8. Document version history#
Version Aster |
Author (s) or Contributor (s), organization |
Description of changes |
\(7.4\) |
J.M. PROIX, O. DIARD, T., T. KANIT, T. EDF /R & D |
Initial Release |
\(8.4\) |
J.EL- GHARIB, J.M. PROIX, J.M., EDF /R & D |
Complements |
\(9.2\) |
J.EL- GHARIB, J.M. PROIX, J.M., EDF /R & D |
Added behavior MONO_DD_KR. |
\(9.4\) |
J.EL- GHARIB, J.M. PROIX, J.M., EDF /R & D |
Behavior Optimization MONO_DD_KR. |
\(10.2\) |
J.M. PROIX |
note on the interaction matrix |
\(10.3\) |
|
Add behavior DD_CFC |
\(10.5\) |
|
Addition of crystal lattice rotation |
\(11.1\) |
|
Adding major deformations |
\(11.1\) |
|
Files 16373 (convergence criterion) and 17422: correction of POLYCRISTAL internal variables. |
\(11.1\) |
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Sheet 14586: Add behavior DD_FAT |
\(11.2\) |
J.M. PROIX |
Sheet 18398: resorption of ALGO_C_PLAN |
\(11.2\) |
J.M. PROIX |
Sheet 18692 added to DD_CC |
\(11.3\) |
J.M. PROIX |
Form 19021 adding internal variables for DD_CC_IRRA |
\(12.2\) |
J.M. PROIX |
Fiches 22563: simplification of the law DD_CC and 22491 coefficient \({\mu }^{\mathit{loca}}\) for homogenization, |
Jacobian expression for integrated elasto-visco-plastic equations
The system to be solved is of the form:
\(R(Y)=R(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})=\{\begin{array}{}s(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ e(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ {n}_{s}\left\{\begin{array}{}a(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ g(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ p(\Sigma ,\Delta {E}^{\text{vp}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\end{array}\right\}\end{array}\)
\(=\{\begin{array}{}{\Lambda }^{\text{-1}}\Sigma -({\Lambda }_{\text{-}}^{\text{- 1}}){\Sigma }^{\text{-}}-(\Delta E-\Delta {E}^{\text{th}}-\Delta {E}^{\text{vp}})\\ \Delta {E}^{\text{vp}}-{\sum }_{s}{\mu }_{s}\Delta {\gamma }_{s}\\ {n}_{s}\left\{\begin{array}{}\Delta {\alpha }_{s}-h({\tau }_{s}^{\text{+}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ \Delta {\gamma }_{s}-g({\tau }_{s}^{\text{+}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\\ {\mathrm{\Delta p}}_{s}-f({\tau }_{s}^{\text{+}},{\alpha }_{s}^{\text{+}},{\gamma }_{s}^{\text{+}},{p}_{s}^{\text{+}})\end{array}\right\}\end{array}\)
\(=0\text{avec}{\tau }_{s}^{\text{+}}={\Sigma }^{\text{+}}:{\mu }_{s}\)
So let’s evaluate the terms of the Jacobian hypermatrix \(J\) at the moment \(t+\Delta t\)
\(J=\left[\begin{array}{ccccc}\frac{\partial s}{\partial \text{ΔΣ}}& \frac{\partial s}{\partial {\mathrm{\Delta E}}^{\text{vp}}}& \frac{\partial s}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial s}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial s}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial e}{\partial \text{ΔΣ}}& \frac{\partial e}{\partial {\mathrm{\Delta E}}^{\text{vp}}}& \frac{\partial e}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial e}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial e}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial a}{\partial \text{ΔΣ}}& \frac{\partial a}{\partial {\mathrm{\Delta E}}^{\text{vp}}}& \frac{\partial a}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial a}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial a}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial g}{\partial \text{ΔΣ}}& \frac{\partial g}{\partial {\mathrm{\Delta E}}^{\text{vp}}}& \frac{\partial g}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial g}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial g}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial p}{\partial \text{ΔΣ}}& \frac{\partial p}{\partial {\mathrm{\Delta E}}^{\text{vp}}}& \frac{\partial p}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial p}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial p}{\partial {\mathrm{\Delta p}}_{s}}\end{array}\right]\)
