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Version Aster	Author (s) Organization (s)	Description of changes
7.4	R.Fernandes, C.Chavant EDF -R&D/ AMA	Initial text

Recalibration of the criterion on triaxial compression

Taking the general expression of the criterion under the conditions of a triaxial in compression, we find:

\(f={(\frac{g(s)}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}-(-\frac{m({\gamma }^{p})k({\gamma }^{p})}{\sqrt{6}{\sigma }_{c}}\frac{g(s)}{{h}_{c}^{0}}-\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}{I}_{1}+s({\gamma }_{p})\mathrm{.}k({\gamma }^{p}))\) \(={(\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣\mathrm{.}h}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}+\frac{1}{{\sigma }_{c}}(\frac{m({\gamma }^{p})k({\gamma }^{p})}{{\sigma }_{c}\sqrt{6}}\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣\mathrm{.}h}{{\sigma }_{c}{h}_{c}^{0}}+\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})k({\gamma }^{p})}{{\sigma }_{c}\sqrt{6}}\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣+\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}∣{\sigma }_{1}-{\sigma }_{3}∣+\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{3}-{\sigma }_{1})+\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{{\sigma }_{c}}({\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}-{\sqrt{\frac{2}{3}}}^{\frac{1}{a({\gamma }^{p})}}(\frac{m({\gamma }^{p})}{{\sigma }_{c}}(-{\sigma }_{3})+s({\gamma }^{p}))\)

\(={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}\left[{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}-{({\sigma }_{c})}^{\frac{1}{a({\gamma }^{p})}}(\frac{m({\gamma }^{p})}{{\sigma }_{c}}(-{\sigma }_{3})+s({\gamma }^{p}))\right]\)

Normalization of Q

\(Q=\frac{1}{{h(\theta )}^{5}}\left[(1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta ))\frac{s}{{s}_{\text{II}}}+\frac{{\gamma }_{\text{cjs}}\sqrt{54}}{{\mathrm{6.s}}_{\text{II}}^{2}}\text{dev}(\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s})\right]\)

We put \(t=\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s}\) and \(t=\text{dev}(\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s})\) (see Reference document CJS R7.01.13)

\({Q}_{\text{II}}^{2}=\mathrm{Q.}Q=\frac{1}{{h(\theta )}^{10}}\left[{1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta )}^{2}+\frac{3}{2}\mathrm{.}\frac{{{\gamma }_{\text{cjs}}}^{2}}{{{s}_{\text{II}}}_{4}}{t}^{d}\mathrm{.}{t}^{d}+\frac{{\gamma }_{\text{cjs}}\sqrt{54}}{{\mathrm{3.s}}_{\text{II}}^{3}}(1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta ))s\mathrm{.}{t}^{d}\right]\)

To evaluate this expression, we place ourselves in the case where

is diagonal for the sake of simplifying the calculations.

So: \(\begin{array}{cc}s=& \left[\begin{array}{}{s}_{1}\\ {s}_{2}\\ {s}_{3}\\ 0\\ 0\\ 0\end{array}\right]\end{array}\) and \(\begin{array}{cc}{t}^{d}=\frac{1}{3}& \left[\begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}\right]\end{array}\)

Using the property of

:, we \(s={s}_{1}+{s}_{2}+{s}_{3}\) shows that \({s}_{\text{II}}^{4}=({s}_{1}^{2}{s}_{2}^{2}+{s}_{1}^{2}{s}_{3}^{2}+{s}_{2}^{2}{s}_{3}^{2})\) and therefore:

\(\begin{array}{cccc}{t}^{\mathrm{d.}}{t}^{d}=\frac{1}{9}& \mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& \mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& =\frac{{{s}_{\text{II}}}^{4}}{6}\end{array}\)

We also show from property \({s}_{1}+{s}_{2}+{s}_{3}=0\) that \({s}_{1}^{3}{s}_{2}^{3}{s}_{3}^{3}={\mathrm{3s}}_{1}{s}_{2}{s}_{3}=3.\text{det}(s)\) and therefore:

