3. Digital integration#

In this paragraph, we briefly return to the numerical integration of the proposed thermomechanical model. The model was integrated according to two schemes: an explicit schema and an implicit schema. We only give the broad outlines of these two diagrams without necessarily detailing all the quantities involved. In what follows, we will exceptionally use the \(\mathrm{\sigma }\) notation to refer to the effective stress tensor. This makes it possible not to weigh down the ratings. We will not detail the expression of the derivatives that occur within the set of equations.

3.1. Explicit integration diagram#

The explicit integration scheme (ALGO_INTE =” SPECIFIQUE “) consists of a classical elastic prediction phase followed by a (visco) plastic correction phase if one of the two, or both, mechanism (s) is/are activated during this*elastic shot.* The input variables for the resolution algorithm are:

\(t\): time at the moment « + »;
\({t}^{-}\): time at the moment « -« ;
\(T\): temperature at the moment « + »;
\({T}^{-}\): temperature at the moment « -« ;
\({T}_{0}\): reference temperature;
\({\mathrm{\sigma }}^{-}\): stress tensor effective at the moment « -« ;
\({\mathrm{\xi }}^{\mathit{vp}-}\): work-hardening variable of the viscoplastic mechanism at the moment « -« ;
\({\mathrm{\xi }}^{p-}\): variable for the work-hardening of the plastic mechanism at the moment « -« ;
\(\mathrm{\Delta }\mathrm{\epsilon }\): total deformation increment.

The output variables are:

\(\mathrm{\sigma }\): stress tensor effective at the moment « + »;
\({\mathrm{\xi }}^{\mathit{vp}}\): work-hardening variable of the viscoplastic mechanism at the time « + »;
\({\mathrm{\xi }}^{p}\): variable for working the plastic mechanism at the moment « + »;
\(\frac{\partial {\mathrm{\sigma }}_{\mathit{ij}}}{\partial \mathrm{\Delta }{\mathrm{\epsilon }}_{\mathit{kl}}}\): tangent operator.

From the input data, we do the following:

Elastic prediction: \({\mathrm{\sigma }}^{e}={\mathrm{\sigma }}^{-}+D\mathrm{:}\left(\mathrm{\Delta }\mathrm{\epsilon }-\mathrm{\alpha }\mathrm{\Delta }T\right)\)
Active viscoplastic mechanism?
- Calculation of \({F}^{\mathit{vp}}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{\mathit{vp}-},T\right)\)
- If \({F}^{\mathit{vp}}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{\mathit{vp}-},T\right)<0\), we are elastic:
  \(\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}=0\)
  
  \(\mathrm{\Delta }{\mathrm{\xi }}^{\mathit{vp}}=\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}=0\)
  
  \({\mathrm{\xi }}^{\mathit{vp}}={\mathrm{\xi }}^{\mathit{vp}-}\)
  
  \({\mathrm{\gamma }}^{\mathit{vp}}={\mathrm{\gamma }}^{\mathit{vp}-}\)
- If \({F}^{\mathit{vp}}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{\mathit{vp}-},T\right)\ge 0\), you are viscoplastic:
  \(\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}={A}_{v}^{T}{\left(\frac{⟨{F}^{\mathit{vp}}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{\mathit{vp}},T\right)⟩}{{P}_{a}}\right)}^{{n}_{v}}{G}^{\mathit{vp}-}\mathrm{\Delta }t\)
  
  \(\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}=\sqrt{\frac{2}{3}\mathrm{\Delta }{e}^{\mathit{vp}}\mathrm{:}\mathrm{\Delta }{e}^{\mathit{vp}}}\) with \(\mathrm{\Delta }{e}^{\mathit{vp}}=\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}-\mathit{Tr}\frac{\left(\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}\right)}{3}\mathrm{\delta }\)
  
  \(\mathrm{\Delta }{\mathrm{\xi }}^{\mathit{vp}}=\mathit{min}(\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}},{\mathrm{\xi }}_{[5]}-{\mathrm{\xi }}^{\mathit{vp}-})\)
  
  \({\mathrm{\xi }}^{\mathit{vp}}={\mathrm{\xi }}^{\mathit{vp}-}+\mathrm{\Delta }{\mathrm{\xi }}^{\mathit{vp}}\)
  
  \({\mathrm{\gamma }}^{\mathit{vp}}={\mathrm{\gamma }}^{\mathit{vp}-}+\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}\)
  
  \({\mathrm{\epsilon }}^{\mathit{vp}}={\mathrm{\epsilon }}^{\mathit{vp}-}+\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}\)
Active plastic mechanism?
- Position of the stress state in relation to the characteristic threshold:
  If \({F}^{\mathit{vp}}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}_{[5]},T\right)\ge 0\), there is coupling between the two mechanisms
  
  Otherwise, there is no coupling between the two mechanisms
- Calculation of \({F}^{p}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{p-},T\right)\)
- If \({F}^{p}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{p-},T\right)<0\), we are elastic:
  \(\mathrm{\Delta }{\mathrm{\epsilon }}^{p}=0\)
  
  \(\mathrm{\Delta }{\mathrm{\gamma }}^{p}=0\)
  
  \({\mathrm{\gamma }}^{p}={\mathrm{\gamma }}^{p-}\)
  
  \({\mathrm{\xi }}^{p}={\mathrm{\xi }}^{p-}+\mathrm{\Delta }{\mathrm{\xi }}^{p}\) with:
  
  \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=0\) if there is no coupling between the two mechanisms
  
  \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}\) if there is a coupling between the two mechanisms
  
  Update of constraints:

\(\mathrm{\sigma }={\mathrm{\sigma }}^{-}+D\left(\mathrm{\Delta }\mathrm{\epsilon }-\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}-\mathrm{\alpha }\mathrm{\Delta }T\right)\)

- If \({F}^{p}\left({\mathrm{\sigma }}^{e},{\mathrm{\xi }}^{p-},T\right)\ge 0\), you are plastic:
  - Calculation of \(\mathrm{\Delta }\mathrm{\lambda }\) (equation)
  - \(\mathrm{\Delta }\mathrm{\epsilon }=\mathrm{\Delta }\mathrm{\lambda }{G}^{p-}\)
  - \(\mathrm{\Delta }{\mathrm{\gamma }}^{p}=\sqrt{\frac{2}{3}\mathrm{\Delta }{e}^{p}\mathrm{:}\mathrm{\Delta }{e}^{p}}\) with \(\mathrm{\Delta }{e}^{p}=\mathrm{\Delta }{\mathrm{\epsilon }}^{p}-\mathit{Tr}\frac{\left(\mathrm{\Delta }{\mathrm{\epsilon }}^{p}\right)}{3}\mathrm{\delta }\)
  - \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}\) if there is no coupling between the two mechanisms
  - \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}+\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}\) if there is a coupling between the two mechanisms
  - \({\mathrm{\xi }}^{p}={\mathrm{\xi }}^{p-}+\mathrm{\Delta }{\mathrm{\xi }}^{p}\)
  - Update of constraints:

\(\mathrm{\sigma }={\mathrm{\sigma }}^{-}+D\left(\mathrm{\Delta }\mathrm{\epsilon }-\mathrm{\Delta }{\mathrm{\epsilon }}^{\mathit{vp}}-\mathrm{\Delta }{\mathrm{\epsilon }}^{p}-\mathrm{\alpha }\mathrm{\Delta }T\right)\)

Calculation of the tangent operator \(\frac{\partial {\mathrm{\sigma }}_{\mathit{ij}}}{\partial \mathrm{\Delta }{\mathrm{\epsilon }}_{\mathit{kl}}}\) (see § 3.1.2)

3.1.1. Determining the plastic multiplier increment#

The plastic multiplier increment \(\mathrm{\Delta }\mathrm{\lambda }\) is determined by applying the Kuhn-Tucker condition, i.e. applying the following condition:

(3.1)#\[ {F} ^ {p}\ left (\ mathrm {\ sigma}, {\ mathrm {\ xi}} ^ {p}, T\ right) =0\]

Or in incremental form:

