8. Detailed presentation of the algorithm#
We use a formulation that is implicit in relation to the criterion and explicit in relation to the direction of flow: the criterion must be verified at the end of the step, while the direction of flow will be that calculated at the beginning of the step (and therefore the value of the dilatance will also be that calculated at the beginning of the step of time).
We place ourselves at a material point, and we consider that the following are given:
The deformation increase tensor \(\Delta \varepsilon\) from which we deduce \(\Deltae\) and; \(\Delta {\varepsilon }_{\nu }\)
The constraints at the start of the \({\sigma }^{\text{-}}\) step from where \({s}^{\text{-}}\) and \({I}_{1}^{\text{-}}\) are deduced;
The values of the internal variables at the start of the time step (only the cumulative plastic deformation \({\gamma }^{{p}^{\text{-}}}\) is necessary).
It is a question of calculating:
The constraints at the end of time step \(\sigma\);
The internal variables at the end of the time step (\({\gamma }^{p}\), \({\varepsilon }_{\nu }^{p}\), the domains of behavior);
Tangent behavior at the end of the step: \(\frac{\partial \sigma }{\partial \varepsilon }\)
8.1. Calculation of the elastic solution#
\(\begin{array}{}\Delta {\varepsilon }^{e}=\Delta {\varepsilon }^{\text{-}}-\alpha \Deltat \\ {s}^{e}={s}^{\text{-}}+2\mu \Deltae \\ {I}_{}^{e}={I}_{1}^{\text{-}}+\mathrm{3K}\Delta {\varepsilon }_{\nu }\end{array}\)
8.2. Calculation of the elastic criterion#
Calculation of \({g}^{e}={s}_{\text{II}}^{e}h({\theta }^{e})\)
Calculation of, \({m}^{\text{-}}=m({\gamma }^{{p}^{\text{-}}})\) \({s}^{\text{-}}=s({\gamma }^{{p}^{\text{-}}})\), \({a}^{\text{-}}=a({\gamma }^{{p}^{\text{-}}})\), and \({k}^{\text{-}}=k({a}^{\text{-}})\)
Calculation of \({u}^{e}=-\frac{{m}^{\text{-}}{k}^{\text{-}}}{\sqrt{6}{\sigma }_{c}}\frac{{g}^{e}}{{h}_{c}^{0}}-\frac{{m}^{\text{-}}{k}^{\text{-}}}{3{\sigma }_{c}}{I}_{1}^{e}+{s}^{\text{-}}\mathrm{.}{k}^{\text{-}}\)
Calculation of \({f}^{e}={(\frac{{g}^{e}}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{\text{-}}}}-{u}^{e}\)
8.3. Algorithm#
If \({f}^{e}>0\)
Calculation of:
\(\begin{array}{}{I}_{1}^{\text{0-}}=\frac{3{\sigma }_{c}\mathrm{.}{s}^{\text{-}}}{{m}^{\text{-}}};\text{}{g}^{\text{-}}=g({s}^{\text{-}})\\ {\varphi }_{0}^{\text{-}}={\varphi }_{0}({m}^{\text{-}},{s}^{\text{-}},{a}^{\text{-}})\text{;}{C}_{0}^{\text{-}}={C}_{0}({m}^{\text{-}},{s}^{\text{-}},{a}^{\text{-}})\text{;}{\sigma }_{\text{0}}^{\text{-}}={\sigma }_{\text{t0}}({\varphi }_{0}^{\text{-}},{C}_{0}^{\text{-}})\\ \alpha {\text{'}}^{\text{-}}=\alpha \text{'}({I}_{1}^{\text{-}},{g}^{\text{-}},{\sigma }_{\text{t0}}^{\text{-}})\text{;}{\psi }^{\text{-}}=\psi (\alpha {\text{'}}^{\text{-}})\text{;}\beta {\text{'}}^{\text{-}}=\beta \text{'}({\psi }^{\text{-}})\end{array}\)
A priory calculation of the projection at the top
\(s=0\); Calculating \({\gamma }^{p}={\gamma }^{{p}^{\text{-}}}+\frac{1}{2\mu }\sqrt{\frac{2}{3}}{s}_{\text{II}}^{e}={\gamma }^{{p}^{\text{sommet}}}\) and \({I}_{1}=\frac{3{\sigma }_{c}\mathrm{.}s({\gamma }^{p})}{m({\gamma }^{p})}={I}_{1}^{\text{sommet}}\).