\(\begin{array}{ccccc}\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{\mathrm{3.s}}_{\text{II}}^{3}}\mathrm{s.}{t}^{d}=\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{\mathrm{9.s}}_{\text{II}}^{3}}& & \mid \begin{array}{}{s}_{1}\\ {s}_{2}\\ {s}_{3}\\ 0\\ 0\\ 0\end{array}& \text{.}\mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& =\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{s}_{\text{II}}^{3}}\text{det}(s)={\gamma }_{s}\mathrm{.}\mathrm{cos}(3\theta )\end{array}\)

We deduce from this as follows:

\({Q}_{\text{II}}^{2}=\frac{1}{{h(\theta )}^{10}}\left[{(1+\frac{{\gamma }_{\text{cjs}}}{2}\text{cos}(3\theta ))}^{2}+\frac{{{\gamma }_{\text{cjs}}}^{2}}{4}+{\gamma }_{\text{cjs}}\text{cos}(3\theta )(1+\frac{{\gamma }_{\text{cjs}}}{2}\text{cos}(3\theta ))\right]\)

Projection angle framing

We remind you that \(\mathrm{cos}{\phi }_{s}\underset{s\to 0}{\to }\frac{3}{(\beta {\text{'}}^{2}+3)\sqrt{{(\frac{3}{\beta {\text{'}}^{2}+3})}^{2}-\frac{1}{4}+\frac{1}{2(1+{\gamma }_{\text{cjs}}\mathrm{cos}(3\theta ))}+\frac{{\gamma }_{\text{cjs}}^{2}-1}{{4(1+{\gamma }_{\text{cjs}}\mathrm{cos}(3\theta ))}^{2}}}}\)

We pose: \(X(\psi )=\frac{1}{2(1+{\gamma }_{\text{cjs}}\mathrm{cos}(\psi ))}+\frac{{\gamma }_{\text{cjs}}^{2}-1}{{4(1+{\gamma }_{\text{cjs}}\mathrm{cos}(\psi ))}^{2}}\) where \(\psi \in [\mathrm{0,2}\pi [\)

Note that: \(X(-\psi )=X(\psi )\), the function

Being even we restricted the study interval to \(\psi \in [\mathrm{0,2}\pi [\).

The resolution of \(\frac{\mathrm{dX}}{d\psi }=0\) results in \(\frac{{\gamma }_{\text{cjs}}\text{sin}(\psi )}{{2(1+{\gamma }_{\text{cjs}}\text{cos}(\psi ))}^{3}}\mathrm{.}{\gamma }_{\text{cjs.}}({\gamma }_{\text{cjs}}+\mathrm{cos}(3\psi ))=0\)

From this we deduce that the lower and upper limits of function \(X\) are:

\(\{\begin{array}{}X(\psi =0)=\frac{1}{4}\\ X({\psi }_{\text{cjs}})=\frac{1}{4(1-{\gamma }_{\text{cjs}}^{2})}\text{où}{\psi }_{\text{cjs}}\text{est tel que cos}({\psi }_{\text{cjs}})=-{\gamma }_{\text{cjs}}\end{array}\)

We can thus give the following framework for \(\mathrm{cos}{\phi }_{s}\): \(\mathrm{cos}{\phi }_{s}^{\text{min}}\le \mathrm{co}{\phi }_{s}\le \mathrm{cos}{\phi }_{s}^{\text{max}}\) with:

\(\{\begin{array}{}\mathrm{cos}{\phi }_{s}^{\text{min}}=\frac{3}{(\beta {\text{'}}^{2}+3)\sqrt{{(\frac{3}{\beta {\text{'}}^{2}+3})}^{2}+\frac{{{\gamma }_{\text{cjs}}}^{2}}{4(1-{\gamma }_{\text{cjs}}^{2})}}}\\ \mathrm{cos}{\phi }_{s}^{\text{max}}=1\end{array}\)