(3.2)#\[ {F} ^ {p}\ left ({\ mathrm {\ sigma}}} _ {\ mathit {ij}}} ^ {-} + {D} _ {\ mathit {ijkl}}\ mathrm {:}:}\ left (\ mathrm {\ sigma}}} _ {\ mathit {ijkl}}}\ mathrm {:}}\ mathrm {\ sigma}}\ left (\ mathrm {\ sigma}}\ left (\ mathrm {\ delta}}} -\ mathrm {\ delta}}\ mathrm {\ lambda} {G} _ {\ mathit {kl}}} {\ mathit {kl}}}} ^ {\ mathit {kl}} _ {\ mathit {kl}}} ^ {\ mathit {VP} -}\ mathrm {VP} -}\ mathrm {\ delta}} -}\ mathrm {\ delta} t- {\ mathrm {\ alpha}} _ {\ mathit {kl}}} ^ {\ mathit {kl}} ^ {\ mathit {VP} -} -}\ mathrm {VP} -} -}\ mathrm {\ Delta} t- {\ mathrm {\ alpha}} _ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {VP}} T\ right), {\ mathrm {\ xi}}} ^ {p-}} +\ mathrm {\ Delta} {\ mathrm {\ xi}} ^ {p}, {T} ^ {-}} +\ mathrm {\ Delta} T\ right) =0\]

We choose to make an explicit resolution by developing Euler:

(3.3)#\[\begin{split} \ begin {array} {c} {F} ^ {p}\ left ({\ mathrm {\ sigma}}} _ {\ mathit {ij}}} ^ {-} + {D} _ {\ mathit {ijkl}} {\ mathit {ijkl}}}}\ mathrm {:}\ left (\ mathrm {\ delta}}} _ {\ mathit {ijkl}}} -\ mathrm {\ Delta}\ mathrm {\ lambda} {G} {G}} _ {\ mathit {kl}}} ^ {p-} -\ mathrm {\ phi} ⟩ {G} _ {\ mathit {kl}}} _ {\ mathit {kl}}} ^ {\ mathit {kl}} ^ {\ mathit {kl}} ^ {\ mathit {kl} -} ^ {\ mathit {kl} -} ^ {\ mathit {lp} -}\ mathrm {\ delta} t- {\ mathrm {\ alpha}}} _ {\ mathit {kl}}}\ mathrm {\ Delta} T\ right), {\ mathrm {\ xi}}} ^ {p-}, {T} ^ {-}\ right) -\\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ lambda}} {\ mathrm {\ lambda}} {\ lambda}} {\ left (\ frac {\ partial {F}} ^ {p}}} {\ partial {\ mathrm {\ sigma}}\ mathrm {\ delta}}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ lambda}} {\ lambda} it {ij}}}\ right)}} _ {{\ mathrm {\ sigma}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:}\ left ({D} _ {D} _ {\ mathit {ijkl}}}\ mathrm {:} {G } _ {\ mathit {kl}} ^ {p-}\ right) +\ mathrm {\ Delta} {\ mathrm {\ xi}} ^ {p} {\ left (\ frac {\ partial {F}} ^ {f} ^ {p}} ^ {p}}\ right)} _ {{\ mathrm {\ xi}}} ^ {\ mathrm {\ xi}} ^ {xi}} ^ {p-}} +\ mathrm {\ Delta} T {\ delta} T {\ left (\ frac {\ partial {F} ^ {p}} {\ partial T}\ right)} _ {{T} ^ {-}} =0\ hfill\ end {array}\end{split}\]

We note:

(3.4)#\[ \begin{align}\begin{aligned} {\ left (\ frac {\ partial {F} ^ {p}}} {\ partial {\ mathrm {\ sigma}} _ {\ mathit {ij}}}\ right)} _ {{\ mathrm {\ sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {\ sigma}} _ {\ mathit {ij}}} {\ mathit {ij}}} {\ mathit {ij}}} ^ {\ sigma}} _ {\ mathit {ij}}} {\ mathit {ij}}} ^ {\ sigma}} _ {\ mathit {ij}}} ^ {\ sigma}} _ {\ mathit {ij}}} ^ {\ sigma}} _ {\ mathit {ij}}} {\ mathit {ij}}} ^ {thrm {\ sigma}} _ {\ mathit {ij}}} ^ {-}}\ right)\\ \ textrm {;}\\ {\ left (\ frac {\ partial {F}} ^ {p}}} {\ partial {\ mathrm {\ xi}}} ^ {p}}\ right)} _ {{\ mathrm {\ xi}}} ^ {p-}}} ^ {p-}} =\ left (\ frac {\ partial {F}}} ^ {p-}} =\ left (\ frac {\ partial {F}}} ^ {p-}}\ right) `; :math:` {\ left (\ frac {\ partial {F} ^ {p}} {\ partial T}\ right)} _ {{T} ^ {-}} =\ left (\ left (\ frac {\ partial {F}} ^ {f} ^ {p}}} {\ partial {T} ^ {-}}} =\ left (\ frac {\ partial {F}} ^ {-}}\ right)\end{aligned}\end{align} \]

Two cases should be distinguished.

If there is coupling between the plastic and viscoplastic mechanisms (stress state above the characteristic threshold). We have \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}+\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}\), and so:

(3.5)#\[\begin{split} \ begin {array} {c} {F} ^ {p}\ left ({\ mathrm {\ sigma}}} _ {\ mathit {ij}}} ^ {-} + {D} _ {\ mathit {ijkl}} {\ mathit {ijkl}}}}\ mathrm {:}\ left (\ mathrm {\ delta}} {\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} - {\ mathit {ijkl}}} -\ mathrm {:}}\ left (\ mathrm {\ delta}} {\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} -\ mathrm {\ phi} ⟩ {G} _ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ alpha}} _ {\ mathit {kl}}}\ mathrm {\ delta} T\ right), {\ mathrm {\ delta} T\ right), {\ mathrm {\ xi}}} ^ {p-}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}}} _ {\ mathit {kl}}}} _ {\ mathit {kl}}}} _ {\ mathit -}\ right) -\\ mathrm {\ delta}}\ mathrm {\ delta}\ mathrm {\ lambda}\ left (\ frac {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {\ sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {ij}}} _ {\ mathit {ij}}} _ {\ mathit {ij}}}} ^ {-}}\ right)\ mathrm {:}\ left ({D} _ {\ mathit {sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {ijkl}}\ mathrm {:} {G} _ {\ mathit {kl}}} ^ {p-}\ right) +\ left (\ mathrm {\ Delta}\ mathrm {\ lambda}\ sqrt {\ mathit {kl}}} {l}}\ sqrt {\ frac {2} {l}}} {\ mathit {II}} ^ {p-} +\ mathrm {\ Delta} {\ mathrm {\ delta}} {\ mathrm {\ gamma}}\ right)\ frac {\ partial {F} ^ {p}} +\ frac {\ partial {F} ^ {p-}} +\ mathrm {\ delta} T\ left (\ frac {\ partial {p}}}} +\ mathrm {\ Delta} T\ left (\ frac {\ partial {p}}} {\ partial}} F} ^ {p}} {\ partial {T} ^ {-}}\ right) =0\ hfill\ end {array}\end{split}\]

And:

(3.6)#\[\begin{split} \ begin {array} {c} {F} ^ {p}\ left ({\ mathrm {\ sigma}}} _ {\ mathit {ij}}} ^ {-} + {D} _ {\ mathit {ijkl}} {\ mathit {ijkl}}}}\ mathrm {:}\ left (\ mathrm {\ delta}} {\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} - {\ mathit {ijkl}}} -\ mathrm {:}}\ left (\ mathrm {\ delta}} {\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} -\ mathrm {\ phi} ⟩ {G} _ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ alpha}} _ {\ mathit {kl}}}\ mathrm {\ delta} T\ right), {\ mathrm {\ delta} T\ right), {\ mathrm {\ xi}}} ^ {p-}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}}} _ {\ mathit {kl}}}} _ {\ mathit {kl}}}} _ {\ mathit -}\ right) =\\ {F} ^ {p}\ left ({\ mathrm {\ sigma}} _ {\ mathit {ij}} ^ {-}, {\ mathrm {\ xi}}} ^ {\ xi}}} ^ {p-}}} ^ {p}} ^ {-}\ right) +\ frac {\ partial {\ mathrm {\ sigma}} {\ partial {\ mathrm {\ sigma}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {-}}\ mathrm {:}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:}\ left (\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ epsilon}} _ {\ mathit {kl}} } -⟨mathrm {\ phi} ⟩ {G} _ {\ mathit {kl}} _ {\ mathit {kl}}} ^ {\ mathit {VP} -}\ mathrm {\ Delta} t- {\ mathrm {\ alpha}}} _ {\ mathit {alpha}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}}\ mathrm {\ delta} T\ right)\ right)\ hfill\ end {array}} _ {\ mathit {alpha}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}}\ mathrm {\ delta} T\ right)\ hfill\ end {array}}\end{split}\]

