA|initial kinematic hardening modulus	= param (2*NPI+KPAR+1, ilaw)|
|GA|rate at which the kinematic hardening modulus decrease with increasing plastic deformation \\= param (2*NPI+ KPAR+2, ilaw) with KPAR= 6 (NPOINT= -1) or KPAR= 9 (NPOINT= -2)|

**For IND_KH = 11 or 21:  Components of the initial back-stress** \\
^Line 1 (3G10.0)^^
|BS1|3 components (directions: XX, YY, XY) //= param (2*NPI+KPAR+3 → 5, ilaw)// \\ with KPAR= 6 (NPOINT= -1) or KPAR= 9 (NPOINT= -2)|
|BS2|:::|
|BS3|:::|
===== Results =====
==== Stresses (for each of the 4 IP of the shell) ====
|SIG(1)|$N_x$, normal effort in the local X‑direction $=\int_0^{th}\sigma_{xx} dZ$|
|SIG(2)|$N_y$, normal effort in the local Y‑direction $=\int_0^{th}\sigma_{yy} dZ$|
|SIG(3)|$N_{xy}$, normal effort in the local X-Y plan $=\int_0^{th}\sigma_{xy} dZ$|
|SIG(4)|$M_x$, moment associated to the local X‑direction|
|SIG(5)|$M_y$, moment associated to the local Y‑direction|
|SIG(6)|$M_{xy}$, moment associated to the local X-Y plan|

==== State variables (for each of the 4 IP of the shell) ====
|Q(1)| thick, the actual thickness for this IP.|
|Q(2)| $\sigma_y^e$, the equivalent VM type stress integrated through the thickness of this IP|
|Q(3)| Yield, index of stress state for this IP|
|:::|= 0, the stress state in this IP is elastic.|
|:::|= 1, it is elastoplastic.|
|Q(4)|Limit, index of deformation limit for this IP|
|:::|= 0, under the limit.|
|:::|= 1, reached the limit.|

**__Repeat NPI times__** (NPI number of integration points across the thickness)\\
(IPI is the first/second/third/fourth integration points of the shell)
\[
K = 4 + (IPI-1)*Nvar
\]
  * Nvar = 10 if IND_FULL_SAVING = 0 and IND_KH = 0
  * Nvar = 13 if IND_FULL_SAVING  0 and IND_KH = 0 
  * Nvar = 13 if IND_FULL_SAVING = 0 and IND_KH $\neq$ 0
  * Nvar = 16 if IND_FULL_SAVING  0 and IND_KH $\neq$ 0

|Q(K+1)|$\sigma_y$, the yield stress at this IP in the thickness.|
|Q(K+2)|$\sigma_y^{vm}$, the equivalent VM type stress at this IP in the thickness|
|Q(K+3)| Yield, index of stress state for this IP in the thickness.|
|Q(K+4)|$\varepsilon_{ep}^p$, equivalent plastic GREEN strain at this IP in the thickness.|
|Q(K+5)|$\sigma_{11}$, the local stress at this IP in the local direction x.|
|Q(K+6)|$\sigma_{22}$, the local stress at this IP in the local direction y.|
|Q(K+7)|$\sigma_{12}$, the local stress at this IP in the local direction xy.|
|Q(K+8)|$\alpha_{11}$, the local anisotropic coefficient for this IP.|
|Q(K+9)|$\alpha_{12}$, idem.|
|Q(K+10)|$\alpha_{22}$, idem.|
|Q(K+11)|$\alpha_{33}$, idem.|

**__and__**, only if IND_FULL_SAVING $\neq$ 0
|Q(K+ 12)|$\varepsilon_{xx}$ (GREEN STRAIN RATE TENSOR)|
|Q(K+ 13)|$\varepsilon_{yy}$ (GREEN STRAIN RATE TENSOR)|
|Q(K+14)| $\varepsilon_{xy}$ (GREEN STRAIN RATE TENSOR)|

**__and__**, only if IND_KH $\neq$ 0: **__variable part__** of the backstress
|Q(K+ (12 or 15)) |XXI1 backstress component: XX \\ (to add to the BS1 component IF IND_KH=11 OR 21)|
|Q(K+ (13 or 16))| XXI2 backstress component: YY \\ (to add to the BS2 component IF IND_KH=11 OR 21)|
|Q(K+(14 or 17))|XXI3 backstress component: XY \\ (to add to the BS3 component IF IND_KH=11 OR 21)|

The total number of state variables is equal to 4 + NPI*(10 or 14 or 17)
(__for each of the 4 IP of the shell__)
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