7. Description of document versions#

Version Aster	Author (s) or Contributor (s), organization	Description of changes
5	P. Schoenberger EDF /R &D/ MMN	Initial text, Chaboche’s law
7	E.Lorentz, J.M.Proix EDF /R &D/ AMA	Add laws VMIS_CIN1_CHAB, VMIS_CIN2_CHAB
8	P. de Bonnières, J.M.Proix EDF /R &D/ AMA	Addition of viscosity: laws VISC_CIN1_CHAB and VISC_CIN2_CHAB, and removal of law CHABOCHE.
9.3	J.M. Proix EDF /R &D/ AMA	Addition of the VMIS/VISC_CIN2_MEMO law, taking into account the memory effect of maximum work hardening.
11 3	J.M. Proix EDF /R &D/ AMA	Addition of the law VMIS/VISC_CIN2_NRAD, taking into account the effect of non-proportionality of the load.
12.1	J.M. Proix EDF /R &D/ AMA	Addition of the note on the positivity of the coefficients k and w, sheet 21019

Tangent behavior matrix

To obtain the tangent behavior in the elastoplastic case, it is necessary to calculate \(\frac{d\Delta {\varepsilon }^{p}}{d\tilde{{\sigma }^{e}}}\) [éq 2.3-7].

To do this, we use the expression \(\Delta {\varepsilon }^{p}\) according to \(\tilde{{\sigma }^{e}}\) and \(p\), which is written in the form:

\(\Delta {\varepsilon }^{p}\mathrm{=}\frac{3\Delta p}{\mathrm{2D}(p)}{\tilde{\sigma }}_{e}+{B}_{1}^{\text{*}}(p){\alpha }_{1}^{\mathrm{-}}+{B}_{2}^{\text{*}}(p){\alpha }_{2}^{\mathrm{-}}\)

with

\({B}_{i}^{\text{*}}(p)\mathrm{=}\mathrm{-}\Delta p\frac{{M}_{i}(p)}{D(p)}\)

\({M}_{i}(p)\mathrm{=}\frac{{C}_{i}(p)}{1+{\delta }_{i}{\gamma }_{i}(p)\Delta p}\)

\(D(p)\mathrm{=}R(p)+(3\mu +{M}_{1}(p){N}_{1}(p,{\beta }_{1})+{M}_{2}(p){N}_{2}(p,{\beta }_{2}))\Delta p+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\)

The following definitions are recalled:

\(\begin{array}{c}R(p)\mathrm{=}{R}_{\mathrm{\infty }}+({R}_{0}\mathrm{-}{R}_{\mathrm{\infty }}){e}^{\mathrm{-}\text{bp}}\\ {C}_{i}(p)\mathrm{=}{C}_{i}^{\mathrm{\infty }}(1+(k\mathrm{-}1){e}^{\mathrm{-}\text{wp}})\\ {\gamma }_{i}(p)\mathrm{=}{\gamma }_{i}^{0}({a}_{\mathrm{\infty }}+(1\mathrm{-}{a}_{\mathrm{\infty }}){e}^{\mathrm{-}\text{bp}})\end{array}\)

So:

\(\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\frac{3\Delta p}{\mathrm{2D}(p)}\text{Id}+\frac{\delta (\frac{3\Delta p}{\mathrm{2D}(p)})}{\delta {\tilde{\sigma }}^{e}}\mathrm{\otimes }{\tilde{\sigma }}^{e}+\frac{\delta {B}_{1}^{\text{*}}(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{\otimes }{\alpha }_{1}^{\mathrm{-}}+\frac{\delta {B}_{2}^{\text{*}}(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{\otimes }{\alpha }_{2}^{\mathrm{-}}\)

Quantities of type \(\frac{\delta A(p)}{\delta {\tilde{\sigma }}^{e}}\) are calculated using: \(\frac{\delta A(p)}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\frac{\delta A(p)}{\delta p}\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}\)

These different terms are expressed by:

\(\frac{\delta (\frac{3\Delta p}{\mathrm{2D}(p)})}{\delta p}\mathrm{=}\frac{3}{2}I(p)\) with \(I(p)\mathrm{=}\frac{1}{D(p)}\mathrm{-}\frac{{D}^{\text{'}}(p)}{{D}^{2}(p)}\Delta p\)

:math:`frac{delta {B}_{i}^{text{}}(p)}{delta p}mathrm{=}mathrm{-}frac{{M}_{i}^{text{“}}(p)}{D(p)}Delta pmathrm{-}{M}_{i}(p)text{.}I(p)mathrm{=}{H}_{i}(p)`

Let’s detail the calculation of \({D}^{\text{'}}\):

