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Archive for Rational Mechanics and Analysis

, Volume 226, Issue 1, pp 1–31 | Cite as

A Minimal Integrity Basis for the Elasticity Tensor

  • M. Olive
  • B. Kolev
  • N. AuffrayEmail author
Article

Abstract

We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.

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References

  1. 1.
    Abud M., Sartori G.: The geometry of spontaneous symmetry breaking. Ann. Phys. 150(2), 307–372 (1983)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ahmad F.: Invariants and structural invariants of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 55(4), 597–606 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Annin B. D., Ostrosablin N. I.: Anisotropy of the elastic properties of materials. Prikl. Mekh. Tekhn. Fiz. 49(6), 131–151 (2008)MathSciNetGoogle Scholar
  4. 4.
    Ashman R., Cowin S., Van Buskirk W., Rice J.: A continuous wave technique for the measurement of the elastic properties of cortical bone. J. Biomech. 17(5), 349–361 (1984)CrossRefGoogle Scholar
  5. 5.
    Auffray,N., Kolev, B., Olive, M.: Handbook of bidimensional tensors. Part I: Harmonic decomposition and symmetry classes. Math. Mech. Solids, 2016. doi: 10.1177/1081286516649017
  6. 6.
    Auffray N., Kolev B., Petitot M.: On anisotropic polynomial relations for the elasticity tensor. J. Elast. 115(1), 77–103 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    AuffrayN. Ropars P.: Invariant-based reconstruction of bidimensional elasticity tensors. Int. J. Solids Struct. 87, 183–193 (2016)CrossRefGoogle Scholar
  8. 8.
    Backus G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. 8(3), 633–671 (1970)ADSCrossRefGoogle Scholar
  9. 9.
    Baerheim R.: Harmonic decomposition of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 46(3), 391–418 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bedratyuk L.: On complete system of covariants for the binary form of degree 8. Mat. Visn. Nauk. Tov. Im. Shevchenka 5, 11–22 (2008)zbMATHGoogle Scholar
  11. 11.
    Bedratyuk L.: A complete minimal system of covariants for the binary form of degree 7. J. Symb. Comput. 44(2), 211–220 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Betten, J.: Irreducible invariants of fourth-order tensors. Math. Model. 8, 29–33, 1987. Mathematical modelling in science and technology (Berkeley, CA, 1985)Google Scholar
  13. 13.
    Betten, J., Helisch, W.: Integrity bases for a fourth-rank tensor. In: IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, pp 37–42. Springer, 1995Google Scholar
  14. 14.
    Blinowski A., Ostrowska-Maciejewska J., Rychlewski J.: Two-dimensional Hooke’s tensors—isotropic decomposition, effective symmetry criteria. Arch. Mech. (Arch. Mech. Stos.) 48(2), 325–345 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Boehler J. P.: On irreducible representations for isotropic scalar functions. Z. Angew. Math. Mech. 57(6), 323–327 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Boehler J.-P.: Lois de comportement anisotrope des milieux continus. J. Mécanique 17(2), 153–190 (1978)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Boehler, J.-P.: Application of Tensor Functions in Solid Mechanics. CISM Courses and Lectures. Springer, Wien, 1987Google Scholar
  18. 18.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bóna A., Bucataru I., Slawinski M.: Characterization of elasticity-tensor symmetries using SU(2). J. Elast. 75(3), 267–289 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bóna A., Bucataru I., Slawinski M.: Space of SO(3)-orbits of elasticity tensors. Arch. Mech. 60(2), 123–138 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Brouwer, A. E.: Invariants of binary forms, 2015. http://www.win.tue.nl/~aeb/math/invar.html.
  22. 22.