Hence the expression for the plastic multiplier:

(3.7)#\[\begin{split} \ begin {array} {c}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ lambda} =\ frac {{F} ^ {p}\ left ({\ mathrm {\ sigma}}}} _ {\ mathit {ij}}} _ {\ mathit {ij}}}} _ {\ mathit {ij}}}} ^ {ij}}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {\ partial {F} ^ {p}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:}\ left ({D} _ {\ mathit {ijkl} _ {\ mathit {ijkl}}} {\ mathit {ijkl}}}\ mathrm {ijkl}}}\ mathrm {ijkl}}}\ mathrm {:}\ left (\ mathrm {\ delta}} {\ epsilon}}\ left (\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} {\ mathit {ijkl}}}\ mathrm {ijkl}}}\ mathrm {ijkl}}} {kl}} -\ mathrm {\ phi} ⟩ {G} ⟩ {G} _ {\ mathit {kl}}} ^ {\ mathit {VP} -}\ mathrm {\ Delta} t\ right)\ right)\ right) +\ mathrm {\ delta}} +\ mathrm {\ Delta}} {\ mathrm {\ delta}} {\ partial {\ mathrm {\ xi}}} ^ {p-}}}} {\ left (\ frac {\ partial {F} ^ {p}} {\ partial {\ mathrm {\ sigma}}}} _ {\ mathit {ij}}} ^ {-}}\ right)\ mathrm {:}\ left ({D}}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:} {G} _ {\ mathit {kl}}}} ^ {p-}\\ right) -\ sqrt {p-}\ right) -\ sqrt {\ frac {2} {\ frac {3}} {3}} {g} _ {\ mathit {II}}} _ {\ mathit {II}}} ^ {p-}} ^ {p-}\ right) -\ sqrt {p-}\ right) -\ sqrt {\ frac {2} {3}} {3}} {g} _ {\ mathit {II}}}} ^ {p-}} ^ {{F} ^ {p}} {\ partial {\ partial {\ mathrm {\ xi}}} ^ {p-}}} +\\\ frac {\ left (\ frac {\ partial {F} ^ {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {\ mathrm {\ p}}} {\ partial {\ mathrm {\ p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {\ mathrm {\ p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {mathit {ij}} ^ {-}}\ mathrm {:} {\ mathrm {\ alpha}} _ {\ mathit {ij}}\ right)\ mathrm {\ Delta} T} {\ frac {\ frac {\ partial {F}} {:} {\ partial {F}} ^ {F} ^ {F} ^ {F} ^ {F} ^ {F} ^ {f} ^ {p}}} {\ partial {F} ^ {p}}} {\ partial {\ alpha}}} _ {\ mathit {ij}}} {\ mathit {ij}} T} {\ frac {\ partial {F}} ^ {F} ^ {F} ^ {F} ^ {F} ^ {f} ^ {p}}} {\ partial {F} ^ {m {:}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:} {G} _ {\ mathit {kl}} ^ {p-}\ right) -\ sqrt {\ frac {2} {2} {3}} {3}}} {3}} {g}} {g} _ {\ mathit {II}}} ^ {p-}\ right) -\ sqrt {\ frac {2} {2} {3}} {3}} {3}} {g}} {g} _ {\ mathit {II}}} ^ {p-}\ right) -\ sqrt {\ frac {2} {2} {3}} partial {F} ^ {p}} {\ partial {\ partial {\ mathrm {\ xi}}} ^ {p-}}}\ end {array}\end{split}\]

With:

(3.8)#\[ summarize\ mathrm {\ phi} ●= {A} _ {v} _ {v}} ^ {v} ^ {v}} {\ mathrm {\ sigma}}} ^ {e}} ^ {e}, {\ mathrm {\ xi}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathit {VP} -}} ({\ mathrm {\ sigma}}}} ^ {{a}}\ right)} ^ {{n} _ {v}} = {A} _ {v}} ^ {v}} ^ {V} ^ {{n}} ^ {\ mathit {VP}} ({\ mathrm {\ sigma}}} ^ {-}} ^ {-} {-} + {v}} + {D} _ {\ D} _ {\ left}\ mathrm {:}\ left (\ mathrm {\ delta}}} ^ {-}\ sigma}} ^ {-}} ^ {-}} ^ {-} + {v}} + {D} _ {\ D} _ {\ mathit {ijkl}}}\ mathrm {:}\ left (\ mathrm {\ delta}}} ^ {\ sigma}} ^ {-}\ sigma}} ^ {-} {\ epsilon}} _ {\ mathit {kl}}} - {\ mathrm {\ alpha}}} _ {\ mathit {kl}}\ mathrm {\ Delta} T\ right), {\ mathrm {\ xi}}}} {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}} ^ {\ mathrm {\ xi}}}\]

Quantity \({A}_{v}^{T}\) represents quantity \({A}_{v}\) calculated at temperature \(T\) (instant « + »).

If there is no coupling between the two mechanisms (state of constraint below the characteristic threshold), we therefore have \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}\), we have the following expression for the plastic multiplier:

(3.9)#\[\begin{split} \ begin {array} {c}\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ lambda} =\ frac {{F} ^ {p}\ left ({\ mathrm {\ sigma}}}} _ {\ mathit {ij}}} _ {\ mathit {ij}}}} _ {\ mathit {ij}}}} ^ {ij}}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {ij}}} ^ {\ partial {F} ^ {p}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:}\ left ({D} _ {\ mathit {ijkl} _ {\ mathit {ijkl}}} {\ mathit {ijkl}}}\ mathrm {ijkl}}}\ mathrm {ijkl}}}\ mathrm {:}\ left (\ mathrm {\ delta}} {\ epsilon}}\ left (\ mathrm {\ epsilon}} _ {\ mathit {ijkl}}} {\ mathit {ijkl}}}\ mathrm {ijkl}}}\ mathrm {ijkl}}} {kl}} -\ mathrm {\ phi} ⟩ {G} ⟩ {G} _ {\ mathit {kl}}} ^ {\ mathit {VP} -}\ mathrm {\ Delta} t\ right)\ right)\ right)} {\ right)}} {\ left)} {\ left)} {\ left (\ frac {\ partial {F}} ^ {p}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}}} {\ mathit {ij}}} {\ mathit {ij}}} ^ {p}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}}} {\ mathit {ij}}} {\ mathit {ij}}} ^ {p}} {-}}\ right)\ mathrm {:}\ left ({D} _ {\ mathit {ijkl}}\ mathrm {:} {G} _ {\ mathit {kl}}} ^ {p-}\ left ({D-}\ right) -\ sqrt {\ frac {2} {3} } {g} _ {\ mathit {II}}} ^ {p-}} ^ {p-}\ frac {\ partial {F}} {\ partial {\ mathrm {\ xi}}} ^ {p-}}}} +\\ frac {\ frac {\ frac {\ left (\ frac {\ partial {F}} ^ {-}}} -K\\ frac {\ partial {-}}} -K\\ frac {\ left (\ frac {\ partial {F}} ^ {-}}} -K\\ frac {\ partial {-}}} -K\\ frac {\ partial {-}}} -K\\ frac {\ partial {-}}} -K\\ frac {\ partial {-}} {F} ^ {p}} {\ partial {\ mathrm {\ mathrm {\ sigma}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:} {\ mathrm {\ alpha}}}} _ {\ mathit {ij}}}} _ {\ mathit {ij}}} {\ mathit {ij}}}\ right)\ mathrm {\ delta}} {\ delta} T} {\ frac {\ partial {F} ^ {p}} _ {\ mathit {ij}}} _ {\ mathit {ij}}} {\ mathit {ij}}} {\ mathit {ij}}}\ right)\ mathrm {\ delta}} T} {\ frac {\ partial {F}} ^ {alpha}} {\ sigma}} _ {\ mathit {ij}}} ^ {-}} ^ {-}}\ mathrm {:}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:} {G}} _ {G} _ {G} _ {G} _ {G} _ {l}} _ {kl}}} _ {\ mathit {kl}}}} {\ mathit}}} {\ mathit {kl}}} ^ {p-}\ right) -\ sqrt {\ frac {2} {3}}} {g} {g} {G} _ {\ mathit {kl}} _ {\ mathit {kl}}} {g} _ {\ mathit {kl}}} II}} ^ {p-}\ frac {\ partial {F} ^ {p}} {\ partial {\ mathrm {\ xi}}} ^ {p-}}}}\ end {array}}}\ end {array}\end{split}\]