If \(\{\begin{array}{}({I}_{1}^{e}-{I}_{1}^{\text{sommet}})<-\frac{\mathrm{3K}}{2\mu }\beta \text{'}{s}_{n}^{e}\mathrm{cos}{\phi }_{s}^{\text{max}};\text{si}\beta {\text{'}}^{\text{-}}<0\\ ({I}_{1}^{e}-{I}_{1}^{\text{sommet}})<-\frac{\mathrm{3K}}{2\mu }\beta \text{'}{s}_{n}^{e}\mathrm{cos}{\phi }_{s}^{\text{min}};\text{si}\beta {\text{'}}^{\text{-}}\ge 0\end{array}\)
The projection to the top is not used a priory step. We calculate the regular solution.
\({Q}^{\text{-}}=\{\begin{array}{}Q({\sigma }^{\text{-}})\text{si}{\sigma }^{\text{-}}\ne 0\\ Q({\sigma }^{\text{e}})\text{si}{\sigma }^{\text{-}}=0\end{array}\) \({n}^{f}=\{\begin{array}{}n(\beta {\text{'}}^{\text{-}},{\sigma }^{\text{-}})\text{si}{\sigma }^{\text{-}}\ne 0\\ n(\beta {\text{'}}^{\text{e}},{\sigma }^{\text{e}})\text{si}{\sigma }^{\text{-}}=0\end{array}\) \({G}^{f}=\{\begin{array}{}G(\beta {\text{'}}^{\text{-}},{\sigma }^{\text{-}})\text{si}{\sigma }^{\text{-}}\ne 0\\ G(\beta {\text{'}}^{\text{e}},{\sigma }^{\text{e}})\text{si}{\sigma }^{\text{-}}=0\end{array}\)
If \({\gamma }^{{p}^{\text{-}}}=0\)
Initialization \(\Delta {\lambda }^{0}=0;{\gamma }^{{p}^{0}}={\gamma }^{{p}^{\text{-}}};{s}^{0}={s}^{e};{I}_{1}^{0}={I}_{1}^{e};{f}^{0}={f}^{e}\)
And \(\{\begin{array}{}\Delta {\gamma }^{{p}^{1}}=\frac{1}{10}\text{max}\mid \Delta {\varepsilon }_{\text{ij}}^{e}\mid \\ \delta {\lambda }^{{p}^{1}}=\frac{\Delta {\gamma }^{{p}^{1}}}{\tilde{{G}_{\text{II}}^{{f}_{b}}}}\sqrt{\frac{2}{3}}\end{array}\)
Sinon
Calculation of the increase in the plastic multiplier \(\Delta \lambda\) by Newton:
Initialization \(\Delta {\lambda }^{0}=0;{\gamma }^{{p}^{0}}={\gamma }^{{p}^{\text{-}}};{s}^{0}={s}^{e};{I}_{1}^{0}={I}_{1}^{e};{f}^{0}={f}^{e}\)
\({\frac{\partial u}{\partial \sigma }}^{0}={\frac{\partial u}{\partial \sigma }}^{\text{-}}=-\frac{{m}^{\text{-}}}{\sqrt{6}{\sigma }_{c}}\frac{{k}^{\text{-}}}{{h}_{c}^{0}}{Q}^{\text{-}}-{k}^{\text{-}}\frac{{m}^{\text{-}}}{3{\sigma }_{c}}I\)
\({\frac{\partial u}{\partial {\gamma }^{p}}}^{0}=-\frac{1}{\sqrt{6}{\sigma }_{c}}\frac{\partial (\mathrm{km})}{{\gamma }^{p}}({\gamma }^{{p}^{\text{-}}})\frac{{g}^{e}}{{h}_{c}^{0}}-\frac{1}{3{\sigma }_{c}}\frac{\partial (\mathrm{km})}{\partial {\gamma }^{p}}({\gamma }^{{p}^{\text{-}}}){I}_{1}^{e}+\frac{\partial (\mathrm{ks})}{\partial {\gamma }^{p}}({\gamma }^{{p}^{\text{-}}})\)
