Lagamine

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====== LICHA ======
===== Description =====
Constitutive law defining distributed loads on a line or a surface.
Implemented by: S. Cescotto - mai 1986

==== The model ====
Definition of a uniformly distributed load (whether normal or tangent, whether in global axis) on a line ([[elements:licha|LICHA]] element) or on a surface ([[elements:sucha|SUCHA]] element).
==== Files ====
Write here the names of the main subroutines of the law (those called by loi2 for Lagamine) \\
Prepro: LLICHA.F \\
Lagamine: LICHAB.F/SUCHAB.F (element subroutine)

===== Availability =====
|Plane stress state| YES |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| YES |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 95|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Case 1: 2-D state (LICHA or SUCHA) ====
=== 1.1 Uniform functions in local or global axes ===
^Line 1 (2I5)^^
|IVAL |= 1 in local axis|
|:::|= 5 in global axis|
|IMLIC| = 0 normal load for are-length method|
|:::|= 1 dead load|
^Line 2 (4G10.0)^^
|**If IVAL = 1**|| 
|PRESSF| Normal pressure |
|TAUF|Tangent load|
|PRESSD |Normal pressure|
|TAUD| Tangent load|
|**If IVAL = 5**|| 
|SIGXF| Total stress in X axis|
|SIGYF| Total stress in Y axis|
|SIGXD| Total stress in X axis| 
|SIGYD| Total stress in Y axis|
{{ :laws:licha_1.png?300  |}}
During the non-linear analysis, PRESSF and TAUF will be multiplied by the load factor ALAMBF (= FMULT cumulated) while PRESSD and TAUD will be multiplied by the load factor ALAMBD (= DMULT cumulated).
=== 1.2 Linear functions in local or global axis ===
^Line 1 (2I5)^^
|IVAL |= 2 in local axis|
|:::|= 6 in global axis|
|IMLIC| = 0 normal load for are-length method|
|:::|= 1 dead load|
**If IVAL = 2**
^Line 2 (4G10.0)^^
|PRESSF1| Normal pressure for node 1|
|TAUF1|Tangent load for node 1|
|PRESSD1 |Normal pressure for node 1|
|TAUD1| Tangent load for node 1|
^Line 3 (4G10.0)^^
|PRESSF2| Normal pressure for node 2|
|TAUF2|Tangent load for node 2|
|PRESSD2 |Normal pressure for node 2|
|TAUD2| Tangent load for node 2|
**If IVAL = 6**
^Line 2 (4G10.0)^^
|SIGXF1| Total stress in X axis for node 1|
|SIGYF1| Total stress in Y axis for node 1|
|SIGXD1| Total stress in X axis for node 1| 
|SIGYD1| Total stress in Y axis for node 1|
^Line 3 (4G10.0)^^
|SIGXF2| Total stress in X axis for node 2|
|SIGYF2| Total stress in Y axis for node 2|
|SIGXD2| Total stress in X axis for node 2| 
|SIGYD2| Total stress in Y axis for node 2|
{{ :laws:licha_2.png?600 |}}
__Remarks__: \\
If IVAL = 2 and NNODE = 3: Bilinear functions: The nodes N1 and N2 are the two nodes on the extremities of the LICHA element. The distributed load on node N3 is an linear interpolation of the distributed load on nodes N1 and N2. \\ \\
During the non-linear analysis, PRESSF and TAUF will be multiplied by the load factor ALAMBF (= FMULT cumulated) while PRESSD and TAUD will be multiplied by the load factor ALAMBD (= DMULT cumulated).

