Skip to main content
SpringerLink
Log in

Harmonic Decomposition, Irreducible Basis Tensors, and Minimal Representations of Material Tensors and Pseudotensors

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

A Correction to this article was published on 04 August 2023

This article has been updated

Abstract

We propose a general and efficient method to derive various minimal representations of material tensors or pseudotensors for crystals. By a minimal representation we mean one that pertains to a specific Cartesian coordinate system under which the number of independent components in the representation is the smallest possible. The proposed method is based on the harmonic and Cartan decompositions and, in particular, on the introduction of orthonormal irreducible basis tensors in the chosen harmonic decomposition. For crystals with non-trivial point group symmetry, we demonstrate by examples how deriving restrictions imposed by symmetry groups (e.g., \(C_{2}\), \(C_{s}\), \(C_{3}\), etc.) whose symmetry elements do not completely specify a coordinate system could possibly miss the minimal representations, and how the Cartan decomposition of SO(3)-invariant irreducible tensor spaces could lead to coordinate systems under which the representations are minimal. For triclinic materials, and for material tensors and pseudotensors which observe a sufficient condition given herein, we describe a procedure to obtain a coordinate system under which the explicit minimal representation has its number of independent components reduced by three as compared with the representation with respect to an arbitrary coordinate system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 05 August 2023

    The original online version of this article was revised: Equation (50) as published in the original article, published online date 11 April 2023, has been updated.

  • 04 August 2023

    A Correction to this paper has been published: https://doi.org/10.1007/s10659-023-10030-z

Notes

  1. Here we follow Sirotin and Shaskolskaya [4, p. 47] in calling the 32 geometric crystal classes (as defined by the point groups of crystals) “symmetry classes”. Later we shall also refer to “symmetry classes” of elements in material tensor or pseudotensor space, which carry a completely different meaning; see Sect. 4.2.

  2. As pointed out by Forte and Vianello [7], Schouten [8, p. 162] in a book published in 1951 writes down two assertions in a footnote with no further elaboration: “The number of constants [for the triclinic and monoclinic “sub-systems”] could be reduced to 18 and 12 respectively. No reduction is possible in the other systems.” The second assertion is incorrect.

  3. Harmonic tensors are defined in Sect. 2.3. There is a theorem which asserts that any \(r\)-th order tensor \(\boldsymbol {A}\) can be expressed in terms of harmonic tensors of orders not higher than \(r\), the second-order identity tensor \(\boldsymbol {I}\), and the third-order alternating tensor \(\boldsymbol {\varepsilon }\). Each such expression is called a harmonic decomposition of tensor \(\boldsymbol {A}\). See [9, Section 17.4] and [10] for details.

  4. See [9] and the references therein for details.

  5. In exhibiting the matrix \([D^{l}_{mn}(\boldsymbol {R})]\) we follow the standard practice in the physics literature (see, e.g., Biedenharn and Louck [14, p. 46]). We let the indices \(m\) and \(n\) run from \(l\) to \(-l\) when reading the matrix from top to bottom and when reading from left to right, respectively.

  6. The adjective “harmonic” comes from the fact that there is a linear isomorphism between \(\mathscr{H}^{r}\) and the space \(\mathcal {H}^{r}\) of complex-valued homogeneous polynomials of degree \(r\) that satisfy the Laplace equation, whose solutions are called harmonic functions. The elements of \(\mathcal {H}^{r}\) are called homogeneous harmonic polynomials of degree \(r\). See, e.g., [9, Chap. 17] and the references therein for details.

  7. This assertion will be justified in Sect. 3.4, where the relationship between (30) with restriction (31) and the classical harmonic decomposition of tensor spaces will be laid bare.

  8. See Weyl [26, pp. 52–56], Gurevich [27, Section 16].

  9. Sirotin and Shaskolskaya [4] call the independent components in a minimal representation “independent invariants”, a term that we shall also use.

  10. By (21) the inversion \(\boldsymbol {\mathcal {I}}\) is a symmetry operation for all even-order material tensors. Hence a crystal with \(G_{\mathrm{cr}} = C_{2h}\) or \(C_{s}\) has the same minimal representations as those of a crystal with \(G_{\mathrm{cr}} = C_{2}\). In fact, the Hall-effect tensors of \(C_{2}\), \(C_{2h}\), and \(C_{s}\) crystals all belong to the symmetry class \([C_{2h}]\).

