====== ENDC ======

===== Description =====

Endochronic (internal time) model coupled with damage for elasto-plastic cyclic loading analysis in plane state at constant temperature.

==== The model ====

This law is used for mechanical analysis of 2-D continuum element undergone large deformation by using endochronic (internal time) theory coupled with damage model for elasto-plastic cyclic loading.

==== Files ====

Prepro: LENDC.F \\
Lagamine: ENDC2D.F
===== Availability =====

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

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

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

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 85|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (2I5) ^^
|NINTV| Number of sub-steps used to integrate numerically the constitutive equation in a time step |
|NPOINT| The function form |
|:::| = 1 : Linear expansion (for 2-terms kernal function) |
|:::| = 2 : Saturated expansion (for 2-terms kernal function) |
|:::| \[\rho(\zeta)=\rho_0+\rho_1(\zeta)\]\[\rho_1(\zeta)=\frac{E_1}{E}\;e^{-\alpha_1.\zeta}+\frac{E_2}{E}\] |
|:::| = 3 : Linear expansion (for 3-terms kernal function) |
|:::| = 4 : Saturated expansion (for 3-terms kernal function) |
|:::| \[\rho(\zeta)=\rho_0+\rho_1(\zeta)\]\[\rho_1(\zeta)=\frac{E_1}{E}\;e^{-\alpha_1.\zeta}+\frac{E_2}{E}+\frac{E_3}{E}\;e^{-\alpha_3.\zeta}\] |

==== Real parameters ====

^ Line 1 (7G10.0) ^^
|E| YOUNG's elastic modulus |
|$\nu$| POISSON's ratio |
|$\sigma_o$| Initial yield limit |
|$\sigma_f$| Yield limit at finite strain |
|$E_{to}$| Initial tangent modulus |
|$E_{tf}$| Tangent modulus at finite strain |
|:::| Thickness = 1.0 (by default) |
|$\varepsilon_f$| Equivalent plastic strain at the unloading point |
^ Line 2 (7G10.0) ^^
|$2\sigma_y$| Stress drop during the elastic unloading |
|:::| = 2$\sigma_0$ at initial yielding point |
|$A_o$| Ratio of saturated and initial stress |
|:::| Only for the choice of saturated expansion form, meaning NPOINT = 2 or 4) |
|$\sigma_{po}$| Yield limit for three-terms kernal function |
|$E_{tpo}$| Yield limit for three-terms kernal function |
|Dam| Parameter of damage model |
|Edam| Parameter of damage model |
|Rdam| Parameter of damage model \[D(\zeta)=Dam\;e^{-Edam\;.\;\zeta\;.\;Rdam}\]|

===== Stresses =====

==== Number of stresses ====

4 for plane state

==== Meaning ====

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

|SIG(1)|$\sigma_{xx}$|
|SIG(2)|$\sigma_{yy}$|
|SIG(3)|$\sigma_{xy}$|
|SIG(4)|$\sigma_{zz}$|

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

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

21

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

|Q(1)| = 1 : Plane strain |
|:::| Circumferential strain rate ($\dot{\varepsilon}_{\theta}$) in axisymmetrical state |
|Q(2)| Current yield limit in tension |
|Q(3)| = 0 : Current stress state is elastic |
|:::| = 1 : Current stress state is plastic |
|Q(4)| = 0 : Loading occurs |
|:::| = 1 : Neutral loading occurs |
|Q(5)| Equivalent plastic strain ($\bar{\varepsilon}^p$) |
|Q(6)| Internal time $\zeta$ |
|Q(7)| Current kernal function's value $f_n$ |
|Q(8)| Current derived kernal function's value $df_n$ |
|Q(9)$\rightarrow$Q(20)| Current back stresses |
|Q(21)| Current damage coefficient |

{{ :laws:endc1.png?600 |}}
{{ :laws:endc2.png?600 |}}
{{:laws:endc3.png?300|}} {{ :laws:endc4.png?300|}}