=== 1.3 Parabolic functions in local or global axes ===
^Line 1 (2I5)^^
|IVAL |= 3 in local axis|
|:::|= 7 in global axis|
|IMLIC| = 0 normal load for are-length method|
|:::|= 1 dead load|
**If IVAL = 3**
^Line 2 (4G10.0)^^
|PRESSF1| Normal pressure for node 1|
|TAUF1|Tangent load for node 1|
|PRESSD1 |Normal pressure for node 1|
|TAUD1| Tangent load for node 1|
^Line 3 (4G10.0)^^
|PRESSF2| Normal pressure for node 2|
|TAUF2|Tangent load for node 2|
|PRESSD2 |Normal pressure for node 2|
|TAUD2| Tangent load for node 2|
^Line 4 (4G10.0)^^
|PRESSF3| Normal pressure for node 3|
|TAUF3|Tangent load for node 3|
|PRESSD3 |Normal pressure for node 3|
|TAUD3| Tangent load for node 3|
**If IVAL = 7**
^Line 2 (4G10.0)^^
|SIGXF1| Total stress in X axis for node 1|
|SIGYF1| Total stress in Y axis for node 1|
|SIGXD1| Total stress in X axis for node 1| 
|SIGYD1| Total stress in Y axis for node 1|
^Line 3 (4G10.0)^^
|SIGXF2| Total stress in X axis for node 2|
|SIGYF2| Total stress in Y axis for node 2|
|SIGXD2| Total stress in X axis for node 2| 
|SIGYD2| Total stress in Y axis for node 2|
^Line 4 (4G10.0)^^
|SIGXF3| Total stress in X axis for node 3|
|SIGYF3| Total stress in Y axis for node 3|
|SIGXD3| Total stress in X axis for node 3| 
|SIGYD3| Total stress in Y axis for node 3|
{{ :laws:licha_3.png?550 |}}
__Remarks__: \\
If IVAL = 3 and NNODE = 2: Impossible \\ \\
During the non-linear analysis, PRESSF and TAUF will be multiplied by the load factor ALAMBF (= FMULT cumulated) while PRESSD and TAUD will be multiplied by the load factor ALAMBD (= DMULT cumulated).

=== 1.4 Y-dependent functions – Type 1 ===
^Line 1 (I5)^^
|IVAL |= ± 4 in local axis|
|:::|= ± 8 in global axis|
^Line 2 (4G10.0)^^
|PRESSF|  Normal pressure = PRESSA + PRESSB*y + PRESSC*y² + PRESSD*DMULT + PRESSF*FMULT   |
|TAUF|:::|
|PRESSD |:::|
|TAUD| :::|
^Line 3 (6G10.0)^^
|PRESSA|  Tangent load = TAUA + TAUB*y + TAUC*y2 + TAUD*DMULT + TAUF*FMULT \\ y is the ordinate of the node where these stresses are applied  |
|TAUA|:::|
|PRESSB |:::|
|TAUB| :::|
|PRESSC |:::|
|TAUC| :::|
If $F_n(y).F_n(y=0) \leq 0$ and IVAL = + 4 => $F_n= 0$ and $F_r= 0$ \\
If $F_n(y).F_n(y=0) > 0$ and IVAL = - 4 => $F_n= 0$ and $F_r= 0$ \\
{{ :laws:licha_4.png?500 |}}
During the non-linear analysis, PRESSF and TAUF will be multiplied by the load factor ALAMBF (= FMULT cumulated) while PRESSD and TAUD will be multiplied by the load factor ALAMBD (= DMULT cumulated).

=== 1.5 Y-dependent functions – Type 2 ===
^Line 1 (I5)^^
|IVAL |= ± 10 in local axis|
|:::|= ± 11 in global axis|
^Line 2 (4G10.0)^^
|PRESSF|  Normal pressure = (PRESSF + PRESSA + PRESSB*y + PRESSC*y²)*FMULT + PRESSD*DMULT   |
|TAUF|:::|
|PRESSD |:::|
|TAUD| :::|
^Line 3 (6G10.0)^^
|PRESSA|  Tangent load = (TAUF + TAUA + TAUB*y + TAUC*y2)*FMULT + TAUD*DMULT \\ y is the ordinate of the node where these stresses are applied  |
|TAUA|:::|
|PRESSB |:::|
|TAUB| :::|
|PRESSC |:::|
|TAUC| :::|
If $F_n(y).F_n(y=0) < 0$ and IVAL = + 10 => $F_n= 0$ and $F_r= 0$ \\
If $F_n(y).F_n(y=0) > 0$ and IVAL = - 10 => $F_n= 0$ and $F_r= 0$ \\
During the non-linear analysis, PRESSF and TAUF will be multiplied by the load factor ALAMBF (= FMULT cumulated) while PRESSD and TAUD will be multiplied by the load factor ALAMBD (= DMULT cumulated).