In the case of the memory effect, simply modify the term :math:`{R}^{text{“}}(p)`. Like :math:`Rmathrm{=}{R}^{mathrm{-}}+Delta Rmathrm{=}{R}^{mathrm{-}}+bfrac{(Q(Delta p)mathrm{-}{R}^{mathrm{-}})}{1+bDelta p}Delta pmathrm{=}{R}^{mathrm{-}}+frac{bDelta p}{1+bDelta p}({Q}_{M}+({Q}_{0}mathrm{-}{Q}_{M}){e}^{mathrm{-}2mu q}mathrm{-}{R}^{mathrm{-}})` :math:`{R}^{text{“}}(p)mathrm{=}frac{b}{1+bDelta p}(frac{Qmathrm{-}{R}^{mathrm{-}}}{1+bDelta p}mathrm{-}2mu Delta p(Qmathrm{-}{Q}_{M})frac{mathrm{partial }Delta q}{mathrm{partial }Delta p})mathrm{=}frac{b}{1+bDelta p}(frac{Qmathrm{-}{R}^{mathrm{-}}}{1+bDelta p}mathrm{-}mathrm{2mu Delta p}({Q}_{0}mathrm{-}{Q}_{M})frac{mathrm{partial }Delta q}{mathrm{partial }Delta p})` or :math:`Delta qmathrm{=}eta (frac{2}{3}{J}_{2}({varepsilon }^{p}mathrm{-}{xi }^{mathrm{-}})mathrm{-}{q}^{mathrm{-}})` so :math:`frac{mathrm{partial }Delta q}{mathrm{partial }Delta p}mathrm{=}eta frac{{varepsilon }^{p}mathrm{-}{xi }^{mathrm{-}}}{{J}_{2}({varepsilon }^{p}mathrm{-}{xi }^{mathrm{-}})}frac{mathrm{partial }{varepsilon }^{p}}{mathrm{partial }Delta p}` and :math:`frac{delta Delta {varepsilon }^{p}}{delta Delta p}mathrm{=}frac{delta (frac{3Delta p}{mathrm{2D}(p)})}{delta Delta p}{tilde{sigma }}^{e}+frac{delta {B}_{1}^{text{}}(p)}{delta Delta p}{alpha }_{1}^{mathrm{-}}+frac{delta {B}_{2}^{text{*}}(p)}{delta Delta p}{alpha }_{2}^{mathrm{-}}mathrm{=}frac{3}{2}I(Delta p){tilde{sigma }}^{e}+{H}_{1}^{text{*}}(Delta p){alpha }_{1}^{mathrm{-}}+{H}_{2}^{text{*}}(Delta p){alpha }_{2}^{mathrm{-}}`

In the case of non-proportionality (\({\delta }_{1}\mathrm{\ne }1\) or \({\delta }_{2}\mathrm{\ne }1\)), some derivatives are modified:

\({M}_{i}^{\text{'}}(p)\mathrm{=}\frac{{C}_{i}^{\text{'}}(p)}{1+{\delta }_{i}{\gamma }_{i}(p)\Delta p}\mathrm{-}\frac{{C}_{i}(p)}{{(1+{\delta }_{i}{\gamma }_{i}(p)\Delta p)}^{2}}({\gamma }_{i}^{\text{'}}{\delta }_{i}\Delta p+{\gamma }_{i}{\delta }_{i})\)

\({D}^{\text{'}}\mathrm{=}{R}^{\text{'}}+\frac{K}{N\Delta t}{(\frac{\Delta p}{\Delta t})}^{\frac{1}{N}\mathrm{-}1}+3\mu +{M}_{1}{N}_{1}+{M}_{2}{N}_{2}+\Delta p({M}_{1}^{\text{'}}{N}_{1}+{M}_{2}^{\text{'}}{N}_{2}+{M}_{1}{N}_{1}^{\text{'}}+{M}_{2}{N}_{2}^{\text{'}})\)

with \({N}_{i}^{\text{'}}\mathrm{=}\frac{1+{\gamma }_{i}^{\text{'}}({\delta }_{i}\Delta p+({\delta }_{i}\mathrm{-}1){\beta }_{i})+{\gamma }_{i}({\delta }_{i}+({\delta }_{i}\mathrm{-}1){\beta }_{i}^{\text{'}})\mathrm{-}{N}_{i}({\gamma }_{i}+{\gamma }_{i}^{\text{'}}\Delta p)}{1+{\gamma }_{i}\Delta p}\)

It remains to calculate: \(\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}\)

So we use, following [éq 2.3-8]: \(\frac{\delta p}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\mathrm{-}\frac{{\tilde{F}}_{,{\tilde{\sigma }}^{e}}(p,{\tilde{\sigma }}^{e})}{{\tilde{F}}_{,p}(p,{\tilde{\sigma }}^{e})}\)