    Brouwer A. E., Popoviciu M.: The invariants of the binary decimic. J. Symb. Comput. 45(8), 837–843 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Brouwer A. E., Popoviciu M.: The invariants of the binary nonic. J. Symb. Comput. 45(6), 709–720 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bruns W., Ichim B.: Normaliz: algorithms for affine monoids and rational cones. J. Algebra 324(5), 1098–1113 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Cartan, É.: The Theory of Spinors. Dover Publications, Inc., New York, 1981. With a foreword by Raymond Streater, A reprint of the 1966 English translation, Dover Books on Advanced MathematicsGoogle Scholar
  26. 26.
    Cayley A.: A seventh memoir on quantics. Philos. Trans. R. Soc. Lond. 151, 277–292 (1861)CrossRefGoogle Scholar
  27. 27.
    Cowin S. C.: Properties of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 42, 249–266 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Cowin S. C.: Continuum Mechanics of Anisotropic Materials. Springer, New York (2013)zbMATHCrossRefGoogle Scholar
  29. 29.
    Cowin, Mehrabadi, M.: On the identification of material symmetry for anisotropic elastic materials. Q. J. Mech. Appl. Math. 40, 451–476, 1987Google Scholar
  30. 30.
    Cröni, H. L.: Zur Berechnung von Kovarianten von Quantiken. PhD thesis, 2002.Google Scholar
  31. 31.
    Derksen H.: Computation of invariants for reductive groups. Adv. Math. 141(2), 366–384 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Derksen H., Kemper G.: Computing invariants of algebraic groups in arbitrary characteristic. Adv. Math., 217(5), 2089–2129 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Derksen, H., Kemper, G.: Computational Invariant Theory, volume 130 of Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, enlarged edition, 2015. With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov, Invariant Theory and Algebraic Transformation Groups, VIIIGoogle Scholar
  34. 34.
    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
  35. 35.
    Forte S., Vianello M.: Symmetry classes for elasticity tensors. J. Elast. 43(2), 81–108 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Forte S., Vianello M.: Restricted invariants on the space of elasticity tensors. Math. Mech. Solids 11(1), 48–82 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Forte S., Vianello M.: A unified approach to invariants of plane elasticity tensors. Meccanica 49(9), 2001–2012 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    François D., Pineau A., Zaoui A.: Mechanical Behaviour of Materials. Springer, New York (1998)zbMATHGoogle Scholar
  39. 39.
    François M.: A damage model based on Kelvin eigentensors and Curie principle. Mech. Mater. 44, 23–34 (2012)CrossRefGoogle Scholar
  40. 40.
    Golubitsky, M., Stewart, I., Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory. Vol. II, Volume 69 of Applied Mathematical Sciences. Springer, New York, 1988Google Scholar
  41. 41.
    Goodman, R., Wallach, N. R.: Symmetry, Representations, and Invariants, volume 255 of Graduate Texts in Mathematics. Springer, Dordrecht, 2009Google Scholar
  42. 42.
    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 Angewandte Math. 69, 323–354, 1868Google Scholar
  43. 43.
    Grace, J. H., Young, A.: The Algebra of Invariants. Cambridge Library Collection. Cambridge University Press, Cambridge, 2010. Reprint of the 1903 originalGoogle Scholar
  44. 44.
    Grédiac M.: On the direct determination of invariant parameters governing the bending of anisotropic plates. Int. J. Solids Struct. 33(27), 3969–3982 (1996)zbMATHCrossRefGoogle Scholar
  45. 45.
    Gurevich, G. B.: Foundations of the Theory of Algebraic Invariants. P. Noordhoff Ltd., Groningen, 1964Google Scholar
  46. 46.
    Gurtin, M.: The linear theory of elasticity. In: Linear Theories of Elasticity and Thermoelasticity, pp. 1–295. Springer, New York, 1973Google Scholar
  47. 47.
    Hahn H.: A derivation of invariants of fourth rank tensors. J. Compos. Mater. 8(1), 2–14 (1974)ADSCrossRefGoogle Scholar
  48. 48.
    He Q.-C.: Characterization of the anisotropic materials capable of exhibiting an isotropic young or shear or area modulus. Int. J. Eng. Sci. 42, 2107–2118 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Helbig K., Thomsen L.: 75-plus years of anisotropy in exploration and reservoir seismics: a historical review of concepts and methods. Geophysics 70(6), 9–23 (2005)ADSCrossRefGoogle Scholar
