7. Appendix: Eigenvalues of the strain tensor

We consider an orthonormal \({({e}_{i})}_{i=\mathrm{1,2}\mathrm{,3}}\) base of three-dimensional Euclidean space, and a \(\varepsilon\) tensor of order 2, which is symmetric, and therefore diagonalizable. We denote \({\varepsilon }_{j}^{i}\) the mixed components of the \(\varepsilon ={\varepsilon }_{j}^{i}\text{.}{e}_{i}\otimes {e}^{\text{*}j}\) tensor in the base \({({e}_{i})}_{i=\mathrm{1,2}\mathrm{,3}}\). We use the Einstein convention on repeated mixed indices.

We note: \({Q}_{k},{\eta }_{k}\) the triplet of the normalized eigenvectors and associated eigenvalues of the problem:

(7.11)\[ \ varepsilon\ text {.} {Q} _ {k} = {\ eta} _ {k} _ {k} {Q} _ {k}\ iff {\ varepsilon} _ {j} ^ {i}\ text {.} {({Q} _ {k})} ^ {j} {e} {e} _ {e} _ {i} = {\ eta} _ {k}} {({Q} _ {k})} ^ {i} {e} _ {i}\ text {i}\ text {for} k=\ mathrm {1,2}\ mathrm {0,3}\]

Note:

We note that: \((\varepsilon +\xi \text{Id})\text{.}{Q}_{k}=({\lambda }_{k}+\xi )\text{.}{Q}_{k},\forall \xi \in ℝ\) , so adding to \(\varepsilon\) any diagonal tensor does not change the proper directions of \(\varepsilon\) .

We know that the eigenvectors \({({Q}_{k})}^{j}{e}_{j}\) form an orthonormal base (main coordinate system):

(7.11)\[ {({Q} _ {k})}} _ {i} {e} ^ {\ text {*} i}\ text {.} {({Q} _ {l})}} ^ {j} {j} {e} {e} _ {j} = {\ delta} _ {\ mathrm {kl}}}\ text {.} {\ delta} ^ {{i} _ {j}}}\ Rightarrow {({Q} _ {k})} _ {j} {({Q} _ {\ ell})}} ^ {j})} ^ {j} = {\ delta}} = {\ delta} _ {\ delta} _ {\ mathrm {kl}}\]

Let’s differentiate between these two relationships:

(7.11)\[ {(d\ varepsilon)} _ {j}} ^ {i}\ text {.} {({Q} _ {k})}} ^ {j} + {\ varepsilon} _ {j} ^ {i}\ text {.} {(d {Q} _ {k})} ^ {k})} ^ {j} =d {\ eta} = d {\ eta}} _ {k})} ^ {i} + {\ eta} _ {k} {k} {(d {Q}}} {j} =d {Q}} =d {\ eta}} = {\ eta} _ {k})} ^ {Q} _ {k})} ^ {Q} _ {k})} ^ {Q} _ {k})} ^ {i}\ text {for} k=\ mathrm {1,2}\ mathrm {3,3} {\]

(7.11)\[ {(d {Q} _ {k})}} _ {j}\ text {.} {({Q} _ {l})} ^ {j}} + {({Q} _ {k})} _ {j}\ text {.} {(d {Q} _ {l})}} ^ {j} =0\]

Let’s project equation () onto the eigenvector \({({Q}_{l})}^{i}{e}_{i}\) and use equation ():

(7.11)\[\begin{split} \ begin {array} {} {(d\ varepsilon)} _ {j} ^ {i}\ text {.} {({Q} _ {k})} ^ {j} {j} {({Q} _ {l})} _ {i} + {\ varepsilon} _ {j} ^ {i} {i}\ text {.} {(d {Q} _ {k})} ^ {j} {j} {({q} _ {l})} _ {i} =d {\ eta} _ {k} {k})} ^ {i})} ^ {i}} {i} {i} {i}} _ {l})} _ {i} = d {Q} _ {k})} ^ {k})} ^ {k})} ^ {k})} ^ {k})} ^ {i}} {({Q} _ {l})} _ {i}\ text {for} k, l=\ mathrm {1,2}\ mathrm {0,3}\\ iff {(d\ varepsilon)} _ {i}\ text {.} {d\ varepsilon)} _ {varepsilon)} _ {j} ^ {i}\ text {.} {({Q} _ {k})} ^ {j} {j} {({Q} _ {l})} _ {i} + {\ eta} _ {l}\ text {.} {(d {Q} _ {k})}} ^ {j} {({Q} _ {l})} _ {j} =d {\ eta} _ {k} _ {k}\ text {.} {\ delta} _ {\ mathrm {kl}}} + {\ eta}} _ {k} {(d {Q} _ {k})} ^ {i} {({Q} _ {l})} _ {i}\\\\\\\\\\\\)}\\\\\\)}\\\\\)}\\\ iff {(d\ varepsilon)}\\ iff {(d\ varepsilon)}\\ iff {(d\ varepsilon)}\\ iff {(d\ varepsilon)}\ j} ^ {i}\ text {.} {({Q} _ {k})}} ^ {j} {({Q} _ {l})} _ {i} =d {\ eta} _ {k} _ {k}\ text {.} {\ delta} _ {\ mathrm {kl}}} + ({\ eta} _ {k} - {\ eta} _ {l})\ text {.} {(d {Q} _ {k})} ^ {i} {i} {({Q} _ {l})} _ {i}\ text {for} k, l=\ mathrm {1.2}\ mathrm {1,2}\ mathrm {2,3}\ end {array}\end{split}\]

