Journal of Elasticity

, Volume 132, Issue 1, pp 67–101 | Cite as

Harmonic Factorization and Reconstruction of the Elasticity Tensor

  • M. Olive
  • B. Kolev
  • B. DesmoratEmail author
  • R. Desmorat


In this paper, we study anisotropic Hooke’s tensor: we propose a factorization of its fourth-order harmonic part into second-order tensors. We obtain moreover explicit equivariant reconstruction formulas, using second-order covariants, for transverse isotropic and orthotropic fourth-order harmonic tensors, and for trigonal and tetragonal fourth-order harmonic tensors up to a cubic fourth order covariant remainder.


Anisotropy Sylvester theorem Harmonic factorization Harmonic product Tensorial reconstruction Covariant tensors 

Mathematics Subject Classification

74E10 15A72 74B05 


  1. 1.
    Auffray, N., Kolev, B., Petitot, M.: On anisotropic polynomial relations for the elasticity tensor. J. Elast. 115(1), 77–103 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Backus, G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. 8(3), 633–671 (1970) ADSCrossRefGoogle Scholar
  3. 3.
    Baerheim, R.: Classification of symmetry by means of Maxwell multipoles. Q. J. Mech. Appl. Math. 51, 73–103 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Betten, J., Helisch, W.: Integrity bases for a fourth-rank tensor. In: Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, Nottingham, 1994. Solid Mech. Appl, vol. 39, pp. 37–42. Kluwer, Dordrecht (1995) CrossRefGoogle Scholar
  5. 5.
    Boehler, J.-P.: A simple derivation of representations for nonpolynomial constitutive equations in some case of anisotropy. Z. Angew. Math. Mech. 59(4), 157–167 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boehler, J.-P.: Introduction to the invariant formulation of anisotropic constitutive equations. In: Applications of Tensor Functions in Solid Mechanics. CISM Courses and Lectures, vol. 292, pp. 13–30. Springer, Vienna (1987) CrossRefGoogle Scholar
  7. 7.
    Boehler, J.-P., Kirillov, A.A. Jr., Onat, E.T.: On the polynomial invariants of the elasticity tensor. J. Elast. 34(2), 97–110 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer, New York (1995). Translated from the German manuscript, Corrected reprint of the 1985 translation zbMATHGoogle Scholar
  9. 9.
    Burr, A., Hild, F., Leckie, F.A.: Micro-mechanics and continuum damage mechanics. Arch. Appl. Mech. 65, 437–456 (1995) ADSzbMATHGoogle Scholar
  10. 10.
    Cayley, A.: A seventh memoir on quantics. Philos. Trans. R. Soc. Lond. 151, 277–292 (1861) CrossRefGoogle Scholar
  11. 11.
    Chaboche, J.-L.: Le concept de contrainte effective appliqué à l’élasticité et à la viscoplasticité en présence d’un endommagement anisotrope. In: Boehler, J.-P. (ed.) Colloque Int. CNRS 295, Villard de Lans, pp. 737–760. Springer (Martinus Nijhoff Publishers and Editions du CNRS), Boston (1982, 1979) Google Scholar
  12. 12.
    Cordebois, J., Sidoroff, F.: Endommagement anisotrope en élasticité et plasticité. J. Méc. Théor. Appl. Special Volume, 45–65 (1982) zbMATHGoogle Scholar
  13. 13.
    Cormery, F., Welemane, H.: A stress-based macroscopic approach for microcracks unilateral effect. Compos. Mater. Sci. 47, 727–738 (2010) CrossRefGoogle Scholar
  14. 14.
    Desmorat, B., Desmorat, R.: Tensorial polar decomposition of 2D fourth order tensors. C. R., Méc. 343, 471–475 (2015) CrossRefGoogle Scholar
  15. 15.