=== 1.6 Imposition gravity by SUCHA ===
^Line 1 (4I5)^^
|IVAL |= 9|
|IMLIC| = 0 normal load for are-length method|
|:::|= 1 dead load|
|ISRW| = Formulation index for Sr,w|
|IMULT| = FMULT/DMULT multiplicator index|
|:::|= 0 the gravity will be multiplied by FMULT|
|:::|= 1 the gravity will be multiplied by DMULT (Used only if ISRW ≠ 0)|
**If ISRW = 0**
^Line 2 (2G10.0) ^^
|PRESSF | Normal pressure|
|PRESSD | Normal pressure|
**If ISRW ≠ 0**
^Line 2 (3G10.0)^^
|GAMMAS|= $\rho_s.g$ [N/m<sup>3</sup>]|
|GAMMAW| = $\rho_w.g$ [N/m<sup>3</sup>]|
|POROS| = Soil porosity n|
^Line 3 (7G10.0)^^
|CSR1| = 1st coefficient of the function S<sub>r,w</sub>|
|CSR2| = 2nd coefficient of the function S<sub>r,w</sub>|
|CSR3| = 3rd coefficient of the function S<sub>r,w</sub>|
|CSR4 |= 4th coefficient of the function S<sub>r,w</sub>|
|SRES |= Residual saturation degree ( = S<sub>res</sub> )|
|SRFIELD |= Field saturation degree ( = S<sub>r,field</sub> )|
|AIREV |= Air entry value [Pa]|
__Remarks__: \\
With IVAL = 9 and ISRW = 0, the gravity ($\rho g$) is imposed by mean of "PRESSF" multiplied by ALAMBF or "PRESSD" multiplied by ALAMBD. \\
With IVAL = 9 and ISRW ≠ 0, the gravity ($\rho g= (1-n) \rho_s*g+S_r*n*\rho_w *g$) is imposed and multiplied by ALAMBF if IMULT = 0, or by ALAMBD if IMULT = 1. \\
The obtained value (positive) will be used to impose gravity forces in the –Y direction. The SUCHA element is used so of course in a 2D analysis.
==== Case 2: 3-D state (SUCHA) ====
The ξ and η directions correspond to the intrinsic coordinates of the SUCHA element. \\
Reminder: for SUCHA element, the positive axis of the tangential stresses is in the opposite direction of the axes ξ (positively oriented from IP (integration point) 1 to IP 2) and η (positively oriented from IP 1 to IP 3). The positive axis of the pressure is also in the opposite direction of the positive normal to the element (given by the corkscrew rule).
{{ :elements:sucha_1.png?400 |}}

