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On Commutativity and Near Commutativity of Translational and Rotational Averages: Analytical Proofs and Numerical Examinations

Abstract

We show that, in general, the translational average over a spatial variable—discussed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), and referred to as the equivalent-medium average—and the rotational average over a symmetry group at a point—discussed by Gazis et al. (Acta Crystallogr. 16(9):917–922, 1963), and referred to as the effective-medium average—do not commute. However, they do commute in special cases of particular symmetry classes, which correspond to special relations among the elasticity parameters. We also show that this noncommutativity is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the square of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology, where the weak-anisotropy assumption results in empirically adequate models.

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Fig. 1

Notes

  1. Readers interested in formulation of matrix (14) might also refer to Bóna et al. [2].

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Acknowledgements

We wish to acknowledge discussions with Theodore Stanoev. The numerical examination was motivated by a discussion with Robert Sarracino. This research was performed in the context of The Geomechanics Project supported by Husky Energy. Also, this research was partially supported by the Natural Sciences and Engineering Research Council of Canada, grant 238416-2013.

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Appendices

Appendix A: Proofs of Lemmas

A.1 Lemma 1

Proof

For discrete symmetries, we can write integral (2) as a sum,

$$ \widetilde{C}^{\mathrm{sym}}=\frac{1}{n} \bigl( \tilde{A}_{1}^{\mathrm{sym}}\,C\,\tilde{A}_{1}^{\mathrm{sym}} \,{}^{{T}}+\cdots + \tilde{A}_{n}^{\mathrm{sym}}\,C\, \tilde{A}_{n}^{\mathrm{sym}}\,{}^{{T}} \bigr) , $$
(13)

where \(\widetilde{C}^{\mathrm{sym}}\) is expressed in Kelvin’s notation, in view of Thomson [13, p. 110], as discussed in Chapman [5, Sect. 4.4.2].

To write the elements of the monoclinic symmetry group as \(6\times 6\) matrices, we must consider orthogonal transformations in \(\mathbb{R} ^{3}\). Transformation \(A\in \mathit{SO}(3)\) of \(c_{\mathit{ijk}\ell }\) corresponds to transformation of \(C\) given by

$$\begin{aligned} \tilde{A} &=\left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} A_{11}^{2} & A_{12}^{2} & A_{13}^{2} & \sqrt{2}A_{12}A_{13} \\ A_{21}^{2} & A_{22}^{2} & A_{23}^{2} & \sqrt{2}A_{22}A_{23} \\ A_{31}^{2} & A_{32}^{2} & A_{33}^{2} & \sqrt{2}A_{32}A_{33} \\ \sqrt{2}A_{21}A_{31} & \sqrt{2}A_{22}A_{32} & \sqrt{2}A_{23}A _{33} & A_{23}A_{32}+A_{22}A_{33} \\ \sqrt{2}A_{11}A_{31} & \sqrt{2}A_{12}A_{32} & \sqrt{2}A_{13}A _{33} & A_{13}A_{32}+A_{12}A_{33} \\ \sqrt{2}A_{11}A_{21} & \sqrt{2}A_{12}A_{22} & \sqrt{2}A_{13}A _{23} & A_{13}A_{22}+A_{12}A_{23} \end{array}\displaystyle \right . \\ & \hspace{1.5in} \left . \textstyle\begin{array}{c@{\quad }c} \sqrt{2}A_{11}A_{13} & \sqrt{2}A_{11}A_{12} \\ \sqrt{2}A_{21}A_{23} & \sqrt{2}A_{21}A_{22} \\ \sqrt{2}A_{31}A_{33} & \sqrt{2}A_{31}A_{32} \\ A_{23}A_{31}+A_{21}A_{33} & A_{22}A_{31}+A_{21}A_{32} \\ A_{13}A_{31}+A_{11}A_{33} & A_{12}A_{31}+A_{11}A_{32} \\ A_{13}A_{21}+A_{11}A_{23} & A_{12}A_{21}+A_{11}A_{22} \end{array}\displaystyle \right ] , \end{aligned}$$
(14)

which is an orthogonal matrix, \(\tilde{A}\in \mathit{SO}(6)\) (Slawinski [11, Sect. 5.2.5]).Footnote 1