With regard to the first row of the matrix, independently of the work-hardening and flow equations, we have:
\(\begin{array}{cccc}\frac{\partial s}{\partial \text{ΔΣ}}={D}^{-1}& \frac{\partial s}{\partial {\mathrm{\Delta E}}^{\text{vp}}}=\text{Id}& \frac{\partial s}{\partial {\mathrm{\Delta \alpha }}_{s}}=& \frac{\partial s}{\partial {\mathrm{\Delta \gamma }}_{s}}=\frac{\partial s}{\partial {\mathrm{\Delta p}}_{s}}\end{array}=0\)
The second line can also be written independently of the flow and the workings:
\(\begin{array}{ccc}\begin{array}{ccc}\frac{\partial e}{\partial \text{ΔΣ}}=0& \frac{\partial e}{\partial {\mathrm{\Delta E}}^{\text{vp}}}=\text{Id}& \frac{\partial e}{\partial {\mathrm{\Delta \alpha }}_{s}}=0\end{array}& \frac{\partial e}{\partial {\mathrm{\Delta \gamma }}_{s}}=-{\mu }_{s}& \frac{\partial e}{\partial {\mathrm{\Delta p}}_{s}}=0\end{array}\)
The first column of the rows corresponding to equations (a), (g), and (p) is written:
\(\begin{array}{c}\frac{\partial a}{\partial \text{ΔΣ}}=\frac{\partial a}{\partial {\mathrm{\Delta \tau }}_{s}}\frac{{\mathrm{\Delta \tau }}_{s}}{\text{ΔΣ}}\\ \frac{\partial g}{\partial \text{ΔΣ}}=\frac{\partial g}{\partial {\mathrm{\Delta \tau }}_{s}}\frac{{\mathrm{\Delta \tau }}_{s}}{\text{ΔΣ}}\\ \frac{\partial p}{\partial \text{ΔΣ}}=\frac{\partial p}{\partial {\mathrm{\Delta \tau }}_{s}}\frac{{\mathrm{\Delta \tau }}_{s}}{\text{ΔΣ}}\end{array}\)
with
\(\frac{{\mathrm{\Delta \tau }}_{s}}{\text{ΔΣ}}={({\mu }_{s})}^{T}\)
The second column is identically zero (because of equation (e)): the flow and work hardening relationships can only be expressed as a function of \(\Delta {\gamma }_{s}\) and not of \({\mathrm{\Delta E}}^{\text{vp}}\).
The last block of equations, for its part, depends on the behaviors chosen:
\(\begin{array}{ccc}\frac{\partial a}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial a}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial a}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial g}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial g}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial g}{\partial {\mathrm{\Delta p}}_{s}}\\ \frac{\partial p}{\partial {\mathrm{\Delta \alpha }}_{s}}& \frac{\partial p}{\partial {\mathrm{\Delta \gamma }}_{s}}& \frac{\partial p}{\partial {\mathrm{\Delta p}}_{s}}\end{array}\)
Example
Let’s choose the viscoplastic flow relationship MONO_VISC1
\(\begin{array}{}(g){\mathrm{\Delta \gamma }}_{s}-{\mathrm{\Delta p}}_{s}\frac{{\tau }_{s}-{\mathrm{c\alpha }}_{s}}{\mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid }=0\\ (p){\mathrm{\Delta p}}_{s}-\mathrm{\Delta t}\text{.}{\langle \frac{\mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid -{R}_{s}({p}_{s})}{k}\rangle }^{n}=0\end{array}\)
with isotropic work hardening MONO_ISOT1 : \({R}_{s}({p}_{s})={R}_{0}+Q(\sum _{r=1}^{N}{h}_{\text{sr}}(1-{e}^{-{\text{bp}}_{r}}))\),
and kinematic work hardening defined by MONO_CINE1
\((a){\mathrm{\Delta \alpha }}_{s}-{\mathrm{\Delta \gamma }}_{s}+{\mathrm{d\alpha }}_{s}{\mathrm{\Delta p}}_{s}=0\)
so:
\(\begin{array}{c}\frac{\partial a}{\partial {\mathrm{\Delta \tau }}_{s}}=0\\ \frac{\partial g}{\partial {\mathrm{\Delta \tau }}_{s}}=0\\ \frac{\partial p}{\partial {\mathrm{\Delta \tau }}_{s}}=\frac{-\mathrm{n\Delta t}}{{K}^{n}}{\langle \mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid -{R}_{s}({p}_{s})\rangle }^{n-1}\frac{{\tau }_{s}-\mathrm{c\alpha }}{\mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid }\end{array}\)
