1. Introduction

When the loading path is no longer monotonic, isotropic and kinematic work hardening is no longer equivalent. In particular, it is possible to expect to have simultaneously a kinematic part and an isotropic part. If one seeks to accurately describe the effects of cyclical loading, it is desirable to adopt sophisticated (but easy to use) models such as the Taheri model, for example, see [R5.03.05]. On the other hand, for less complex loading paths, it may be desirable to include only linear kinematic work hardening, all the non-linearities of work hardening being carried by the term isotropic. This makes it possible to accurately describe a traction curve, while still representing phenomena such as the Bauschinger effect [1] (see for example the figure).

The characteristics of work-hardening are then given by a traction curve and a constant, called Prager’s, for the term linear kinematic work hardening. They are introduced in command DEFI_MATERIAU:

Isotropic linear work hardening	Nonlinear isotropic work hardening
DEFI_MATERIAU / ECRO_LINE SY: limite d’élasticité D_SIGM_EPSI: pente de la courbe de traction PRAGER: (C: constante de Prager )	material_challenge traction: (SIGM: pull curve prager: (C: Prager constant)

These characteristics can also depend on the temperature, provided that the keywords factors ECMI_LINE_FO and ECMI_TRAC_FO are then used instead of ECRO_LINE and TRACTION. The use of these laws of behavior is available in commands STAT_NON_LINE or DYNA_NON_LINE:

Isotropic linear work hardening	Nonlinear isotropic work hardening
STAT_NON_LINE COMPORTEMENT: RELATION:‘VMIS_ECMI_LINE’	STAT_NON_LINE COMPORTEMENT: RELATION: “VMIS_ECMI_TRAC”

In the rest of this document, the combined work hardening model is described in detail. The details of its numerical integration in connection with the construction of the coherent tangent matrix are then presented. Finally, a uniaxial tensile compression test illustrates the identification of the characteristics of the material.