The required symmetry-group elements are

$$ A_{1}^{\mathrm{mono}}= \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right ] \mapsto \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\displaystyle \right ] = \tilde{A}_{1}^{\mathrm{mono}} $$

and

$$ A_{2}^{\mathrm{mono}}= \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right ] \mapsto \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\displaystyle \right ] = \tilde{A}_{2}^{\mathrm{mono}}. $$

For the monoclinic case, expression (13) can be stated explicitly as

$$ \widetilde{C}^{\mathrm{mono}}= \frac{ (\tilde{A}_{1}^{ \mathrm{mono}} )\,C\, (\tilde{A}_{1}^{\mathrm{mono}} ) ^{T}+ (\tilde{A}_{2}^{\mathrm{mono}} )\,C\, (\tilde{A} _{2}^{\mathrm{mono}} )^{T}}{2}. $$

Performing matrix operations, we obtain

$$ \widetilde{C}^{\mathrm{mono}} =\left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} c_{1111} & c_{1122} & c_{1133} & 0 & 0 & \sqrt{2}c_{1112} \\ c_{1122} & c_{2222} & c_{2233} & 0 & 0 & \sqrt{2}c_{2212} \\ c_{1133} & c_{2233} & c_{3333} & 0 & 0 & \sqrt{2}c_{3312} \\ 0 & 0 & 0 & 2c_{2323} & 2c_{2313} & 0 \\ 0 & 0 & 0 & 2c_{2313} & 2c_{1313} & 0 \\ \sqrt{2}c_{1112} & \sqrt{2}c_{2212} & \sqrt{2}c_{3312} & 0 & 0 & 2c_{1212} \end{array}\displaystyle \right ] , $$
(15)

which exhibits the form of the monoclinic tensor in its natural coordinate system. In other words, \(\widetilde{C}^{\mathrm{mono}}=C ^{\mathrm{mono}}\), in accordance with Corollary 1. □

A.2 Lemma 2

Proof

For orthotropic symmetry,

\(\tilde{A}_{1}^{\mathrm{ortho}}=\tilde{A}_{1}^{\mathrm{mono}}\), \({\tilde{A}_{2}^{\mathrm{ortho}}=\tilde{A}_{2}^{\mathrm{mono}}}\),

$$ A_{3}^{\mathrm{ortho}}= \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\displaystyle \right ] \mapsto \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\displaystyle \right ] = \tilde{A}_{3}^{\mathrm{ortho}} , $$

and

$$ A_{4}^{\mathrm{ortho}}= \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\displaystyle \right ] \mapsto \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\displaystyle \right ] = \tilde{A}_{4}^{\mathrm{ortho}}. $$

For the orthotropic case, expression (13) can be stated explicitly as

$$\begin{aligned} \widetilde{C}^{\mathrm{ortho}}= &\, \bigl[ \bigl(\tilde{A}_{1}^{ \mathrm{ortho}} \bigr)\,C\, \bigl(\tilde{A}_{1}^{\mathrm{ortho}} \bigr) ^{T} + \bigl(\tilde{A}_{2}^{\mathrm{ortho}} \bigr)\,C\, \bigl(\tilde{A} _{2}^{\mathrm{ortho}} \bigr)^{T} \\ &{}+ \bigl(\tilde{A}_{3}^{\mathrm{ortho}} \bigr)\,C\, \bigl( \tilde{A} _{3}^{\mathrm{ortho}} \bigr)^{T}+ \bigl( \tilde{A}_{4}^{ \mathrm{ortho}} \bigr)\,C\, \bigl(\tilde{A}_{4}^{\mathrm{ortho}} \bigr) ^{T} \bigr]/4 . \end{aligned}$$

