Introduction
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Multiaxial fatigue endurance models under periodic loading are models of the following type:

:math:`{\text{VAR}}_{\text{amplitude}}+a\mathrm{\times }{\text{VAR}}_{\text{moyenne}}<b`,

where :math:`b` is the endurance threshold under simple shear, and *a* is a dimensionless positive constant. :math:`{\text{VAR}}_{\text{amplitude}}` is a certain definition of the magnitude (half of the variation) of the loading cycle and :math:`{\text{VAR}}_{\text{moyenne}}` is a variable in relation to the average stress (or sometimes deformation) or stresses (or sometimes deformations). The models are distinguished by different definitions of :math:`{\text{VAR}}_{\text{amplitude}}` and :math:`{\text{VAR}}_{\text{moyenne}}`.

To go from endurance to the accumulation of damage, an equivalent constraint is introduced defined by:

:math:`{\sigma }_{\text{eq}}\mathrm{=}{\text{VAR}}_{\text{amplitude}}+a\mathrm{\times }{\text{VAR}}_{\text{moyenne}}`.

This equivalent stress gives us a unit damage on the fatigue curve. Since the second member of inequality :math:`b` corresponds to the shear threshold, a shear fatigue curve is needed. But shear fatigue curves are rare because they are difficult to obtain, so an attempt is made to use fatigue curves in alternating tensile compression. To do this, it is necessary to multiply the equivalent stress by a corrective coefficient of the order of :math:`\sqrt{3}`.

The macroscopic models of MATAKE (critical plane) and micro macro models of DANG - VAN are described. We show that under certain hypotheses the DANG - VAN model is similar to the macroscopic model of MATAKE. The only difference is in the variable :math:`{\text{VAR}}_{\text{moyenne}}`: DANG - VAN uses hydrostatic pressure, while MATAKE uses normal stress in terms of maximum shear amplitude.

After defining the shear plane, we express the shear stress in this plane. The shear planes are then explored using a method described in reference [:ref:`bib4 <bib4>`] which consists in cutting the surface of a sphere into pieces of equal sizes.

Since the normal vectors are known, we then determine for each plane the points which are the most distant from each other. Among these we find the two points that are farthest from each other. This being done, we use, if necessary, the circle method passing through three points in order to obtain the circle circumscribed to the loading path.

In the first part of this document we present endurance models in multiaxial fatigue under periodic loading, as well as the concept of accumulation of damage. The transition from stamina to damage accumulation is also discussed.

In the second part, the criteria of MATAKE and DANG - VAN are then presented under the aspects of endurance limit and accumulation of damage under periodic loading.

The third part is devoted to the definition of the shear plane, the expression of shear stresses in this plane and finally, how to explore shear planes.

The fourth part is dedicated to the determination of the circle circumscribed to the shear path in the plane of the same name. Finally we describe the criteria and quantities that are introduced in*Code_Aster*.

After extending the models of MATAKE and DANG - VAN to the accumulation of damage under periodic loading, we present the adaptation of these models to the accumulation of damage under non-periodic loading. Thus, the fifth part is devoted to the definition of the elementary equivalent stress. We also describe the modified FATEMI - SOCIE criterion.

The sixth part is devoted to how to select the axis (or both axes) onto which the history of the split is projected.

The seventh part is dedicated to the actual projection of the tip of the cission vector on this axis or these two axes. Finally, concerning the criteria of MATAKE and DANG - VAN formulated in the accumulation of damage under non-periodic loading, we describe the quantities that are introduced in*Code_Aster.*