Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 33–45 | Cite as

Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation

  • Emilio BarchiesiEmail author
  • Gregor Ganzosch
  • Christian Liebold
  • Luca Placidi
  • Roman Grygoruk
  • Wolfgang H. Müller
Original Article


Due to the latest advancements in 3D printing technology and rapid prototyping techniques, the production of materials with complex geometries has become more affordable than ever. Pantographic structures, because of their attractive features, both in dynamics and statics and both in elastic and inelastic deformation regimes, deserve to be thoroughly investigated with experimental and theoretical tools. Herein, experimental results relative to displacement-controlled large deformation shear loading tests of pantographic structures are reported. In particular, five differently sized samples are analyzed up to first rupture. Results show that the deformation behavior is strongly nonlinear, and the structures are capable of undergoing large elastic deformations without reaching complete failure. Finally, a cutting edge model is validated by means of these experimental results.


Out-of-plane buckling Pantographic structures Higher gradient continua Experimental analysis 


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  1. 1.
    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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andreaus, U., Baragatti, P.: Experimental damage detection of cracked beams by using nonlinear characteristics of forced response. Mech. Syst. Signal Process. 31, 382–404 (2012)ADSGoogle Scholar
  4. 4.
    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 (2017)Google Scholar
  5. 5.
    Battista, A., Cardillo, C., Del Vescovo, D., Rizzi, N.L., Turco, E.: Frequency shifts induced by large deformations in planar pantographic continua. Nanomech. Sci. Technol.: Int. J. 6(2), 161–178 (2015)Google Scholar
  6. 6.
    Battista, A., Del Vescovo, D., Rizzi, N.L., Turco, E.: Frequency shifts in natural vibrations in pantographic metamaterials under biaxial tests. Technische Mechanik 37(1), 1–17 (2017)Google Scholar
  7. 7.
    Bertram, A., Glüge, R.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro–macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Caprino, S., Esposito, R., Marra, R., Pulvirenti, M.: Hydrodynamic limits of the vlasov equation. Commun. Partial Differ. Equ. 18(5–6), 805–820 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Hydrodynamic limit in a particle system with topological interactions. Arab. J. Math. 3(4), 381–417 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Super-hydrodynamic limit in interacting particle systems. J. Stat. Phys. 155(5), 867–887 (2014)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Chatzigeorgiou, G., Javili, A., Steinmann, P.: Multiscale modelling for composites with energetic interfaces at the micro-or nanoscale. Math. Mech. Solids 20(9), 1130–1145 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cuomo, M., Contrafatto, L., Greco, L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cuomo, M.: Forms of the dissipation function for a class of viscoplastic models. Math. Mech. Complex Syst. 5(3), 217–237 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    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)ADSMathSciNetzbMATHGoogle Scholar
  16. 16.
    De Masi, A., Olla, S.: Quasi-static hydrodynamic limits. J. Stat. Phys. 161(5), 1037–1058 (2015)ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Del Vescovo, D., Giorgio, I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. 80, 153–172 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887–928 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    dell’Isola, F., Della Corte, A., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    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)ADSGoogle Scholar
  21. 21.
    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 (2015)Google Scholar
  22. 22.
    dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation. Proc. Eston. Acad. Sci. 64(3), 219–225 (2015)Google Scholar
  23. 23.
    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), 23 (2016)Google Scholar
  24. 24.
    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, 3473–3498 (2015)ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. In: Variational Models and Methods in Solid and Fluid Mechanics, pp. 1–15. Springer (2011)Google Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    dell’Isola, F., Steigmann, D.J.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 18, 113–125 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to gabrio piola’s peridynamics and generalized continuum theories. In: Generalized continua as models for classical and advanced materials, pp. 77–128. Springer (2016)Google Scholar
  29. 29.
