1. Introduction#
The mechanisms that cause ductile failure in steels are associated with the development of cavities within the material. There are generally three phases:
germination: this is the initiation of cavities, at sites that preferentially correspond to defects 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 interaction of cavities with each other to create macroscopic cracks.
In what follows, we only deal with the growth and coalescence phases. Rousselier [bib1] proposed a law of behavior capable of accounting for these two phases. Compared to this original formulation, Lorentz et al. [bib2] introduced several modifications mainly relating to the treatment of large deformations (multiplicative decomposition), the evolution of porosity (a function of total deformation) and the expression of the flow law at the singular point of the threshold surface.
More precisely, the model is based on hypotheses that introduce a microstructure consisting of a cavity and a rigid plastic matrix that is therefore isochoric. In this case, 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 deformation by:
\(J=\text{det}F=\frac{1-{f}_{0}}{1-f}\text{avec}f=\frac{{V}^{c}}{V}\iff \dot{f}=(1-f)\text{tr}D\) eq 1-1
where \({f}_{0}\) refers to the initial porosity, \(F\) the transformation gradient tensor, the transformation gradient tensor, \(J\) the volume change and \(\mathrm{nitalicD}\) the deformation rate.
To build the law of cavity growth, Rousselier was inspired by a phenomenological analysis that led him to the following ingredients:
large plastic deformations,
irreversible volume changes,
isotropic work hardening.
These considerations led him to write plasticity criterion \(F\) in the following form:
\(F(\tau ,R)={\tau }_{\text{eq}}+{\sigma }_{1}Df\text{exp}(\frac{{\tau }_{H}}{{\sigma }_{1}})-R(p)-{\sigma }_{y}\) eq 1-2
where \(\tau\) is the Kirchhoff stress, \(R\) the isotropic work hardening as a function of the cumulative plastic deformation \(p\) and \({\sigma }_{1}\), \(D\) and \({\sigma }_{y}\) of the material parameters. The presence of hydrostatic stress \({\tau }_{H}\) in the plasticity criterion allows changes in plastic volume. It should also be noted that this model does not include a specific damage variable because the only microstructural information retained is porosity, directly linked to macroscopic deformation by the equation [éq 1-1].
As for the treatment of large deformations, the theory of Simo and Miehe is adopted but in a slightly modified formulation. The approximations provided make it possible to make the numerical integration of the law of behavior easier but also to place Simo and Miehe’s theory within the framework of generalized standard materials.
Subsequently, some notions of mechanics in large deformations are briefly given, then the theory of Simo and Miehe and the modifications made are recalled. Finally, the behavioral relationships of the Rousselier model and its numerical integration are presented.