\(\begin{array}{ccc}\frac{\partial a}{\partial {\mathrm{\Delta \alpha }}_{s}}& =& 1+{\mathrm{d\Delta p}}_{s}\\ \frac{\partial g}{\partial {\mathrm{\Delta \alpha }}_{s}}& =& 0\\ \frac{\partial p}{\partial {\mathrm{\Delta \alpha }}_{s}}& =& \frac{\text{nc}\mathrm{\Delta t}}{{K}^{n}}{\langle \mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid -{R}_{s}({p}_{s})\rangle }^{n-1}\frac{{\tau }_{s}-{\mathrm{c\alpha }}_{s}}{\mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid }\end{array}\)
\(\begin{array}{ccc}\frac{\partial a}{\partial {\mathrm{\Delta \gamma }}_{s}}& =& -1\\ \frac{\partial g}{\partial {\mathrm{\Delta \gamma }}_{s}}& =& 1\\ \frac{\partial p}{\partial {\mathrm{\Delta \gamma }}_{s}}& =& 0\end{array}\)
\(\begin{array}{}\begin{array}{ccc}\frac{\partial a}{\partial {\mathrm{\Delta p}}_{s}}& =& {\mathrm{d\alpha }}_{s}\\ \frac{\partial g}{\partial {\mathrm{\Delta p}}_{s}}& =& \frac{{\tau }_{s}-{\mathrm{c\alpha }}_{s}}{\mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid }\\ \frac{\partial p}{\partial {\mathrm{\Delta p}}_{s}}& =& 1+\frac{\mathrm{n\Delta t}}{{K}^{n}}{\langle \mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid -{R}_{s}({p}_{s})\rangle }^{n-1}\frac{{\text{dR}}_{s}({p}_{s})}{{\mathrm{d\Delta p}}_{s}}\end{array}\\ \frac{{\text{dR}}_{s}({p}_{s})}{{\mathrm{d\Delta p}}_{s}}={\text{Qbh}}_{\text{ss}}{e}^{-{\text{bp}}_{s}}\end{array}\)
and, concerning the interaction between sliding systems, there is only one non-zero term:
\(\begin{array}{}\frac{\partial p}{\partial {\mathrm{\Delta p}}_{r}}=1+\frac{\mathrm{n\Delta t}}{{K}^{n}}{\langle \mid {\tau }_{s}-{\mathrm{c\alpha }}_{s}\mid -{R}_{s}({p}_{s})\rangle }^{n-1}\frac{{\text{dR}}_{s}({p}_{s})}{{\mathrm{d\Delta p}}_{r}}\\ \frac{{\text{dR}}_{s}({p}_{s})}{{\mathrm{d\Delta p}}_{r}}={\text{Qbh}}_{\text{sr}}{e}^{-{\text{bp}}_{r}}\end{array}\)
Consistent tangent operator evaluation
It is a question of finding the coherent tangent operator, i.e. calculated from the solution of \((R(Y)=0)\) at the end of the increment. For a small variation of \(R\), this time considering \(\Delta E\) as a variable and not as a parameter, we get:
\(\frac{\mathrm{\partial }R}{\mathrm{\partial }\Sigma }\delta \Sigma +\frac{\mathrm{\partial }R}{\mathrm{\partial }\Delta E}\delta \Delta E+\frac{\mathrm{\partial }R}{\mathrm{\partial }\Delta {\gamma }_{s}}\delta \Delta {\gamma }_{s}\mathrm{=}0\)
This system can be written as:
\(\frac{\mathrm{\partial }R}{\mathrm{\partial }Y}\delta (Y)\mathrm{=}X\) with \(Y\mathrm{=}\left[\begin{array}{c}\Sigma \\ \Delta {\gamma }_{s}\end{array}\right]\) and \(X\mathrm{=}\left[\begin{array}{c}\delta \Delta E\\ 0\end{array}\right]\)
By writing the Jacobian matrix in the form:
\(J\text{.}\delta Y\mathrm{=}\left[\begin{array}{cc}{Y}_{0}& {Y}_{1}\\ {Y}_{2}& {Y}_{3}\end{array}\right]\left[\begin{array}{c}\Sigma \\ \Delta Z\end{array}\right]\)
With:
\(\begin{array}{}{Y}_{0}={\Lambda }^{-1}\\ \Delta Z=\left\{\Delta {\gamma }_{s}\right\}\times {n}_{s}\end{array}\)
The dimensions of the sub-matrices are:
\(\begin{array}{}\text{dim}({Y}_{0}={\Lambda }^{-1})=\left[\mathrm{6,6}\right]\\ \text{dim}{Y}_{1}=\left[\mathrm{6,}{n}_{s}\right]\\ \text{dim}{Y}_{2}=\left[{n}_{s}\mathrm{,6}\right]\\ \text{dim}{Y}_{3}=\left[{n}_{s},{n}_{s}\right]\end{array}\)
By operating by successive removals and substitutions, the third block of the equation system gives:
\(\begin{array}{c}\Delta Z\mathrm{=}\mathrm{-}{({Y}_{3})}^{\mathrm{-}1}{Y}_{2}\Sigma \\ ({Y}_{0}\mathrm{-}{Y}_{1}{({Y}_{3})}^{\mathrm{-}1}{Y}_{2})\Sigma \mathrm{=}\Delta E\end{array}\)
the tangent operator sought can therefore be written directly:
\({(\frac{\mathrm{\partial }\Sigma }{\mathrm{\partial }E})}_{t+\Delta t}\mathrm{=}{(\frac{\mathrm{\partial }\Sigma }{\mathrm{\partial }\Delta E})}_{t+\Delta t}\mathrm{=}{({Y}_{0}\mathrm{-}{Y}_{1}{Y}_{3}^{\mathrm{-}1}{Y}_{2})}^{\mathrm{-}1}\)