\(\frac{\partial {f}^{0}}{\partial \sigma }=\frac{1}{{a}^{\text{-}}}{(\frac{1}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{\text{-}}}}({g}^{e})\frac{1-{a}^{\text{-}}}{{a}^{\text{-}}}{Q}^{\text{-}}-{\frac{\partial u}{\partial \sigma }}^{0}\ne {\frac{\partial f}{\partial \sigma }}^{\text{-}}\)
\(\frac{\partial {f}^{0}}{\partial {\gamma }^{p}}=-{(\frac{1}{{a}^{\text{-}}})}^{2}{(\frac{{g}^{e}}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{\text{-}}}}\text{log}(\frac{{g}^{e}}{{\sigma }_{c}{h}_{c}^{0}})\frac{\partial a}{\partial {\gamma }^{p}}({\gamma }^{{p}^{\text{-}}})-\frac{\partial {u}^{0}}{\partial {\gamma }^{p}}\ne \frac{\partial {f}^{\text{-}}}{\partial {\gamma }^{p}}\)
\(\frac{\partial {f}^{{0}^{\text{*}}}}{\partial \Delta \lambda }=-\frac{\partial {f}^{0}}{\partial \sigma }\mathrm{.}(2\mu \tilde{{G}^{f}}+{\mathrm{KG}}^{f}I)+\frac{\partial {f}^{0}}{\partial {\gamma }^{p}}\sqrt{\frac{2}{3}}\tilde{{G}_{\text{II}}^{f}}\)
Iteration n loop
\(\frac{\partial {f}^{n}}{\partial \Delta \lambda }\delta {\lambda }^{n+1}=-{f}^{n}\)
\(\Delta {\lambda }^{n+1}=\Delta {\lambda }^{n}+\delta {\lambda }^{n+1}\)
\(\Delta {\gamma }^{{p}^{n+1}}={\Delta \gamma }^{n+1}\sqrt{\frac{2}{3}}\tilde{{G}_{\text{II}}^{f}}\text{;}\Delta {\varepsilon }_{v}^{p}={\Delta \lambda }^{n+1}{G}^{f}\)
\({s}^{n+1}={s}^{e}-2\mu \Delta {\lambda }^{{p}^{n+1}}\tilde{{G}^{f}}\text{;}{I}_{1}^{n+1}={I}_{1}^{e}-\mathrm{3K}\Delta {\lambda }^{{p}^{n+1}}{G}^{f}\)
If \(\Delta {\gamma }^{{p}^{n+1}}<0\) Non convergence
Calculation \({Q}^{n+1}\)
\({g}^{n+1}=g({s}^{n+1})\text{;}{m}^{n+1}=m({\gamma }^{{p}^{n+1}})\text{;}{s}^{n+1}=s({\gamma }^{{p}^{n+1}})\text{;}{a}^{n+1}=a({\gamma }^{{p}^{n+1}})\text{;}{k}^{n+1}=k({a}^{n+1})\)
\({u}_{n+1}=-\frac{{m}^{n+1}{k}^{n+1}}{\sqrt{6}{\sigma }_{c}}\frac{{g}^{n+1}}{{h}_{c}^{0}}-\frac{{m}^{n+1}{k}^{n+1}}{3{\sigma }_{c}}{I}_{1}^{n+1}+{s}^{n+1}\mathrm{.}{k}^{n+1}\)
\({f}^{n+1}={(\frac{{g}^{n+1}}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{n+1}}}-{u}^{n+1}\)
\({\frac{\partial u}{\partial \sigma }}^{n+1}=-\frac{m}{\sqrt{6}{\sigma }_{c}}\frac{{k}^{n+1}}{{h}_{c}^{0}}{Q}^{n+1}-{k}^{n+1}\frac{{m}^{n+1}}{3{\sigma }_{c}}I\)