From where:

(7.11)\[\begin{split} \ {\ begin {array} {} d {\ eta} _ {k} = {(d\ varepsilon)} _ {j} ^ {i}\ text {.} {({Q} _ {k})} ^ {j} {j} {({Q} _ {k})} _ {i}\ text {for} k=\ mathrm {1,2}\ mathrm {0,3}\\ mathrm {0,3}\\\}\\ ({\ eta} _ {3}}\ mathrm {0,3}}\\ mathrm {0,3}}\\ ({\ eta} _ {k})\ mathrm {0,3}\\ ({\ eta} _ {l})\ mathrm {3,3}\\ mathrm {3,3}\\\ ({\ eta} _ {k})\ {(d {Q} _ {k})}} ^ {i} {({Q} _ {l})} _ {i} = {(d\ varepsilon)} _ {j}} ^ {i}\ text {.} {({Q} _ {k})} ^ {j} {({j}} {j}} {j} {j} {l})} _ {i}\ text {for} k\ne l=\ mathrm {1,2}\ mathrm {1,2}\ mathrm {1,2}\ mathrm {3,3}\ mathrm {0,3}\ end {array}\end{split}\]

Note \({\tilde{\varepsilon }}_{j}^{i}\) the mixed components of a tensor in base \({({Q}_{k})}_{k=\mathrm{1,2}\mathrm{,3}}\). So:

(7.11)\[\begin{split} \ {\ begin {array} {} d {\ eta} _ {k} _ {k} = {(d\ tilde {\ varepsilon})} _ {k}\ text {for} k=\ mathrm {1.2}\ mathrm {1,2}\\ mathrm {1,2}\\\\}\\ {\ eta} _ {k}\ = {1.2}\\\}\\ mathrm {1,2}\\ mathrm {2,3} (\ text {3}} (\ text {no summation on} k)\\ ({\ eta}} _ {k} - {k} - {\ eta} _ {k} - {\ eta} _ {k} l})\ text {.} {(d {Q} _ {k})} ^ {i} {i} {({Q} _ {l})} _ {i} = {(d\ tilde {\ varepsilon})} _ {l} ^ {k}\ text {for} k\ne l=\ mathrm {1,2}\ mathrm {0,3}\ end {array}\ end {array}\end{split}\]

Of course, we check on the track.

of the deformation tensor (which is independent of the coordinate system chosen):

(7.11)\[ \ sum _ {k=\ mathrm {1.2}\ mathrm {2,3}} d {\ eta} _ {k} =\ mathrm {tr} (d\ tilde {\ varepsilon}) =\ mathrm {tr}) =\ mathrm {tr} (d\ varepsilon) =d (\ mathrm {tr}\ varepsilon)\]

Let us consider the isotropic elasticity free energy density:

(7.11)\[ \ phi (\ varepsilon) =\ frac {1} {2} {2}\ lambda {(\ mathrm {tr}\ varepsilon)} ^ {2} +\ mu\ sum _ {k=\ mathrm {1.2} {k=\ mathrm {1.2}}\ mathrm {1,2}}\ mathrm {0,3}}} {(\ eta} _ {k})} ^ {2})} ^ {2}\]

then state law gives the stress tensor:

(7.11)\[ \ sigma = {\ phi} _ {,\ epsilon} (\ varepsilon) =\ lambda (\ mathrm {tr}\ varepsilon)\ frac {d\ mathrm {tr}\ varepsilon}\ varepsilon} {d\ varepsilon} {d\ varepsilon} {d\ varepsilon} {d\ varepsilon} {d\ varepsilon} {d\ varepsilon} +\ mu\ sum _ {k=\ mathrm {1,2}\ mathrm {0,3}} {\ eta} _ {k}\ frac {d {\ eta} _ {k}} {k}} {d\ varepsilon} =\ lambda (\ mathrm {tr}\ varepsilon)\ mathrm {Id} +2\ mu\ sum} +2\ mu\ sum} _ {id} +2\ mu\ sum} _ {k} +2\ mu\ sum _ {k=\ mathrm {1.2}}\ mathrm {2}} {\ eta} _ {k}\ mathrm {}. {({Q} _ {k})}} ^ {j} {({Q} _ {k})} _ {i}\ mathrm {.} {e} _ {j}\ otimes {e} ^ {\ text {*} i}\]

Applying the remark made above, the natural coordinate system of the stress tensor \(\sigma\) is therefore identical to that of the deformations \(\varepsilon\).

The main constraints are therefore naturally in the main frame of reference \({({Q}_{k})}^{j}{e}_{j}\):

(7.11)\[ {s} _ {k} = {\ tilde {\ sigma}}} _ {k} ^ {k} =\ lambda\ text {tr} (\ varepsilon) +2\ mu {\ eta} _ {k}\]