Model description
=====================

The behavioral relationship proposed by Taheri [:ref:`bib5 <bib5>`] makes it possible to describe the response of austenitic steels under cyclic stresses: it is in fact well suited to represent the phenomenon of progressive deformation. Before setting out the equations themselves, we can specify that this model differs from classical plasticity (VonMises criterion with kinematic and isotropic work hardening) by two particularities, which are sources of difficulties in numerical formulation. On the one hand, the evolution of dissipative variables is based on two load criteria instead of one: the first, conventional, conditions the appearance of plastic deformation, the second makes it possible to keep track of the "maximum" work hardening achieved by the material in order to account for the ratchet phenomenon. On the other hand, to represent the progressive deformation in a satisfactory manner, a semi-discrete internal variable has been introduced. Constant when the behavior is dissipative, it only changes in the elastic regime of the material. Although original in appearance, this model is nevertheless based on physical bases, which are still shown in Taheri [:ref:`bib5 <bib5>`], [:ref:`bib7 <bib7>`]. It is accessible, in an extended viscoplastic version (necessary to describe the behavior under high temperatures), by the STAT_NON_LINE command under the keyword RELATION: VISC_TAHERI.




Plastic behavior
----------------------

A detailed description of the law of behavior is given in Taheri et al. [:ref:`bib6 <bib6>`], [:ref:`bib7 <bib7>`]. Briefly, the state of the material is described by its state of deformation, its temperature and four internal variables:


.. csv-table::

    ":math:`\varepsilon` ", "total deformation tensor"
    ":math:`T` ", "temperature"
    ":math:`p` ", "cumulative plastic deformation"
    ":math:`{\varepsilon }^{p}` ", "plastic deformation tensor"
    ":math:`{\sigma }^{p}` ", "peak stress, memory of maximum work hardening"
    ":math:`{\varepsilon }_{n}^{p}` ", "plastic deformation tensor at the last discharge (semi-discrete variable)."


The equations of state that express the associated thermodynamic forces as a function of state variables are written as:


.. _RefEquation 1.1-1.1-1:

:math:`\sigma \mathrm{=}K\text{Tr}(\varepsilon \mathrm{-}{\varepsilon }^{\mathit{th}})\mathrm{Id}+2\mu (\tilde{\varepsilon }\mathrm{-}{\varepsilon }^{p})\text{}{\varepsilon }^{\mathit{th}}\mathrm{=}\alpha (T\mathrm{-}{T}^{\text{réf}})\text{Id}` eq 1.1-1.1-1

:math:`R\mathrm{=}\text{D}\left[{R}^{0}+{(\frac{2}{3})}^{a}A{({\varepsilon }^{p}\mathrm{-}{\varepsilon }_{n}^{p})}_{\text{eq}}^{a}\right]\text{}\text{D}\mathrm{=}1\mathrm{-}m{e}^{\mathrm{-}bp(1\mathrm{-}\frac{{\sigma }^{p}}{S})}` eq 1.1-1.1-2

:math:`\mathrm{X}\mathrm{=}\text{C}\left[S{\varepsilon }^{p}\mathrm{-}{\sigma }^{p}{\varepsilon }_{n}^{p}\right]\text{}\text{C}\mathrm{=}{C}_{\mathrm{\infty }}+{C}_{1}{e}^{\mathrm{-}bp(1\mathrm{-}\frac{{\sigma }^{p}}{S})}` eq 1.1-1.1-3


.. csv-table::

    ":math:`\tilde{a}` ", "deviatoric part of a :math:`a` tensor"
    ":math:`R` ", "isotropic work hardening variable"
    ":math:`X` ", "kinematic work hardening variable"
    ":math:`K,\mu` ", "compressibility and shear modules"
    ":math:`\alpha` ", "thermal expansion coefficient"
    ":math:`{T}^{\text{réŽf}}` ", "reference temperature"
    ":math:`S` ", "ratchet constraint"
    ":math:`b,{R}^{0},A,a,m,{C}_{\infty },{C}_{1}` ", "other work-hardening characteristics of the material"


Note that the elasticity modules and the thermal expansion coefficient are entered by the user by the command DEFI_MATERIAU, keyword ELAS, while the characteristics of the work hardening are set by the keyword TAHERI. These characteristics may depend on temperature, using the keywords ELAS_FO and TAHERI_FO. It should also be noted that an example of identifying work-hardening characteristics on uniaxial situations is given in Geyer [:ref:`bib2 <bib2>`].

