Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams

  • A. Battista
  • A. Della Corte
  • F. dell’Isola
  • P. Seppecher


In the present paper we study a natural nonlinear generalization of Timoshenko beam model and show that it can describe the homogenized deformation energy of a 1D continuum with a simple microstructure. We prove the well posedness of the corresponding variational problem in the case of a generic end load, discuss some regularity issues and evaluate the critical load. Moreover, we generalize the model so as to include an additional rotational spring in the microstructure. Finally, some numerical simulations are presented and discussed.


Nonlinear elasticity Generalized Timoshenko beam Microstructured beam Non-convex variational problems 

Mathematics Subject Classification

74B20 49J45 


  1. 1.
    Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti (1744). chapter Additamentum 1, E065Google Scholar
  2. 2.
    Bernoulli, D.: The 26th letter to Euler. In: Correspondence Mathématique et Physique, vol. 2. P. H. Fuss (1742)Google Scholar
  3. 3.
    Bernoulli, J.: Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura. Die Werke von Jakob Bernoulli, pp. 223–227 (1692)Google Scholar
  4. 4.
    Lagrange, J.L.: Mécanique analytique, vol. 1-2. Mallet-Bachelier, Paris (1744)Google Scholar
  5. 5.
    Mora, M. G., Müller, S.: A nonlinear model for inextensible rods as a low energy \(\Gamma \)-limit of three-dimensional nonlinear elasticity. In: Annales de l’IHP Analyse non linéaire, vol. 21, pp. 271–293 (2004)Google Scholar
  6. 6.
    Pideri, C., Seppecher, P.: Asymptotics of a non-planar rod in non-linear elasticity. Asymptot. Anal. 48(1, 2), 33–54 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eugster, S.R.: Geometric Continuum Mechanics and Induced Beam Theories, vol. 75. Springer, New York (2015)zbMATHGoogle Scholar
  8. 8.
    Eugster, S., Glocker, C.: Determination of the transverse shear stress in an Euler–Bernoulli beam using non-admissible virtual displacements. PAMM 14(1), 187–188 (2014)CrossRefGoogle Scholar
  9. 9.
    Timoshenko, S.P.: On the correction factor for shear of the differential equation for transverse vibrations of prismatic bar. Philos. Mag. 6(41), 744 (1921)CrossRefGoogle Scholar
  10. 10.
    Plantema, F.J.: Sandwich construction; the bending and buckling of sandwich beams, plates, and shells. Wiley, London (1966)Google Scholar
  11. 11.
    Turco, E., Barcz, K., Pawlikowski, M., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Zeitschrift für angewandte Mathematik und Physik 67(5), 122 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Birsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B Eng. 43(3), 1315–1328 (2012)CrossRefGoogle Scholar
  13. 13.
    Eugster, S.R.: Augmented nonlinear beam theories. In: Geometric Continuum Mechanics and Induced Beam Theories, pp. 101–115. Springer, Berlin (2015)Google Scholar
  14. 14.
    Piccardo, G., Ferrarotti, A., Luongo, A.: Nonlinear generalized beam theory for open thin-walled members. Math. Mech. Solids 22(10), 1907–1921 (2016). 2017MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Luongo, A., Zulli, D.: Mathematical Models of Beams and Cables. Wiley, New York (2013)CrossRefGoogle Scholar
  16. 16.
    Ruta, G.C., Varano, V., Pignataro, M., Rizzi, N.L.: A beam model for the flexural-torsional buckling of thin-walled members with some applications. Thin-Walled Struct. 46(7), 816–822 (2008)CrossRefGoogle Scholar
  17. 17.
    Hamdouni, A., Millet, O.: An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section. Int. J. Nonlin. Mech. 41(3), 396–416 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grillet, L., Hamdouni, A., Millet, O.: An asymptotic non-linear model for thin-walled rods. Comptes Rendus Mécanique 332(2), 123–128 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Grillet, L., Hamdouni, A., Millet, O.: Justification of the kinematic assumptions for thin-walled rods with shallow profile. Comptes Rendus Mécanique 333(6), 493–498 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hamdouni, A., Millet, O.: An asymptotic linear thin-walled rod model coupling twist and bending. Int. Appl. Mech. 46(9), 1072–1092 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2016)CrossRefGoogle Scholar
  22. 22.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils, Paris (1909)zbMATHGoogle Scholar
  23. 23.
    Forest, S.: Mechanics of Cosserat Media—An Introduction, pp. 1–20. Ecole des Mines de Paris, Paris (2005)Google Scholar
  24. 24.
    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010)CrossRefzbMATHGoogle Scholar
  25. 25.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Found. Micropolar Mech. Springer, New York (2012)Google Scholar
  26. 26.
    Riey, G., Tomassetti, G.: A variational model for linearly elastic micropolar plate-like bodies. J. Convex Anal. 15(4), 677–691 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Riey, G., Tomassetti, G.: Micropolar linearly elastic rods. Commun. Appl. Anal. 13(4), 647–658 (2009)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kannan, R., Krueger, C.K.: Advanced Analysis: On the Real Line. Springer, New York (2012)zbMATHGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)zbMATHGoogle Scholar