    Desmorat, B., Desmorat, R.: Second order tensorial framework for 2D medium with open and closed cracks. Eur. J. Mech. A, Solids 58, 262–277 (2016) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elast. 43(2), 81–108 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Forte, S., Vianello, M.: A unified approach to invariants of plane elasticity tensors. Meccanica 49(9), 2001–2012 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Francois, M., Berthaud, Y., Geymonat, G.: Une nouvelle analyse des symétries d’un matériau élastique anisotrope. Exemple d’utilisation à partir de mesures ultrasonores. C. R. Acad. Sci., Sér. IIb 322, 87–94 (1996) zbMATHGoogle Scholar
  19. 19.
    Geymonat, G., Weller, T.: Symmetry classes of piezoelectric solids. C. R. Acad. Sci., Sér. I 335, 847–8524 (2002) zbMATHGoogle Scholar
  20. 20.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Applied Mathematical Sciences., vol. 69. Springer, Berlin (1988) CrossRefzbMATHGoogle Scholar
  21. 21.
    Gordan, P.: Beweis, dass jede Covariante und Invariante einer Bineren Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist. J. Reine Angew. Math. 69, 323–354 (1868) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gordan, P.: Über das Formensystem Binaerer Formen (1875) Google Scholar
  23. 23.
    Gordan, P.: Vorlesungen über Invariantentheorie 2nd edn. Chelsea, New York (1987). Erster Band: Determinanten, vol. I: Determinants; Zweiter Band: Binäre Formen, vol. II: Binary forms, Edited by Georg Kerschensteiner zbMATHGoogle Scholar
  24. 24.
    Grace, J.H., Young, A.: The Algebra of Invariants. Cambridge Library Collection. Cambridge University Press, Cambridge (2010). Reprint of the 1903 original CrossRefzbMATHGoogle Scholar
  25. 25.
    Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Chichester (2000) zbMATHGoogle Scholar
  26. 26.
    Kachanov, M.: On continuum theory of medium with cracks. Mech. Solids 7, 54–59 (1972) Google Scholar
  27. 27.
    Kanatani, K.: Distribution of directional data and fabric tensors. Int. J. Eng. Sci. 22, 14–164 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Krajcinovic, D.: Damage Mechanics. Applied Mathematics and Mechanics. North-Holland, Amsterdam (1996) Google Scholar
  29. 29.
    Ladevèze, P.: Sur une théorie de l’endommagement anisotrope. Technical report, Internal report 34 of LMT-Cachan (1983) Google Scholar
  30. 30.
    Leckie, F.A., Onat, E.T.: Tensorial nature of damage measuring internal variables. In: Hult, J., Lemaitre, J. (eds.) Physical Non-Linearities in Structural Analysis, pp. 140–155. Springer, Berlin (1980) Google Scholar
  31. 31.
    Lemaitre, J., Chaboche, J.-L.: Mécanique des Matériaux Solides. Dunod Malakoff (1985). English translation: Mechanics of Solid Materials. Cambridge University Press (1990) Google Scholar
  32. 32.
    Lemaitre, J., Desmorat, R.: Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer, Berlin (2005) Google Scholar
  33. 33.
    Lercier, R., Ritzenthaler, C.: Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. J. Algebra 372, 595–636 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, I.-S.: On representations of anisotropic invariants. Int. J. Eng. Sci. 20(10), 1099–1109 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Luque, J.-G.: Invariants des hypermatrices (2007). Available at:
  36. 36.
    Man, C.-S.: Remarks on isotropic extension of anisotropic constitutive functions via structural tensors. In: XXIV ICTAM, 21–26 August 2016, pp. 21–26 (2016) Google Scholar
  37. 37.
    Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965) CrossRefGoogle Scholar
  38. 38.