  11. Both Khatkevich and Federov adopt the passive view of rotations.

References

  1. Nye, J.F.: Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, Oxford (1985)

    MATH  Google Scholar 

  2. Newnham, R.E.: Properties of Materials: Anisotropy, Symmetry, Structure. Oxford University Press, Oxford (2005)

    Google Scholar 

  3. Malgrange, C., Ricolleau, C., Schlenker, M.: Symmetry and Physical Properties of Crystals. Springer, Dordrecht (2014)

    Book  MATH  Google Scholar 

  4. Sirotin, Yu.I., Shaskolskaya, M.P.: Fundamentals of Crystal Physics. Mir, Moscow (1982)

    Google Scholar 

  5. Khatkevich, A.G.: The elastic constants of crystals. Kristallografiya 6, 700–703 (1961) [in Russian]. Sov. Phys. – Crystallogr. 6, 561–563 (1962) [English translation]

    Google Scholar 

  6. Federov, F.I.: Theory of Elastic Waves in Crystals. Plenum, New York (1968). Translated from Russian by J.E.S. Bradley

    Book  Google Scholar 

  7. Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elast. 43, 81–108 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Schouten, J.A.: Tensor Analysis for Physicists. Oxford University Press, Oxford (1951)

    MATH  Google Scholar 

  9. Man, C.-S.: Crystallographic texture and group representations. J. Elast. 149, 3–445 (2022). https://doi.org/10.1007/s10659-022-09882-8

    Article  MathSciNet  MATH  Google Scholar 

  10. Spencer, A.J.M.: A note on the decomposition of tensors into traceless symmetric tensors. Int. J. Eng. Sci. 8, 475–481 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  11. Olive, M.: Effective computations of SO(3) and O(3) linear representation symmetry classes. Math. Mech. Complex Syst. 7, 203–237 (2019)

    Article  MathSciNet  Google Scholar 

  12. Forte, S., Vianello, M.: Symmetry classes and harmonic decomposition for photoelasticity tensors. J. Int. Eng. Sci. 35, 1317–1326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Roman, S.: Advanced Linear Algebra, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  14. Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics: Theory and Application. Addison-Wesley, Reading (1981)

    MATH  Google Scholar 

  15. Roe, R.-J.: Description of crystallite orientation in polycrystalline materials. iii. General solution to pole figures. J. Appl. Phys. 36, 2024–2031 (1965)

    Article  Google Scholar 

  16. Du, W., Man, C.-S.: Material tensors and pseudotensors of weakly-textured polycrystals with orientation distribution function defined on the orthogonal group. J. Elast. 127, 197–233 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Backus, G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. Space Phys. 8, 633–671 (1970)

    Article  Google Scholar 

  18. Man, C.-S., Du, W.: Recasting classical expansion of orientation distribution function as tensorial Fourier expansion. J. Elast. (2022). https://doi.org/10.1007/s10659-022-09917-0

    Article  Google Scholar 

  19. Jahn, H.A.: Note on the Bhagavantam-Suryanarayana method of enumerating the physical constants of crystals. Acta Crystallogr. 2, 30–33 (1949)

    Article  MathSciNet  Google Scholar 

  20. Sirotin, Yu.I.: Decomposition of material tensors into irreducible parts. Sov. Phys. Crystallogr. 19, 565–568 (1975)

    MATH  Google Scholar 

  21. Man, C.-S., Huang, M.: A representation theorem for material tensors of weakly-textured polycrystals and its applications in elasticity. J. Elast. 106, 1–42 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1988)

    Book  MATH  Google Scholar 

  23. Mochizuki, E.: Spherical harmonic decomposition of an elastic tensor. Geophys. J. 93, 521–526 (1988)

    Article  MATH  Google Scholar 

  24. Cowin, S.C.: Properties of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 42, 249–266 (1989). Corrigendum, 46, 541–542 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Baerheim, R.: Harmonic decomposition of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 46, 391–418 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

  27. Gurevich, G.B.: Foundations of the Theory of Algebraic Invariants. Noordhoff, Groningen (1964). Translated by J.R.M. Radok and A.J.M. Spencer

    MATH  Google Scholar 

  28. Kearsley, E.A., Fong, J.T.: Linearly independent sets of isotropic Cartesian tensors of ranks up to eight. J. Res. Natl. Bur. Stand. B, Math. Sci. 79B, 49–58 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. Man, C.-S.: Material tensors of weakly-textured polycrystals. In: Chien, W.-Z., et al. (eds.) Proceedings of the Third International Conference on Nonlinear Mechanics (ICNM-III), pp. 87–94. Shanghai University Press, Shanghai (1998)

