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 & 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. doi: 10.1093/qjmam/55.4.597. Google Scholar

[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. doi: 10.1016/j.ijsolstr.2008.09.009. Google Scholar

[3]

G. Backus, A geometrical picture of anisotropic elastic tensors,, Rev. Geophys. Spacephys., 8 (1970), 633. doi: 10.1029/RG008i003p00633. Google Scholar

[4]

J. Betten, Integrity basis for a second-order and a fourth-order tensor,, International Journal of Math. Science, 5 (1982), 87. doi: 10.1155/S0161171282000088. Google Scholar

[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. doi: 10.1007/BF00041187. Google Scholar

[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. Google Scholar

[7]

I. Bucataru and M. A. Slawinski, Invariant properties for finding distance in space of elasticity tensors,, Journal of Elasticity, 94 (2009), 97. doi: 10.1007/s10659-008-9186-9. Google Scholar

[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. doi: 10.1016/S0022-5096(01)00064-3. Google Scholar

[9]

S. C. Cowin, Properties of the anisotropic elasticity tensors,, Q. J. Mech. Appl. Math., 42 (1989), 249. doi: 10.1093/qjmam/42.2.249. Google Scholar

[10]

J. Dieudonné, "Eléments d'Analyse. Tome III: Chapitres XVI et XVII,", Cahiers Scientifiques, (1970). Google Scholar

[11]

S. Forte and M. Vianello, Symmetry classes for elasticity tensors,, J. of Elast., 43 (1996), 81. doi: 10.1007/BF00042505. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0020-7683(94)00183-W. Google Scholar

[14]

D. Hilbert, Ueber die Theorie der algebraischen Formen,, Math. Ann., 36 (1890), 473. doi: 10.1007/BF01208503. Google Scholar

[15]

D. Hilbert, Ueber die vollen Invariantensysteme,, Math. Ann., 42 (1893), 313. doi: 10.1007/BF01444162. Google Scholar

[16]

M .N. Jones, "Spherical Harmonics and Tensors for Classical Field Theory,", Electronic & Electrical Engineering Research Studies: Applied and Engineering Mathematics Series, 2 (1985). Google Scholar

[17]

M. Mehrabadi and S. Cowin, Eigentensors of linear anisotropic elastic materials,, Quarterly Journal of Mechanics and Applied Mathematics, 43 (1990), 15. doi: 10.1093/qjmam/43.1.15. Google Scholar

[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. doi: 10.1016/0020-7683(94)00112-A. Google Scholar

[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. Google Scholar

[20]

O. Mohr, Welche Umstände bedingen die Elastizitätgrenze und den Bruch eines Materials?,, Z. Ver. Dtsch. Ing., 44 (1900), 1524. Google Scholar

[21]

O. Mohr, "Abhandlungen aus dem Gebiete der Technischen Mechanik,", 2nd edition, (1914). Google Scholar

[22]

P. J. Olver, Canonical elastic moduli,, Journal of Elasticity, 19 (1998), 189. doi: 10.1007/BF00045616. Google Scholar

[23]

E. T. Onat, Effective properties of elastic materials that contain penny shaped voids,, Int. J. Engng Sci., 22 (1984), 1013. doi: 10.1016/0020-7225(84)90102-2. Google Scholar

[24]

N. I. Ostrasablin, On invariants of the fourth-rank tensor of elastic moduli,, Sib. Zh. Indust. Mat., 1 (1998), 155. Google Scholar

[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. doi: 10.1007/s10808-006-0090-4. Google Scholar

[26]

J. Pratz, Décomposition canonique des tenseurs de rang 4 de l'é1asticité,, Journal de Mécanique Théorique et Appliquée, 2 (1983), 893. Google Scholar

[27]

J. Rychlewski, On Hooke's law,, J. Appl. Math. Mech., 48 (1984), 303. doi: 10.1016/0021-8928(84)90137-0. Google Scholar

[28]

J. Rychlewski, Unconventional approach to linear elasticity,, Arch. Mech. (Arch. Mech. Stos.), 47 (1995), 149. Google Scholar

