# American Institute of Mathematical Sciences

December  2013, 6(6): 1525-1537. doi: 10.3934/dcdss.2013.6.1525

## Structure of the space of 2D elasticity tensors

 1 Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, bâtiment Boussinesq, Cité Scientifique, 59655 Villeneuve d'Ascq cedex, France 2 Institut Pprime, UPR CNRS 3346, Bd M. et P. Curie, téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil cedex, France

Received  June 2012 Revised  September 2012 Published  April 2013

In this paper, we present a geometric representation of the 2D elasticity tensors using the representation theory of linear groups. We use Kelvin's representation in which $\mathbb{O}(2)$ acts on the 2D stress tensors as subgroup of $\mathbb{O}(3)$. We present the method in the simple case of the stress tensors and we recover Mohr's circle construction. Next, we apply it to the elasticity tensors. We explicitly give a linear frame of the elasticity tensor space in which the representation of the rotation group is decomposed into irreducible subspaces. Thanks to five independent invariants choosen among six, an elasticity tensor in 2D can be represented by a compact line or, in degenerated cases, by a circle or a point. The elasticity tensor space, parameterized with these invariants, consists in the union of a manifold of dimension $5$, two volumes and a surface. The complet description requires six polynomial invariants, two linear, two quadratic and two cubic.
Citation: Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525
##### References:
 [1] F. Ahmad, Invariants and structural invariants of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math., 55 (2002), 597-606. doi: 10.1093/qjmam/55.4.597. [2] N. Auffray, R. Bouchet and Y. Bréchet, Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior, International Journal of Solids and Structures, 46 (2009), 440-454. doi: 10.1016/j.ijsolstr.2008.09.009. [3] G. Backus, A geometrical picture of anisotropic elastic tensors, Rev. Geophys. Spacephys., 8 (1970), 633-671. doi: 10.1029/RG008i003p00633. [4] J. Betten, Integrity basis for a second-order and a fourth-order tensor, International Journal of Math. Science, 5 (1982), 87-96. doi: 10.1155/S0161171282000088. [5] J.-P. Boehler, A. A. Kirillov, Jr. and E. T. Onat, On the polynomial invariants of the elastic tensor, Journal of Elasticity, 34 (1994), 97-110. doi: 10.1007/BF00041187. [6] A. Bóna, I. Bucataru and M. A. Slawinski, Space of SO(3)-orbits of elastic tensors, Arch. Mech. (Arch. Mech. Stos.), 60 (2008), 123-138. [7] I. Bucataru and M. A. Slawinski, Invariant properties for finding distance in space of elasticity tensors, Journal of Elasticity, 94 (2009), 97-114. doi: 10.1007/s10659-008-9186-9. [8] P. Chadwick, M. Vianello and S. C. Cowin, A new proof that the number of linear elastic symmetries is eight, Journal of the Mechanics and Physics of Solids, 49 (2001), 2471-2492. doi: 10.1016/S0022-5096(01)00064-3. [9] S. C. Cowin, Properties of the anisotropic elasticity tensors, Q. J. Mech. Appl. Math., 42 (1989), 249-266. doi: 10.1093/qjmam/42.2.249. [10] J. Dieudonné, "Eléments d'Analyse. Tome III: Chapitres XVI et XVII," Cahiers Scientifiques, Fasc. XXXIII, Gauthier-Villars Éditeur, Paris, 1970. [11] S. Forte and M. Vianello, Symmetry classes for elasticity tensors, J. of Elast., 43 (1996), 81-108. doi: 10.1007/BF00042505. [12] M. François, G. Geymonat and Y. Berthaud, Determination of the symmetries of an experimentally determined stiffness tensor; application to acoustic measurements, Int. J. Solids and Structures, 35 (1998), 31-32. [13] Q.-C. He and A. Curnier, A more fundamental approach to damaged elastic stress-strain relations, International Journal of Solids and Structures, 32 (1995), 1433-1457. doi: 10.1016/0020-7683(94)00183-W. [14] D. Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann., 36 (1890), 473-534. doi: 10.1007/BF01208503. [15] D. Hilbert, Ueber die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373. doi: 10.1007/BF01444162. [16] M .N. Jones, "Spherical Harmonics and Tensors for Classical Field Theory," Electronic & Electrical Engineering Research Studies: Applied and Engineering Mathematics Series, 2, Research Studies Press, Ltd., Chichester; John Wiley & Sons, Inc., New York, 1985. [17] M. Mehrabadi and S. Cowin, Eigentensors of linear anisotropic elastic materials, Quarterly Journal of Mechanics and Applied Mathematics, 43 (1990), 15-41. doi: 10.1093/qjmam/43.1.15. [18] M. Mehrabadi, S. Cowin and J. Jarić, Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions, International Journal of Solids and Structures, 32 (1995), 439-449. doi: 10.1016/0020-7683(94)00112-A. [19] O. Mohr, Über die Darstellung des spannungszustandes und des Deformationszustandes eines Körperelementes und über die Andwendung derselben in der Festigkeitslehre, Civilingenieur, 28 (1882), 112-158. [20] O. Mohr, Welche Umstände bedingen die Elastizitätgrenze und den Bruch eines Materials?, Z. Ver. Dtsch. Ing., 44 (1900), 1524-1530. [21] O. Mohr, "Abhandlungen aus dem Gebiete der Technischen Mechanik," 2nd edition, Berlin, 1914. [22] P. J. Olver, Canonical elastic moduli, Journal of Elasticity, 19 (1998), 189-212. doi: 10.1007/BF00045616. [23] E. T. Onat, Effective properties of elastic materials that contain penny shaped voids, Int. J. Engng Sci., 22 (1984), 1013-1021. doi: 10.1016/0020-7225(84)90102-2. [24] N. I. Ostrasablin, On invariants of the fourth-rank tensor of elastic moduli, Sib. Zh. Indust. Mat., 1 (1998), 155-163. [25] N. I. Ostrasablin, On affine transformations of the equations of the linear theory of elasticity, Journal of Applied Mechanics and Technical Physics, 47 (2006), 564-572. doi: 10.1007/s10808-006-0090-4. [26] J. Pratz, Décomposition canonique des tenseurs de rang 4 de l'é1asticité, Journal de Mécanique Théorique et Appliquée, 2 (1983), 893-913. [27] J. Rychlewski, On Hooke's law, J. Appl. Math. Mech., 48 (1984), 303-314. doi: 10.1016/0021-8928(84)90137-0. [28] J. Rychlewski, Unconventional approach to linear elasticity, Arch. Mech. (Arch. Mech. Stos.), 47 (1995), 149-171. [29] A. J. M. Spencer, A note on the decomposition of tensors into traceless symmetric tensors, Int. J. Engng. Sci., 8 (1970), 475-481. doi: 10.1016/0020-7225(70)90024-8. [30] S. Sternberg, "Group Theory and Physics," Cambridge University press, Cambridge, 1994. [31] A. Thionnet and Ch. Martin, A new constructive method using the theory of invariants to obtain material behavior laws, International Journal of Solids and Structures, 43 (2006), 325-345. doi: 10.1016/j.ijsolstr.2005.05.021. [32] T. C. T. Ting, Invariants of anisotropic elastic constants, Q. J. Mech. Appl. Math., 40 (1987), 431-448. doi: 10.1093/qjmam/40.3.431. [33] W. Thomson (Lord Kelvin), Elements of a mathematical theory of elasticity, Philos. Trans. R. Soc., 156 (1856), 481-498. [34] W. Thomson (Lord Kelvin), "Mathematical and Physical Papers. Elasticity, Heat, Electromagnetism, Vol. 3," 2nd edition, Cambridge University Press, Cambridge, 1890. [35] P. Vannucci, Plane anisotropy by the polar method, Meccanica, 40 (2005), 437-454. doi: 10.1007/s11012-005-2132-z. [36] P. Vannucci and G. Verchery, Anisotropy of plane complex elastic bodies, Int. J. of Solids and Structures, 47 (2010), 1154-1166. doi: 10.1016/j.ijsolstr.2010.01.002. [37] G. Verchery, Les invariants des tenseurs d'ordre 4 du type de l'élasticité, in, "Proceedings of the Euromech Colloquium 115 Villard-de-Lans, 1979," Paris, (1983), 93-104. doi: 10.1007/978-94-009-6827-1_7. [38] W. Voigt, "Lehrbuch der Kristallphysics," Teubner, Leipzig, 1910. [39] L. J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables, Proc. R. Soc. Lond. Ser. A, 391 (1984), 149-179. doi: 10.1098/rspa.1984.0008.