Performing matrix operations, we obtain

$$ \widetilde{C}^{\mathrm{ortho}} =\left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} c_{1111} & c_{1122} & c_{1133} & 0 & 0 & 0 \\ c_{1122} & c_{2222} & c_{2233} & 0 & 0 & 0 \\ c_{1133} & c_{2233} & c_{3333} & 0 & 0 &0 \\ 0 & 0 & 0 & 2c_{2323} & 0 & 0 \\ 0 & 0 & 0 & 0 & 2c_{1313} & 0 \\ 0& 0 & 0 & 0 & 0 & 2c_{1212} \end{array}\displaystyle \right ], $$
(16)

which exhibits the form of the orthotropic tensor in its natural coordinate system. In other words, \(\widetilde{C}^{\mathrm{ortho}}=C ^{\mathrm{ortho}}\), in accordance with Corollary 2. □

Appendix B: Evaluation of Jacobian

$$\begin{aligned} \mathscr{C}_{3} =& \Biggl[\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{c _{3333}^{i}} \Biggr]^{-1} \Biggl[ \frac{1}{n}\sum\limits _{i=1}^{n} \frac{c _{3312}^{i}}{c_{3333}^{i}} \Biggr]^{2} - \Biggl[\frac{1}{n}\sum \limits _{i=1}^{n}\frac{(c_{3312}^{i})^{2}}{c_{3333}^{i}} \Biggr]. \\ \frac{\partial \mathscr{C}_{3}}{\partial c_{2323}^{\,j}} =&\frac{ \partial \mathscr{C}_{3}}{\partial c_{1313}^{\,j}} =\frac{\partial \mathscr{C}_{3}}{\partial c_{2313}^{\,j}}=0. \\ \frac{\partial \mathscr{C}_{3}}{\partial c_{3312}^{\,j}} =&2 \Biggl[\frac{1}{n} \sum\limits _{i=1}^{n}\frac{1}{c_{3333}^{i}} \Biggr]^{-1} \Biggl[ \frac{1}{n} \sum\limits _{i=1}^{n} \frac{c_{3312}^{i}}{c_{3333}^{i}} \Biggr] \biggl(\frac{1}{n} \biggr) \biggl( \frac{1}{c_{3333}^{\,j}} \biggr)-\frac{2}{n}\frac{c_{3312} ^{\,j}}{c_{3333}^{\,j}}. \\ \frac{\partial \mathscr{C}_{3}}{\partial c_{3333}^{\,j}} =&- \Biggl[\frac{1}{n} \sum\limits _{i=1}^{n}\frac{1}{c_{3333}^{i}} \Biggr]^{-2} \biggl[ \frac{1}{n} \biggl(\frac{-1}{(c_{3333}^{\,j})^{2}} \biggr) \biggr] \Biggl[ \frac{1}{n} \sum\limits _{i=1}^{n} \frac{c_{3312}^{i}}{c_{3333}^{i}} \Biggr]^{2} \\ &{} + \Biggl[\frac{1}{n}\sum\limits _{i=1}^{n} \frac{1}{c_{3333} ^{i}} \Biggr]^{-1} \Biggl[\frac{2}{n}\sum \limits _{i=1}^{n}\frac{c _{3312}^{i}}{c_{3333}^{i}} \Biggr] \biggl[ \frac{1}{n} \biggl(\frac{-c _{3312}^{\,j}}{(c_{3333}^{\,j})^{2}} \biggr) \biggr] + \frac{1}{n} \frac{ (c_{3312}^{\,j} )^{2}}{ (c_{3333}^{\,j} )^{2}} . \end{aligned}$$

Examining the above two equations—where for \(x=a\), \(c_{2313}^{\,j}=c _{3312}^{\,j}=0\), with \(j=1,\ldots ,n\)—we see that

$$ \frac{\partial \mathscr{C}_{3}}{\partial c_{3312}^{\,j}} \bigg\vert _{x=a} =\frac{\partial \mathscr{C}_{3}}{\partial c_{3333}^{\,j}} \vert _{x=a}=0. $$

