Advertisement

Journal of Elasticity

, Volume 134, Issue 2, pp 193–217 | Cite as

Intrinsic Formulation and Lagrange Duality for Elastic Cable Networks with Geometrical Nonlinearity

  • Yoshihiro KannoEmail author
Article
  • 52 Downloads

Abstract

The intrinsic approach to elasticity is a variational problem that characterizes the static equilibrium state and is formulated only in terms of the unknown strain tensor field. This paper establishes an intrinsic approach to elastic cable networks undergoing large deformations. In this formulation, the potential energy function attains a global minimum at strains in any equilibrium state. It is clearly explained that both the classical displacement formulation and the intrinsic strain formulation can be derived as the Lagrange dual problems of the stress formulation, and that the difference between these two formulations stems from the variety of perturbation functions used to derive the Lagrangian in the Lagrange duality theory.

Keywords

Duality Lagrangian Legendre–Fenchel transform Second-order cone programming Elasticity Cable network 

Mathematics Subject Classification

90C25 90C46 49N15 

Notes

Acknowledgement

This work is partially supported by JSPS KAKENHI 17K06633.

References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program., Ser. B 95, 3–51 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization. Springer, New York (2012) zbMATHGoogle Scholar
  3. 3.
    Antman, S.S.: Ordinary differential equations of non-linear elasticity. I: Foundations of the theories of non-linearly elastic rods and shells. Arch. Ration. Mech. Anal. 61, 307–351 (1976) CrossRefzbMATHGoogle Scholar
  4. 4.
    Atai, A.A., Steigmann, D.J.: On the nonlinear mechanics of discrete networks. Arch. Appl. Mech. 67, 303–319 (1997) ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Belmont (2009) zbMATHGoogle Scholar
  6. 6.
    Buchholdt, H.A., Davies, M., Hussey, M.J.L.: The analysis of cable nets. IMA J. Appl. Math. 4, 339–358 (1968) CrossRefGoogle Scholar
  7. 7.
    Cannarozzi, M.: A minimum principle for tractions in the elastostatics of cable networks. Int. J. Solids Struct. 23, 551–568 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chien, W.-Z.: The intrinsic theory of thin shells and plates, part I: general theory. Q. Appl. Math. 1, 297–327 (1944) CrossRefGoogle Scholar
  9. 9.
    Chvátal, V.: Linear Programming. Freeman, New York (1983) zbMATHGoogle Scholar
  10. 10.
    Ciarlet, P.G., Ciarlet, P. Jr.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models Methods Appl. Sci. 15, 259–271 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ciarlet, P.G., Ciarlet, P. Jr.: Direct computation of stresses in planar linearized elasticity. Math. Models Methods Appl. Sci. 19, 1043–1064 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G., Ciarlet, P. Jr., Sauter, S.A., Simian, C.: Intrinsic finite element methods for the computation of fluxes for Poisson’s equation. Numer. Math. 132, 433–462 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ciarlet, P.G., Geymonat, G., Krasucki, F.: Legendre–Fenchel duality in elasticity. C. R. Acad. Sci., Ser. I 349, 597–602 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ciarlet, P.G., Geymonat, G., Krasucki, F.: A new duality approach to elasticity. Math. Models Methods Appl. Sci. 22, 1150003 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ciarlet, P.G., Gratie, L., Mardare, C.: Intrinsic methods in elasticity: a mathematical survey. Discrete Contin. Dyn. Syst. 23, 136–164 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ciarlet, P.G., Mardare, C.: The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems. C. R. Acad. Sci., Ser. I 347, 677–683 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ciarlet, P.G., Mardare, C.: Existence theorems in intrinsic nonlinear elasticity. J. Math. Pures Appl. 94, 229–243 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ciarlet, P.G., Mardare, C.: Boundary conditions in intrinsic nonlinear elasticity. J. Math. Pures Appl. 101, 458–472 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ciarlet, P.G., Mardare, C.: Intrinsic formulation of the displacement-traction problem in linearized elasticity. Math. Models Methods Appl. Sci. 24, 1197–1216 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005) zbMATHGoogle Scholar
  21. 21.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976); SIAM, Philadelphia (1999) zbMATHGoogle Scholar
  22. 22.
    Jungnickel, D.: Graphs, Networks and Algorithms, 3rd edn. Springer, Berlin (2008) CrossRefzbMATHGoogle Scholar
  23. 23.
    Kanno, Y.: Nonsmooth Mechanics and Convex Optimization. CRC Press, Boca Raton (2011) CrossRefzbMATHGoogle Scholar
  24. 24.
    Kanno, Y., Ohsaki, M.: Minimum principle of complementary energy of cable networks by using second-order cone programming. Int. J. Solids Struct. 40, 4437–4460 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kanno, Y., Ohsaki, M.: Minimum principle of complementary energy for nonlinear elastic cable networks with geometrical nonlinearities. J. Optim. Theory Appl. 126, 617–641 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kanno, Y., Ohsaki, M., Ito, J.: Large-deformation and friction analysis of nonlinear elastic cable networks by second-order cone programming. Int. J. Numer. Methods Eng. 55, 1079–1114 (2002) CrossRefzbMATHGoogle Scholar
  27. 27.
    Koiter, W.T.: On the complementary energy theorem in non-linear elasticity theory. In: Fichera, G. (ed.) Trends in Applications of Pure Mathematics of Mechanics, pp. 207–232. Pitman, London (1976) Google Scholar
  28. 28.
    Lewis, W.J.: Tension Structures: Form and Behavior. Thomas Telford, London (2003) CrossRefGoogle Scholar
  29. 29.
    Opoka, S., Pietraszkiewicz, W.: Intrinsic equations for non-linear deformation and stability of thin elastic shells. Int. J. Solids Struct. 41, 3275–3292 (2004) CrossRefzbMATHGoogle Scholar
  30. 30.
    Panagiotopoulos, P.D.: A variational inequality approach to the inelastic stress-unilateral analysis of cable-structures. Comput. Struct. 6, 133–139 (1976) CrossRefzbMATHGoogle Scholar
  31. 31.
    Qiu, W., Wang, M., Zhang, J.: Direct computation of stresses in linear elasticity. J. Comput. Appl. Math. 292, 363–368 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd edn. Wiley, Hoboken (2002) Google Scholar
  33. 33.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematics and Informatics CenterThe University of TokyoTokyoJapan

Personalised recommendations