3.1.2. Tangent operator#

The expression for the stress tensor at the moment « + » is:

\[\]

: label: eq-62

{mathrm {sigma}} _ {mathit {ijkl}}} = {mathrm {sigma}} _ {mathit {ij}} ^ {-} + {D} _ {mathit {ijkl}} {mathit {ijkl}}}mathrm {ijkl}}}mathrm {ijkl}}}mathrm {:}left (mathrm {delta}} {mathrm {epsilon}} _ {mathit {ijkl}}} -mathrm {Delta}mathrm {lambda} {G} {G} _ {mathit {kl}} ^ {p-} -mathrm {phi} ⟩ {G} _ {mathit {kl}} _ {mathit {kl}}}} ^ {mathit {kl}}} ^ {mathit {VP} -}mathrm {Delta} t- {mathrm {alpha}} __ {mathit {kl}}} _ {mathit {kl}}} ^ {mathit {VP} -}mathrm {Delta} t- {mathrm {alpha}} kl}}mathrm {Delta} Tright)

The tangent operator is defined by:

(3.10)#\[ \ frac {\ partial\ mathrm {\ Delta} {\ delta} {\ mathrm {\ sigma}}} _ {\ mathit {ij}}} {\ partial\ mathrm {\ Delta} {\ mathrm {\ epsilon}}}} _ {\ mathit {epsilon}}}} _ {\ mathit {kl}}}} = {\ mathit {kl}}}} = {D} _ {\ mathit {ijkl}}} = {D} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijkl}}} mn}}\ mathrm {:} {G} _ {\ mathit {mn}}} ^ {p-}\ right)\ otimes\ frac {\ partial\ mathrm {\ Delta}\ mathrm {\ lambda}} {\ partial\ lambda}} {\ partial\ mathrm {\ lambda}}} {\ partial\ mathrm {\ lambda}}} {\ partial\ mathrm {\ lambda}}} {\ partial\ mathrm {\ delta}}} -\ left ({D} _ {\ mathit {ijmn}}\ mathrm {:} {G}} _ {\ mathit {mn}} ^ {\ mathit {mp} -}\ right)\ otimes\ frac {\ partial ⇒\ partial\ mathrm {\ phi} ⟩} {\ mathrm {\ phi} ⟩}} {\ partial\ mathrm {\ php}}} _ {\ mathit {kl}}}\ mathrm {\ Delta} t\]

The same two cases should be distinguished. When there is no coupling between the two mechanisms, we therefore have \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}\). Starting with (), the first term is equal to:

(3.11)#\[ \ frac {\ partial\ mathrm {\ Delta}\ mathrm {\ delta}}\ mathrm {\ lambda}} {\ partial\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}} =\ frac {\ frac {\ frac {\ partial {\ lambda}}} {\ partial {\ mathrm {\ epsilon}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:}\ left ({D}} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijmn}}\ mathrm {::} {G}} _ {\ left}} _ {\ mathit {mn} _ {\ mathit {mn} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijmn}}} _ {\ mathit {ijmn}}} -\ left ({D} _ {\ mathit {ijmn}}}}\ mathrm {}}}\ mathrm {::} {G} _ {G} _ {G} _ {G} _ {\ mathit {mn} _ {\ mathit {mn}}\ phi} ⟩} {\ partial\ mathrm {\ delta} {\ delta} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}\ mathrm {\ Delta} t\ right)} {\ frac {\ partial {\ delta}\ right)}} {\ frac {\ partial {F}} {\ partial {F}} ^ {F} ^ {F} ^ {F} ^ {F} ^ {f} ^ {f} ^ {p}}} {\ partial {f} ^ {p}}} {\ partial {\ epsilon}}} _ {\ mathit {ij}}} {\ mathit {ij}}\ right)} {\ frac {\ partial {F}} Delta {\ Delta} t\ right)}} {\ frac {:}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:} {G} _ {\ mathit {kl}}} ^ {p-}\ right) -\ right) -\ sqrt {\ frac {2} {3}} {3}} {g}} _ {\ mathit {II}}} ^ {p-}\ frac {\ partial {F} ^ {p}} {\ partial {\ mathrm {\ mathrm {\ xi}}} ^ {p-}}}\]

And the second term:

(3.12)#\[ \begin{align}\begin{aligned} \ frac {\ partial →\ mathrm {\ phi} ⟩}} {\ partial\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}} =\ frac {{A}} _ {v} _ {v}} ^ {T} {n} _ {v}} {v}} {{P}}} {\ left (\ frac {{A}} _ {F}} {\ left (\ frac {{A}} _ {F}} ^ {\ mathit {VP}} ({\ mathrm {\ sigma}}} ^ {\ sigma}} ^ {\ mathrm {\ xi}}} ^ {\ mathit {VP} -}, T) ⟩} {{P} _ {a}}}\ right)}} ^ {{a}}}\ right)} ^ {\ mathit {a}}}\ right)} ^ {{n}}}\ right)} ^ {{n}} _ {v} -1}\ frac {\ partial {F}} ^ {\ mathit {VP}}} {{P} _ {a}}\ right)} ^ {\ mathit {VP}}}\ right)} ^ {{n}}}\ right partial {\ mathrm {\ sigma}} _ {\ mathit {ij}}} {\ mathit {ij}}} ^ {e}}\ mathrm {:}\ frac {\ partial {\ sigma}} _ {\ mathit {ij}}} _ {\ mathit {ij}}}} _ {\ mathit {ij}}}} ^ {e}}} ^ {e}}} {\ mathit {ij}}} ^ {e}}} {\ mathit {ij}}} ^ {e}}} {\ mathit {ij}}} ^ {e}}} {\ mathit {ij}}}\\ \ frac {\ partial →\ mathrm {\ phi} ⟩}} {\ partial\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}} =\ frac {{A}} _ {v} _ {v}} ^ {T} {n} _ {v}} {v}} {{P}}} {\ left (\ frac {{A}} _ {F}} {\ left (\ frac {{A}} _ {F}} ^ {\ mathit {VP}} ({\ mathrm {\ sigma}}} ^ {\ sigma}} ^ {\ mathrm {\ xi}}} ^ {\ mathit {VP} -}, T) ⟩} {{P} _ {a}}}\ right)}} ^ {{a}}}\ right)} ^ {\ mathit {a}}}\ right)} ^ {{n}}}\ right)} ^ {{n}} _ {v} -1}\ frac {\ partial {F}} ^ {\ mathit {VP}}} {{P} _ {a}}\ right)} ^ {\ mathit {VP}}}\ right)} ^ {{n}}}\ right partial {\ mathrm {\ sigma}} _ {\ mathit {ij}}} ^ {e}}\ mathrm {:} {D} _ {\ mathit {ijkl}}\end{aligned}\end{align} \]

The resulting tangent matrix is therefore equal to:

(3.13)#\[\begin{split} \ begin {array} {c}\ frac {\ partial\ mathrm {\ mathrm {\ delta}} {\ mathrm {\ sigma}} _ {\ mathit {ij}}} {\ partial\ mathrm {\ Delta} {\ delta} {\ delta}} {\ mathrm {\ mathrm {\ epsilon}}}} _ {\ mathit {kl}}}} = {D} _ {\ mathit {ijkl}}}} {\ mathit {ijkl}}} -\ left ({D}}} _ {\ mathit {ijmn}}\ mathrm {:} {G}} _ {\ mathit {mn}} ^ {p-}\ right)\ otimes\ frac {\ frac {\ partial {F} ^ {F} ^ {f} ^ {p}}} {\ partial}} {\ partial {\ p}}} {\ partial {\ p}}} {\ partial {\ p}}} {\ partial {\ p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial {p}}} {\ partial}} ^ {-}}\ mathrm {:} ({D} _ {\ mathit {ijkl}}} -\ left ({D}} _ {\ mathit {ijmn}}\ mathrm {:} {G} _ {\ mathit {mn}}}} ^ {\ mathit {mp}}}} ^ {\ mathit {VP} -} -}\ right)\ left ({D} _ {D} _ {\ mathit {ijkl}}} ^ {\ mathit {ivp} -} -}\ right)\ otimes\ frac {\ partial {ijmn}}} {\ mathit {ijkl}}} ^ {\ mathit {mp}}} ^ {\ mathit {VP} -} -}\ right)\ otimes\ frac {\ partial {ijmn}} {\ mathrm {\ epsilon}} _ {\ mathit {kl}}}}\ mathrm {\ Delta} t\ right)} {\ frac {\ partial {F}} ^ {p}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}}} ^ {-}}\ mathrm {:}\ left ({D} _ {\ mathit {ijkl}}}\ mathrm {:} {:} {G} _ {G} _ {\ mathit {j}} _ {g} _ {\ mathit {kl}}} {g} {g}} {g} _ {\ mathit {II}} ^ {p-}}\ frac {\ partial {F}} ^ {p}} {\ partial {\ mathrm {\ xi}} ^ {p-}}} -\\ frac {{a}}}\ frac {{A} _ {a} _ {v}} {v}} {\ left (\ frac {A}} _ {F}} {\ left (\ frac {A}} _ {F}} ^ {\ mathit {VP}} ({\ mathrm {\ sigma}}} ^ {e}} ^ {\ mathrm {\ xi}}} ^ {\ mathit {VP} -}, T) ⟩} {{P}} _ {P} _ {a}}}\ right)} ^ {{n}}}\ right)} ^ {{n} _ {v} -1}\ left ({D} _ {ijmn}}} {\ mathit {ijmn}}}\ mathrm:} {G} _ {\ mathit {mn}}} ^ {\ mathit {vm}}}} ^ {m {m}}}\ right)\ otimes\ left (\ frac {\ partial {F}} ^ {\ mathit {VP}}}}} {\ mathit {m}}} {\ mathit {m}} ^ {e}} ^ {e}} ^ {e}} ^ {e}} ^ {e}} ^ {e}} ^ {e}} ^ {e}} ^ {D}}}} {\ mathit {D}}}} {\ partial {VP}}}}} {\ mathrm {VP}}}}} {\ mathit {V}}}} {\ mathit {V}}}} {\ mathit {V}}}} Ijkl} }\ right)\ mathrm {\ Delta} t\ end {array}\end{split}\]

When there is a coupling between the two mechanisms, we therefore have \(\mathrm{\Delta }{\mathrm{\xi }}^{p}=\mathrm{\Delta }{\mathrm{\gamma }}^{p}+\mathrm{\Delta }{\mathrm{\gamma }}^{\mathit{vp}}\). Starting with (), the first term is equal to:

(3.14)#\[ \ frac {\ partial\ mathrm {\ Delta}\ mathrm {\ delta}}\ mathrm {\ lambda}} {\ partial\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}} =\ frac {\ frac {\ frac {\ partial {\ lambda}}} {\ partial {\ mathrm {\ epsilon}}} _ {\ mathit {ij}} ^ {-}}\ mathrm {:}\ left ({D}} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijmn}}\ mathrm {::} {G}} _ {\ left}} _ {\ mathit {mn} _ {\ mathit {mn} _ {\ mathit {ijkl}}} -\ left ({D} _ {\ mathit {ijmn}}} _ {\ mathit {ijmn}}} -\ left ({D} _ {\ mathit {ijmn}}}}\ mathrm {}}}\ mathrm {::} {G} _ {G} _ {G} _ {G} _ {\ mathit {mn} _ {\ mathit {mn}}\ phi} ⟩} {\ partial\ mathrm {\ delta} {\ delta} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}}\ mathrm {\ Delta} t\ right) +\ frac {\ partial {F}} {\ partial {F}} {\ partial {F}} ^ {\ partial\ mathrm}}\ frac {\ partial\ mathrm {\ partial\ mathrm {\ mathrm}}}\ frac {\ partial\ mathrm {\ Delta} {\ mathrm {\ gamma}}}} ^ {\ mathit {VP}}} {\ partial\ mathrm {\ Delta} {\ Delta} {\ mathrm {\ epsilon}} _ {\ mathit {kl}}} ^ {\ mathit {lp}}}\ mathrm {:}\ frac {\ partial\ mathrm {\ Delta} {\ delta} {\ mathrm {\ epsilon}}} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}} _ {\ mathit {lp}}} {\ partial {\ mathrm {\ epsilon}}} _ {\ mathrm {\ epsilon}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {kl}}} _ {\ mathit {lp}}} {\ partial {\ mathrm {\ epsilon}}} _ {\ mathit {ij}} ^ {e}}\ mathrm {:} {D}} _ {\ mathit {ijkl}}} {\ frac {\ partial {F} ^ {p}} {\ partial {\ partial {\ mathrm {\ sigma}}}\ mathrm {\ mathit {\ sigma}}} _ {\ mathit {sigma}}} _ {\ mathit {ij}}} {\ mathit {ijkl}}\ mathrm {:} {G} _ {\ mathit {G}} _ {\ mathit {kl}}}} ^ {p-}\ right) -\ sqrt {\ frac {2} {3}} {g} _ {\ mathit {II}}} _ {\ mathit {II}}} _ {\ mathit {II}}}} ^ {p-}} ^ {p-}\ frac {\ partial {F}} {\ partial}} {\ mathrm {\ xi}} _ {\ mathit {II}}}} {\ mathit {II}}}} ^ {p-}} ^ {p-}}}\]

With the following two terms:

(3.15)#\[\begin{split} \ frac {\ partial\ mathrm {\ Delta} {\ Delta} {\ mathrm {\ delta}} {\ mathrm {\ Delta} {\ mathrm {\ epsilon}} {\ mathrm {\ epsilon}}} _ {\ mathit {epsilon}}}} _ {\ mathrm {\ epsilon}}}} _ {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}} ^ {\ mathit {lp}}}} =\ frac {2} {3}\ frac {\ epsilon}}} _ {\ mathit {epsilon}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} ^ {\ mathit {VP}}}} =\\ Delta} {e} _ {\ mathit {ij}}}} ^ {\ mathit {ij}}} {\ mathrm {\ delta} {\ mathrm {\ gamma}}} ^ {\ mathit {VP}}}}\ mathrm {:}} ^ {\ mathit {ij}}} ^ {\ mathit {v2}}} ^ {\ mathit {VP}}}}\ mathrm {:}}}\ mathrm {:}\ left ({\ mathrm {\ delta}} _ {\ mathit {ik}}\ otimes {\ mathrm {VP}}}}}\ mathrm {:}}\ mathrm {:}\ left ({\ mathrm {\ delta}}} _ {\ mathit {ik}}\ otimes} _ {\ mathit {jl}} -\ frac {1} {3} {3} {\ mathrm {\ delta}} _ {\ mathit {ij}}\ otimes {\ mathrm {\ delta}}} -\ frac {\ delta}}} _ {\ mathit {kl}}\ right)\end{split}\]

And:

(3.16)#\[ \ frac {\ partial\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {\ mathit {kl}}} ^ {\ mathit {VP}}} {\ partial {\ mathrm {\ delta}}} {\ mathrm {\ sigma}}} _ {\ sigma}}} _ {v}} _ {v}} _ {v}} {\ partial {\ mathrm {\ sigma}}}} _ {\ sigma}}} _ {v}} _ {v}} {{P} _ {a}} {\ left (\ frac {{F}} ^ {\ mathit {VP}}} ({\ mathrm {\ sigma}} ^ {e}, {\ mathrm {\ xi}}}}} ^ {\ mathrm {\ xi}}}}} ^ {xi}}}} ^ {\ mathrm {\ xi}}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}} ^ {xi}}}\ frac {\ partial {F} ^ {\ mathit {VP}}}} {\ partial {\ mathrm {\ sigma}} _ {\ mathit {ij}} ^ {e}}\ otimes {G}}\ otimes {G}} _ {\ mathit {VP}} _ {\ mathit {VP}}}\ mathrm {\ Delta}}}\ otimes {G}} _ {\ mathit {VP}}}\ otimes {G}} _ {\ mathit {VP}}}\ otimes {G}} _ {\ mathit {V}}}\ otimes {G} _ {\ mathit {G}} _ {\ mathit {\]

This explicit schema has been removed from the current version of the model because it is not widely used and not very robust in the face of the following implicit integration.