Next, let us examine \(\mathscr{C}_{1}\) and \(\mathscr{C}_{2}\). First, note that

$$ \frac{\partial \mathscr{C}_{1}}{\partial c_{3333}^{\,j}}=\frac{ \partial \mathscr{C}_{2}}{\partial c_{3333}^{\,j}} =\frac{\partial \mathscr{C}_{1}}{\partial c_{3312}^{\,j}}=\frac{\partial \mathscr{C} _{2}}{\partial c_{3312}^{\,j}}=0. $$

We let

$$\begin{aligned} &f=\frac{1}{n}\sum\limits _{i=1}^{n} \frac{c_{2323}^{i}}{2 (c_{2323} ^{i} c_{1313}^{i}- [c_{2313}^{i} ]^{2} )}, \\ &g=\frac{1}{n}\sum\limits _{i=1}^{n} \frac{c_{1313}^{i}}{2 (c_{2323} ^{i} c_{1313}^{i}- [c_{2313}^{i} ]^{2} )} \end{aligned}$$

and

$$ h=\frac{1}{n}\sum\limits _{i=1}^{n} \frac{c_{2313}^{i}}{2 (c_{2323} ^{i} c_{1313}^{i}- [c_{2313}^{i} ]^{2} )}, $$

which leads to

$$ \mathscr{C}_{1}=\frac{f}{2 [fg-h^{2} ]}- \Biggl(\frac{1}{n} \sum \limits _{i=1}^{n}\frac{1}{c_{2323}^{i}} \Biggr)^{-1} . $$

Thus,

$$\begin{aligned} \frac{\partial \mathscr{C}_{1}}{\partial c_{2323}^{\,j}} =&\frac{ \partial f}{\partial c_{2323}^{\,j}} \frac{1}{2 [fg-h^{2} ]} -\frac{f}{2} \bigl[fg-h^{2} \bigr] ^{-2} \biggl[g\frac{\partial f}{\partial c_{2323}^{\,j}}+ f \frac{ \partial g}{\partial c_{2323}^{\,j}}-2h\frac{\partial h}{\partial c _{2323}^{\,j}} \biggr] \\ & {}+ \Biggl(\frac{1}{n}\sum\limits _{i=1}^{n} \frac{1}{c_{2323}^{i}} \Biggr) ^{-2}\frac{1}{n}\frac{-1}{ [c_{2323}^{\,j} ]^{2}}, \\ \frac{\partial f}{\partial c_{2323}^{\,j}} =& \frac{1}{n}\frac{1}{2 (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313}^{\,j} ] ^{2} )} -\frac{c_{2323}^{\,j}}{2n}c_{1313}^{\,j} \bigl(c_{2323} ^{\,j} c_{1313}^{\,j} - \bigl[c_{2313}^{\,j} \bigr]^{2} \bigr)^{-2} , \\ \frac{\partial g}{\partial c_{2323}^{\,j}} =& \frac{- [c_{1313} ^{\,j} ]^{2}}{2n} \bigl(c_{2323}^{\,j} c_{1313}^{\,j} - \bigl[c _{2313}^{\,j} \bigr]^{2} \bigr)^{-2}, \\ \frac{\partial h}{\partial c_{2323}^{\,j}} =& \frac{c_{2313}^{\,j}}{2n} \bigl(2c_{1313}^{\,j} \bigr) \bigl(c_{2323} ^{\,j} c_{1313}^{\,j} - \bigl[c_{2313}^{\,j} \bigr]^{2} \bigr)^{-2} . \\ \frac{\partial f}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a} =& \frac{1}{2n c_{2323}^{\,j} c_{1313}^{\,j}} - \frac{c_{2323}^{\,j} c _{1313}^{\,j}}{2n (c_{2323}^{\,j} c_{1313}^{\,j} )^{2}}=0 . \\ \frac{\partial g}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a} =& \frac{- (c_{1313}^{\,j} )^{2}}{2n (c_{2323}^{\,j} ) ^{2} (c_{1313}^{\,j} )^{2}}= \frac{-1}{2n (c_{2323} ^{\,j} )^{2}}. \\ \frac{\partial h}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a} =&0. \\ \frac{\partial \mathscr{C}_{1}}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a} =& 0-\frac{f}{2} \bigl[fg-h^{2} \bigr]^{-2} \biggl[0-\frac{f}{2n(c _{2323}^{\,j})^{2}}-0 \biggr] \\ &{} + \Biggl(\frac{1}{n}\sum\limits _{i=1} ^{n} \frac{1}{c_{2323}^{i}} \Biggr)^{-2}\frac{1}{n}\frac{-1}{ (c _{2323}^{\,j} )^{2}}. \\ \frac{f^{2}}{4n [fg-h^{2} ]^{2}} \bigg\vert _{x=a} =& \frac{ [\frac{1}{n}\sum _{i=1}^{n} (\frac{1}{2c_{1313} ^{i}} ) ]^{2}}{4n (\frac{1}{4n^{2}}\sum _{i=1} ^{n} \frac{1}{c_{1313}^{i}}\sum _{i=1}^{n} \frac{1}{c_{2323} ^{i}} )^{2}} = \frac{n}{ (\sum _{i=1}^{n}\frac{1}{c _{2323}^{i}} )^{2}}. \end{aligned}$$