\({\frac{\partial u}{\partial {\gamma }^{p}}}^{n+1}=-\frac{1}{\sqrt{6}{\sigma }_{c}}\frac{\partial (\mathrm{km})}{\partial {\gamma }^{p}}({\gamma }^{{p}^{n+1}})\frac{{g}^{n+1}}{{h}_{c}^{0}}-\frac{1}{3{\sigma }_{c}}\frac{\partial (\mathrm{km})}{\partial {\gamma }^{p}}({\gamma }^{{p}^{n+1}}){I}_{1}^{n+1}+\frac{\partial (\mathrm{ks})}{\partial {\gamma }^{p}}({\gamma }^{{p}^{n+1}})\)
\(\frac{\partial {f}^{n+1}}{\partial \sigma }=\frac{1}{{a}^{n+1}}{(\frac{A}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{n+1}}}{({g}^{n+1})}^{\frac{A-{a}^{n+1}}{{a}^{n+1}}}{Q}^{n+1}-{\frac{\partial u}{\partial \sigma }}^{n+1}\)
\(\frac{\partial {f}^{n+1}}{\partial {\gamma }^{p}}=-{(\frac{1}{{a}^{n+1}})}^{2}{(\frac{{g}^{n+1}}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{{a}^{n+1}}}\text{log}(\frac{{g}^{n+1}}{{\sigma }_{c}{h}_{c}^{0}})\frac{\partial \alpha }{\partial {\gamma }^{p}}({\gamma }^{{p}^{n+1}})-\frac{\partial {u}^{n+1}}{\partial {\gamma }^{p}}\)
\(\frac{\partial {f}^{{n+1}^{\text{*}}}}{\partial \Delta \lambda }=-\frac{\partial {f}^{n+1}}{\partial \sigma }\mathrm{.}(2\mu \tilde{{G}^{f}}+{\mathrm{KG}}^{f}I)+\frac{\partial {f}^{n+1}}{\partial {\gamma }^{p}}\sqrt{\frac{2}{3}}\tilde{{G}_{\text{II}}^{f}}\)
If \(\mid {f}^{n+1}\text{/}{\sigma }_{c}\mid >{\varepsilon }_{\text{prec}}\)
n=n+1
If n > max internal number
If \(\{\begin{array}{}({I}_{1}^{e}-{I}_{1}^{\text{sommet}})>-\frac{\mathrm{3K}}{2\mu }\beta \text{'}{s}_{\text{II}}^{e}\mathrm{cos}{\phi }_{s}^{\text{min}}\text{;si}\beta \text{'}<0\\ ({I}_{1}^{e}-{I}_{1}^{\text{sommet}})>-\frac{\mathrm{3K}}{2\mu }\beta \text{'}{s}_{\text{II}}^{e}\mathrm{cos}{\phi }_{s}^{\text{max}}\text{;si}\beta \text{'}\ge 0\end{array}\)
We remember the projection at the top: \(s=0;{I}_{1}={I}_{1}^{\text{sommet}};{\gamma }^{p}={\gamma }^{{p}^{\text{sommet}}}\)
Sinon
Non-convergence
Sinon
Non-convergence
Sinon
Convergence
If FULL_MECA
Calculation of:
\(\frac{{\partial \sigma }^{n+1}}{\partial \varepsilon }=H+\frac{H\mathrm{.}{G}^{f}\mathrm{.}{(\frac{\partial {f}^{n+1}}{\partial \sigma })}_{T}H}{\sqrt{\frac{2}{3}}\frac{\partial {f}^{n+1}}{\partial {\gamma }^{p}}\tilde{{G}_{\text{II}}^{f}}-{(\frac{\partial {f}^{n+1}}{\partial \sigma })}^{T}H{G}^{f}}\)
Mechanical symmetrization:
\(\frac{\partial {\sigma }_{\text{sym}}^{n+1}}{\partial \varepsilon }=\frac{1}{2}(\frac{\partial {\sigma }^{n+1}}{\partial \varepsilon }+\frac{\partial {\sigma }^{{n+1}^{T}}}{\partial \varepsilon })\)