2. Formulation of the model

2.1. Theoretical framework

In the initial formulation proposed in [bib1], two distinct damage variables, each with its own kinetics, are proposed. A variable \(\phi\) is in particular associated with changes in the microstructure depending solely on time (i.e. static aging of the material), while the variable \(\omega\) 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 \(\phi\) is retained. In practice, its identification is difficult, and it is possible not to involve microstructural aging by setting the coefficient \({k}_{c}\) to zero (cf. equations in section 2.2). Moreover, the variable \(\omega\) is renamed \(D\) because its law of evolution is different from that proposed by Hayhurst (cf. [bib1]): the law is here in hyperbolic sine. The benefits of this formulation are detailed in [bib2].

2.2. Model equations

The model equations are written as:

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

where:

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

Note Bene:

The parameters of model \(K,{\mathrm{\epsilon }}_{\mathrm{0,}}{\mathrm{\sigma }}_{\mathrm{0,}}{h}_{\mathrm{1,}}{h}_{\mathrm{2,}}{A}_{\mathrm{0,}}{\mathrm{\alpha }}_{D},\text{et}{k}_{c}\) can be temperature functions (in \(°C\)). In [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 \({A}_{0}\) that is zero. In this case, it is nevertheless a question of entering a value for \({\mathrm{\sigma }}_{0}\), 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:

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

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

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

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