1. Notations#
\(\sigma\) refers to the effective stress tensor in small disturbances, noted in the form of the following vector:
\(\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ \sqrt{2}{\sigma }_{12}\\ \sqrt{2}{\sigma }_{13}\\ \sqrt{2}{\sigma }_{23}\end{array}\)
We note:
\({D}^{e}\) |
elasticity tensor |
\({I}_{1}=\text{tr}(\sigma )\) |
first constraint invariant |
\(s=\sigma -\frac{{I}_{1}}{3}\mathrm{Id}\) |
deviatory stress tensor |
\({s}_{\mathrm{II}}=\sqrt{s\mathrm{.}s}\) |
second invariant of the deviatory stress tensor |
\({\sigma }_{\text{eq}}\mathrm{=}\sqrt{\frac{3}{2}{s}_{\mathit{ij}}{s}_{\mathit{ij}}}\) |
equivalent stress |
\({I}_{1}^{\mathrm{el}}\) |
elastic stress prediction trace |
\({s}^{\mathrm{el}}={\sigma }^{\mathrm{el}}-\frac{{I}_{1}^{\mathrm{el}}}{3}\mathrm{Id}\) |
deviatory stress tensor for elastic stress prediction |
\({\sigma }_{\text{eq}}^{\mathit{el}}\mathrm{=}\sqrt{\frac{3}{2}{s}_{\mathit{ij}}^{\mathit{el}}{s}_{\mathit{ij}}^{\mathit{el}}}\) |
equivalent stress of elastic stress prediction |
\(\tilde{\varepsilon }=\varepsilon -\frac{\text{tr}(\varepsilon )}{3}\mathrm{Id}\) |
deformation deviator |
\({\varepsilon }_{v}=\text{tr}(\varepsilon )\) |
volume deformation |
\(\dot{p}=\sqrt{\frac{2}{3}{\dot{\tilde{\varepsilon }}}_{\mathrm{ij}}^{\text{vp}}{\dot{\tilde{\varepsilon }}}_{\mathrm{ij}}^{\text{vp}}}\) |
cumulative viscoplastic deviatory deformations |
\(f\) |
viscoplastic load surface |
\(G\) |
viscoplastic flow potential |
\({\alpha }_{0}\), \({R}_{0}\) and \({\beta }_{0}\) |
work-hardening parameters corresponding to the elasticity threshold (\(p\mathrm{=}0\)) |
\({\alpha }_{\text{pic}}\), \({R}_{\text{pic}}\) and \({\beta }_{\text{pic}}\) |
work-hardening parameters corresponding to peak \((p\mathrm{=}{p}_{\mathit{pic}})\) |
\({\alpha }_{\text{ult}}\), \({R}_{\text{ult}}\) and \({\beta }_{\text{ult}}\) |
work-hardening parameters corresponding to the ultimate threshold \((p\mathrm{=}{p}_{\mathit{ult}})\) |
\(\Phi\) |
magnitude of the speed of irreversible deformations |
\(A\) |
creep parameter |
\(n\) |
power of the law of creep |
\({P}_{\mathit{ref}}\) |
reference pressure |