1. Field of application#

1.1. Purpose#

The law of behavior VMIS_ISOT_TRAC makes it possible to account for any form of non-linear isotropic work hardening via the data of the stress-deformation response in a simple tensile test, in the form of a sampled function. However, identifying this response may be simpler by giving yourself a pre-established function that depends on a small number of parameters. The user is then also exempt from the sampling phase, which is sometimes delicate when the function varies rapidly. The performance of integrating the law of behavior also benefits, especially when the sampling required would have been particularly fine.

This is the whole purpose of the law of behavior VMIS_ISOT_NL (plasticity) or VISC_ISOT_NL (viscoplasticity), dedicated to metallic materials. The work hardening function \(R(\kappa )\) consists in the assembly of several elementary terms:

(1.1)#\[\begin{split}R(\kappa )=\{\begin{array}{cc}\widehat{R}({\epsilon }_{L}^{p})& \text{si}\kappa \le {\epsilon }_{L}^{p}\\ \widehat{R}(\kappa )& \text{si}\kappa \ge {\epsilon }_{L}^{p}\end{array}\end{split}\]

where \(\kappa\) designates the work-hardening variable (which generally coincides with the cumulative plastic deformation), \({\epsilon }_{L}^{p}\) sets the limit of the Lüders plateau (zero in the absence of a plate) and:

(1.2)#\[\widehat{R}(\kappa )={R}_{0}+{R}_{H}\kappa +{R}_{1}(1-{e}^{-{\gamma }_{1}\kappa })+{R}_{2}(1-{e}^{-{\gamma }_{2}\kappa })+{R}_{K}{({p}_{0}+\kappa )}^{{\gamma }_{K}}\]

The cancellation of the affine term, of each exponential term or of the term in power is carried out by fixing respectively \({R}_{H}=0\), \({R}_{1}=0\) or \({R}_{2}=0\) and finally \({R}_{K}=0\), which allows access to a simpler work hardening function if necessary.

This behavior model is compatible with:

  • with a kinematics of small disturbances (HPP) or large deformations (GDEF_LOG);

  • with a plastic response independent of the loading speed (VMIS_ISOT_NL) or viscoplastic (VISC_ISOT_NL) based on a Norton law;

  • with a local formulation or with an internal variable gradient in which the gradient of the work-hardening variable intervenes, cf. [Lorentz & Andrieux, 1999] and [Zhang et al., 2018].

Finally, it should be noted that the elastic part of the model is expressed explicitly, that is to say that the stress is expressed as a function of deformation and plastic deformation in the current state. There is therefore no incremental evolution of the constraint; it does not contribute to defining the mechanical state of the system, it is a consequence of it (in particular, the constraint does not enter into the definition of an initial state of the system).

Note: the practical consequence of this formalism is that an initial constraint given in ETAT_INIT for STAT_NON_LINE will be ignored.

1.2. Material parameters#

The modular expression of the behavioral relationship makes it possible to group the various parameters of the model by categories:

  • the elastic part is isotropic and depends on the Young’s modulus \(E\) and the Poisson’s ratio \(\nu\) defined under the keyword factor ELAS (E, NU) of the command DEFI_MATERIAU;

  • the isotropic work hardening parameters () are defined under the keyword factor ECRO_NL (R0, RH, R1, GAMMA_1, R2, GAMMA_2, RK, P0, P0, GAMMA_K, EPSP_LUDERS);

  • the parameters of Norton’s viscosity law, when present, are defined under the keyword factor NORTON (coefficient K and exponent N);

  • in non-local formulation [R5.04.01], you must also define the corresponding parameters under the keyword factor NON_LOCAL (C_GRAD_VARI and COEF_PENA_LAGR)

The parameters may depend on the temperature (keywords factor *_FO). The law of behavior is expressed in mechanical deformations, that is to say it is compatible with the usual shrinkage terms, in particular thermal deformation.

1.3. Internal variables#

The law of behavior is based on eight internal variables: on the one hand, the work-hardening level \(\kappa\) and the six components of plastic deformation (which it is therefore not useful to calculate during post-treatment); and on the other hand, for post-processing purposes only, the state of the current time step is also maintained, the state of the current time step is also maintained, with values 0=elastic, 1=plastic with regular flow, 2=plastic with singular flow, 2=plastic with singular flow, and 2=plastic with singular flow, and 2=plastic with singular flow, we also maintain the state of the current time step, (we will see later what this is about).

The internal variables are grouped together in the following table:

EPSEQ

V1

Work hardening variable \(\kappa\)

INDIPLAS

V2

State of the current time step (0, 1, or 2)

EPSPXX - EPSPYZ

V3 - V8

Components of plastic deformation