1. Introduction#
The mechanisms that cause the ductile failure of metals are associated with the development of cavities within the material. There are generally three phases:
germination: this is the priming or nucleation of cavities, at sites that preferentially correspond to the second phase particles present in the material,
growth: this is the phase that corresponds to the actual development of cavities, controlled essentially by the plastic flow of the metal matrix that surrounds these cavities,
coalescence: this is the phase that corresponds to the location of the deformation between the cavities to create macroscopic cracks.
The Rousselier model [bib1], [bib2], [bib3] presented here is based on microstructural hypotheses that introduce a microstructure consisting of cavities and a matrix whose elastic deformations are negligible compared to plastic deformations. In this case, and in the absence of nucleation of new cavities, porosity \(f\), defined as the ratio between the volume of the cavity \({V}^{c}\) and the total volume \(V\) of the representative elementary volume, is directly linked to the macroscopic plastic deformation by:
\(\frac{{\rho }_{0}}{\rho }\mathrm{=}\frac{1\mathrm{-}{f}_{0}}{1\mathrm{-}f}\) with \(f\mathrm{=}\frac{{V}^{c}}{V}\mathrm{\iff }\dot{f}\mathrm{=}(1\mathrm{-}f)\mathrm{tr}{\dot{\varepsilon }}^{p}\) eq 1-1
where \({f}_{0}\) designates the initial porosity, \({\mathrm{\rho }}_{o}\) and \(\mathrm{\rho }\) are respectively the density in the initial and current configurations (we take \({\mathrm{\rho }}_{o}=1\) below) and \({\dot{\varepsilon }}^{p}\) the plastic deformation rate of the total volume \(V\).
The construction of the model is based on a thermodynamic and phenomenological analysis which leads to the writing of the plastic potential \(F\) in the following form:
\(F(\tau ,p,f)={\tau }_{\text{eq}}+{\sigma }_{1}{D}_{1}f\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})-R(p)\) eq 1-2
where \(\tau \mathrm{=}\sigma \mathrm{/}\rho\) is the Kirchhoff stress, \(\sigma\) is the Cauchy stress, is the Cauchy stress, \(R\) the isotropic work hardening function of the cumulative plastic deformation \(p\), \({\mathrm{\sigma }}_{1}\) and \({D}_{1}\) of the material parameters. The presence of hydrostatic stress \({\mathrm{\tau }}_{m}\) in the plastic potential allows changes in plastic volume.
In the case of nucleation of new cavities, it is considered that the volume fraction created is proportional to the cumulative plastic deformation. So all you have to do is replace \(f\) by \(f+{A}_{n}p\) in the model equations. \({A}_{n}\) is a material parameter. Equation [éq 1-1] is not changed.
In the viscoplastic case, we write the viscoplastic potential \({F}^{\text{vp}}\) as a function of the plastic potential \(F\):
\({F}^{\text{vp}}\mathrm{=}\Lambda (F,p,f)\) eq 1-3
We will only consider the particular case such as:
\(\dot{p}=\frac{\partial \mathrm{\Lambda }}{\partial F}={\dot{\mathrm{\varepsilon }}}_{0}{\left[\text{sh}(\frac{F}{{\mathrm{\sigma }}_{0}})\right]}^{m}\) eq 1-4
which is reduced to a power function (Norton law) when the two parameters of the material \({\dot{\varepsilon }}_{0}\) and \({\sigma }_{0}\) are very large.
Next, the behavioral relationships of the Rousselier model and its numerical integration are presented.