8. Algorithm#

8.1. Internal variables#

The modified Hoek-Brown law of behavior is governed by the following three internal variables:

  1. the work-hardening parameter \(\mathrm{\gamma }\) corresponding to the major irreversible deformation.

  2. the cumulative plastic volume deformation \({\mathrm{\varepsilon }}_{\mathrm{\nu }}^{p}\) whose law of evolution is given by \(d{\varepsilon }_{\nu }^{p}=3\eta d\lambda =\frac{3\eta }{\eta +1}d\gamma\).

  3. the state of plasticization; it is 0 if the Gauss point is in elastic charge or under discharge, and 1 if the Gauss point is in plastic charge.

8.2. Algorithm#

An implicit formulation with respect to the criterion and to the direction of flow is retained. We place ourselves at a material point and we assume we know \({t}^{\text{-}}\):

  • The deformation increase tensor \(\Delta \varepsilon\) from which we deduce \(\Delta e\), \(\Delta {\varepsilon }_{\nu }\)

  • The constraints at the start of time step \({\sigma }^{\text{-}}\) from which we deduce \({s}^{\text{-}},{I}_{1}^{\text{-}}\)

  • The value of the internal variables \({\gamma }^{\text{-}}\) and \({\varepsilon }_{\nu }^{p\text{-}}\) at the start of the time step that give us

\((S{\sigma }_{c}^{2}{)}^{\text{-}},(m{\sigma }_{c}{)}^{\text{-}},{b}^{\text{-}},{\phi }^{\text{-}}\)

The aim of the algorithm is then to calculate:

  • Constraints at the end of a time step \(\sigma\)

  • The internal variables at the end of the time step

  • Tangent behavior at the end of a time step: \(\frac{\partial \sigma }{\partial \varepsilon }\) if the law is in effective constraints

  • Tangent behavior at the end of the time step: \(\frac{\partial \sigma }{\partial \varepsilon }\) and \(\frac{\partial \sigma }{\partial {\sigma }_{p}}\) if the law is under total constraints.

Algorithm:

Calculation of the elastic solution:

If the law is under effective constraints:

\(\{\begin{array}{c}{s}^{e}\text{=}{s}^{\text{-}}\text{+}2\mu \Delta e\\ {I}_{1}^{e}\text{=}{I}_{1}^{\text{-}}+\mathrm{3K}\Delta {\varepsilon }_{\nu }\end{array}\)

If the law is under total constraints:

\(\{\begin{array}{c}{s}^{e}\text{=}{s}^{\text{-}}\text{+}2\mu \Delta e\\ {I}_{1}^{e}\text{=}{I}_{1}^{\text{-}}+\mathrm{3K}\Delta {\varepsilon }_{\nu }\text{+}3\Delta {\sigma }_{p}\end{array}\)

Calculation of the elastic criterion \(F({\sigma }^{e},{\gamma }^{-})\). If \(S{\sigma }_{c}^{2}-{\sigma }_{3}^{e}\text{.}m{\sigma }_{c}<0\), the function \(F\) is not defined in point \(({\sigma }^{e},{\gamma }^{-})\). We then consider that we are in the plastic case.

Resolution: calculation of \(\sigma ,\gamma\)

If \(F({\sigma }^{e},{\gamma }^{-})\le 0\), then \(\Delta \varepsilon =\Delta {\varepsilon }^{e},{\gamma }^{p}\text{=}{\gamma }^{{p}^{\text{-}}},\Delta \sigma \text{=}H\Delta \varepsilon\).

Otherwise, we’re looking for \(\sigma ,\gamma\) such as \(F(\sigma ,\gamma )\le 0\), which is the same as looking for \(\Delta \gamma\) such as \(F(\Delta \gamma )=\stackrel{ˉ}{F}(\Delta \gamma )=0\). This problem is solved using a Newton method on \(\stackrel{ˉ}{F}\).

Newton algorithm:

Initialization: \({\mathrm{\Delta \gamma }}^{0}=0\)

After each iteration:

  • if \(\Delta {\gamma }^{n\text{+}1}\le 0\), there was no convergence: we subdivide the time step

  • if \(\Delta {\gamma }^{n\text{+}1}\le \frac{{\sigma }_{\text{eq}}^{e}}{3\mu [:ref:\)eta (Delta {gamma }^{n+1})+1 <eta (Delta {gamma }^{n+1})+1>`]} , there is no solution (see paragraph :ref:`4.2 <Ref101847972>): we subdivide the time step

Update variables: constraints, internal variables

Calculation of the coherent tangent matrix \(\frac{\partial \sigma }{\partial \varepsilon }\) if the law is in effective constraints and \(\frac{\partial \sigma }{\partial \varepsilon }\) and \(\frac{\partial \sigma }{\partial {\sigma }_{p}}\) if the law is in total constraints for the option RIGI_MECA_TANG or FULL_MECA.