So,

$$ \frac{\partial \mathscr{C}_{1}}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a}= \frac{n}{ (\sum _{i=1}^{n} \frac{1}{c_{2323}^{i}} )^{2} (c_{2323}^{\,j} )^{2}} - \frac{n}{ (\sum _{i=1}^{n} \frac{1}{c_{2323}^{i}} ) ^{2} (c_{2323}^{\,j} )^{2}}=0. $$

Similarly, by symmetry of the equations,

$$ \frac{\partial \mathscr{C}_{2}}{\partial c_{1313}^{\,j}} \bigg\vert _{x=a}=0. $$

Next, we consider the derivative with respect to \(c_{1313}^{\,j}\).

$$\begin{aligned} \frac{\partial \mathscr{C}_{1}}{\partial c_{1313}^{\,j}} =&\frac{ \partial f}{\partial c_{1313}^{\,j}} \frac{1}{2 [fg-h^{2} ]}-\frac{f}{2} \bigl[fg-h^{2} \bigr] ^{-2} \biggl[g\frac{\partial f}{\partial c_{1313}^{\,j}}+ f \frac{ \partial g}{\partial c_{1313}^{\,j}}-2h\frac{\partial h}{\partial c _{1313}^{\,j}} \biggr]. \\ \frac{\partial f}{\partial c_{1313}^{\,j}} =& \frac{- [c_{2323} ^{\,j} ]^{2}}{2n} \bigl(c_{2323}^{\,j} c_{1313}^{\,j} - \bigl[c _{2313}^{\,j} \bigr]^{2} \bigr)^{-2}, \\ \frac{\partial g}{\partial c_{1313}^{\,j}} =& \frac{1}{n}\frac{1}{2 (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313}^{\,j} ] ^{2} )} -\frac{c_{1313}^{\,j}}{2n}c_{2323}^{\,j} \bigl(c_{2323} ^{\,j} c_{1313}^{\,j} - \bigl[c_{2313}^{\,j} \bigr]^{2} \bigr)^{-2} , \\ \frac{\partial h}{\partial c_{1313}^{\,j}} =& \frac{c_{2313}^{\,j}}{2n} \bigl(2c_{2323}^{\,j} \bigr) \bigl(c_{2323} ^{\,j} c_{1313}^{\,j} - \bigl[c_{2313}^{\,j} \bigr]^{2} \bigr)^{-2} . \end{aligned}$$