  50. 50.
    Hilbert D.: Theory of Algebraic Invariants. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  51. 51.
    Hobson E. W.: The Theory of Spherical and Ellipsoidal Harmonics. Chelsea Publishing Company, New York (1955)zbMATHGoogle Scholar
  52. 52.
    Ihrig E., Golubitsky M.: Pattern selection with O(3) symmetry. Phys. D 13(1-2), 1–33 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Itin Y.: Quadratic invariants of the elasticity tensor. J. Elast. 125(1), 1–24 (2015)MathSciNetGoogle Scholar
  54. 54.
    Kraft, H., Procesi, C.: Classical Invariant Theory, a Primer. Lectures notes avaiable at http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf, 2000
  55. 55.
    Lemaitre J., Chaboche J.-L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  56. 56.
    Lercier, R., Olive, M.: Covariant algebra of the binary nonic and the binary decimic. In: Bassa, A., Couvreur, A., Kohel, D. (eds.) Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, vol. 686. American Mathematical Society, 2017. doi: 10.1090/conm/686
  57. 57.
    Lercier R., Ritzenthaler C.: Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. J. Algebra 372, 595–636 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Liu S.: On representations of anisotropic invariants. Int. J. Eng. Sci. 20, 1099–1109 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Luque, J.-G.: Invariants des hypermatrices. http://tel.archives-ouvertes.fr/tel-00250312, 2007
  60. 60.
    Mehrabadi M. M., Cowin S. C.: Eigentensors of linear anisotropic elastic materials. Q. J. Mech. Appl. Math. 43(1), 15–41 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Milton G.: The Theory of Composites. Cambridge University Press, Cambridge (2002)zbMATHCrossRefGoogle Scholar
  62. 62.
    Norris A. N.: Quadratic invariants of elastic moduli. Q. J. Mech. Appl. Math. 60(3), 367–389 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Olive, M.: Géométrie des espaces de tenseurs : une approche effective appliquée à la mécanique des milieux continus. PhD thesis, University of Aix-Marseille, Nov. 2014Google Scholar
  64. 64.
    Olive, M.: About Gordan’s algorithm for binary forms. Found. Comput. Math. 2016. doi: 10.1007/s10208-016-9324-x
  65. 65.
    Olver, P. J.: Classical Invariant Theory, Volume 44 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999Google Scholar
  66. 66.
    Onat E.: Effective properties of elastic materials that contain penny shaped voids. Int. J. Eng. Sci. 22(8), 1013–1021 (1984)zbMATHCrossRefGoogle Scholar
  67. 67.
    Ostrosablin N. I.: On invariants of a fourth-rank tensor of elasticity moduli. Sib. Zh. Ind. Mat. 1(1L), 155–163 (1998)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Pessard E., Morel F., Morel A.: The anisotropic fatigue behavior of forged steel. Adv. Eng. Mater. 11(9), 732–735 (2009)CrossRefGoogle Scholar
  69. 69.
    Pierce J. F.: Representations for transversely hemitropic and transversely isotropic stress–strain relations. J. Elast. 37(3), 243–280 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Rivlin R.: Further remarks on the stress–deformation relation for isotropic materials. J. Ration. Mech. Anal. 4, 681–701 (1955)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Rosi G., Auffray N.: Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016)CrossRefGoogle Scholar
  72. 72.
    Rosi G., Nguyen V.-H., Naili S.: Numerical investigations of ultrasound wave propagating in long bones using a poroelastic model. Math. Mech. Solids 21(1), 119–133 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Rychlewski J.: On Hooke’s law. J. Appl. Math. Mech. 48(3), 303–314 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Salençon J.: Handbook of Continuum Mechanics: General Concepts Thermoelasticity. Springer, New York (2012)zbMATHGoogle Scholar
  75. 75.
