Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Lazarus, A., Maurini, C., Neukirch, S.: “Stability of discretized nonlinear elastic systems,” In: Extremely Deformable Structures. Springer, pp. 1–53 (2015)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

  28. dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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. Baldacci, R.: Scienza delle Costruzioni, vol. I. UTET, Torino (1997)

    Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Challamel, N., Lanos, C., Casandjian, C.: Strain-based anisotropic damage modelling and unilateral effects. Int. J. Mech. Sci. 47(3), 459–473 (2005)

    Article  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Desmorat, B., Desmorat, R.: Topology optimization in damage governed low cycle fatigue. C.R. Mec. 336(5), 448–453 (2008)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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. 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)

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  66. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21(5), 562–577 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  Google Scholar 

  72. Goriely, A., Tabor, M.: Nonlinear dynamics of filaments II. Nonlinear analysis. Physica D 105(1–3), 45–61 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università di Roma “La Sapienza”, Rome, Italy

    Ugo Andreaus

  2. CNRS, LSPM UPR3407, Université Paris 13, Sorbonne Paris Cité, 93430, Villetaneuse, France

    Mario Spagnuolo

  3. International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100, L’Aquila, Italy

    Mario Spagnuolo & Tomasz Lekszycki

  4. Institute of Mechanics and Printing, Warsaw University of Technology, 85 Narbutta Street, 02-524, Warsaw, Poland

    Tomasz Lekszycki

  5. Department of Experimental Physiology and Pathophysiology, Medical University of Warsaw, 1b Banacha Street, 02-097, Warsaw, Poland

    Tomasz Lekszycki

  6. Institute for Nonlinear Mechanics (INM), Universität Stuttgart, Stuttgart, Germany

    Simon R. Eugster

Authors
  1. Ugo Andreaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mario Spagnuolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tomasz Lekszycki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Simon R. Eugster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Spagnuolo.

Additional information

Communicated by Francesco dell’Isola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Andreaus, U., Spagnuolo, M., Lekszycki, T. et al. A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Continuum Mech. Thermodyn. 30, 1103–1123 (2018). https://doi.org/10.1007/s00161-018-0665-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-018-0665-3

Keywords

Search

Navigation