1. Chaboche elasto-visco-plastic models available in Code_Aster#
For the calculation of structures subject to cyclic loading, conventional isotropic (linear or not) and linear kinematics [R5.03.02] and [R5.03.16] workings are no longer sufficient. In particular, it is not possible to correctly describe the stabilized cycles obtained experimentally on a tensile test subject to an alternating imposed deformation or a traction-compression.
If one seeks to precisely describe the effects of cyclical loading, it is desirable to adopt more sophisticated (but easy to use) models such as the Saïd Taheri model, for example, cf. [R5.03.05], or else if the number of cycles is limited, the model by Jean-Louis Chaboche which is presented here.
In reality, the Chaboche model can be more or less sophisticated. The models developed in Code_Aster include either one kinematic variable (VMIS_CIN1_CHAB and VISC_CIN1_CHAB) or two (VISC_CIN2_CHAB and VMIS_CIN2_CHAB), and isotropic work hardening.
The choice to use two kinematic variables certainly complicates the model, but makes it possible to correctly identify uniaxial tests in a wider range of deformations [bib2], [bib7]. A number of identifications of the parameters of this model have been carried out mainly for A316 and A304 stainless steels ([bib7], [bib8]).
The models have 8 parameters (a single kinematic variable) or 10 (two kinematic variables), introduced in command DEFI_MATERIAU:
CIN1_CHAB (CIN1_CHAB_FO) = _F (
♦ R_0 = R_0, ◊ R_I = R_I, (useless if B=0) ◊ B = b, (default: 0.)
♦ C_I = C_I, ◊ K = k, (default: 1.) ◊ W = w, (default: 0.)
♦ G_0 = G_0, ◊ A_I = A_I, (default: 0.) )
CIN2_CHAB (CIN2_CHAB_FO) = _F (
♦ R_0 = R_0, ◊ R_I = R_I,
useless if B=0 or if memory effect)
◊ B = b, (default: 0.)
♦ C1_I = C1_I, ♦ C2_I = C2_I, ◊ K = k, (default: 1.) ◊ W = w, (default: 0.)
♦ G1_0 = G1_0, ♦ G2_0 = G2_0, ◊ A_I = A_I, (default: 0.) )
The 8 or 10 parameters are real constants. All these parameters can depend on the temperature (keywords CIN1_CHAB_FO or CIN2_CHAB_FO) and the expected values are of the function type.
In the case where it is desired to introduce viscosity in addition to viscosity (models VISC_CIN1_CHAB and VISC_CIN2_CHAB), it is also necessary to provide in the command DEFI_MATERIAU, under the keyword LEMAITRE (or LEMAITRE_FO), the parameters N and UN_SUR_K, which may depend on the temperature.
LEMAITRE (LEMAITRE_FO) = _F (
♦ N = n, ♦ UN_SUR_K = 1/K
The parameter UN_SUR_M of the LEMAITRE keyword (respectively LEMAITRE_FO) must be set to zero (respectively to the function identically null).
It is also possible to take into account a memory effect of the greatest plastic deformation using the models (VISC_CIN2_MEMO and VMIS_CIN2_MEMO). The keywords to fill in are:
MEMO_ECRO (MEMO_ECRO_FO) = _F (
♦ Q_M = Qm, ♦ Q_0 = Q0, ♦ MU = mu, ♦ ETA = eta, (default: 0.5)
In case of non-proportional loading, it is necessary to enrich the model with the data of two additional parameters:
CIN2_NRAD = _F (
◊ DELTA1 = :math:`{\delta }_{1}` (default= 1.E+0),
◊ DELTA2 = :math:`{\delta }_{2}` (default= 1.E+0),
with :math:`0\mathrm{\le }{\delta }_{1}\mathrm{\le }1`, :math:`0\mathrm{\le }{\delta }_{2}\mathrm{\le }1`
The laws of behavior are available in all commands using the COMPORTEMENT keyword with the following relationships:
VISC_CIN1_CHAB, VISC_CIN2_CHAB, VISC_CIN2_MEMO, VISC_CIN2_NRAD, VISC_MEMO_NRAD,, VMIS_CIN1_CHAB, VMIS_CIN2_CHAB, VMIS_CIN2_MEMO, VMIS_CIN2_NRAD, VMIS_MEMO_NRAD.
Note: the VISCOCHAB [R5.03.14] model also allows you to represent the effects described in this document. It also includes additional restoration and work-hardening terms. But its use in structural calculations is more expensive in terms of calculation time (because one must solve a system of 27 equations with 27 unknowns either by the Runge-Kutta method or by the Newton method). In addition, it poses robustness problems when the time step is large, because Newton’s method can fail. This results in numerous subdivisions of the time step.
The models described in this document are optimized, insofar as they lead to solving a single scalar resolution, and the resolution method used is very robust (Brent method or secant, cf. [R5.03.14]); it is therefore a model capable of rapidly integrating large time steps.
In the rest of this document, the characteristics of the various models are described. The details of their numerical integration in connection with the construction of the coherent tangent matrix are then presented. Finally, some elements are also given for identifying the characteristics of the material.