    Google Scholar 

  30. Man, C.-S., Huang, M.: A simple explicit formula for the Voigt-Reuss-Hill average of elastic polycrystals with arbitrary crystal and texture symmetries. J. Elast. 105, 29–48 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shubnikov, A.V., Koptsik, V.A.: Symmetry in Science and Art. Plenum, New York (1974)

    Book  Google Scholar 

  32. Brandmüller, J.: An extension of the Neumann–Minnigerode–Curie principle. Comput. Math. Appl. 12B, 97–100 (1986)

    Article  MathSciNet  Google Scholar 

  33. Koptsik, V.A., Sirotin, Yu.I.: The symmetry of piezoelectric and elastic tensors and the symmetry of the physical properties of crystals. Kristallografiya 6, 766–768 (1961) [in Russian]. Sov. Phys. – Crystallogr. 6, 612–614 (1962) [English translation]

    Google Scholar 

  34. Cowin, S.C.: On the number of distinct elastic constants associated with certain anisotropic elastic symmetries. Z. Angew. Math. Phys. 46, S210–S224 (1995). Special Issue

    MathSciNet  MATH  Google Scholar 

  35. Keyes, R.W.: The effects of elastic deformations on the electrical conductivity of semiconductors. In: Seitz, F., Turnbull, D. (eds.) Solid State Physics, vol. 11. Academic, New York (1960)

    Google Scholar 

  36. Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Encyclopedia of Physics, vol. VIa/2, pp. 1–295. Springer, Berlin (1972)

    Google Scholar 

Download references

Acknowledgements

We thank two anonymous reviewers for their comments, which helped us improve the presentation of our paper.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Chi-Sing Man

  2. Department of Science and Mathematics, Glenville State University, Glenville, WV, 26351, USA

    Wenwen Du

Authors
  1. Chi-Sing Man
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenwen Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to the research. Man wrote the first draft. Both authors reviewed and revised the manuscript.

Corresponding author

Correspondence to Chi-Sing Man.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

In honor of Roger Fosdick, Teacher and Friend

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original online version of this article was revised: Equation (50) as published in the original article, published online date 11 April 2023, has been updated.

Appendix: Lists of Irreducible Basis Tensors

Appendix: Lists of Irreducible Basis Tensors

In what follows we assume that in the physical space a right-handed orthonormal triad \(\boldsymbol {e}_{1}\), \(\boldsymbol {e}_{2}\), and \(\boldsymbol {e}_{3}\) has been chosen, which defines a Cartesian coordinate system. The components of every tensor \(\boldsymbol {H}= H_{i_{1} i_{2} \cdots i_{r}} \boldsymbol {e}_{i_{1}} \otimes \boldsymbol {e}_{i_{2}} \otimes \cdots \otimes \boldsymbol {e}_{i_{r}} \in V^{\otimes r}_{c}\), where the summation convention is in force, are referred to this coordinate system.

For each set of irreducible basis tensors, we show only those \(\boldsymbol {H}^{k,s}_{m}\) with \(m \geq 0\). Each \(\boldsymbol {H}^{k,a}_{\bar{m}}\) can be written down easily by applying (29) to \(\boldsymbol {H}^{k,s}_{m}\).

1.1 A.1 Irreducible Basis Tensors for the Second-Order Hall-Effect Tensor

The tensor space in question is \(V^{\otimes 2}_{c}\). Each tensor \(\boldsymbol {\rho }^{(H)}\) in this space is represented by a \(3 \times 3\) matrix as usual.

$$\begin{gathered} \boldsymbol {H}^{0}_{0} = \frac{1}{\sqrt{3}}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}^{1}_{0} = \frac{1}{\sqrt{2}}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}^{1}_{1} = \frac{1}{2}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & -i \\ 0 & 0 & -1 \\ i & 1 & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}^{2}_{0} = \frac{1}{\sqrt{6}}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}^{2}_{1} = \frac{1}{2}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 1 \\ 0 & 0 & -i \\ 1 & -i & 0 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}^{2}_{2} = \frac{1}{2}\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} -1 & i & 0 \\ i & 1 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ). \end{gathered}$$
(122)