    Shioda T.: On the graded ring of invariants of binary octavics. Am. J. Math. 89, 1022–1046 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Smith G.: On isotropic integrity bases. Arch. Ration. Mech. Anal. 18(4), 282–292 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Smith G.: On isotropic integrity bases. Arch. Ration. Mech. Anal. 18(4), 282–292 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Smith G.: On a fundamental error in two papers of C.C. Wang On representations for isotropic functions, part I and II. Arch. Rational Mech. Anal. 36, 161–165 (1970)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Smith G.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci., 9, 899–916 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Smith G.: Constitutive Equations for Anisotropic and Isotropic Materials. North-Holland, Amsterdam (1994)Google Scholar
  81. 81.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Spencer A.: Part III. Theory of invariants. Continuum Phys. 1, 239–353 (1971)Google Scholar
  83. 83.
    Spencer, A. J. M., Rivlin, R. S.: Finite integrity bases for five or fewer symmetric \({3 \times 3}\) matrices. Arch. Ration. Mech. Anal. 2, 435–446, 1958/1959Google Scholar
  84. 84.
    Spencer A. J. M., Rivlin R. S.: Isotropic integrity bases for vectors and second-order tensors. I. Arch. Ration. Mech. Anal. 9, 45–63 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Sternberg S.: Group Theory and Physics. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  86. 86.
    Sturmfels, B.: Algorithms in invariant theory. In: Texts and Monographs in Symbolic Computation, 2nd ed. Springer, Vienna, 2008Google Scholar
  87. 87.
    Thomson W. L. K.: On six principal strains of an elastic solid. Philos. Trans. R. Soc. Lond. 166, 495–498 (1856)Google Scholar
  88. 88.
    Ting T. C. T.: Invariants of anisotropic elastic constants. Q. J. Mech. Appl. Math. 40(3), 431–448 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Verchery, G.: Les invariants des tenseurs d’ordre 4 du type de l’élasticité. In Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes, pp. 93–104. Springer, 1982Google Scholar
  90. 90.
    Vianello M.: An integrity basis for plane elasticity tensors. Arch. Mech. (Arch. Mech. Stos.) 49(1), 197–208 (1997)MathSciNetzbMATHGoogle Scholar
  91. 91.
    von Gall.: Ueber das simultane Formensystem einer Form 2ter und 6ter Ordnung. Jahresbericht über das Gymnasium zu Lengo, 1874Google Scholar
  92. 92.
    von Gall.: Ueber das vollständige System einer binären Form achter Ordnung. Math. Ann., 17(1), 139–152, 1880Google Scholar
  93. 93.
    von Gall.: Das vollstandige formensystem der binaren form 7ter ordnung. Math. Ann. 31), 318–336, 1888Google Scholar
  94. 94.
    Wang, C.-C.: A new representation theorem for isotropic functions: an answer to professor G.F. Smith’s criticism of my papers on representations for isotropic functions, part I. Arch. Ration. Mech. Anal. 36, 166–197, 1970Google Scholar
  95. 95.
    Weyl, H.: The Classical Groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Their invariants and representations, Fifteenth printing, Princeton PaperbacksGoogle Scholar
  96. 96.
    Wineman A., Pipkin A.: Material symmetry restrictions on constitutive equations. Arch. Ration. Mech. Anal. 17, 184–214 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Xiao H.: On isotropic invariants of the elasticity tensor. J. Elast. 46(2), 115–149 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Young A.: The irreducible concomitants of any number of binary quartics. Proc. Lond. Math. Soc. 1(1), 290–307 (1898)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Zheng Q.-S.: Theory of representations for tensor functions—a unified invariant appr oach to constitutive equations. Appl. Mech. Rev. 47, 545–587 (1994)ADSCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.LMT-Cachan (ENS Cachan, CNRS, Université Paris Saclay)Cachan CedexFrance
  2. 2.CNRS, Centrale Marseille, I2M, UMR 7373Aix Marseille UniversitéMarseilleFrance
  3. 3.Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd DescartesMSME, Université Paris-EstMarne-la-ValléeFrance

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