1. Ratings#
\(\sigma\) refers to the effective stress tensor in small disturbances defined as being the difference between total stresses and water pressure in the case of saturated soils, noted in the form of the following vector:
- left (
begin {array} {c} {sigma} _ {sigma} _ {text {11}}}\ {sigma} _ {text {33}} _ {text {33}}\ sqrt {2} {sigma} _ {sigma} _ {text {11}}}\sigma} _ {text {23}}}\ sqrt {2} {sigma} _ {text {23}}}\ sqrt {2} {sigma} _ {text {23}}}\ sqrt {2}} {sigma} _ {text {31}}}end {array}
right)
We note:
\(\underline{\underline{P\mathrm{=}\mathrm{-}\frac{1}{3}\text{tr}(\sigma )}}\) |
lockdown constraint |
\(s\mathrm{=}\sigma +\text{PI}\) |
constraint deviator |
\({I}_{2}=\frac{1}{2}\text{tr}(s\text{.}s)\) |
second constraint invariant |
\(Q\mathrm{=}{\sigma }_{\text{eq}}\mathrm{=}\sqrt{{\mathrm{3I}}_{2}}\) |
equivalent stress |
\(\epsilon =\frac{1}{2}(\nabla u+{\nabla }^{T}u)\) |
total deformation |
\(\varepsilon \mathrm{=}{\varepsilon }_{e}+{\varepsilon }_{p}+{\varepsilon }_{\text{th}}\) |
partition of deformations (elastic, plastic, thermal) |
\(\underline{\underline{{\varepsilon }_{v}\mathrm{=}\mathrm{-}\text{tr}(\varepsilon )+3\alpha (T\mathrm{-}{T}_{0})}}\) |
total volume deformation |
\({\varepsilon }_{V}^{p}\mathrm{=}\mathrm{-}\text{tr}({\varepsilon }^{p})\) |
volume plastic deformation |
\(\tilde{\varepsilon }\mathrm{=}\varepsilon +\frac{1}{3}{\varepsilon }_{v}I\) |
deformation deviator |
\({\tilde{\varepsilon }}^{e}\mathrm{=}\tilde{\varepsilon }\mathrm{-}{\tilde{\varepsilon }}^{p}\) |
deflector of elastic deformations |
\({\tilde{\varepsilon }}^{p}\mathrm{=}{\varepsilon }^{p}+\frac{1}{3}{\varepsilon }_{v}^{p}I\) |
deviatoric plastic deformation |
\({\varepsilon }_{\text{eq}}^{e}\mathrm{=}\sqrt{\frac{2}{3}\text{tr}({\tilde{\varepsilon }}^{e}\text{.}{\tilde{\varepsilon }}^{e})}\) |
equivalent elastic deformation |
\({\varepsilon }_{\text{eq}}^{p}\mathrm{=}\sqrt{\frac{2}{3}\text{tr}({\tilde{\varepsilon }}^{p}\text{.}{\tilde{\varepsilon }}^{p})}\) |
equivalent plastic deformation |
\(e\) material void index (ratio of pore volume to solid grain volume)
\({e}_{0}\) initial void index
\(\phi\) porosity (ratio of pore volume to total volume)
\(\kappa\) swelling coefficient (elastic slope in a hydrostatic compression test)
\(M\) slope of the critical state line
\({k}_{0}\mathrm{=}\frac{(1+{e}_{0})}{\kappa }\)
\({P}_{\text{cr}}\) internal variable of the model, critical pressure equal to half of the consolidation pressure \({P}_{\text{cons}}\)
\(\lambda\) compressibility coefficient (plastic slope in a hydrostatic compression test)
\(k=\frac{(1+{e}_{0})}{(\lambda -\kappa )}\)
\(\mu\) elastic shear coefficient (Lamé coefficient)
\(f\) charging surface
\(\Lambda\) plastic multiplier
\({I}^{d}\) 2nd order unit tensor whose current term is \({\delta }_{\text{ij}}\)
\({I}_{4}^{d}\) fourth-order unit tensor whose current term is \({\delta }_{\text{ijkl}}\)