Formulation of the model
=====================




Theoretical framework
---------------

In the initial formulation proposed in [:ref:`bib1 <bib1>`], two distinct damage variables, each with its own kinetics, are proposed. A variable :math:`\phi` is in particular associated with changes in the microstructure depending solely on time (i.e. static aging of the material), while the variable :math:`\omega` describes the cavitation mechanisms developing under the combined influence of viscoplastic deformation and the triaxiality of the stresses.

In the modeling selected in Code_Aster, the variable :math:`\phi` is retained. In practice, its identification is difficult, and it is possible not to involve microstructural aging by setting the coefficient :math:`{k}_{c}` to zero (cf. equations in section 2.2). Moreover, the variable :math:`\omega` is renamed :math:`D` because its law of evolution is different from that proposed by Hayhurst (cf. [:ref:`bib1 <bib1>`]): the law is here in hyperbolic sine. The benefits of this formulation are detailed in [:ref:`bib2 <bib2>`].




Model equations
-------------------

The model equations are written as:

:math:`\mathrm{\{}\begin{array}{c}\mathit{élasticité}\mathrm{:}\\ \sigma \mathrm{=}(1\mathrm{-}D)C{\varepsilon }^{e}\text{et}{\varepsilon }^{e}\mathrm{=}\varepsilon \mathrm{-}{\varepsilon }^{\mathit{th}}\mathrm{-}{\varepsilon }^{p}\\ \text{}\\ \mathit{viscoplasticité}\mathrm{:}\\ \dot{{\varepsilon }^{p}}\mathrm{=}\frac{3}{2}\dot{p}\frac{\tilde{\sigma }}{{\sigma }_{\mathit{eq}}}\text{avec}\dot{p}\mathrm{=}{\varepsilon }_{0}\text{sinh}(\frac{{\sigma }_{\mathit{eq}}(1\mathrm{-}H)}{K(1\mathrm{-}D)(1\mathrm{-}\phi )})\\ \dot{\phi }\mathrm{=}\frac{{k}_{c}}{3}{(1\mathrm{-}\phi )}^{4}\\ \text{}\\ \mathit{écrouissage}\mathrm{:}\\ H\mathrm{=}{H}_{1}+{H}_{2}\\ \dot{{H}_{i}}\mathrm{=}\frac{{h}_{i}}{{\sigma }_{\mathit{eq}}}({H}_{i}^{\text{*}}\mathrm{-}{\delta }_{i}{H}_{i})\dot{p}\text{pour}i\mathrm{=}\mathrm{1,2}\\ \text{}\\ \mathit{endommagement}\mathrm{:}\\ \text{si}{\alpha }_{\sigma }\mathrm{=}0\dot{D}\mathrm{=}{A}_{0}\text{sinh}(\frac{{\alpha }_{D}\text{<}{\sigma }_{I}{\text{>}}_{\text{+}}+{\sigma }_{\mathit{eq}}(1\mathrm{-}{\alpha }_{D})}{{\sigma }_{0}})\\ \text{si}{\alpha }_{\sigma }\mathrm{=}1\dot{D}\mathrm{=}{A}_{0}\text{sinh}(\frac{{\alpha }_{D}\text{<}\mathit{tr}(\sigma ){\text{>}}_{\text{+}}+{\sigma }_{\mathit{eq}}(1\mathrm{-}{\alpha }_{D})}{{\sigma }_{0}})\end{array}`

where:


.. csv-table::

    ":math:`\varepsilon`, :math:`{\varepsilon }^{\text{e}}`, :math:`{\varepsilon }^{\text{th}}` and :math:`{\varepsilon }^{\text{p}}` ", "are the total, elastic, thermal, and plastic deformations respectively,"
    ":math:`\text{<}x{\text{>}}_{\text{+}}` ", "is the positive part of :math:`x`,"
    ":math:`{\sigma }_{1}` ", "is the maximum principal stress,"
    ":math:`\tilde{\sigma }\mathrm{=}\sigma –\frac{1}{3}\mathit{Tr}(\sigma )I` ", "is the deviatoric part of the stress tensor,"
    ":math:`{\sigma }_{\mathit{eq}}\mathrm{=}\sqrt{\frac{3}{2}\tilde{{\sigma }_{\mathit{ij}}}\tilde{{\sigma }_{\mathit{ij}}}}` ", "is the Von-Mises deviatoric constraint,"
    ":math:`C` ", "is the elastic stiffness tensor,"
    ":math:`p` ", "is the cumulative plastic deformation,"
    ":math:`H`, :math:`{H}_{1}`, :math:`{H}_{2}` ", "are the viscoplastic isotropic work hardening variables,"
    ":math:`D` ", "is the isotropic damage scalar variable,"
    ":math:`\phi` ", "is the microstructural damage scalar variable,"
    ":math:`{\alpha }_{\sigma }` ", "is the parameter allowing you to choose to calculate the damage compared to :math:`({\sigma }_{\mathit{eq}},{\sigma }_{1})` for :math:`{\alpha }_{\sigma }\mathrm{=}0` or :math:`({\sigma }_{\text{eq}},\text{Tr}(\sigma ))` for :math:`{\alpha }_{\sigma }\mathrm{=}1`,"
    ":math:`{\alpha }_{\text{D}}` ", "is the parameter for adjusting the sensitivity to triaxiality :math:`({\alpha }_{\text{D}}\mathrm{=}1)` or the sensitivity to the maximum principal stress :math:`({\alpha }_{\text{D}}\mathrm{=}0)` for the calculation of the damage,"
    ":math:`{\delta }_{i}` ", "is 0 or 1 depending on whether linear or non-linear isotropic work hardening is desired, respectively."


**Note Bene:**

The parameters of model :math:`K,{\mathrm{\epsilon }}_{\mathrm{0,}}{\mathrm{\sigma }}_{\mathrm{0,}}{h}_{\mathrm{1,}}{h}_{\mathrm{2,}}{A}_{\mathrm{0,}}{\mathrm{\alpha }}_{D},\text{et}{k}_{c}` can be temperature functions (in :math:`°C`). In [:ref:`bib2 <bib2>`], the parameters identified vary according to the temperature according to an Arrhenius law.

In order to prevent damage from progressing, the user can use a value of :math:`{A}_{0}` that is zero. In this case, it is nevertheless a question of entering a value for :math:`{\mathrm{\sigma }}_{0}`, which occurs in the hyperbolic sine, a value that is not zero and is sufficiently large.

**Note:**

The preceding system of equations can be reduced: in fact, those relating to the evolution of work-hardening are integrated as follows:

:math:`{H}_{i}\mathrm{=}\frac{{H}_{i}^{\text{*}}}{{\delta }_{i}}\left[1\mathrm{-}\mathrm{exp}(\frac{\mathrm{-}{h}_{i}{\delta }_{i}}{{\sigma }_{\mathit{eq}}}p)\right]`

and the equation relating to micro-structural damage amounts to:

:math:`\varphi =1-\frac{1}{{(1+{k}_{c}t)}^{1/3}}`

It is this expression that will be used in the rest of the document.