Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 1103–1123 | Cite as

A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams

  • Ugo Andreaus
  • Mario SpagnuoloEmail author
  • Tomasz Lekszycki
  • Simon R. Eugster
Original Article


We present a finite element discrete model for pantographic lattices, based on a continuous Euler–Bernoulli beam for modeling the fibers composing the pantographic sheet. This model takes into account large displacements, rotations and deformations; the Euler–Bernoulli beam is described by using nonlinear interpolation functions, a Green–Lagrange strain for elongation and a curvature depending on elongation. On the basis of the introduced discrete model of a pantographic lattice, we perform some numerical simulations. We then compare the obtained results to an experimental BIAS extension test on a pantograph printed with polyamide PA2200. The pantographic structures involved in the numerical as well as in the experimental investigations are not proper fabrics: They are composed by just a few fibers for theoretically allowing the use of the Euler–Bernoulli beam theory in the description of the fibers. We compare the experiments to numerical simulations in which we allow the fibers to elastically slide one with respect to the other in correspondence of the interconnecting pivot. We present as result a very good agreement between the numerical simulation, based on the introduced model, and the experimental measures.


Pantographic structures Nonlinear Euler–Bernoulli beam Ritz method Finite element method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 20150790 (2016)ADSCrossRefGoogle Scholar
  2. 2.
    Bersani, A.M., Della Corte, A., Piccardo, G., Rizzi, N.L.: An explicit solution for the dynamics of a taut string of finite length carrying a traveling mass: the subsonic case. Zeitschrift für angewandte Mathematik und Physik 67(4), 108 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lazarus, A., Maurini, C., Neukirch, S.: “Stability of discretized nonlinear elastic systems,” In: Extremely Deformable Structures. Springer, pp. 1–53 (2015)Google Scholar
  4. 4.
    Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2d models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67(5), 121 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Andreaus, U., dell’Isola, F., Giorgio, I., Placidi, L., Lekszycki, T., Rizzi, N.L.: Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5), 114 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on the static response and wave propagation. Proceedings of the Estonian Academy of Sciences 64(3), 219 (2015)CrossRefGoogle Scholar
  9. 9.
    Di Carlo, A., Rizzi, N.L., Tatone, A.: One-dimensional continuum model of a modular lattice: identification of the constitutive functions for the contact and inertial actions. Meccanica 25, 168–174 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    dell’Isola, F., Della Corte, A., Giorgio, I., Scerrato, D.: Pantographic 2D sheets: Discussion of some numerical investigations and potential applications. Int. J. Non-Linear Mech. 80, 200–208 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.H., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.J.: Buckling modes in pantographic lattices. C.R. Mec. 344(7), 487–501 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    Battista, A., Rosa, L., dellErba, R., Greco, L.: “Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena,” Mathematics and Mechanics of Solids, p. 1081286516657889, (2016)Google Scholar
  17. 17.
    Della Corte, A., Battista, A., et al.: Referential description of the evolution of a 2d swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non-Linear Mech. 80, 209–220 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum. uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys. 133(2), 281–345 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in potts models in the continuum. J. Stat. Phys. 134(2), 243–306 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Forest, S., Sab, K.: Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models. Mathematics and Mechanics of Solids, p. 1081286517720844, (2017)Google Scholar
  21. 21.
    Wang, Z.-P., Poh, L.H., Dirrenberger, J., Zhu, Y., Forest, S.: Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Comput. Methods Appl. Mech. Eng. 323, 250–271 (2017)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. part i: The two-dimensional case. Comput. Methods Appl. Mech. Eng. 198(27), 2198–2208 (2009)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Auffray, N., Dirrenberger, J., Rosi, G.: A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int. J. Solids Struct. 69, 195–206 (2015)CrossRefGoogle Scholar