3.2. Implicit integration diagram#

The schema is selected when ALGO_INTE = “NEWTON”.

3.2.1. Elastic prediction phase#

This step is the same as in paragraph § 3.1.

3.2.2. Correction phase: nonlinear equations to be solved#

This step consists in solving the system of nonlinear equations established on the basis of viscoplastic and/or plastic mechanisms. The unknowns of the system of nonlinear equations are the effective stresses at the time « + », \(\mathrm{\sigma }\), the plastic multiplier \(\mathrm{\Delta }\mathrm{\lambda }\), the work-hardening variable of the plastic mechanism at the time « + », \({\mathrm{\xi }}^{p}\), and the work-hardening variable of the viscoplastic mechanism at the time « + », \({\mathrm{\xi }}^{\mathit{vp}}\). The vector of unknowns is therefore, at most, nine (3D) in size. The nonlinear equations to be solved are:

The incremental equation of state:

(3.17)#\[ {E} _ {1}\ mathrm {:}\ phantom {\ rule {4em} {0ex}}\ mathrm {\ sigma} - {\ mathrm {\ sigma}} ^ {-} -D\ left (\ mathrm {\ Delta}\ phantom {\ rule {4em}} {0ex}}\ mathrm {\ sigma}} - {\ mathrm {\ sigma}}} ^ {-} -D\ left (\ mathrm {\ delta}} {G} ^ {p} -\ mathrm {\ phi} ({F}} ({F} ^ {\ mathit {VP}}) ⟩ {G} ^ {\ mathit {VP}} -\ mathrm {\ alpha}\ alpha}\ alpha}\ mathrm {\ alpha}\ alpha}\ mathrm {\ alpha}\ alpha}\ mathrm {\ alpha}\ mathrm {\ alpha}\ mathrm {\ alpha}\ mathrm {\ alpha}\ mathrm {\ alpha}\ mathrm {\\]

The Kuhn-Tucker condition:

\[\]

: label: eq-71

<0text {alors}mathrm {Delta}mathrm {lambda} =0\text {Si} {F} ^ {p} =0text {alors}mathrm {Delta}mathrm {lambda} >{ E} _ {2}mathrm {:}phantom {rule {4em} {4em} {0ex}}{begin {array} {c}text {Si} {F} {F} ^ {f} ^ {p 0end {array}

The incremental evolution of the work-hardening variable of the plastic mechanism:

(3.18)#\[ {E} _ {3}\ mathrm {:}\ phantom {\ rule {4em} {\ rule {4em} {0ex}} {\ mathrm {\ xi}} - {\ mathrm {\ xi}}} ^ {xi}}} ^ {xi}} ^ {xi}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {xi}}} ^ {xi}} ^ {\]

The incremental evolution of the work-hardening variable of the viscoplastic mechanism:

(3.19)#\[ {E} _ {4}\ mathrm {:}\ phantom {\ rule {4em} {4em} {0ex}} {\ mathrm {\ xi}} ^ {\ mathit {VP}}} - {\ mathrm {\ xi}}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm {\ xi}} - {\ mathrm} =0\]

These unknowns form a square system, \(R(\mathrm{\Delta }Y)\) or \(\mathrm{\Delta }Y=(\mathrm{\Delta }\mathrm{\sigma },\mathrm{\Delta }\mathrm{\lambda },\mathrm{\Delta }{\mathrm{\xi }}^{p},\mathrm{\Delta }{\mathrm{\xi }}^{\mathit{vp}})\). At iteration \(k\) of the Newton local correction loop, the following matrix equation, with unknown \(\mathrm{\delta }(\mathrm{\Delta }{Y}^{k+1})\), is solved:

(3.20)#\[ \ frac {\ mathit {dR} (\ mathrm {\ Delta} {Y} ^ {k})} {d (\ mathrm {\ Delta} {Y} ^ {k})}\ mathrm {\ delta} (\ mathrm {\ delta} {\ delta} {Y} {Y} ^ {k}) =-R ({\ mathrm {\ Delta} Y}\ delta}} delta} (\ mathrm {\ delta}}} delta} (\ mathrm {\ delta}})\]

The Jacobian matrix \(\frac{\mathit{dR}(\mathrm{\Delta }{Y}^{k})}{d(\mathrm{\Delta }{Y}^{k})}\), which is not symmetric, is constructed as follows:

(3.24)#\[\begin{split} \ frac {\ mathit {dR} (\ mathrm {\ Delta} {Y} {Y} ^ {k})} {d (\ mathrm {\ Delta} {Y} ^ {k})} =\ left (\ begin {array} {\ array} {cccc} {cccc}}\ frac {\ partial {E}}) {\ partial}} {\ partial {({\ mathrm {\ sigma}}} _ {\ mathit}} _ {\ mathit {ij}})} ^ {k}}} &\ frac {\ partial {E}} _ {1}} {\ partial {(\ mathrm {\ Delta}\ mathrm {\ lambda})}} ^ {k}}} &\ frac {\ partial {E}}} &\ frac {\ partial {E} _ {1}}} {1}} {\ partial}} {1}} {\ partial {1}} {\ partial {(\ mathrm {\ delta}}\ mathrm {\ lambda})} ^ {k}})} ^ {k}} &\ frac {\ partial {E} _ {1}} {1}} {\ partial}} {1}} {\ partial {1}} {\ partial {1}}} {\ partial {} &\ frac {\ partial {E} _ {1}} {\ partial}} {\ partial {({\ mathrm {\ xi}}} ^ {k}})} ^ {k}}\\ frac {\ partial {E} _ {1}}} {\ partial {2}}} {\ partial {2}}} {\ partial {2}}} {\ mathrm {\ sigma}}} _ {\ mathit {ij}}})} ^ {k}})} ^ {k}} &\ frac {\ partial {E} _ {2}}} {\ partial {(\ mathrm {\ Delta}\ mathrm {\ lambda})} ^ {k}}} &\ frac {\ partial {E}} _ {2}} {\ partial {({\ mathrm {\ xi}}} ^ {p}})} ^ {p})}} ^ {k}} &\ frac {\ partial {\ mathrm {\ xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {p}})} ^ {k}} &\ frac {\ partial {E} _ {3}} {\ mathrm {\ xi}}} ^ {\ mathrm {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {p}})} ^ {\ mathit {VP}})} ^ {\ mathit {VP}})} ^ {k}} &\ frac {\ partial {E} _ {3}} {\ mathrm {\ xi}}} ^ {\ m {\ sigma}} _ {\ mathit {ij}})})}} ^ {k}}} &\ frac {\ partial {E} _ {3}} {\ partial {(\ mathrm {\ Delta}\ mathrm {\ Delta}\ mathrm {\ lambda}})}} ^ {\ lambda})}} ^ {k}}} &\ frac {\ partial {E} _ {3}} {\ partial {(\ mathrm {\ delta}}\ mathrm {\ lambda})}} ^ {\ lambda})}} ^ {k}}} &\ frac {\ partial {E} _ {3}} {\ partial {(\ partial {\ delta}}\ mathrm {\ delta}\ mathrm {\ lambda})} ^ {p})} ^ {k}}} &\ frac {\ partial {E}} _ {3}} {\ partial {({\ mathrm {\ xi}}} ^ {\ mathit {VP}})}} ^ {k}}} ^ {k}}}} {\ k}}}\\\ frac {\ partial {E} _ {4}} {\ partial {({\ mathrm {\ sigma}}})} ^ {k}}} ^ {k}}}} ^ {k}}}} ^ {k}}}} ^ {k}}}} ^ {k}}}} ^ {k}}}} ^ {k}}}} {\ mathit {ij}})} ^ {k}}} &\ frac {\ partial {E} _ {4}} {\ partial {(\ mathrm {\ Delta}\ mathrm {\ lambda})})}} ^ {\ lambda})}} ^ {k}}} &\ frac {\ partial {E} _ {4}} {\ partial {({\ mathrm {\ xi}}} ^ {p})} ^ {k}} &\ frac {\ partial {E} _ {4}} {\ partial {}} {\ partial {({\ partial {({\ mathrm {\ xi}}}} ^ {\ mathit {xi}}} ^ {k}}}\ end {array} _ {4}}} {\ partial {4}}} {\ partial {4}}} {\ partial {4}}} {\ partial {({\ partial {{array} {({({\ mathrm {\ xi}}}} ^ {xi}}) ^ {k}}\ end {array}\ right)\end{split}\]