\(\tilde{F}(p,{\tilde{\sigma }}^{e})\mathrm{=}{S}_{\text{eq}}(p,{\tilde{\sigma }}^{e})\mathrm{-}R(p)\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\mathrm{=}G(p,{\tilde{\sigma }}^{e})\mathrm{-}R(p)\mathrm{-}K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\)

with \(S\mathrm{=}A{\tilde{\sigma }}^{e}+{B}_{1}{\alpha }_{1}^{\mathrm{-}}+{B}_{2}{\alpha }_{2}^{\mathrm{-}}A\mathrm{=}\frac{R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}}{D(p)}{B}_{i}\mathrm{=}\mathrm{-}\frac{2}{3}\frac{{M}_{i}(p)(R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N})}{D(p)}\)

So, by asking \({R}_{v}(p)\mathrm{=}R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\):

\(\begin{array}{c}\begin{array}{c}\frac{\mathit{\delta p}}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\mathrm{-}\frac{{G}_{,{\tilde{\sigma }}^{e}}(p,{\tilde{\sigma }}^{e})}{{G}_{,p}(p,{\tilde{\sigma }}^{e})\mathrm{-}{R}_{v}^{\text{'}}(p)}\mathrm{=}\mathrm{-}\frac{\frac{3}{2}\frac{{R}_{v}(p)}{D(p)}\frac{S}{{S}_{\text{eq}}}}{\frac{3}{2}\frac{S}{{S}_{\text{eq}}}\mathrm{:}{S}_{,p}\mathrm{-}{R}_{v}^{\text{'}}(p)}\mathrm{=}\mathrm{-}\frac{3}{2}\frac{\frac{{R}_{v}}{{\mathit{DS}}_{\text{eq}}}(A{\tilde{\sigma }}^{e}+{B}_{1}{\alpha }_{1}^{\mathrm{-}}+{B}_{2}{\alpha }_{2}^{\mathrm{-}})}{\frac{3}{2}\frac{S}{{S}_{\text{eq}}}\mathrm{:}{S}_{,p}\mathrm{-}{R}_{v}^{\text{'}}(p)}\\ \mathrm{=}\mathrm{-}\frac{3}{2}\frac{{L}_{1}(p)\text{.}{\tilde{\sigma }}^{e}+{L}_{\text{21}}(p){\alpha }_{1}^{\mathrm{-}}+{L}_{\text{22}}(p){\alpha }_{2}^{\mathrm{-}}}{{L}_{3}(p)}\end{array}\\ \end{array}\)

with

\(\begin{array}{c}{L}_{1}(p)\mathrm{=}\frac{{R}_{v}^{2}(p)}{{D}^{2}(p)\mathrm{\times }{S}_{\text{eq}}}\mathrm{=}\frac{{A}^{2}(p)}{{S}_{\text{eq}}}\\ {L}_{\text{21}}(p)\mathrm{=}\frac{{R}_{v}(p)}{D(p)}{B}_{1}(p)\frac{1}{{S}_{\text{eq}}}{L}_{\text{22}}(p)\mathrm{=}\frac{{R}_{v}(p)}{D(p)}{B}_{2}(p)\frac{1}{{S}_{\text{eq}}}\\ {L}_{3}(p)\mathrm{=}\frac{3}{2}\frac{S}{{S}_{\text{eq}}}\mathrm{:}({A}^{\text{'}}(p){\tilde{\sigma }}^{e}+{B}_{1}^{\text{'}}(p){\alpha }_{1}^{\mathrm{-}}+{B}_{2}^{\text{'}}(p){\alpha }_{2}^{\mathrm{-}})\mathrm{-}{R}^{\text{'}}(p)\mathrm{-}\frac{K}{N\Delta t}{(\frac{\Delta p}{\Delta t})}^{\frac{1}{N}\mathrm{-}1}\end{array}\)

Finally, \(\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\) is in the form:

\(\begin{array}{c}\frac{\delta \Delta {\varepsilon }^{p}}{\delta {\tilde{\sigma }}^{e}}\mathrm{=}\frac{3}{2}\frac{\Delta p}{D(p)}\text{Id}+\frac{3}{2}({I}_{S}(p){\tilde{\sigma }}^{e}+{I}_{\mathit{a1}}(p){\alpha }_{1}^{\mathrm{-}}+{I}_{\mathit{a2}}(p){\alpha }_{2}^{\mathrm{-}})\mathrm{\otimes }{\tilde{\sigma }}^{e}\\ +({H}_{s}^{1}{\tilde{\sigma }}^{e}+{H}_{\mathit{a1}}^{1}{\alpha }_{1}^{\mathrm{-}}+{H}_{\mathit{a2}}^{1}{\alpha }_{2}^{\mathrm{-}})\mathrm{\otimes }{\alpha }_{1}^{\mathrm{-}}\\ +({H}_{s}^{2}{\tilde{\sigma }}^{e}+{H}_{\mathit{a1}}^{2}{\alpha }_{1}^{\mathrm{-}}+{H}_{\mathit{a2}}^{2}{\alpha }_{2}^{\mathrm{-}})\mathrm{\otimes }{\alpha }_{2}^{\mathrm{-}}\end{array}\)

with:

\(\begin{array}{c}\begin{array}{cc}{I}_{S}(p)\text{=-}\frac{3}{2}I(p)\text{.}\frac{{L}_{1}(p)}{{L}_{3}(p)}& {I}_{\mathit{a1}}(p)\text{=-}\frac{3}{2}\frac{I(p){L}_{\text{21}}(p)}{{L}_{3}(p)}\\ & {I}_{\mathit{a2}}(p)\text{=-}\frac{3}{2}\frac{I(p){L}_{\text{22}}(p)}{{L}_{3}(p)}\end{array}\\ {H}_{s}^{1}(p)\text{=-}\frac{3}{2}\frac{{H}_{1}(p)\text{.}{L}_{1}(p)}{{L}_{3}(p)}{H}_{\mathit{a1}}^{1}(p)\text{=-}\frac{3}{2}\frac{{H}_{1}(p){L}_{\text{21}}(p)}{{L}_{3}(p)}{H}_{\mathit{a2}}^{1}(p)\text{=-}\frac{3}{2}\frac{{H}_{1}(p){L}_{\text{22}}(p)}{{L}_{3}(p)}\\ {H}_{s}^{2}(p)\text{=-}\frac{3}{2}\frac{{H}_{2}(p)\text{.}{L}_{1}(p)}{{L}_{3}(p)}{H}_{\mathit{a1}}^{2}(p)\text{=-}\frac{3}{2}\frac{{H}_{2}(p){L}_{\text{21}}(p)}{{L}_{3}(p)}{H}_{\mathit{a2}}^{2}(p)\text{=-}\frac{3}{2}\frac{{H}_{2}(p){L}_{\text{22}}(p)}{{L}_{3}(p)}\end{array}\)

Solving the equation f (p) = 0

It involves solving a non-linear scalar equation by looking for the solution within a confidence interval. To do this, it is proposed to couple a secant method with a control of the search interval. Let the following equation be solved:

\(f(x)\mathrm{=}0\), \(x\mathrm{\in }\left[a,b\right]\), \(f(a)<0\), \(f(b)>0\) eq A2-1

The secant method consists in building a series of points \({x}^{n}\) that converge towards the solution. It is defined by recurrence (linear approximation of the function by its chord):

\({x}^{n+1}={x}^{n-1}-f({x}^{n-1})\frac{{x}^{n}-{x}^{n-1}}{f({x}^{n})-f({x}^{n-1})}\) eq A2-2

Moreover, if \({x}^{n+1}\) were to leave the interval, then we replace it with the limit of the interval in question:

\(\{\begin{array}{c}\text{si}{x}^{n+1}<a\text{alors}{x}^{n+1}:=a\\ \text{si}{x}^{n+1}>b\text{alors}{x}^{n+1}:=b\end{array}\) eq A2-3

On the other hand, if \({x}^{n+1}\) is in the current interval, then we update the interval:

\(\mathrm{\{}\begin{array}{c}\text{si}{x}^{n+1}\mathrm{\in }\left[a,b\right]\text{et}f({x}^{n+1})<0\text{alors}a\mathrm{=}{x}^{n+1}\\ \text{si}{x}^{n+1}\mathrm{\in }\left[a,b\right]\text{et}f({x}^{n+1})>0\text{alors}b\mathrm{=}{x}^{n+1}\end{array}\) eq A2-4

It is considered to have converged when \(f\) is close enough to 0 (tolerance to be entered). As for the first two points of the sequence, we can choose the limits of the interval, or else, if we have an estimate of the solution, we can adopt this estimate and one of the limits of the interval.

Note:

This method works well if there is only one solution in the range \(\left[a,b\right]\). Without this being formally demonstrated, we can see that \(f(0)>0\) .

We then search \(b\) such as \(f(b)<0\) .

We’re leaving for this of \(b\mathrm{=}\frac{{({\tilde{s}}^{\underline{e}}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}})}_{\text{eq}}\mathrm{-}R({p}^{\mathrm{-}})}{\mathrm{3m}}\)

If \(f(b)>0\) , we multiply \(b\) by 10 and test if \(f(b)>0\) , and so on, until we find a value \(b\) such as \(f(b)<0\) .

We are sure that there is then at least one solution on \(\left[a,b\right]\) .