1.2 A.2 Irreducible Basis Tensors for Piezoelectricity Tensor

The tensor space in question is \(V^{\otimes}_{c} \otimes [V^{\otimes 2}_{c}]\). Each tensor \(\boldsymbol {d}\) in this space is represented by a \(3 \times 6\) matrix (see, e.g., Sirotin and Shaskolskaya [4, p. 606], Newnham [2, p. 88]):

$$ \boldsymbol {d}= \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{array}\displaystyle \right ) := \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} d_{111} & d_{122} & d_{133} & 2d_{123} & 2d_{113} & 2d_{112} \\ d_{211} & d_{222} & d_{233} & 2d_{223} & 2d_{213} & 2d_{212} \\ d_{311} & d_{322} & d_{333} & 2d_{323} & 2d_{313} & 2d_{312} \end{array}\displaystyle \right ). $$
(123)

The irreducible basis tensors are as follows:

$$\begin{gathered} \boldsymbol {H}_{0}^{1,1} =\frac{\sqrt{3}}{3}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{array}\displaystyle \right ), \quad \boldsymbol {H}_{1}^{1,1}=\frac{\sqrt{6}}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-1 & -1 & -1 & 0 & 0 & 0 \\ i & i & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{0}^{1,2} =\frac{\sqrt{15}}{15}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{1}^{1,2}=\frac{\sqrt{30}}{30} \left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-2 & 1 & 1 & 0 & 0 & 3i \\ -i & 2i & -i & 0 & 0 & -3 \\ 0 & 0 & 0 & 3i & -3 & 0 \end{array}\displaystyle \right ), \quad \boldsymbol {H}_{0}^{2} =\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{1}^{2}=\frac{\sqrt{6}}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & -i & i & 0 & 0 & 1 \\ -1 & 0 & 1 & 0 & 0 & i \\ 0 & 0 & 0 & -1 & -i & 0 \end{array}\displaystyle \right ),\quad \boldsymbol {H}_{2}^{2}=\frac{\sqrt{6}}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & -1 & -i & 0 \\ 0 & 0 & 0 & i & -1 & 0 \\ i & -i & 0 & 0 & 0 & 2 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{0}^{3} =\frac{\sqrt{10}}{10}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 1 & 1 & -2 & 0 & 0 & 0 \end{array}\displaystyle \right ), \quad \boldsymbol {H}_{1}^{3}=\frac{\sqrt{30}}{60}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-3 & -1 & 4 & 0 & 0 & 2i \\ i & 3i & -4i & 0 & 0 & -2 \\ 0 & 0 & 0 & -8i & 8 & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{2}^{3} =\frac{\sqrt{3}}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & 2i & -2 & 0 \\ 0 & 0 & 0 & 2 & 2i & 0 \\ -1 & 1 & 0 & 0 & 0 & 2i \end{array}\displaystyle \right ), \quad \boldsymbol {H}_{3}^{3}=\frac{\sqrt{2}}{4}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & -1 & 0 & 0 & 0 & -2i \\ -i & i & 0 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ). \end{gathered}$$
(124)

1.3 A.3 Irreducible Basis Tensors for Elasticity Tensor

The following irreducible basis tensors in \([V^{\otimes 2}_{c}]^{\otimes 2}\)] first appeared in [30]. They are given in the Kelvin notation (see, e.g., [30] and the references therein.