The derived terms associated with \({R}_{1}\), corresponding to the first line of \(\frac{\mathit{dR}(\mathrm{\Delta }{Y}^{k})}{d(\mathrm{\Delta }{Y}^{k})}\), are:

(3.22)#\[\begin{split} \ begin {array} {c}\ frac {d {d {({R} _ {1})}} _ {\ mathit {ij}}} {d {(\ mathrm {\ Delta} {Y} _ {1})}}} _ {\ mathit {1})}}} _ {\ mathit {mn}}} -\ frac {\ partial {D} _ {1})}} _ {\ mathit {mn}}} = {I} _ {\ mathit {ijmn}}} -\ frac {\ partial {D} _ {1})}} _ {\ mathit {n}}}} = {I} _ {\ mathit {ijmn}}} -\ frac {\ partial {D} _ {1})}}} _ {\ mathit {ijkl}}} {\ partial {\ mathrm {\ epsilon}}} _ {\ mathit {sigma}}} _ {\ mathit {kl}}\ mathrm {\ Delta} {\ mathrm {\ epsilon}}} _ {\ mathit {sigma}}} _ {\ mathit {kl}} _ {\ mathit {kl}} -\ mathrm {\ Delta}}\ left (\ mathrm {\ delta}} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta} {\ delta}}} ^ {p} - {⇒\ mathrm {\ phi}} ({F}} ^ {\ mathit {VP}}) ⟩} ^ {+} {G} _ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} _ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {kl}} ^ {kl}} ^ {kl}} ^ {\ mathit {kl}}} ^ {\ mathit {kl}}} ^ {\ mathit {\ Delta}\ mathrm {\ lambda} {D} {D} _ {\ mathit {ijkl}}\ mathrm {:}\ frac {\ partial {G} _ {\ mathit {kl}} {\ mathit {kl}} {\ mathit {kl}}} ^ {p}} {\ partial {\ mathrm {\ sigma}} _ {\ mathit {mn}}}} + {D}} + {D} _ {\ mathit {ijkl}}}\ mathrm {:}\ left ({G} _ {\ mathit {kl}}} ^ {\ mathit {VP}}}\ otimes\ mathit {VP}}}}\ otimes\ frac {\ frac {\ partial {ijkl}}}\ partial {\ mathrm {\ sigma}} _ {\ mathit {mn}}}}\ mathrm {\ delta} t\ right) +\\ {\ mathrm {\ phi} ({F} ^ {p}) ({F} ^ {p}) ⟩}} ^ {p}) ⟩}} ^ {p}) ⟩} ^ {p}) ⟩} ^ {p}) ⟩} ^ {+}}\ ^ {+}\ mathrm {+}\ mathrm {+}\ mathrm {+}\ mathrm {+}\ mathrm {\ frac {+}}\ mathrm {\ frac {\ +}}\ mathrm {\ frac {\ partial {G} _ {\ mathit {kl}}} ^ {\ mathit {lp}}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {mn}}}}}\ end {array}}\end{split}\]

(3.23)#\[ \ frac {d {({R} _ {1})}} _ {\ mathit {ij}}}} {d {(\ mathrm {\ Delta} {Y} _ {2})} _ {\ mathit {mn}}}} = {\ mathit {mn}}}}} = {D}}} = {D} _ {D} _ {\ mathit {mn}}}}} = {D} _ {D} _ {D} _ {\ mathit {mn}}}}} = {D} _ {D} _ {D} _ {d} _ {\ mathit {mn}}}}} = {D} _ {D} _ {D} _ {\ mathit {mn}}}}} = {D} _ {D} _ {D} _ {m\]

(3.24)#\[\]

(3.25)#\[ \ frac {d {({R} _ {1})}} _ {\ mathit {ij}}} {d {(\ mathrm {\ Delta} {Y} _ {3})} _ {\ mathit {mn}}}} =\ mathrm {\ mn}}}} =\ mathrm {\ delta}}} =\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}} {D} _ {\ mathit {ijkl}}}\ mathrm {:}\ frac {\ partial {G} _ {\ mathit {kl}}} ^ {p}} {\ partial {\ mathrm {\ xi}}} ^ {p}}\]

(3.26)#\[ \ frac {d {({R} _ {1})}} _ {\ mathit {ij}}} {d {(\ mathrm {\ Delta} {Y} _ {4})} _ {\ mathit {mn}}}} =\ mathrm {\ mn}}}} =\ mathrm {\ delta}}} =\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}}\ mathrm {\ delta}} {D} _ {\ mathit {ijkl}}}\ mathrm {:}\ left (\ frac {\ partial {⟨mathrm {\ xi}}} {\ mathrm {\ Phi}} ({F}} ^ {VP}}) ⟩} ^ {\ mathrm {\ xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {VP}}} {\ mathrm {\ xi}}} ^ {\ mathrm {\ xi}}} ^ {\ mathrm {\ xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} ^ {\ mathit {xi}}} {\ mathit {kl}}} ^ {\ mathit {VP}}}} {\ partial {\ mathrm {\ xi}} ^ {\ mathit {VP}}} {\ mathrm {\ Phi} ({F}} ({F}} ^ {\ mathit {VP}}}) {\ partial {\ mathrm {VP}}}} ^ {+}\ right)\ mathrm {\ Delta}} ({F} Phi} ({F}} ^ {\ phi}) ({F} ^ {\ mathit {VP}}) ⟩} ^ {+}\ right)\ mathrm {\ Delta}} ({F} Phi} ({F}}\]

The derived terms associated with \({R}_{2}\), corresponding to the second line of \(\frac{\mathit{dR}(\mathrm{\Delta }{Y}^{k})}{d(\mathrm{\Delta }{Y}^{k})}\), are, in elasticity (\({R}_{2}\equiv \mathrm{\Delta }\mathrm{\lambda }\)):

(3.29)#\[ \begin{align}\begin{aligned} \ frac {d ({R} _ {2})}} {d {(\ mathrm {\ Delta} {Y} _ {1})}} _ {\ mathit {ij}}}} = {0}} _ {0} _ {\ mathit {ij}}\\ \ textrm {;}\\ \ frac {d ({R} _ {2})} {d (\ mathrm {\ Delta} {Y} _ {2})} =1`; :math:`\frac{d({R}_{2})}{d(\mathrm{\Delta }{Y}_{3})}=0`; :math: `\ frac {d ({d ({R} ({R} _ {2}))} {d (\ mathrm {\ Delta} {Y} _ {4})} =0\end{aligned}\end{align} \]

And in plasticity (\({R}_{2}\equiv {F}^{p}\)):

\[\]