    Dietrich, L., Lekszycki, T., Turski, K.: Problems of identification of mechanical characteristics of viscoelastic composites. Acta Mech. 126(1), 153–167 (1998)zbMATHGoogle Scholar
  30. 30.
    Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Syst. 3(1), 43–82 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Esposito, R., Pulvirenti, M.: From particles to fluids. Handbook of mathematical fluid dynamics 3, 1–82 (2004)Google Scholar
  32. 32.
    Eugster, S.R., dell’Isola, F.: Exegesis of the introduction and Sect. I from “Fundamentals of the mechanics of continua”** by E. Hellinger. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 97(4), 477–506 (2017)ADSMathSciNetGoogle Scholar
  33. 33.
    Eugster, S.R., dell’Isola, F.: Exegesis of Sect. II and III. A from “Fundamentals of the mechanics of continua”** by E. Hellinger. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik (2017).
  34. 34.
    Ganzosch, G., dell’Isola, F., Turco, E., Lekszycki, T., Müller, W.H.: Shearing tests applied to pantographic structures. Acta Polytechnica CTU Proceedings 7, 1–6 (2016)Google Scholar
  35. 35.
    Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)Google Scholar
  36. 36.
    Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D., Steigmann, D.: Buckling modes in pantographic lattices. C.R. Mec. 344, 487–501 (2016)ADSGoogle Scholar
  37. 37.
    Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A 473(2207), 21 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Giorgio, I., Scerrato, D.: Multi-scale concrete model with rate-dependent internal friction. Eur. J. Environ. Civ. Eng. 21(7–8), 821–839 (2017)Google Scholar
  39. 39.
    Greco, L., Giorgio, I., Battista, A.: In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions. Math. Mech. Solids 22(10), 1950–1975 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Harrison, P.: Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos. A Appl. Sci. Manuf. 81, 145–157 (2016)Google Scholar
  41. 41.
    Kocsis, A., Challamel, N., Károlyi, G.: Discrete and nonlocal models of engesser and haringx elastica. Int. J. Mech. Sci. 130, 571–585 (2017)Google Scholar
  42. 42.
    Melnik, A.V., Goriely, A.: Dynamic fiber reorientation in a fiber-reinforced hyperelastic material. Math. Mech. Solids 18(6), 634–648 (2013)MathSciNetGoogle Scholar
  43. 43.
    Milton, G., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Misra, A.: Effect of asperity damage on shear behavior of single fracture. Eng. Fract. Mech. 69(17), 1997–2014 (2002)Google Scholar
  45. 45.
    Misra, A.: Mechanistic model for contact between rough surfaces. J. Eng. Mech. 123(5), 475–484 (1997)Google Scholar
  46. 46.
    Nadler, B., Steigmann, D.J.: A model for frictional slip in woven fabrics. Comptes Rendus Mec. 331(12), 797–804 (2003)ADSzbMATHGoogle Scholar
  47. 47.
    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)ADSMathSciNetzbMATHGoogle Scholar
  48. 48.
    Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte Mathematik und Physik 66(6), 3699–3725 (2015)ADSMathSciNetzbMATHGoogle Scholar
  49. 49.
    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 (2016)MathSciNetzbMATHGoogle Scholar
  50. 50.
    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)ADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Russo, L.: The Forgotten Revolution: How Science was Born in 300 BC and Why It had to be Reborn. Springer, Berlin (2013)Google Scholar
  52. 52.
    Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer, Berlin (2009)zbMATHGoogle Scholar
  53. 53.
    Scerrato, D., Giorgio, I., Rizzi, N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3), 1–19 (2016)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mechanics Research Communications (2017)Google Scholar
  55. 55.
    Steigmann, D.J., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin. 31(3), 373–382 (2015)ADSMathSciNetzbMATHGoogle Scholar
  56. 56.
    Stigler, S.M.: Stigler’s law of eponymy. Transactions of the New York Academy of Sciences 39(1 Series II), 147–157 (1980)Google Scholar
  57. 57.
    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, 28 (2016)MathSciNetzbMATHGoogle Scholar
  58. 58.
    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)Google Scholar
  59. 59.
    Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. B Eng. 118, 1–14 (2017)Google Scholar
  60. 60.
    Victor, A.: Eremeyev and Wojciech Pietraszkiewicz. Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)MathSciNetGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Università degli studi di Roma “La Sapienza”RomeItaly
  2. 2.Faculty of MechanicsBerlin University of TechnologyBerlinGermany
  3. 3.International Telematic University UninettunoRomeItaly
  4. 4.Institute of Mechanics and PrintingWarsaw University of TechnologyWarsawPoland

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