    Montemurro, M., Vincenti, A., Vannucci, P.: A two-level procedure for the global optimum design of composite modular structures—application to the design of an aircraft wing. J. Optim. Theory Appl. 155(1), 24–53 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Olive, M.: Géométrie des espaces de tenseurs, une approche effective appliquée à la mécanique des milieux continus. PhD thesis (2014) Google Scholar
  40. 40.
    Olive, M., Auffray, N.: Isotropic invariants of a completely symmetric third-order tensor. J. Math. Phys. 55(9), 092901 (2014) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Olive, M., Kolev, B., Auffray, N.: A minimal integrity basis for the elasticity tensor. Arch. Ration. Mech. Anal. 226(1), 1–31 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Onat, E.T.: Effective properties of elastic materials that contain penny shaped voids. J. Eng. Sci. 22, 1013–1021 (1984) CrossRefzbMATHGoogle Scholar
  43. 43.
    Ostrosablin, N.I.: On invariants of a fourth-rank tensor of elasticity moduli. Sib. Zh. Ind. Mat. 1(1), 155–163 (1998) MathSciNetzbMATHGoogle Scholar
  44. 44.
    Schouten, J.A.: Tensor Analysis for Physicists. Clarendon, Oxford (1951) zbMATHGoogle Scholar
  45. 45.
    Smith, G.F.: On isotropic integrity bases. Arch. Ration. Mech. Anal. 18, 282–292 (1965) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Smith, G.F., Bao, G.: Isotropic invariants of traceless symmetric tensors of orders three and four. Int. J. Eng. Sci. 35(15), 1457–1462 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Spencer, A.: A note on the decomposition of tensors into traceless symmetric tensors. Int. J. Eng. Sci. 8, 475–481 (1970) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Spencer, A.: Constitutive theory for strongly anisotropic solids. In: Spencer, A.J.M. (ed.) CISM Courses and Lectures, vol. 1, pp. 1–32. Springer, Vienna (1984). 282 Google Scholar
  49. 49.
    Sternberg, S.: Group Theory and Physics. Cambridge University Press, Cambridge (1994) zbMATHGoogle Scholar
  50. 50.
    Sylvester, J.J.: Note on spherical harmonics. In: Collected Mathematical Papers, vol. 3, pp. 37–51. Cambridge University Press, Cambridge (1909) Google Scholar
  51. 51.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Handbuch für Physik, vol. III/3. Springer, Berlin (1965) zbMATHGoogle Scholar
  52. 52.
    Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40, 437–454 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vannucci, P., Pouget, J.: Laminates with given piezoelectric expansion coefficients. Mech. Adv. Mat. Struct. 13(5), 419–427 (2006) CrossRefGoogle Scholar
  54. 54.
    Vannucci, P., Verchery, G.: Stiffness design of laminates using the polar method. Int. J. Solids Struct. 38, 9281–9894 (2001) CrossRefzbMATHGoogle Scholar
  55. 55.
    Verchery, G.: Les invariants des tenseurs d’ordre 4 du type de l’élasticité. In: Boehler, J.-P. (ed.) Colloque Int. CNRS 295, Villard de Lans, pp. 93–104. Springer, Boston (1982, 1979). Google Scholar
  56. 56.
    Vianello, M.: An integrity basis for plane elasticity tensors. Arch. Mech. 49, 197–208 (1997) MathSciNetzbMATHGoogle Scholar
  57. 57.
    Xiao, H.: On isotropic invariants of the elasticity tensor. J. Elast. 46(2), 115–149 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yang, J.: Special Topics in the Theory of Piezoelectricity. Springer, New York (2009) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.LMT (ENS Cachan, CNRS, UMR 8535, Université Paris Saclay)Cachan CedexFrance
  2. 2.CNRS, Centrale Marseille, I2M, UMR 7373Aix Marseille UniversitéMarseilleFrance
  3. 3.UMPC Univ. Paris 06, CNRS, UMR 7190, Institut d’AlembertSorbonne UniversitéParis Cedex 05France
  4. 4.Univ. Paris Sud 11OrsayFrance

Personalised recommendations