Skip to main content
SpringerLink
Log in

Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In dell’Isola et al. (Zeitschrift für Angewandte Math und Physik 66(6):3473–3498, 2015, Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016), the concept of pantographic sheet is proposed. The aim is to design a metamaterial showing: (i) a large range of elastic response; (ii) an extreme toughness in extensional deformation; (iii) a convenient ratio between toughness and weight. However, these required properties must coexist with non-detrimental mechanical characteristics in the presence of other kinds of imposed displacements. The aim of this paper is to prove via numerical simulations that pantographic sheets may effectively resist to coupled bending and extensional deformations. The four-parameter model introduced shows its versatility as it is able to encompass all the considered types of (large) deformations. The numerical integration scheme which we use is based on the same concepts exploited in Turco et al. (Zeitschrift für Angewandte Math und Physik 67(4):1–28, 2016): They prove that the Hencky-type discretization is very efficient also in nonlinear large deformations and large displacements regimes. In Part II of this paper, we will show that the used models are very effective to describe experimental evidence.

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., 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 Math. und Physik 66(6), 3473–3498 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 1–23 (2016)

  3. 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 Math. und Physik 67(4), 1–28 (2016)

    MathSciNet  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Seppecher, P., Alibert, J.-J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, volume 319(1), p. 012018. IOP Publishing, (2011)

  6. Alibert, J.-J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für Angewandte Math. und Physik 66, 2855–2870 (2015)

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Turco, E., Barcz, K., Rizzi, N.L.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence. Zeitschrift für Angewandte Math. und Physik, 1–16 (2016). doi:10.1007/s00033-016-0714-3

  10. 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 Math. und Physik (ZAMP) 66(6), 3699–3725 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Giorgio I., Grygoruk R., dell’Isola F., Steigmann D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)

    Article  Google Scholar 

  13. Scerrato D., Giorgio I., Rizzi N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für Angewandte Math. und Physik 67(3), 1–19 (2016)

    MathSciNet  MATH  Google Scholar 

  14. 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. Zeitschrift für Angewandte Math. und Mechanik (2016). doi:10.1002/zamm.201600066

  15. D’Agostino M.V., Giorgio I., Greco L., Madeo A., Boisse P.: Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements. Int. J. Solids Struct. 59, 1–17 (2015)

    Article  Google Scholar 

  16. Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model. J. Eng. Math. (2016). doi:10.1007/s10665-016-9856-8

  17. Giorgio, I.: Numerical identification procedure between a micro Cauchy model and a macrosecond gradient model for planar pantographic structures. Zeitschrift für Angewandte Math. und Mechanik (2016). doi:10.1007/s00033-016-0692-5

  18. Placidi, L., Dhaba, A.E.: Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids (2015). doi:10.1177/1081286515616043

  19. dell’Isola F., Giorgio I., Andreaus U.: Elastic pantographic 2d lattices: A numerical analysis on static response and wave propagation. Proc. Estonian Acad. Sci. 64(3), 219–225 (2015)

    Article  Google Scholar 

  20. 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 

  21. 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 Math. und Physik 67(3), 1–23 (2016)

    MathSciNet  MATH  Google Scholar 

  22. 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. (2016). doi:10.1007/s10665-016-9865-7

  23. Andreaus U., Baragatti P., Placidi L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25(3), 375–398 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Andreaus U., Baragatti P., Placidi L.: Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation. Int. J. Non-Linear Mech. 80, 96–106 (2016)

    Article  Google Scholar 

  25. 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 

  26. Cazzani A., Malagù M., Turco E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1), 139–156 (2016)

    Article  MathSciNet  Google Scholar 

  27. 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 

  28. Bilotta A., Formica G., Turco E.: Performance of a high–continuity finite element in three–dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26, 1155–1175 (2010)

    Article  MATH  Google Scholar 

  29. Greco L., Cuomo M.: B-Spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)

    Article  MathSciNet  Google Scholar 

  30. Greco L., Cuomo M.: An implicit G 1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Greco L., Cuomo M.: An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)

    Article  MathSciNet  Google Scholar 

  32. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams. Zeitschrift für Angewandte Math. und Mechanik (2016). doi:10.1002/zamm.201500280

  33. Piccardo G., Ranzi G., Luongo A.: A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Math. Mech. Solids 19(8), 900–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Piccardo G., Ranzi G., Luongo A.: A direct approach for the evaluation of the conventional modes within the gbt formulation. Thin-Walled Struct. 74, 133–145 (2014)

