====== MULTIDAM23D ======

===== Description =====

Elasto(-visco)-plastic damage law of anisotropic materials for solid elements at constant temperature.

==== The model ====

This law is used for mechanical analysis of elasto(-visco)-plastic damage orthotropic solids undergoing large strains, plastic mixed hardening and damage anisotropic hardening are assumed.

==== Files ====

Prepro: LMULTIDAM2.F \\
Lagamine: MULTIDAM23D.F

===== Availability =====

|Plane stress state| NO |
|Plane strain state| NO |
|Axisymmetric state| NO |
|3D state| YES |
|Generalized plane state| NO |

===== Input file =====

==== Parameters defining the type of constitutive law ====

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 545|
|COMMNT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (7I5) ^^
|NINTV| Number of sub-steps used to integrate numerically the constitutive equation in a time step |
|NPOINT| = 2 : (Bilinear evolution of the EVP law) or more (multilinear evolution) |
|IVISC| = 1 for EVP law |
|:::| ≠ 1 for EP law |
|MMATE| = 1 : Brittle material |
|:::| ≠ 1 : Ductile material |
|MNINTV| Maximum of number of sub-steps (0$\rightarrow$100) |
|MITERA| Number of sub-iteration (0$\rightarrow$10) in the plastic and damage correction loop |
|MUTIP| Number of multiplicator for sub-steps (0$\rightarrow$2), when strain variation is found too big, the limit is Deltamin |

==== Real parameters ====

^ Line 1 (4G10.0) ^^
|ECROU| = 0 : Isotropic hardening |
|:::| = 1 : Cinematic hardening |
|:::| $\in$ [0,1] : Mixed hardening |
|DMMAX| = 0 : EP without damage |
|:::| $\in$ [0,1] : maximum damage value at initial fracture |
|:::| Otherwise : 0.95 limit damage value |
|PROC| Precision of iteration (= 0$\rightarrow$1.D-3)|
|DELTAMIN| Maximum of the permitted equivalent strain increment \\ Necessary because of instability in damage resolution |
^Line 2 - Only if IANA ≠ 4 (2D state) (G10.0)^^
|THICK| Thickness for __plane state__ |
^Line 2 or 3 (G10.0)^^
|VISCO| Viscosity parameter (unit : time) |

==== Material parameters ====

^ Line 1 (3G10.0) ^^
|ANU12| POISSON’s ratio in 1-2 plane |
|ANU23| POISSON’s ratio in 2-3 plane |
|ANU13| POISSON’s ratio in 1-3 plane |
^ Line 2 (6G10.0) ^^
|EPSY1| Initial elastic strain limit of uniaxial tension in 1 direction |
|EPSY2| Initial elastic strain limit of uniaxial tension in 2 direction |
|EPSY3| Initial elastic strain limit of uniaxial tension in 3 direction |
|EPSY12| Initial elastic strain limit in 1-2 plane |
|EPSY23| Initial elastic strain limit in 2-3 plane |
|EPSY13| Initial elastic strain limit of 1-3 plane |
^ Line 3 (6G10.0) ^^
|SIGY1| Yield limit of uniaxial tension in 1 direction |
|SIGY2| Yield limit of uniaxial tension in 2 direction |
|SIGY3| Yield limit of uniaxial tension in 3 direction |
|SIGY12| Yield limit in 1-2 plane |
|SIGY23| Yield limit in 2-3 plane |
|SIGY13| Yield limit in 1-3 plane |

==== Effective stress-strain curves ====

__To repeat NPOINT-1 times :__
^ Line 1 (6G10.0) ^^
|EPS1| Strain by uniaxial testing in 1 direction |
|EPS2| Strain by uniaxial testing in 2 direction |
|EPS3| Strain by uniaxial testing in 3 direction |
|EPS12| Strain by testing in plane 1-2 |
|EPS23| Strain by testing in plane 2-3 |
|EPS12| Strain by testing in plane 1-3 |
^ Line 2 (6G10.0) ^^
|SIG1| Stress by uniaxial testing in 1 direction |
|SIG2| Stress by uniaxial testing in 2 direction |
|SIG3| Stress by uniaxial testing in 3 direction |
|SIG12| Stress by testing in 1-2 plane |
|SIG23| Stress by testing in 2-3 plane |
|SIG13| Stress by testing in 1-3 plane |

{{ :laws:multidam2.png?400 |}}

==== Damage parameters ====

^ Line 1 (6G10.0) ^^
|RD01| Initial damage limit in 1 direction |
|RD02| Initial damage limit in 2 direction |
|RD03| Initial damage limit in 3 direction |
|DT1| Damage tangent modulus in 1 direction |
|DT2| Damage tangent modulus in 2 direction |
|DT3| Damage tangent modulus in 3 direction |

===== Stresses =====

==== Number of stresses ====

6 for 3D state \\ 4 for the other cases

==== Meaning ====

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\

For the 3-D state:
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{zz}$|
|SIG(4)|$\sigma_{xy}$|
|SIG(5)|$\sigma_{xz}$|
|SIG(6)|$\sigma_{yz}$|

For the other cases:
|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{xy}$|
|SIG(4)|$\sigma_{zz}$|

===== State variables =====

==== Number of state variables ====

= 32 for 3D state \\ = 30 for the other cases

==== List of state variables ====

N = 32 for 3D state \\ 
N = 30 for the other cases.

|Q(1)| Element thickness ($t$) in plane stress state |
|:::| = 1 : Plane strain state|
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|:::| = 0 : 3D state |
|Q(2)| = 0 : Current state is elastic |
|:::| = 1 : Current state is elasto-plastic |
|Q(3)| = 0 : Current state is not damaged |
|:::| = 1 : Current state is damaged |
|Q(4)| Equivalent plastic strain ($\varepsilon_{eq}$) |
|Q(5)| Equivalent damage ($d_{eq}$) | 
|Q(6)| Plastic hardening level ($R$) |
|Q(7)| Damage hardening level ($B$) |
|Q(8)| Damage in 1 direction of material ($D_1$) |
|Q(9)| Damage in 2 direction of material ($D_2$) |
|Q(10)| Damage in 3 direction of material ($D_3$) |
|Q(11)| Equivalent stress ($\sigma_{eq}$) |
|Q(12)| Plastic work per unit volume ($W_p$) |
|Q(13)| Damage work per unit volume ($W_d$) |
|Q(14)| Total strain energy per unit volume ($W_t$) (elastic + plastic + damage) |
|Q(15)$\rightarrow$Q(20)| Fracture criteria (computed with the real stress-strain evolution) |
|Q(21)$\rightarrow$Q(26)| Strain values at the integration points (x ,y, z, xy,yz and yz directions) |
|Q(27)$\rightarrow$Q(N)| Back stresses for kinematic and mixed hardening |