[29]

A. J. M. Spencer, A note on the decomposition of tensors into traceless symmetric tensors,, Int. J. Engng. Sci., 8 (1970), 475. doi: 10.1016/0020-7225(70)90024-8. Google Scholar

[30]

S. Sternberg, "Group Theory and Physics,", Cambridge University press, (1994). Google Scholar

[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. doi: 10.1016/j.ijsolstr.2005.05.021. Google Scholar

[32]

T. C. T. Ting, Invariants of anisotropic elastic constants,, Q. J. Mech. Appl. Math., 40 (1987), 431. doi: 10.1093/qjmam/40.3.431. Google Scholar

[33]

W. Thomson (Lord Kelvin), Elements of a mathematical theory of elasticity,, Philos. Trans. R. Soc., 156 (1856), 481. Google Scholar

[34]

W. Thomson (Lord Kelvin), "Mathematical and Physical Papers. Elasticity, Heat, Electromagnetism, Vol. 3,", 2nd edition, (1890). Google Scholar

[35]

P. Vannucci, Plane anisotropy by the polar method,, Meccanica, 40 (2005), 437. doi: 10.1007/s11012-005-2132-z. Google Scholar

[36]

P. Vannucci and G. Verchery, Anisotropy of plane complex elastic bodies,, Int. J. of Solids and Structures, 47 (2010), 1154. doi: 10.1016/j.ijsolstr.2010.01.002. Google Scholar

[37]

G. Verchery, Les invariants des tenseurs d'ordre 4 du type de l'élasticité,, in, (1983), 93. doi: 10.1007/978-94-009-6827-1_7. Google Scholar

[38]

W. Voigt, "Lehrbuch der Kristallphysics,", Teubner, (1910). Google Scholar

[39]

L. J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables,, Proc. R. Soc. Lond. Ser. A, 391 (1984), 149. doi: 10.1098/rspa.1984.0008. Google Scholar

show all references

References:
[1]

F. Ahmad, Invariants and structural invariants of the anisotropic elasticity tensor,, Q. J. Mech. Appl. Math., 55 (2002), 597. doi: 10.1093/qjmam/55.4.597. Google Scholar

[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. doi: 10.1016/j.ijsolstr.2008.09.009. Google Scholar

[3]

G. Backus, A geometrical picture of anisotropic elastic tensors,, Rev. Geophys. Spacephys., 8 (1970), 633. doi: 10.1029/RG008i003p00633. Google Scholar

[4]

J. Betten, Integrity basis for a second-order and a fourth-order tensor,, International Journal of Math. Science, 5 (1982), 87. doi: 10.1155/S0161171282000088. Google Scholar

[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. doi: 10.1007/BF00041187. Google Scholar

[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. Google Scholar

[7]

I. Bucataru and M. A. Slawinski, Invariant properties for finding distance in space of elasticity tensors,, Journal of Elasticity, 94 (2009), 97. doi: 10.1007/s10659-008-9186-9. Google Scholar

[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. doi: 10.1016/S0022-5096(01)00064-3. Google Scholar

[9]

S. C. Cowin, Properties of the anisotropic elasticity tensors,, Q. J. Mech. Appl. Math., 42 (1989), 249. doi: 10.1093/qjmam/42.2.249. Google Scholar

[10]

J. Dieudonné, "Eléments d'Analyse. Tome III: Chapitres XVI et XVII,", Cahiers Scientifiques, (1970). Google Scholar

[11]

S. Forte and M. Vianello, Symmetry classes for elasticity tensors,, J. of Elast., 43 (1996), 81. doi: 10.1007/BF00042505. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0020-7683(94)00183-W. Google Scholar

[14]

D. Hilbert, Ueber die Theorie der algebraischen Formen,, Math. Ann., 36 (1890), 473. doi: 10.1007/BF01208503. Google Scholar

[15]

D. Hilbert, Ueber die vollen Invariantensysteme,, Math. Ann., 42 (1893), 313. doi: 10.1007/BF01444162. Google Scholar

[16]

