1. Ratings#
1.1. Generalities#
\(\sigma\) refers to the effective stress tensor in small disturbances, noted in the form of the following vector:
\((\begin{array}{c}{\sigma }_{\text{11}}\\ {\sigma }_{\text{22}}\\ {\sigma }_{\text{33}}\\ \sqrt{2}{\sigma }_{\text{12}}\\ \sqrt{2}{\sigma }_{\text{13}}\\ \sqrt{2}{\sigma }_{\text{23}}\end{array})\)
We note:
\({I}_{1}\mathrm{=}\text{tr}(\sigma )\) |
first constraint invariant |
\(s\mathrm{=}\sigma \mathrm{-}\frac{{I}_{1}}{3}I\) |
deviatory stress tensor |
\({s}_{\mathit{II}}\mathrm{=}\sqrt{s\mathrm{.}s}\) |
second invariant of the deviatory stress tensor |
\({\sigma }_{\text{max}}\) |
major main constraint |
\({\sigma }_{\text{min}}\) |
minor main constraint |
\(\tilde{\varepsilon }\mathrm{=}\varepsilon \mathrm{-}\frac{\text{Tr}(\varepsilon )}{3}I\) |
deformation deviator |
\({\varepsilon }_{V}\mathrm{=}\text{tr}(\varepsilon )\) |
volume deformation |
\(\text{cos}(3\theta )\mathrm{=}{2}^{1\mathrm{/}2}{3}^{3\mathrm{/}2}\frac{\text{det}(s)}{{s}_{\text{II}}^{3}}\) |
|
\({\dot{\gamma }}_{p}\mathrm{=}\sqrt{\frac{2}{3}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{p}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{p}}\) |
cumulative plastic deviatory deformations |
\({\dot{\gamma }}_{\text{vp}}\mathrm{=}\sqrt{\frac{2}{3}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{\text{vp}}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{\text{vp}}}\) |
cumulative viscoplastic deviatory deformations |
\({\xi }_{p}\) |
plastic work hardening parameter |
\({\xi }_{\text{vp}}\) |
viscoplastic work hardening parameter |
\({G}^{\text{visc}}\) |
function controlling the evolution of viscous deformations and describing the flow direction |
\(\tilde{G}\mathrm{=}G\mathrm{-}\frac{\text{Tr}(G)}{3}I\) |
\(G\) deviator |
\(G\mathrm{=}\text{Tr}(G)\) |
trace of \(G\) |
\({\tilde{G}}_{\mathit{II}}\mathrm{=}\sqrt{\tilde{G}\mathrm{\cdot }\tilde{G}}\) |
\(\tilde{G}\) standard |
\(\psi\) |
angle of dilatance |
\({f}^{d}\) |
elastoplastic load surface |
\({f}^{\text{vp}}\) |
viscoplastic load surface |
1.2. Sign convention#
In Code_Aster, the sign convention is that of the mechanics of continuous media:
In compression: \(\sigma <0\); \(\varepsilon \mathrm{=}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}<0\)
In traction: \(\sigma >0\); \(\varepsilon \mathrm{=}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}>0\)
In model LETK, the sign convention is that of soil mechanics:
In compression: \(\sigma >0\)
Contract: \({\varepsilon }_{v}>0\)
In traction: \(\sigma <0\)
Distance: \({\varepsilon }_{v}<0\)
Note:
To integrate this law in*Code_Aster* as it stands, you must change the sign of all fields at the input of the routine corresponding to the law of behavior and at its output.
At the start of the routine:
\(\begin{array}{c}{\sigma }_{\text{L\&K}}^{\mathrm{-}}\mathrm{=}\mathrm{-}{\sigma }^{\mathrm{-}}\\ {\varepsilon }_{\text{L\&K}}^{\mathrm{-}}\mathrm{=}\mathrm{-}{\varepsilon }^{\mathrm{-}}\\ \Delta {\varepsilon }_{\text{L\&K}}\mathrm{=}\mathrm{-}\Delta \varepsilon \end{array}\)
At the end of the routine:
\(\begin{array}{c}\sigma \mathrm{=}\mathrm{-}{\sigma }_{\text{L\&K}}\\ \varepsilon \mathrm{=}\mathrm{-}{\varepsilon }_{\text{L\&K}}\\ \Delta \varepsilon \mathrm{=}\mathrm{-}\Delta {\varepsilon }_{\text{L\&K}}\end{array}\)
1.3. Model parameters#
Notion |
Description |
\({P}_{a}\) |
atmospheric pressure |
\({\sigma }_{c}\) |
resistance in simple compression, intervening in the expression of the criteria |
\({H}_{0}^{\text{ext}}\) |
parameter controlling the resistance in extension, intervening in the expression of the criteria |
\({\sigma }_{\text{point1}}\) |
|
\({x}_{\text{ams}}\) |
non-zero parameter involved in pre-peak work hardening laws |
\(\eta\) |
non-zero parameter involved in post-peak work hardening laws |
\({a}_{0}\) |
value of \(a\) on the damage threshold |
\({m}_{0}\) |
value of \(m\) on the damage threshold |
\({s}_{0}\) |
value of s on the damage threshold |
\({a}_{\text{pic}}\) |
value of \(a\) on the peak threshold |
\({m}_{\text{pic}}\) |
value of m on the peak threshold |
\({\xi }_{\text{pic}}\) |
level of work hardening required at \({\xi }_{p}\) to reach the peak threshold |
\({a}_{e}\) |
value of \(a\) on the cleavage threshold |
\({m}_{e}\) |
value of \(m\) on the cleavage threshold |
\({\xi }_{e}\) |
level of work hardening required at \({\xi }_{p}\) to reach the cleavage threshold |
\({m}_{\text{ult}}\) |
value of \(m\) on the residual threshold |
\({\xi }_{\text{ult}}\) |
level of work hardening required at \({\xi }_{p}\) to reach the residual threshold |
\({m}_{v\mathrm{-}\text{max}}\) |
value of \(m\) on the maximum viscoplastic threshold |
\({\xi }_{v\mathrm{-}\text{max}}\) |
value of \({\xi }_{v}\) for which the maximum viscoplastic criterion is reached |
\({A}_{v}\) |
parameter characterizing the amplitude of the creep speed |
\({n}_{v}\) |
exponent involved in the formula controlling the creep kinetics |
\({\mu }_{\mathrm{0,}v}\) |
parameter relating to pre-peak dilatance |
\({\xi }_{\mathrm{0,}v}\) |
parameter relating to pre-peak dilatance |
\({\mu }_{1}\) |
parameter relating to the dilatance in post peak |
\({\xi }_{1}\) |
parameter relating to post-peak dilatance |