4. Internal variables#

4.1. Single crystal case#

The internal variables in*Code_Aster* are named \({V}_{1}\), \({V}_{2}\), … \({V}_{p}\).

The first six are the 6 components of visco-plastic deformation: \({E}_{\mathrm{ij}}^{\mathrm{vp}}\):

\({E}^{\mathrm{vp}}=\sum _{t}(\Delta {E}^{\mathrm{vp}})\) with \(\Delta {E}^{\mathrm{vp}}=\sum _{s}{\mu }_{s}{\mathrm{\Delta \gamma }}_{s}\)

\({V}_{1}={E}_{\mathrm{xx}}^{\mathrm{vp}}\), \({V}_{2}={E}_{\mathrm{yy}}^{\mathrm{vp}}\), \({V}_{3}={E}_{\mathrm{zz}}^{\mathrm{vp}}\), \({V}_{4}=\sqrt{(2)}{E}_{\mathrm{xy}}^{\mathrm{vp}}\),, \({V}_{5}=\sqrt{(2)}{E}_{\mathrm{xz}}^{\mathrm{vp}}\), \({V}_{6}=\sqrt{(2)}{E}_{\mathrm{yz}}^{\mathrm{vp}}\)

\({V}_{7}\), \({V}_{8}\), \({V}_{9}\) are the values of \({\alpha }_{1}\) \({\gamma }_{1}\) \({p}_{1}\) for the glide system \(s=1\)

\({V}_{10}\), \({V}_{11}\), \({V}_{12}\) correspond to system \(s=2\), and so on, where:

\({\alpha }_{s}\) represents the kinematic variable of the \(s\) system in the case of phenomenological models, and the dislocation density in a model derived from DD;
\({\gamma }_{s}\) represents the plastic sliding of the \(s\) system
\({p}_{1}\) represents the cumulative plastic slippage of the \(s\) system

Taking into account irradiation:

in case DD_CC_IRRA, \({n}_{\mathrm{irra}}=12\) internal variables must be added: \({V}_{6+{\mathrm{3n}}_{s}+1}\) to \({V}_{6+{\mathrm{3n}}_{s}+12}\) contain for each sliding system the density of dislocations linked to irradiation \({\rho }_{s}^{\mathrm{irr}}\)
in case DD_CFC_IRRA, \({n}_{\mathrm{irra}}=24\) internal variables must be added: \({V}_{6+{\mathrm{3n}}_{s}+1}\) to \({V}_{6+{\mathrm{3n}}_{s}+12}\) contain for each sliding system \({\rho }_{s}^{\mathrm{loops}}\ast {b}^{2}\) \({V}_{6+{\mathrm{3n}}_{s}+13}\) to \({V}_{6+{\mathrm{3n}}_{s}+24}\) contain for each sliding system \({\phi }_{s}^{\mathrm{voids}}\)

The cissions for each sliding system are then stored: \({\tau }_{1}\),… \({\tau }_{{n}_{s}}\)

In the case where we take into account the rotation of the crystal lattice, we must add \({n}_{\mathrm{rota}}=16\) internal variables:

\({V}_{6+{\mathrm{3n}}_{s}+1}\) to \({V}_{6+{\mathrm{3n}}_{s}+9}\) are the 9 components of the rotation matrix \(Q\),
\({V}_{6+{\mathrm{3n}}_{s}+10}\) to \({V}_{6+{\mathrm{3n}}_{s}+12}\) are the 3 components of \(\Delta {\omega }^{p}\),
\({V}_{6+{\mathrm{3n}}_{s}+13}\) to \({V}_{6+{\mathrm{3n}}_{s}+15}\) are the 3 components of \(\Delta {\omega }^{e}\),
\({V}_{6+{\mathrm{3n}}_{s}+16}\) represents \(\Theta\)

The antepenultimate internal variable is the cleavage constraint: \(\underset{s}{\text{max}}(\Sigma \text{.}n):n\)