$$ \boldsymbol {H}_{0}^{0,1} =\frac{\sqrt{5}}{15}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}_{0}^{0,2} =\frac{1}{3}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ), $$
$$ \boldsymbol {H}_{0}^{2,1} =\frac{\sqrt{14}}{42}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}2 & -4 & 2 & 0 & 0 & 0 \\ -4 & 2 & 2 & 0 & 0 & 0 \\ 2 & 2 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array}\displaystyle \right ), $$
$$ \boldsymbol {H}_{1}^{2,1} =\frac{\sqrt{42}}{84}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & 4i & 2 & 0 \\ 0 & 0 & 0 & -2i & -4 & 0 \\ 0 & 0 & 0 & -2i & 2 & 0 \\ 4i & -2i & -2i & 0 & 0 & 3\sqrt{2} \\ 2 & -4 & 2 & 0 & 0 & -3\sqrt{2}i \\ 0 & 0 & 0 & 3\sqrt{2} & -3\sqrt{2}i & 0 \end{array}\displaystyle \right ), $$
$$\begin{aligned} \boldsymbol {H}_{2}^{2,1} & =\frac{\sqrt{21}}{42}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-2 & 0 & 2 & 0 & 0 & \sqrt{2}i \\ 0 & 2 & -2 & 0 & 0 & \sqrt{2}i \\ 2 & -2 & 0 & 0 & 0 & -2\sqrt{2}i \\ 0 & 0 & 0 & 3 & 3i & 0 \\ 0 & 0 & 0 & 3i & -3 & 0 \\ \sqrt{2}i & \sqrt{2}i & -2\sqrt{2}i & 0 & 0 & 0 \end{array}\displaystyle \right ), \end{aligned}$$
$$ \boldsymbol {H}_{0}^{2,2} =\frac{1}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}2 & 2 & -1 & 0 & 0 & 0 \\ 2 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & -4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ), \qquad \boldsymbol {H}_{1}^{2,2}=\frac{\sqrt{3}}{6}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & -i & 1 & 0 \\ 0 & 0 & 0 & -i & 1 & 0 \\ 0 & 0 & 0 & -i & 1 & 0 \\ -i & -i & -i & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ), $$
$$ \boldsymbol {H}_{2}^{2,2}=\frac{\sqrt{6}}{12}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-2 & 0 & -1 & 0 & 0 & \sqrt{2}i \\ 0 & 2 & 1 & 0 & 0 & \sqrt{2}i \\ -1 & 1 & 0 & 0 & 0 & \sqrt{2}i \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \sqrt{2}i & \sqrt{2}i & \sqrt{2}i & 0 & 0 & 0 \end{array}\displaystyle \right ), $$
$$ \boldsymbol {H}_{0}^{4} =\frac{\sqrt{70}}{140}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}3 & 1 & -4 & 0 & 0 & 0 \\ 1 & 3 & -4 & 0 & 0 & 0 \\ -4 & -4 & 8 & 0 & 0 & 0 \\ 0 & 0 & 0 & -8 & 0 & 0 \\ 0 & 0 & 0 & 0 & -8 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array}\displaystyle \right ), $$
$$\begin{aligned} \boldsymbol {H}_{1}^{4} & =\frac{\sqrt{14}}{56}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & -\sqrt{2}i & 3\sqrt{2} & 0 \\ 0 & 0 & 0 & -3\sqrt{2}i & \sqrt{2} & 0 \\ 0 & 0 & 0 & 4\sqrt{2}i & -4\sqrt{2} & 0 \\ -\sqrt{2}i & -3\sqrt{2}i & 4\sqrt{2}i & 0 & 0 & 2 \\ 3\sqrt{2} & \sqrt{2} & -4\sqrt{2} & 0 & 0 & -2i \\ 0 & 0 & 0 & 2 & -2i & 0 \end{array}\displaystyle \right ), \\ \boldsymbol {H}_{2}^{4} & =\frac{\sqrt{7}}{28}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}-2 & 0 & 2 & 0 & 0 & \sqrt{2}i \\ 0 & 2 & -2 & 0 & 0 & \sqrt{2}i \\ 2 & -2 & 0 & 0 & 0 & -2\sqrt{2}i \\ 0 & 0 & 0 & -4 & -4i & 0 \\ 0 & 0 & 0 & -4i & 4 & 0 \\ \sqrt{2}i & \sqrt{2}i & -2\sqrt{2}i & 0 & 0 & 0 \end{array}\displaystyle \right ), \end{aligned}$$
$$ \boldsymbol {H}_{3}^{4} =\frac{\sqrt{2}}{8}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 0 & \sqrt{2}i & -\sqrt{2} & 0 \\ 0 & 0 & 0 & -\sqrt{2}i & \sqrt{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \sqrt{2}i & -\sqrt{2}i & 0 & 0 & 0 & 2 \\ -\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 2i \\ 0 & 0 & 0 & 2 & 2i & 0 \end{array}\displaystyle \right ), $$
$$ \boldsymbol {H}_{4}^{4} =\frac{1}{4}\left ( \textstyle\begin{array} [c]{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & -1 & 0 & 0 & 0 & -\sqrt{2}i \\ -1 & 1 & 0 & 0 & 0 & \sqrt{2}i \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ -\sqrt{2}i & \sqrt{2}i & 0 & 0 & 0 & -2 \end{array}\displaystyle \right ). $$
(125)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Man, CS., Du, W. Harmonic Decomposition, Irreducible Basis Tensors, and Minimal Representations of Material Tensors and Pseudotensors. J Elast 154, 3–41 (2023). https://doi.org/10.1007/s10659-023-10010-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-023-10010-3

Keywords

Mathematics Subject Classification (2020)

Search

Navigation