: label: eq-81

frac {d ({R} _ {2})}} {d {(mathrm {Delta} {Y} _ {1})} _ {mathit {ij}}}} =frac {partial {F}} ^ {F}} ^ {p}} {partial {mathrm {sigma}}} _ {mathit {ij}}}} =frac {partial {F}} ^ {F}} ^ {p}}} {partial {p}}}

textrm {;}

frac {d ({R} _ {2})} {d (mathrm {Delta} {Y} _ {2})} =0`; \(\frac{d({R}_{2})}{d(\mathrm{\Delta }{Y}_{3})}=\frac{\partial {F}^{p}}{\partial {\mathrm{\xi }}^{p}}\); :math: `frac {d ({d ({R} ({R} _ {2}))} {d (mathrm {Delta} {Y} _ {4})} =0

The derivative terms associated with \({R}_{3}\), corresponding to the third line of \(\frac{\mathit{dR}(\mathrm{\Delta }{Y}^{k})}{d(\mathrm{\Delta }{Y}^{k})}\) are:

\[\]

: label: eq-82

frac {d ({R} _ {3})}} {d {(mathrm {Delta} {Y} _ {1})}} _ {mathit {ij}}}} ={begin {array} {c}} -mathrm {Delta} -mathrm {Delta}mathrm {Delta}mathrm {delta}mathrm {lambda}sqrt {frac {2} {3}}} ={begin {array} {c} {c} -mathrm {Delta}mathrm {Delta}mathrm {delta}mathrm {lambda}sqrt {frac {2} {3}}} ={begin {array} {{g} _ {mathit {II}}} ^ {p}} {partial {mathrm {sigma}} _ {mathit {ij}}}text {in the contracting case}hfill\-mathrm\-mathrm {Delta}}mathrm {Delta}}mathrm {Delta}mathrm {delta}mathrm {delta}mathrm {delta}mathrm {delta}mathrm {delta}mathrm {delta}mathrm {lambda}sqrt {frac {2} {3}}}frac {partial {g}} _ {mathit {II}} ^ {p}}} {partial {mathrm {sigma}} _ {mathit {ij}}}} -sqrt {frac {2} {3}}}mathrm {Delta}}}}mathrm {Delta}}}mathrm {Delta}}}mathrm {Delta}} tleft ({g}} tleft (g})left ({g}} _ {mathit {II}}} ^ {mathit {VP}}}frac {{partial {3}}}mathrm {Delta}}}mathrm {Delta}} tleft ({g}}left (g}} _ {mathit {II}} thrm {phi} ({F} ^ {mathit {vip}}}) ⟩}} ^ {+}} {partial {mathrm {sigma}}} _ {mathit {ij}}}} +frac {sigma}} _ {mathit {ij}}}} +frac {sigma}}} _ {mathit {ij}}} partial {g} _ {mathit {II}}}} ^ {mathit {VP}}} {partial {mathrm {sigma}} _ {mathit {ij}}}} {mathrm {phi}} ({F} {phi}} ({F}}) ^ {mathit {VP}}}}} {partial {mathrm {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}} {mathit {ij}}}}

\[\]

: label: eq-83

frac {d ({R} _ {3})}} {d (mathrm {Delta} {Y} _ {2})}} =-sqrt {frac {2} {3}} {3}}} {3}}} {g}} {g}} _ {g} _ {mathit {II}}} ^ {p}

(3.32)#\[ \ frac {d ({R} _ {3})}} {d (\ mathrm {\ Delta} {Y} _ {3})} =1-\ mathrm {\ Delta}\ mathrm {\ lambda}\ sqrt {\ frac {2} {3}}} {\ frac {2} {3}}}\ frac {\ partial {g}} _ {\ mathit {II}}} ^ {\ lambda}}\ sqrt {\ frac {2} {3}}} {\ partial}}\ frac {\ partial {g}} _ {\ mathit {II}}} ^ {\ lambda}}\ sqrt {\ frac {2} {3}}} {\ partial}} {\ mathrm {\ xi}}} ^ {p}}\]

(3.29)#\[\]

(3.30)#\[\begin{split} \ frac {d ({R} _ {3})} {d (\ mathrm {\ Delta} {Y} _ {4})} =\ {\ begin {array} {c} 0\ text {in the contracting case}\ hfill\\)}\ hfill\\)}\ hfill\\ -\ sqrt {\ frac {2} {3}}}\ mathrm {\ Delta} t\ left ({g} _ {math\ It {II}} ^ {\ mathit {VP}}}\ frac {\ partial {VP}}}\ frac {\ partial {VP}}} ^ {+}} {\ partial {\ VP}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {\ sigma}}} _ {\ mathit {II}}} _ {\ mathit {II}}}} {\ mathit {II}}} {\ mathit {II}}} ^ {\ mathit {VP}}} {\ partial {\ mathrm {\ sigma}}} _ {\ mathit {ij}}} {01:0\ mathrm {\ phi} ({F} ^ {\ mathit {VP}}) ({\ mathit {VP}}}) ⟩}\ mathit {VP}}) ⟩}}} ^ {+}\ right)\ text {in the dilating case}\ end {array}\end{split}\]

The derived terms associated with \({R}_{4}\), corresponding to the fourth line of \(\frac{\mathit{dR}(\mathrm{\Delta }{Y}^{k})}{d(\mathrm{\Delta }{Y}^{k})}\), are:

(3.31)#\[ \ frac {d ({R} _ {4})}} {d {(\ mathrm {\ Delta} {Y} _ {1})}} _ {\ mathit {ij}}}} =-\ sqrt {\ frac {2} {4})}} {\ frac {2} {2} {3}}}}\ mathrm {2} {3}}}\ mathrm {\ Delta}} t\ left ({g} _ {\ mathit {II}}}} =-\ sqrt {\ frac {2} {2} {3}}}}\ mathrm {2} {3}}}\ mathrm {\ Delta}} t\ left ({g} _ {\ mathit {II}}}} =-\ sqrt {\ frac\ frac {\ partial {\ mathrm {\ sigma}}}} _ {\ mathit {ij}}}} {\ mathrm {\ sigma}}} _ {\ mathit {\ phi}}} _ {\ mathit {ij}}} ({F}}} ({F} ^ {\ mathit {ij}}}} _ {\ mathit {ij}}}} _ {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij}}}} {\ mathit {ij thrm {\ sigma}} _ {\ mathit {ij}}} {\ mathit {ij}}}} {\ mathit {ij}}} {\ mathit {ij}}} {\ right)\ text {si} {\ text {si} {\ si} {\ dot {\ dot {\ mathrm {ij}}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot {\ mathrm {\ ij}}}} {\ dot} - {\ mathrm {\ xi}}} ^ {\ mathit {VP} -}\]

(3.32)#\[\]

\[\]

: label: eq-87

frac {d ({R} _ {4})} {d (mathrm {Delta} {Y} _ {2})} =0

(3.33)#\[ \ frac {d ({R} _ {4})} {d (\ mathrm {\ Delta} {Y} _ {3})} =0\]

(3.34)#\[ \ frac {d ({R} _ {4})} {d (\ mathrm {\ Delta} {\ Delta} {Y} _ {4})} =1-\ sqrt {\ frac {2} {3}}}\ mathrm {\ Delta}}}\ mathrm {\ Delta}}}\ mathrm {\ Delta} t\ left ({g}} t\ left ({g}} _ {g} _ {II}}} ^ {\ mathit {VP}}}\ frac {\ {partial\ mathrm}}\ mathrm {\ Delta}} t\ left ({g}} t\ left (g}} _ {g} _ {G} _ {II}}} ^ {\ mathit {VP}}}\ frac {\ {partial}} thrm {\ phi} ({F} ^ {\ mathit {VP}}}) ⟩}}} ^ {+}} {\ partial {\ mathrm {\ sigma}} _ {\ mathit {ij}}}}} +\ frac {\ partial {g}}} +\ frac {\ partial {j}}}} +\ frac {\ partial {\ mathit {j}}}}} +\ frac {\ partial {\ sigma}}} +\ frac {\ partial {g}} _ {\ partial {g}} _ {\ partial {g}} _ {\ partial {v}}} mathit {ij}}} {\ mathrm {\ phi} ({F} ^ {\ mathit {VP}}) ⟩}} ^ {+}\ right)\]

We will not detail all of the terms in this document.