^Line 1 (I5)^^
|IVAL| = 0, 10 index telling if the function is constant (IVAL=0) or Z-dependent function, in local axis (IVAL=10)|
|:::|= 11, 12 or 13 index telling if the function is constant, in global axis (be careful, at least one component or the normal unit of the SUCHA element must be equal to zero): \\ 11 = SUCHA with normal unit is in YZ plane. \\ 12 = SUCHA with normal unit is in ZX plane. \\ 13 = SUCHA with normal unit is in XY plane.|
|:::|= 14: idem 11, 12, or 13, but no restriction on the orientation of the unit normal vector (more general approach).|
=== 2.1 If IVAL = 0 (constant function, load in local axis) ===
^Line 2 (9G10.0)^^
|PRESSF | normal pressure|
|TKSIF | tangent load in the ξ direction|
|TETAF | tangent load in the η direction|
|PRESSD | normal pressure|
|TKSID | tangent load in the ξ direction|
|TETAD | tangent load in the η direction|
|PRESSC | constant normal pressure|
|TIME1 | time of decreasing beginning|
|TIME2 | time of decreasing end|
These data are reference values for the distributed loads. \\
During the non-linear analysis PRESSF, TKSIF, TETAF will be multiplied by the load factor ALAMBF while PRESSD, TKSID, TETAD will be multiplied by the load factor ALAMBD. The constant normal pressure PRESSC will be also added. \\ 
  * If time < TIME1:
    * PRESS = PRESS
    * TKSI = TKSI
    * TETA = TETA
  * If TIME1 < time < TIME2:
    * PRESS = PRESS*(TIME2-time)/(TIME2-TIME1)
    * TKSI = TKSI *(TIME2-time)/(TIME2-TIME1)
    * TETA = TETA *(TIME2-time)/(TIME2-TIME1)
  * If time > TIME2:
    * PRESS = 0
    * TKSI = 0
    * TETA = 0

=== 2.2 If IVAL = 10 ===
^1 Blank line^^
^Line 3 (2G10.0)^^
|PRESSA | PRESS = (PRESSA + PRESSB*ZINI)* ALAMBF  |
|PRESSB | :::|


=== 2.3 If IVAL = 11, 12, 13 (constant function, load in global axis) ===
^Line 2 (6G10.0)^^
|SIGXF | (*Fmult) Total stress in X axis|
|SIGYF | (*Fmult) Total stress in Y axis|
|SIGZF | (*Fmult) Total stress in Z axis|
|SIGXD | (*Dmult) Total stress in X axis|
|SIGYD | (*Dmult) Total stress in Y axis|
|SIGZD | (*Dmult) Total stress in Z axis|

=== 2.4 If IVAL = 14 (constant function, load in global axis) ===
^Line 2 (6G10.0)^^
|SIGXF | (*Fmult) Total stress in X axis|
|SIGYF | (*Fmult) Total stress in Y axis|
|SIGZF | (*Fmult) Total stress in Z axis|
|SIGXD | (*Dmult) Total stress in X axis|
|SIGYD | (*Dmult) Total stress in Y axis|
|SIGZD | (*Dmult) Total stress in Z axis|
^Line 3 (5G10.0)^^
|SIGXC | Constant total stress in X axis|
|SIGYC | Constant total stress in Y axis|
|SIGZC | Constant Total stress in Z axis|
|TIME1 | Time of decreasing beginning|
|TIME2 | Time of decreasing end|

  * If time < TIME1:
    * SIGX = SIGX
    * SIGY = SIGY
    * SIGZ = SIGZ
  * If TIME1 < time < TIME2:
    * SIGX = SIGX *(TIME2-time)/(TIME2-TIME1) 
    * SIGY = SIGY *(TIME2-time)/(TIME2-TIME1)
    * SIGZ = SIGZ *(TIME2-time)/(TIME2-TIME1)
  * If time > TIME2:
    * SIGX = 0
    * SIGY = 0
    * SIGZ = 0
===== Stresses =====
==== Number of stresses ====
3 for 3D state \\ 2 for the other cases
==== Meaning ====
__For the 3-D state__:
|SIG(1)|current value of the normal pressure|
|SIG(2)|current value of the tangent load in the ξ direction|
|SIG(3)|current value of the tangent load in the η direction|

__For the other cases__:
|SIG(1)|current value of the normal pressure|
|SIG(2)|current value of the tangent load|


===== State variables =====
==== Number of state variables ====
1
==== List of state variables ====
|Q(1)|= 0 (meaningless)|
Lagamine

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