6. Description of document versions#

Document index	Version Aster	Author (s) Organization (s)	Description of changes
B	7.4	P.Badel EDF -R&D/ AMA	Initial text
C	8.5	P.Badel EDF -R&D/ AMA	Sign correction on page 10: a sign was missing — in the second member of equation 2.4.2.2-4 as well as in the following equation 2.4.2.2-5
D	9.4	V.Godard EDF -R&D/ AMA	Modification of control by elastic prediction.
	10.3	F.Voldoire EDF -R&D/ AMA	Addition of a passage on page 5 explaining the slope under uniaxial load.

Demonstration of the natural constraint coordinate system

The term trace in energy does not pose a problem: it is invariant by any change of frame of reference.

Remains the term in \(\mathrm{\sum }_{i}{\varepsilon }_{i}^{2}(H(\mathrm{-}{\varepsilon }_{i})+\frac{1\mathrm{-}d}{1+\gamma d}H({\varepsilon }_{i}))\).

Notion: we write with a index (for example \({\varepsilon }_{i}\)) the \(i\) -th eigenvalue of a tensor which is written (by explaining its two indices) \({\varepsilon }_{\mathit{kl}}\).

If the eigenvalues of the deformation are all distinct, we then show that \(\dot{{\varepsilon }_{i}}\mathrm{=}\dot{{\varepsilon }_{\mathit{ii}}}\), with \(\dot{{\varepsilon }_{\mathit{kl}}}\) the components of \(\dot{\varepsilon }\) in the fixed coordinate system coinciding with the natural deformation coordinate system at the moment in question (in this coordinate system we therefore have \({\varepsilon }_{\mathit{kl}}\mathrm{=}{\varepsilon }_{k}{\delta }_{\mathit{kl}}\)).

Indeed, let’s write the deformations in the form:

\(\varepsilon \mathrm{=}\mathrm{\sum }_{i}{\varepsilon }_{i}{U}_{i}\mathrm{\otimes }{U}_{i}\)

In differentiating this expression, it comes:

\(\dot{\varepsilon }\mathrm{=}\mathrm{\sum }_{i}\dot{{\varepsilon }_{i}}{U}_{i}\mathrm{\otimes }{U}_{i}+{\varepsilon }_{i}\dot{{U}_{i}}\mathrm{\otimes }{U}_{i}+{\varepsilon }_{i}{U}_{i}\mathrm{\otimes }\dot{{U}_{i}}\)

Using the fact that eigenvectors are orthonormal:

\({U}_{i}\mathrm{\cdot }{U}_{j}\mathrm{=}{\delta }_{\mathit{ij}}\mathrm{\Rightarrow }\dot{{U}_{i}}\mathrm{\cdot }{U}_{j}+{U}_{i}\mathrm{\cdot }\dot{{U}_{j}}\mathrm{=}0\)

we obtain the variations of the eigenvalues and the eigenvectors:

\(\dot{{\varepsilon }_{i}}\mathrm{=}\dot{{\varepsilon }_{\mathit{ii}}}\) and \(\dot{{U}_{j}}\mathrm{\cdot }{U}_{k}\mathrm{=}\frac{\dot{{\varepsilon }_{\mathit{jk}}}}{{\varepsilon }_{j}\mathrm{-}{\varepsilon }_{k}}\) for \(j\mathrm{\ne }k\)

This is obviously only valid if the eigenvalues are distinct (as can be clearly seen from the expression of the variations of the eigenvectors). This is because eigenvectors are not continuous functions of matrix elements.

In the case where two eigenvalues of deformations are equal (and apart from the very particular case where they are also zero), they are either positive or negative. Let’s take the case where they are positive (the other case lends itself to being demonstrated in every way similar). The energy concerning these two eigenvalues is then written: \({\mathrm{\sum }}_{i\mathrm{=}2}^{3}{\varepsilon }_{i}^{2}\) (the two equal eigenvalues are considered to have indices 2 and 3). Differentiating this expression, we obtain:

\(2\mathrm{\sum }_{i\mathrm{=}2}^{3}{\varepsilon }_{i}d{\varepsilon }_{i}+2\varepsilon \mathrm{\sum }_{i\mathrm{=}2}^{3}d{\varepsilon }_{i}\) by noting \(\varepsilon\) as the common eigenvalue.

By invariance of the trace of a matrix, in this case the restriction of deformation to the proper plane in question, we obtain:

\(\mathrm{\sum }_{i\mathrm{=}2}^{3}d{\varepsilon }_{i}\mathrm{=}\mathrm{\sum }_{i\mathrm{=}2}^{3}d{\varepsilon }_{\mathit{ii}}\), regardless of how the specific landmark evolved at that moment.

For the remaining eigenvalue (distinct from the other two and with index 1 with the chosen notations), we have: \(d{\varepsilon }_{1}\mathrm{=}d{\varepsilon }_{11}\).

Combining these expressions together, we get:

\(d(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\varepsilon }_{i}^{2}H({\varepsilon }_{i}))\mathrm{=}d({\varepsilon }_{1}^{2}H({\varepsilon }_{1}))+d(\mathrm{\sum }_{i\mathrm{=}1}^{3}{\varepsilon }_{i}^{2}H({\varepsilon }_{i}))\mathrm{=}2\mathrm{\sum }_{i\mathrm{=}1}^{3}{\varepsilon }_{\mathit{ii}}H({\varepsilon }_{\mathit{ii}})d{\varepsilon }_{\mathit{ii}}\)

In conclusion, whether the eigenvalues are distinct or not, we obtain:

\(d({\mathrm{\sum }}_{i}{\varepsilon }_{i}^{2}H({\varepsilon }_{i}))\mathrm{=}2{\mathrm{\sum }}_{i}{\varepsilon }_{\mathit{ii}}H({\varepsilon }_{\mathit{ii}})d{\varepsilon }_{\mathit{ii}}\) with adopted notations.

This reasoning is easily generalized to the case of three equal eigenvalues.

The energy differential at constant damage is then written as:

\(\begin{array}{cc}d\Phi (\varepsilon ,d)\mathrm{\mid }{d\mathrm{=}{C}^{\mathit{te}}}_{}\mathrm{=}& \lambda (\mathit{tr}\varepsilon )d(\mathit{tr}\varepsilon )(H(\mathrm{-}\mathit{tr}\varepsilon )+\frac{1\mathrm{-}d}{1+\gamma d}H(\mathit{tr}\varepsilon ))+\\ & 2\mu {\mathrm{\sum }}_{i}{\varepsilon }_{\mathit{ii}}d{\varepsilon }_{\mathit{ii}}(H(\mathrm{-}{\varepsilon }_{\mathit{ii}})+\frac{1\mathrm{-}d}{1\mathrm{-}\gamma d}H({\varepsilon }_{\mathit{ii}}))\end{array}\)

With this expression, it is clear that the natural deformation coordinate system is also the natural stress coordinate system.