Introduction
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Definition of a proportional load
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Considering a structure subjected to a thermo-mechanical loading in the time interval :math:`[\mathrm{0,}t]`, it will be said that this loading is proportional (or even radial) to the material point :math:`p` if the stress field represented at this point by the tensor :math:`\sigma` is proportional to a tensor independent of the moment in question, the coefficient of proportionality being a monotonic function of time. Formally, this will be expressed as:

:math:`\forall \tau \left[\mathrm{0,}t\right],\sigma (P,\tau )=\alpha (\tau ){\sigma }_{0}(P),\alpha (\tau )>0` monotonic function in :math:`[\mathrm{0,}t]`.

This definition implies, in particular, that the main directions of the stresses remain constant, at the point in question, throughout the loading path (these directions can of course be variable from one point to another).




Importance of proportional loading and the usefulness of indicators
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For plastic materials, mechanical fields depend on the whole story that has passed during the loading journey. Flow laws are therefore incremental in nature and their integration depends on each load case. A notable exception concerns proportional loading, for which the flow law can be integrated once and for all. For example, the Prandtl-Reuss law of plasticity based on the Von Mises criterion can be replaced by a nonlinear elastic law (called the Hencky-Mises law). Cases of strictly proportional loading are quite rare. Indeed, a large number of conditions must be met to carry out such a case [:ref:`bib1 <bib1>`] and these conditions are not often verified for industrial structures. It can even be said that for structures with geometric defects such as cracks, these conditions are never strictly verified.

When the loads are multi-axial, cyclic, or transient thermo-mechanical, some sections of the loading path may be highly non-proportional. It is then useful to identify these sections and to assess the extent of the loss of proportionality, in order for example to adjust the time discretization of the elastoplastic problem for the section in question, or to assess the validity of certain post-treatments (in fracture mechanics for example).




Different types of indicators of loss of proportionality
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It seems difficult to define a single and simple quantity that could detect both spatial areas of loss of proportionality and sections of loading paths (temporal zones) at a material point. This is why we propose scalar quantities that each have their own specificity: two, defined by fields measuring at each point the discharge and the deviation of constraints between two time steps (local indicators), two others of a more global nature, characterizing in a given area of the structure a history of non-proportional loading.

**Note:**

*These indicators are closely linked to the discretization of the problem at time. In particular, if this discretization is too crude, it is very possible not to detect the discharge or the loss of radiality occurring during the time increment.*




elastoplastic discharge indicators
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Another application of discharge indicators consists in alerting the user, if, in the event of significant discharge, the choice of kinematic work hardening would provide a very different solution from the isotropic work hardening used (cf. CR- AMA -11.035 :ref:`[3] <RefNumPara__30524177>`).

This is of practical interest: in several studies, the behavior very often chosen "by default" (because we often only have a traction curve) is VMIS_ISOT_TRAC. However, if local discharges are possible, this can typically lead to overestimating the stresses (with imposed deformation) or underestimating the deformations (with imposed stress). It therefore seems appropriate to warn the user if, when he has used Von Mises laws with isotropic work hardening, he risks obtaining false results when the discharge becomes too large (does not therefore remain in the initial elasticity domain). This may indicate to him that it is necessary to use laws of behavior with kinematic work hardening (VMIS_CINE_LINE, VMIS_ECMI *, VMIS_CIN1_CHAB, etc.).