Chaboche elasto-visco-plastic constitutive model with thermal and cyclic effects for solid elements at constant or variable temperatures with damage computation.
The model is highly adjustable and can be used for very simple laws (elastic, bilinear plasticity) as well as for more complex behaviors (visco-plasticity, isotropic hardening, kinematic hardening, cyclic hardening, …)

Implemented by: Hélène Morch, 2016-2019

Prepro: LCHAB.F
Lagamine: CHAB.F, CHABDAM.F

The Chaboche model is a viscoplastic constitutive model that allows thermo-mechanical cyclic analysis of solids.

The viscosity function of the model is the Norton-Hoff equation: \[\dot{p}=\left<\frac{\sigma_v}{K}\right>^n \] Where:

• $\dot{p}$ is the plastic strain rate norm
• $\left<x\right>= \begin{cases} x & \text{if } x>0 \\ 0 & \text{if } x\leq 0 \end{cases}$
• $\sigma_v=\Vert \underline{\sigma}-\underline{X} \Vert -R-\sigma_Y$

Parameter $K$ can be put to a value close to 0 to model elasto-plasticity (without viscous effects).
$R$ is the isotropic hardening variable, which evolves with the plastic strain rate: \[ \dot{R}=b(Q-R)\dot{p}\]

The back-stress $\underline{X}$ controls kinematic hardening. In the Chaboche model, the back-stress is determined through the summation of Armstrong-Fredericks equations: \[\underline{X}=\displaystyle\sum_{i=1}^{nAF} \underline{X}_i\] \[\underline{\dot{X}}_i=\frac{2}{3}C_i\underline{\dot{\varepsilon}}^p-\gamma_i(\underline{X}_i-\underline{Y}_i)\dot{p}-b_i\|\underline{X}_i\|^{r_i-1}\underline{X}_i+\frac{1}{C_i}\frac{dC_i}{dT}\dot{T}\underline{X}_i\]

The parameter $\gamma_i$ can be made to evolve in order to take into account cyclic hardening of the material: \[\dot{\gamma_i}=D_{\gamma_i}(\gamma_i^0-\gamma_i)\dot{p}\] \[\gamma_i^0=a_{\gamma_i}+b_{\gamma_i}\exp(-c_{\gamma_i}q)\]

Where $q$ is the radius of the plastic strain memory surface.

In the Armstrong-Fredericks equations, a modification tensor $\underline{Y}_i$ can be used to model mean stress evolution. This tensor evolves with respect to the norm of the back-stress: \[\dot{\underline{Y}}_i=-\alpha_{b,i}\left(\frac{3}{2}Y_{st,i}\frac{\underline{X}_i}{\|\underline{X}_i\|}+\underline{Y}_i\right)\|\underline{X}_i\|^{r_i}\]

2+IANISOTH+NAF+NAFcyc+NAFY lines repeated NTEMP times
Parameters must be introduced by increasing temperature order

Parameters in this case follow an exponential law of the type: \[P(T)=A_P(1-B_P*\exp(\frac{T}{C_P}))\]