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##### References:
 [1] F. Ahmad, Invariants and structural invariants of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math., 55 (2002), 597-606. doi: 10.1093/qjmam/55.4.597. [2] N. Auffray, R. Bouchet and Y. Bréchet, Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior, International Journal of Solids and Structures, 46 (2009), 440-454. doi: 10.1016/j.ijsolstr.2008.09.009. [3] G. Backus, A geometrical picture of anisotropic elastic tensors, Rev. Geophys. Spacephys., 8 (1970), 633-671. doi: 10.1029/RG008i003p00633. [4] J. Betten, Integrity basis for a second-order and a fourth-order tensor, International Journal of Math. Science, 5 (1982), 87-96. doi: 10.1155/S0161171282000088. [5] J.-P. Boehler, A. A. Kirillov, Jr. and E. T. Onat, On the polynomial invariants of the elastic tensor, Journal of Elasticity, 34 (1994), 97-110. doi: 10.1007/BF00041187. [6] A. Bóna, I. Bucataru and M. A. Slawinski, Space of SO(3)-orbits of elastic tensors, Arch. Mech. (Arch. Mech. Stos.), 60 (2008), 123-138. [7] I. Bucataru and M. A. Slawinski, Invariant properties for finding distance in space of elasticity tensors, Journal of Elasticity, 94 (2009), 97-114. doi: 10.1007/s10659-008-9186-9. [8] P. Chadwick, M. Vianello and S. C. Cowin, A new proof that the number of linear elastic symmetries is eight, Journal of the Mechanics and Physics of Solids, 49 (2001), 2471-2492. doi: 10.1016/S0022-5096(01)00064-3. [9] S. C. Cowin, Properties of the anisotropic elasticity tensors, Q. J. Mech. Appl. Math., 42 (1989), 249-266. doi: 10.1093/qjmam/42.2.249. [10] J. Dieudonné, "Eléments d'Analyse. Tome III: Chapitres XVI et XVII," Cahiers Scientifiques, Fasc. XXXIII, Gauthier-Villars Éditeur, Paris, 1970. [11] S. Forte and M. Vianello, Symmetry classes for elasticity tensors, J. of Elast., 43 (1996), 81-108. doi: 10.1007/BF00042505. [12] M. François, G. Geymonat and Y. Berthaud, Determination of the symmetries of an experimentally determined stiffness tensor; application to acoustic measurements, Int. J. Solids and Structures, 35 (1998), 31-32. [13] Q.-C. He and A. Curnier, A more fundamental approach to damaged elastic stress-strain relations, International Journal of Solids and Structures, 32 (1995), 1433-1457. doi: 10.1016/0020-7683(94)00183-W. [14] D. Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann., 36 (1890), 473-534. doi: 10.1007/BF01208503. [15] D. Hilbert, Ueber die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373. doi: 10.1007/BF01444162. [16] M .N. Jones, "Spherical Harmonics and Tensors for Classical Field Theory," Electronic & Electrical Engineering Research Studies: Applied and Engineering Mathematics Series, 2, Research Studies Press, Ltd., Chichester; John Wiley & Sons, Inc., New York, 1985. [17] M. Mehrabadi and S. Cowin, Eigentensors of linear anisotropic elastic materials, Quarterly Journal of Mechanics and Applied Mathematics, 43 (1990), 15-41. doi: 10.1093/qjmam/43.1.15. [18] M. Mehrabadi, S. Cowin and J. Jarić, Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions, International Journal of Solids and Structures, 32 (1995), 439-449. doi: 