The evolution of internal variables is defined by two criteria. The former governs traditional plasticity with combined kinematic and isotropic work hardening:


.. _RefEquation 1.1-1.1-4:

:math:`\text{F}={(\tilde{\sigma }-X)}_{\text{eq}}-R\le 0\text{et}{\sigma }^{0}=\frac{\tilde{\sigma }-X}{{(\tilde{\sigma }-X)}_{\text{eq}}}` eq 1.1-1.1-4


.. csv-table::

    ":math:`{(\text{.})}_{\text{eq}}` ", "equivalent standard: :math:`{a}_{\text{eq}}={(\frac{3}{2}\tilde{a}:\tilde{a})}^{\frac{1}{2}}`"
    ":math:`\text{F}` ", "plasticity criterion"
    ":math:`{s}^{0}` ", "normal outside of criterion :math:`\text{F}`"


This criterion is accompanied by the classic charge/discharge condition:


.. _RefEquation 1.1-1.1-5:

:math:`\mathrm{\{}\begin{array}{c}\text{si F}<0\text{ou}\dot{\sigma }\mathrm{:}{\mathrm{s}}^{0}\mathrm{\le }0\\ \text{si F}\mathrm{=}0\text{et}\dot{\sigma }\mathrm{:}{\mathrm{s}}^{0}>0\end{array}\begin{array}{c}\text{}\dot{p}\mathrm{=}0\\ \text{}\dot{p}\mathrm{\ge }0\text{tel que}\dot{\text{F}}\mathrm{=}0\end{array}\begin{array}{c}\text{}(\text{élasticité})\\ (\text{plasticité})\end{array}` eq 1.1-1.1-5

And the flow law associated with criterion :math:`\text{F}` is:


.. _RefEquation 1.1-1.1-6:

:math:`{\dot{\varepsilon }}^{p}\mathrm{=}\frac{3}{2}\dot{p}{\mathrm{s}}^{0}` and therefore :math:`\dot{p}\mathrm{=}\frac{2}{3}{\dot{\varepsilon }}_{\text{eq}}^{p}` eq 1.1-1.1-6

The second criterion governs the evolution of the peak stress. Geometrically in the space of stress deviators, it reflects the fact that the first load surface :math:`(\text{F}=0)`, represented by a sphere with center :math:`X` and radius :math:`R`, remains inside a sphere with center at origin and radius :math:`{\sigma }^{p}`. It is simply written:


.. _RefEquation 1.1-1.1-7:

:math:`\text{G}={X}_{\text{eq}}+R-{\sigma }^{p}\le 0` eq 1.1-1.1-7

:math:`\text{G}` maximum work hardening criterion

According to the previous geometric considerations, the evolution of the peak stress is:


.. _RefEquation 1.1-1.1-8:

:math:`\{\begin{array}{c}\text{si}\text{G}<0\text{ou}{\dot{X}}_{\text{eq}}+\dot{R}\le 0\text{}{\dot{\sigma }}^{p}=0\\ \text{si}\text{G}=0\text{et}{\dot{X}}_{\text{eq}}+\dot{R}>0\text{}{\dot{\sigma }}^{p}\ge 0\text{tel que}\dot{G}=0\end{array}` eq 1.1-1.1-8

It should be noted that in the natural state of the material, the peak stress is not zero but is equal to the initial elastic limit, namely:

:math:`{\sigma }^{p}(\text{initial})=(1-m){R}^{0}`

So far, we have not mentioned the evolution of the semi-discrete internal variable :math:`{\varepsilon }_{n}^{p}`. In fact, it only evolves in an elastic regime. More exactly, this variable takes into account the state of plastic deformation during the last discharge; in other words, at the start of each discharge, this variable should instantly take on the value of the current plastic deformation. However, to maintain continuous behavior, we regulate the evolution of :math:`{\varepsilon }_{n}^{p}` in the following way:

In elastic regime:


.. _RefEquation 1.1-1.1-9:

:math:`{\dot{\varepsilon }}_{n}^{p}=\dot{\xi }({\varepsilon }_{n}^{p}-{\varepsilon }^{p})\{\begin{array}{c}\text{si}{\varepsilon }_{n}^{p}={\varepsilon }^{p}\\ \text{si}{\varepsilon }_{n}^{p}\ge {\varepsilon }^{p}\end{array}\begin{array}{c}\text{}\dot{\xi }=0\\ \text{}\dot{\xi }\ge 0\text{tq}\dot{F}\text{=0}\end{array}\begin{array}{c}(\text{élasticité classique})\\ (\text{pseudo-décharge})\end{array}` eq 1.1-1.1-9

In a plastic regime:

:math:`\dot{{\varepsilon }_{n}^{p}}=0`

The behavior is thus completely determined. Before moving on to the introduction of viscosity, the observation of the two load surfaces calls for an important remark. One might think that surface :math:`\text{G}\mathrm{=}0` is actually activated only in a plastic regime. In practice, this is not the case. For example, we can cite the case of thermal loading: cooling (generally) leads to an expansion of the load surface :math:`\text{F}=0`, so that the peak stress is forced to evolve to preserve :math:`\text{G}\le 0`, even under elastic conditions.




Taking viscosity into account
-------------------------------

To model the behavior of stainless steels under cyclic loading when the temperature is of the order of :math:`550°C`, it is no longer possible to neglect creep terms. To account for these viscosity effects while maintaining the properties of the previous model, a simple method consists in making the evolution of plastic deformation viscous. In other words, viscosity only occurs under plastic conditions: no direct influence on the semi-discrete internal variable or on the load surface :math:`\text{G}=0`. To do this, by following Lemaitre and Chaboche [:ref:`bib3 <bib3>`], we replace the coherence condition [:ref:`éq 1.1-5 <éq 1.1-5>`] by:


.. _RefEquation 1.2-1.2-1:

:math:`\dot{p}={(\frac{\langle \text{F}\rangle }{K{p}^{1/M}})}^{N}` eq 1.2-1.2-1


.. csv-table::

    ":math:`\langle \text{F}\rangle` ", "positive part of F (Macauley brackets)"
    ":math:`K,N,M` ", "material viscosity characteristics"


The viscosity characteristics of the material are entered in command DEFI_MATERIAU, either by the keyword LEMAITRE if they do not depend on the temperature, or by the keyword LEMAITRE_FO otherwise. In the absence of one of these keywords, the behavior is presumed plastic.

All the other equations in the model are left unchanged. It will be seen that such introduction of viscosity only leads to minor modifications of the algorithm for the implicit integration of the law of behavior.




Description of the internal variables calculated by Code_Aster
----------------------------------------

The internal variables calculated by*Code_Aster* are 9 in number. They are arranged in the following order:


.. csv-table::

    "1"," :math:`p` ", "cumulative plastic deformation"
    "2"," :math:`{\sigma }^{p}` ", "peak constraint"
    "3 to 8"," :math:`{\varepsilon }_{n}^{p}` ", "plastic deformation tensor at the last discharge"
    "", "", "(ranked in order :math:`\mathit{xx}`, :math:`\mathit{yy}`,, :math:`\mathit{zz}`,,, :math:`\mathit{xy}`, :math:`\mathit{xz}`, :math:`\mathit{yz}`)"
    "9"," :math:`\chi` ", "charge/discharge indicator (see [:ref:`§2.3 <§2.3>`])"
    "", "", "0 elastic discharge"
    "", "", "1 conventional plastic filler"
    "", "", "2 plastic filler with two surfaces"
    "", "", "3 pseudo-discharge"


As for the tensor of viscoplastic deformations, it is not ranked among the internal variables but can be calculated post-processing using the command CALC_CHAMP, options' EPSP_ELGA 'or' EPSP_ELNO ', (cf. [:external:ref:`U4.61.02 <U4.61.02>`]).