These lead to

$$\begin{aligned} \frac{\partial f}{\partial c_{1313}^{\,j}} \bigg\vert _{x=a} =&\frac{-1}{2n (c_{1313}^{\,j} )^{2}}, \\ \frac{\partial g}{\partial c_{1313}^{\,j}} \bigg\vert _{x=a} =& \frac{1}{2n c_{2323}^{\,j} c_{1313}^{\,j}} - \frac{c_{1313}^{\,j} c _{2323}^{\,j}}{2n (c_{2323}^{\,j} c_{1313}^{\,j} )^{2}}=0 , \\ \frac{\partial h}{\partial c_{1313}^{\,j}} \bigg\vert _{x=a} =&0. \end{aligned}$$

So,

$$ \frac{\partial \mathscr{C}_{1}}{\partial c_{1313}^{\,j}} \bigg\vert _{x=a}=\frac{\partial f}{\partial c_{1313}^{\,j}} \frac{1}{2 [fg-h^{2} ]} \biggl[1-\frac{fg}{fg-h^{2}} \biggr] \bigg\vert _{x=a}=0 $$

and, similarly,

$$ \frac{\partial \mathscr{C}_{2}}{\partial c_{2323}^{\,j}} \bigg\vert _{x=a}=0. $$

Next, we consider the derivative with respect to \(c_{2313}^{\,j}\).

$$\begin{aligned} \frac{\partial \mathscr{C}_{1}}{\partial c_{2313}^{\,j}} =&\frac{ \partial f}{\partial c_{2313}^{\,j}} \frac{1}{2 [fg-h^{2} ]}-\frac{f}{2} \bigl[fg-h^{2} \bigr] ^{-2} \biggl[g\frac{\partial f}{\partial c_{2313}^{\,j}}+ f \frac{ \partial g}{\partial c_{2313}^{\,j}}-2h\frac{\partial h}{\partial c _{2313}^{\,j}} \biggr]. \\ \frac{\partial f}{\partial c_{2313}^{\,j}} \bigg\vert _{x=a} =& \frac{-2 c_{2313}^{\,j} c_{2323}^{\,j}}{2n (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313}^{\,j} ]^{2} )^{2}} \bigg\vert _{x=a}=0, \\ \frac{\partial g}{\partial c_{2313}^{\,j}} \bigg\vert _{x=a} =& \frac{-2 c_{2313}^{\,j} c_{1313}^{\,j}}{2n (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313}^{\,j} ]^{2} )^{2}} \bigg\vert _{x=a}=0, \\ \frac{\partial h}{\partial c_{2313}^{\,j}} \bigg\vert _{x=a} =&\frac{1}{2n (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313} ^{\,j} ]^{2} )} \bigg\vert _{x=a}+ \frac{2 (c_{2313} ^{\,j})^{2} }{2n (c_{2323}^{\,j} c_{1313}^{\,j} - [c_{2313} ^{\,j} ]^{2} )^{2}} \bigg\vert _{x=a} \\ =&\frac{1}{2n c_{2323}^{\,j} c_{1313}^{\,j}}. \end{aligned}$$

Thus,

$$ \frac{\partial \mathscr{C}_{1}}{\partial c_{2313}^{\,j}} \bigg\vert _{x=a}= 0-\frac{f}{2} \bigl[fg-h^{2} \bigr]^{-2} [0+0-0 ]=0 , $$

and, similarly,

$$ \frac{\partial \mathscr{C}_{2}}{\partial c_{2313}^{\,j}} \bigg\vert _{x=a}=0. $$

Hence, \(\mathscr{C}'(a)=[0]\); the Jacobian matrix is zero.

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Bos, L., Dalton, D.R. & Slawinski, M.A. On Commutativity and Near Commutativity of Translational and Rotational Averages: Analytical Proofs and Numerical Examinations. J Elast 136, 189–206 (2019). https://doi.org/10.1007/s10659-018-9703-4

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Keywords

  • Anisotropy
  • Inhomogeneity
  • Symmetry classes
  • Effective medium
  • Equivalent medium
  • Backus average
  • Frobenius norm

Mathematics Subject Classification (2010)

  • 74B02
  • 86A02
  • 41A02
  • 65Z02