M .N. Jones, "Spherical Harmonics and Tensors for Classical Field Theory,", Electronic & Electrical Engineering Research Studies: Applied and Engineering Mathematics Series, 2 (1985). Google Scholar

[17]

M. Mehrabadi and S. Cowin, Eigentensors of linear anisotropic elastic materials,, Quarterly Journal of Mechanics and Applied Mathematics, 43 (1990), 15. doi: 10.1093/qjmam/43.1.15. Google Scholar

[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. doi: 10.1016/0020-7683(94)00112-A. Google Scholar

[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. Google Scholar

[20]

O. Mohr, Welche Umstände bedingen die Elastizitätgrenze und den Bruch eines Materials?,, Z. Ver. Dtsch. Ing., 44 (1900), 1524. Google Scholar

[21]

O. Mohr, "Abhandlungen aus dem Gebiete der Technischen Mechanik,", 2nd edition, (1914). Google Scholar

[22]

P. J. Olver, Canonical elastic moduli,, Journal of Elasticity, 19 (1998), 189. doi: 10.1007/BF00045616. Google Scholar

[23]

E. T. Onat, Effective properties of elastic materials that contain penny shaped voids,, Int. J. Engng Sci., 22 (1984), 1013. doi: 10.1016/0020-7225(84)90102-2. Google Scholar

[24]

N. I. Ostrasablin, On invariants of the fourth-rank tensor of elastic moduli,, Sib. Zh. Indust. Mat., 1 (1998), 155. Google Scholar

[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. doi: 10.1007/s10808-006-0090-4. Google Scholar

[26]

J. Pratz, Décomposition canonique des tenseurs de rang 4 de l'é1asticité,, Journal de Mécanique Théorique et Appliquée, 2 (1983), 893. Google Scholar

[27]

J. Rychlewski, On Hooke's law,, J. Appl. Math. Mech., 48 (1984), 303. doi: 10.1016/0021-8928(84)90137-0. Google Scholar

[28]

J. Rychlewski, Unconventional approach to linear elasticity,, Arch. Mech. (Arch. Mech. Stos.), 47 (1995), 149. Google Scholar

[29]

A. J. M. Spencer, A note on the decomposition of tensors into traceless symmetric tensors,, Int. J. Engng. Sci., 8 (1970), 475. doi: 10.1016/0020-7225(70)90024-8. Google Scholar

[30]

S. Sternberg, "Group Theory and Physics,", Cambridge University press, (1994). Google Scholar

[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. doi: 10.1016/j.ijsolstr.2005.05.021. Google Scholar

[32]

T. C. T. Ting, Invariants of anisotropic elastic constants,, Q. J. Mech. Appl. Math., 40 (1987), 431. doi: 10.1093/qjmam/40.3.431. Google Scholar

[33]

W. Thomson (Lord Kelvin), Elements of a mathematical theory of elasticity,, Philos. Trans. R. Soc., 156 (1856), 481. Google Scholar

[34]

W. Thomson (Lord Kelvin), "Mathematical and Physical Papers. Elasticity, Heat, Electromagnetism, Vol. 3,", 2nd edition, (1890). Google Scholar

[35]

P. Vannucci, Plane anisotropy by the polar method,, Meccanica, 40 (2005), 437. doi: 10.1007/s11012-005-2132-z. Google Scholar

[36]

P. Vannucci and G. Verchery, Anisotropy of plane complex elastic bodies,, Int. J. of Solids and Structures, 47 (2010), 1154. doi: 10.1016/j.ijsolstr.2010.01.002. Google Scholar

[37]

G. Verchery, Les invariants des tenseurs d'ordre 4 du type de l'élasticité,, in, (1983), 93. doi: 10.1007/978-94-009-6827-1_7. Google Scholar

[38]

W. Voigt, "Lehrbuch der Kristallphysics,", Teubner, (1910). Google Scholar

[39]

L. J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables,, Proc. R. Soc. Lond. Ser. A, 391 (1984), 149. doi: 10.1098/rspa.1984.0008. Google Scholar

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