  25. 25.
    Eremeyev, V.A.: On equilibrium of a second-gradient fluid near edges and corner points. In: Advanced Methods of Continuum Mechanics for Materials and Structures. Springer, pp. 547–556 (2016)Google Scholar
  26. 26.
    dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power, Comptes rendus de lAcadémie des sciences. Série IIb, Mécanique, physique, astronomie, p. 7, (1995)Google Scholar
  27. 27.
    Seppecher, P., Alibert, J.-J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, vol. 319, no. 1. IOP Publishing, p. 012018 (2011)Google Scholar
  28. 28.
    dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Madeo, A., Della Corte, A., Greco, L., Neff, P.: Wave propagation in pantographic 2D lattices with internal discontinuities. Proc. Estonian Acad. Sci. 64(3S), 325–330 (2015)CrossRefzbMATHGoogle Scholar
  30. 30.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  31. 31.
    Le, T.N., Battini, J.-M., Hjiaj, M.: Co-rotational dynamicformulation for 2D beams. In: COMPDYN 2011 3rdInternational Conference on Computational Methods in Structural Dynamics & Earthquake Engineering, Corfu, Greece (2011)Google Scholar
  32. 32.
    Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J.: Concepts and Applications of Finite Element Analysis, vol. 4. Wiley, New York (1974)Google Scholar
  33. 33.
    Baldacci, R.: Scienza delle Costruzioni, vol. I. UTET, Torino (1997)Google Scholar
  34. 34.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Turco, E., Barcz, K., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Ppart II: comparison with experimental evidence. Zeitschrift für angewandte Mathematik und Physik 67(5), 123 (2016)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRefGoogle Scholar
  37. 37.
    Cuomo, M., dell’Isola, F., Greco, L.: Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(3), 61 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.: First versus second gradient energies for planar sheets with two families of inextensible fibres: Investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. B Eng. 115, 423–448 (2017)CrossRefGoogle Scholar
  39. 39.
    Turco, E., Golaszewski, M., Cazzani, A., Rizzi, N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete lagrangian model. Mech. Res. Commun. 76, 51–56 (2016)CrossRefGoogle Scholar
  40. 40.
    Marigo, J.-J., Maurini, C., Pham, K.: An overview of the modelling of fracture by gradient damage models. Meccanica 51(12), 3107–3128 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Challamel, N., Lanos, C., Casandjian, C.: Strain-based anisotropic damage modelling and unilateral effects. Int. J. Mech. Sci. 47(3), 459–473 (2005)CrossRefzbMATHGoogle Scholar
  42. 42.
    Catapano, A., Desmorat, B., Vannucci, P.: Invariant formulation of phenomenological failure criteria for orthotropic sheets and optimisation of their strength. Math. Methods Appl. Sci. 35(15), 1842–1858 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Desmorat, B., Desmorat, R.: Topology optimization in damage governed low cycle fatigue. C.R. Mec. 336(5), 448–453 (2008)ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    Carpiuc-Prisacari, A., Poncelet, M., Kazymyrenko, K., Hild, F., Leclerc, H.: Comparison between experimental and numerical results of mixed-mode crack propagation in concrete: Influence of boundary conditions choice. Cem. Concr. Res. 100, 329–340 (2017)CrossRefGoogle Scholar
  45. 45.
    Doitrand, A., Fagiano, C., Hild, F., Chiaruttini, V., Mavel, A., Hirsekorn, M.: Mesoscale analysis of damage growth in woven composites. Compos. A Appl. Sci. Manuf. 96, 77–88 (2017)CrossRefGoogle Scholar
  46. 46.
    Andreaus, U., Sawczuk, A.: Deflection of elastic-plastic frames at finite spread of yielding zones. Comput. Methods Appl. Mech. Eng. 39(1), 21–35 (1983)ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    Andreaus, U., D’Asdia, P.: Displacement analysis in elastic-plastic frames at plastic collapse. Comput. Methods Appl. Mech. Eng. 42(1), 19–35 (1984)ADSCrossRefzbMATHGoogle Scholar
  48. 48.
    Andreaus, U., D’Asdia, P.: Incremental analysis of elastic-plastic frames at finite spread of yielding zones. Eng. Fract. Mech. 21(4), 827–839 (1985)CrossRefGoogle Scholar
  49. 49.
    Andreaus, U., D’Asdia, P.: An incremental procedure for deformation analysis of elastic-plastic frames. Int. J. Numer. Meth. Eng. 26(4), 769–784 (1988)CrossRefzbMATHGoogle Scholar
  50. 50.
    dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    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 (2015)CrossRefGoogle Scholar
  52. 52.
    Rahali, Y., Giorgio, I., Ganghoffer, J., dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Alibert, J.-J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für angewandte Mathematik und Physik 66(5), 2855–2870 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech. Thermodyn. 9(5), 241–257 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Forest, S., Trinh, D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 91(2), 90–109 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to gabrio piolas peridynamics and generalized continuum theories. In: Generalized continua as models for classical and advanced materials. Springer, pp. 77–128 (2016)Google Scholar
  57. 57.
    Russo, L.: The Forgotten Revolution: How Science was Born in 300 BC and Why it had to be Reborn. Springer, New York (2013)Google Scholar
  58. 58.
    dell’Isola, F., Bucci, S., Battista, A.: Against the Fragmentation of Knowledge: The Power of Multidisciplinary Research for the Design of Metamaterials. In: Advanced Methods of Continuum Mechanics for Materials and Structures. Springer, pp. 523–545 (2016)Google Scholar
  59. 59.
    Eugster, S.R., dell’Isola, F.: Exegesis of the Introduction and Sect. I from Fundamentals of the Mechanics of Continua** by E. Hellinger. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 97(4), 477–506 (2017)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Eugster, S.R., dell’Isola, F.: Exegesis of Sect. II and III. A from Fundamentals of the Mechanics of Continua** by E. Hellinger. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, (2017)Google Scholar
  61. 61.
    Franciosi, P., El Omri, A.: Effective properties of fiber and platelet systems and related phase arrangements in n-phase heterogenous media. Mech. Res. Commun. 38(1), 38–44 (2011)CrossRefzbMATHGoogle Scholar
  62. 62.
    Franciosi, P.: Laminate system schemes for effective property estimates of architectured composites with co-(dis) continuous phases. Mech. Res. Commun. 45, 70–76 (2012)CrossRefGoogle Scholar
  63. 63.
    Franciosi, P.: The boundary-due terms in the green operator of inclusion patterns from distant to contact and to connected situations using radon transforms: Illustration for spheroid alignments in isotropic media. Int. J. Solids Struct. 47(2), 304–319 (2010)CrossRefzbMATHGoogle Scholar
  64. 64.
    Franciosi, P., Barboura, S., Charles, Y.: Analytical mean green operators/eshelby tensors for patterns of coaxial finite long or flat cylinders in isotropic matrices. Int. J. Solids Struct. 66, 1–19 (2015)CrossRefGoogle Scholar
  65. 65.
    Franciosi, P.: Mean and axial green and eshelby tensors for an inclusion with finite cylindrical 3d shape. Mech. Res. Commun. 59, 26–36 (2014)CrossRefGoogle Scholar
  66. 66.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21(5), 562–577 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Cazzani, A., Malagù, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21(2), 182–209 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    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)ADSCrossRefzbMATHGoogle Scholar
  69. 69.
    Eremeyev, V., Altenbach, H., Morozov, N.: The influence of surface tension on the effective stiffness of nanosize plates. In: Doklady Physics, vol. 54, no. 2. Springer, pp. 98–100 (2009)Google Scholar
  70. 70.
    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Generalized Continua from the Theory to Engineering Applications. Springer, pp. 179–248 (2013)Google Scholar
  71. 71.
    Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 96(11), 1268–1279 (2016)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Goriely, A., Tabor, M.: Nonlinear dynamics of filaments II. Nonlinear analysis. Physica D 105(1–3), 45–61 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ugo Andreaus
    • 1
  • Mario Spagnuolo
    • 2
    • 3
    Email author
  • Tomasz Lekszycki
    • 3
    • 4
    • 5
  • Simon R. Eugster
    • 6
  1. 1.Università di Roma “La Sapienza”RomeItaly
  2. 2.CNRS, LSPM UPR3407, Université Paris 13VilletaneuseFrance
  3. 3.International Research Center M&MoCSUniversità degli Studi dell’AquilaL’AquilaItaly
  4. 4.Institute of Mechanics and PrintingWarsaw University of TechnologyWarsawPoland
  5. 5.Department of Experimental Physiology and PathophysiologyMedical University of WarsawWarsawPoland
  6. 6.Institute for Nonlinear Mechanics (INM)Universität StuttgartStuttgartGermany

Personalised recommendations