The penultimate internal variable contains the global cumulative plastic deformation, defined by:

\({V}_{p-1}=\sum \Delta {E}_{\mathrm{eq}}^{\mathrm{vp}}\) with \(\Delta {E}_{\mathrm{eq}}^{\mathrm{vp}}=\sqrt{\frac{2}{3}(\Delta {E}^{\mathrm{vp}}:\Delta {E}^{\mathrm{vp}})}\)

The last internal variable, \(\mathit{Vp}\), (\(p=6+{\mathrm{3n}}_{s}+{n}_{\mathrm{rota}}+3\), \({n}_{s}\) being the total number of sliding systems) is an indicator of plasticity (threshold exceeded in at least one sliding system at the current time step). If it is zero, there was no increase in internal variables at the current moment. Otherwise, it contains the number of local Newton iterations (for an implicit resolution) that were required to obtain convergence.

4.2. Polycrystal case#

The internal variables in*Code_Aster* are named \({V}_{1}\), \({V}_{2}\), … \({V}_{p}\).

The number of internal variables is \(p\mathrm{=}7+\mathrm{6m}+\mathrm{\sum }_{g\mathrm{=}\mathrm{1,}m}({\mathrm{3n}}_{s}(g))+\mathrm{6m}+1\), \(m\) being the number of phases and \({n}_{s}(g)\) being the number of phase sliding systems (\(g\)).

If irradiation is taken into account, the total number of internal variables is:

\(p\mathrm{=}7+\mathrm{6m}+\mathrm{\sum }_{g\mathrm{=}\mathrm{1,}m}({\mathrm{3n}}_{s}(g))+\mathrm{12m}+\mathrm{6m}+1\)

The first six internal variables are the components of macroscopic viscoplastic deformation \({E}^{\mathrm{vp}}\):

the seventh is the macroscopic cumulative equivalent viscoplastic deformation \(P\):

\({V}_{7}=\sum \Delta {E}_{\mathrm{eq}}^{\mathrm{vp}}\) with \(\Delta {E}_{\mathrm{eq}}^{\mathrm{vp}}=\sqrt{\frac{2}{3}(\Delta {E}^{\mathrm{vp}}:\Delta {E}^{\mathrm{vp}})}\);

then, for each phase, we find the 6 components of the viscoplastic deformations or of the \(\beta\) tensor of the phase: \({\left\{{\varepsilon }_{\mathrm{xx}}^{\mathrm{vp}}(g),{\varepsilon }_{\mathrm{yy}}^{\mathrm{vp}}(g),{\varepsilon }_{\mathrm{zz}}^{\mathrm{vp}}(g),\sqrt{(2)}{\varepsilon }_{\mathrm{xy}}^{\mathrm{vp}}(g),\sqrt{(2)}{\varepsilon }_{\mathrm{xz}}^{\mathrm{vp}}(g),\sqrt{(2)}{\varepsilon }_{\mathrm{yz}}^{\mathrm{vp}}(g)\right\}}_{g=\mathrm{1,}m}\);
then, for each phase, and for each phase sliding system, we find the values of \({\alpha }_{s}\) \({\gamma }_{s}\) \({p}_{s}\);
if irradiation is taken into account, for each phase, we find the 12 dislocation densities associated with irradiation \({\rho }_{\mathit{irr}}^{s}\);
then, for each phase, we find the 6 components of the phase constraints: \({\left\{{\sigma }_{\mathrm{xx}}(g),{\sigma }_{\mathrm{yy}}(g),{\sigma }_{\mathrm{zz}}(g),\sqrt{(2)}{\sigma }_{\mathrm{xy}}(g),\sqrt{(2)}{\sigma }_{\mathrm{xz}}(g),\sqrt{(2)}{\sigma }_{\mathrm{yz}}(g)\right\}}_{g=\mathrm{1,}m}\);
the last internal variable is an indicator of plasticity (threshold exceeded in at least one sliding system at the current time step).