1. Reference problem

1.1. Geometry and boundary conditions

Figure 1.1-a : geometry of the studied structure

The structure studied is a \(1\mathit{mm}\) long bar with a \(0.1\times 0.1\mathit{mm}\) square cross section. One face is embedded, the opposite face is subjected to a \(\mathit{FZ}\) force varying progressively from \(1\) to \(5N\).

1.2. Material properties

Law of behavior	Elastic behaviour	Damaging behavior (healthy part)	Damaging behavior (weakened zone)
ENDO_ISOT_BETON	\(E\mathrm{=}20000\mathit{MPa}\) \(\nu \mathrm{=}0\)	\({\sigma }_{y}\mathrm{=}6\mathit{MPa}\) \({E}_{T}\mathrm{=}\mathrm{-}10000\mathit{MPa}\)	\({\sigma }_{y}\mathrm{=}5\mathit{MPa}\) \({E}_{T}\mathrm{=}\mathrm{-}10000\mathit{MPa}\)
ROUSS_PR and ROUSSELIER	\(E\mathrm{=}206400\mathit{MPa}\) \(\nu \mathrm{=}0.3\)	\(D\mathrm{=}2\) \({\sigma }_{1}\mathrm{=}490\mathit{MPa}\) \({f}_{0}\mathrm{=}5E-04\) traction curve: \(p\) : cumulative plastic deformation \({r}_{i}\mathrm{=}1500\mathit{MPa}\) \({r}_{o}\mathrm{=}520\mathit{MPa}\) \(b\mathrm{=}2.4\)	\(D\mathrm{=}2\) \({\sigma }_{1}\mathrm{=}400\mathit{MPa}\) \({f}_{0}\mathrm{=}5E-04\) traction curve: \(p\) : cumulative plastic deformation \({r}_{i}\mathrm{=}1500\mathit{MPa}\) \({r}_{o}\mathrm{=}520\mathit{MPa}\) \(b\mathrm{=}2.4\)
BETON_DOUBLE_DP	\(E\mathrm{=}31000\mathit{MPa}\) \(\nu \mathrm{=}0.22\)	\({f}_{c}\mathrm{=}38.3\mathit{MPa}\) \({f}_{t}\mathrm{=}4.0\mathit{MPa}\) \(\beta \mathrm{=}1.16\) \({G}_{t}=2.83E-04\mathit{Nmm}/{\mathit{mm}}^{2}\) \({G}_{c}=2.83E-02\mathit{Nmm}/{\mathit{mm}}^{2}\)	\({f}_{c}=38.3\mathit{MPa}\) \({f}_{t}=3.0\mathit{MPa}\) \(\beta =1.16\) \({G}_{t}=2.83E-04\mathit{Nmm}/{\mathit{mm}}^{2}\) \({G}_{c}=2.83E-02\mathit{Nmm}/{\mathit{mm}}^{2}\)
ENDO_LOCA_EXP	\(E=34129\mathit{MPa}\) \(\nu =0\)	\(\kappa =5.842\) \(p=2\) \({\sigma }_{c}=3.034\ast 1.5\mathit{MPa}\) \({\sigma }_{0}=0.827\ast 1.5\mathit{MPa}\)	\(\kappa =5.842\) \(p=2\) \({\sigma }_{c}=3.034\mathit{MPa}\) \({\sigma }_{0}=0.827\mathit{MPa}\)