10.1016/0020-7683(94)00112-A. [19] O. Mohr, Über die Darstellung des spannungszustandes und des Deformationszustandes eines Körperelementes und über die Andwendung derselben in der Festigkeitslehre, Civilingenieur, 28 (1882), 112-158. [20] O. Mohr, Welche Umstände bedingen die Elastizitätgrenze und den Bruch eines Materials?, Z. Ver. Dtsch. Ing., 44 (1900), 1524-1530. [21] O. Mohr, "Abhandlungen aus dem Gebiete der Technischen Mechanik," 2nd edition, Berlin, 1914. [22] P. J. Olver, Canonical elastic moduli, Journal of Elasticity, 19 (1998), 189-212. doi: 10.1007/BF00045616. [23] E. T. Onat, Effective properties of elastic materials that contain penny shaped voids, Int. J. Engng Sci., 22 (1984), 1013-1021. doi: 10.1016/0020-7225(84)90102-2. [24] N. I. Ostrasablin, On invariants of the fourth-rank tensor of elastic moduli, Sib. Zh. Indust. Mat., 1 (1998), 155-163. [25] N. I. Ostrasablin, On affine transformations of the equations of the linear theory of elasticity, Journal of Applied Mechanics and Technical Physics, 47 (2006), 564-572. doi: 10.1007/s10808-006-0090-4. [26] J. Pratz, Décomposition canonique des tenseurs de rang 4 de l'é1asticité, Journal de Mécanique Théorique et Appliquée, 2 (1983), 893-913. [27] J. Rychlewski, On Hooke's law, J. Appl. Math. Mech., 48 (1984), 303-314. doi: 10.1016/0021-8928(84)90137-0. [28] J. Rychlewski, Unconventional approach to linear elasticity, Arch. Mech. (Arch. Mech. Stos.), 47 (1995), 149-171. [29] A. J. M. Spencer, A note on the decomposition of tensors into traceless symmetric tensors, Int. J. Engng. Sci., 8 (1970), 475-481. doi: 10.1016/0020-7225(70)90024-8. [30] S. Sternberg, "Group Theory and Physics," Cambridge University press, Cambridge, 1994. [31] A. Thionnet and Ch. Martin, A new constructive method using the theory of invariants to obtain material behavior laws, International Journal of Solids and Structures, 43 (2006), 325-345. doi: 10.1016/j.ijsolstr.2005.05.021. [32] T. C. T. Ting, Invariants of anisotropic elastic constants, Q. J. Mech. Appl. Math., 40 (1987), 431-448. doi: 10.1093/qjmam/40.3.431. [33] W. Thomson (Lord Kelvin), Elements of a mathematical theory of elasticity, Philos. Trans. R. Soc., 156 (1856), 481-498. [34] W. Thomson (Lord Kelvin), "Mathematical and Physical Papers. Elasticity, Heat, Electromagnetism, Vol. 3," 2nd edition, Cambridge University Press, Cambridge, 1890. [35] P. Vannucci, Plane anisotropy by the polar method, Meccanica, 40 (2005), 437-454. doi: 10.1007/s11012-005-2132-z. [36] P. Vannucci and G. Verchery, Anisotropy of plane complex elastic bodies, Int. J. of Solids and Structures, 47 (2010), 1154-1166. doi: 10.1016/j.ijsolstr.2010.01.002. [37] G. Verchery, Les invariants des tenseurs d'ordre 4 du type de l'élasticité, in, "Proceedings of the Euromech Colloquium 115 Villard-de-Lans, 1979," Paris, (1983), 93-104. doi: 10.1007/978-94-009-6827-1_7. [38] W. Voigt, "Lehrbuch der Kristallphysics," Teubner, Leipzig, 1910. [39] L. J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables, Proc. R. Soc. Lond. Ser. A, 391 (1984), 149-179. doi: 10.1098/rspa.1984.0008.
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