  30. 30.
    Cupini, G., Guidorzi, M., Marcelli, C.: Necessary conditions and non-existence results for autonomous nonconvex variational problems. J. Differ. Equ. 243(2), 329–348 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. Springer, New York (2007)zbMATHGoogle Scholar
  32. 32.
    Della Corte, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations. M3AS (2017), (2016)
  33. 33.
    Pipkin, A.C.: Some developments in the theory of inextensible networks. Q. Appl. Math. 38(3), 343–355 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Steigmann, D.J., Pipkin, A.C.: Equilibrium of elastic nets. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 335(1639), 419–454 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. In: Proceedings of the Royal Society of London A, vol. 472, no. 2185, p. 20150790. The Royal Society (2016)Google Scholar
  36. 36.
    Ferretti, M., D’Annibale, F., Luongo, A.: Flexural-torsional flutter and buckling of braced foil beams under a follower force. Math. Prob. Eng. (2017).
  37. 37.
    Luongo, A., D’Annibale, F.: Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. Int. J. Nonlin. Mech. 55, 128–139 (2013)CrossRefGoogle Scholar
  38. 38.
    Luongo, A., D’Annibale, F.: Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces. Int. J. Modern Phys. B 26(25), 1246015 (2012)CrossRefGoogle Scholar
  39. 39.
    Di Egidio, A., Luongo, A., Paolone, A.: Linear and nonlinear interactions between static and dynamic bifurcations of damped planar beams. Int. J. Nonlin. Mech. 42(1), 88–98 (2007)CrossRefzbMATHGoogle Scholar
  40. 40.
    Goriely, A., Vandiver, R., Destrade, M.: Nonlinear euler buckling. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol. 464, no. 2099, pp. 3003–3019. The Royal Society (2008)Google Scholar
  41. 41.
    Ball, J.M., Mizel, V.J.: One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. In: Analysis and Thermomechanics, pp. 285-348. Springer, Berlin, Heidelberg (1987)Google Scholar
  42. 42.
    Fertis, D.G.: Nonlinear Structural Engineering. Springer, Berlin, Heidelberg (2006)zbMATHGoogle Scholar
  43. 43.
    Lawrie, I.D.: Phase transitions. Contemp. Phys. 28(6), 599–601 (1987)CrossRefGoogle Scholar
  44. 44.
    De Masi, A., Presutti, E., Tsagkarogiannis, D.: Fourier law, phase transitions and the stationary Stefan problem. Arch. Ration. Mech. Anal. 201(2), 681–725 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    McBride, A.T., Javili, A., Steinmann, P., Bargmann, S.: Geometrically nonlinear continuum thermomechanics with surface energies coupled to diffusion. J. Mech. Phys. Solids 59(10), 2116–2133 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Eremeyev, V.A., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Steigmann, D.J.: Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. J. Elast. 111(1), 91–107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Steigmann, D.J.: A concise derivation of membrane theory from three-dimensional nonlinear elasticity. J. Elast. 97(1), 97–101 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Forest, S., Sievert, R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43(24), 7224–7245 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Ladevèze, P.: Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation. Springer, New York (2012)Google Scholar
  51. 51.
    Rivlin, R.S.: Networks of inextensible cords. In: Collected Papers of RS Rivlin, pp. 566–579. Springer, New York (1997)Google Scholar
  52. 52.
    Pipkin, A.C.: Plane traction problems for inextensible networks. Q. J. Mech. Appl. Math. 34(4), 415–429 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3), 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67(4), 95 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Turco, E., Rizzi, N.L.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. 77, 65–69 (2016)CrossRefGoogle Scholar
  57. 57.
    Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 103(1), 1–21 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Wave dynamics and composite mechanics for microstructured materials and metamaterials, pp. 239–258. Springer, Singapore (2017)Google Scholar
  59. 59.
    Turco, E., Golaszewski, M., Giorgio, I., Placidi, L.: Can a Hencky-type model predict the mechanical behaviour of pantographic lattices? In: Mathematical Modelling in Solid Mechanics, pp. 285–311. Springer, Singapore (2017)Google Scholar
  60. 60.
    Baker, G.L., Blackburn, J.A.: The Pendulum: A Case Study in Physics. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  61. 61.
    De Masi, A., Dirr, N., Presutti, E.: Interface instability under forced displacements. Ann. Henri Poincaré 7(3), 471–511 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Uiversité de La RochelleLa RochelleFrance
  2. 2.M&MoCS, Research CenterUniversity of L’AquilaL’AquilaItaly
  3. 3.University La SapienzaRomeItaly
  4. 4.IMATH-Université de ToulonToulonFrance

Personalised recommendations