    Article  Google Scholar 

  35. Steigmann, D.J., Faulkner, M.G.: Variational theory for spatial rods. J. Elast. 33(1), 1–26 (1993)

  36. Misra A., Poorsolhjouy P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1), 215–234 (2016)

    Article  MathSciNet  Google Scholar 

  37. 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  MATH  Google Scholar 

  38. Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mechan. Solids (2015). (doi:10.1177/1081286515582862)

  39. Dos Reis F., Ganghoffer J.F.: Construction of micropolar continua from the asymptotic homogenization of beam lattices. Comput. Struct. 112, 354–363 (2012)

    Article  Google Scholar 

  40. Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)

  41. Braides A., Solci M.: Interfacial energies on penrose lattices. Math. Models Methods Appl. Sci. 21(5), 1193–1210 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tomic A., Grillo A., Federico S.: Poroelastic materials reinforced by statistically oriented fibres - numerical implementation and application to articular cartilage. IMA J. Appl. Math. 79, 1027–1059 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Caggegi C., Pensée V., Fagone M., Cuomo M., Chevalier L.: Experimental global analysis of the efficiency of carbon fiber anchors applied over cfrp strengthened bricks. Constr. Build. Mater. 53, 203–212 (2014)

    Article  Google Scholar 

  44. Tedesco, F., Bilotta, A., Turco, E.: Multiscale 3D mixed FEM analysis of historical masonry constructions. Eur. J. Environ. Civil Eng. (2016). doi:10.1080/19648189.2015.1134676

  45. Turco, E., Rizzi, N.L.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. (submitted) (2016)

  46. Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Environ. Civil Eng. (2016). doi:10.1080/19648189.2015.1134673

  47. Lekszycki, T., Olhoff, N., Pedersen, J.J.: Modelling and identification of viscoelastic properties of vibrating sandwich beams. Compos. Struct. 22(1), 51–31 (1992)

  48. Bilotta A., Turco E.: A numerical study on the solution of the Cauchy problem in elasticity. Int. J. Solids Struct. 46, 4451–4477 (2009)

    Article  MATH  Google Scholar 

  49. Bilotta A., Morassi A., Turco E.: Reconstructing blockages in a symmetric duct via quasi-isospectral horn operators. J. Sound Vib. 366, 149–172 (2016)

    Article  Google Scholar 

  50. Bilotta, A., Turco, E.: Numerical sensitivity analysis of corrosion detection. Math. Mech. Solids (2014). doi:10.1177/1081286514560093

  51. Alessandrini G., Bilotta A., Formica G., Morassi A., Rosset E., Turco E.: Evaluating the volume of a hidden inclusion in an elastic body. J. Comput. Appl. Math. 198(2), 288–306 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Alessandrini G., Bilotta A., Morassi A., Turco E.: Computing volume bounds of inclusions by EIT measurements. J. Sci. Comput. 33(3), 293–312 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Turco E.: Identification of axial forces on statically indeterminate pin–jointed trusses by a nondestructive mechanical test. Open Civil Eng. J. 7, 50–57 (2013)

    Article  Google Scholar 

  54. Buffa, F., Cazzani, A., Causin, A., Poppi, S., Sanna, G.M., Solci, M., Stochino, F., Turco, E.: The Sardinia Radio Telescope: a comparison between close range photogrammetry and FE models. Math. Mech. Solids (2015) doi:10.1177/1081286515616227

  55. Stochino, F., Cazzani, A., Poppi, S., Turco, E.: Sardinia Radio Telescope finite element model updating by means of photogrammetric measurements. Math. Mech. Solids (2015) doi:10.1177/1081286515616046

  56. 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)

    Article  MathSciNet  Google Scholar 

  57. Cazzani A., Stochino F., Turco E.: On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Zeitschrift für Angewandte Math. und Physik 67(24), 1–30 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Cazzani A., Stochino F., Turco E.: On the whole spectrum of Timoshenko beams. Part II: further applications. Zeitschrift für Angewandte Math. und Physik 67(25), 1–21 (2016)

    MathSciNet  MATH  Google Scholar 

  59. Turco E., dell’Isola F., Rizzi N.L., Grygoruk R., Müller W. N., Liebold C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016)

    Article  Google Scholar 

  60. D’Annibale F., Rosi G., Luongo A.: Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces. Meccanica 50(3), 825–839 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  61. Gabriele, S., Rizzi, N.L., Varano, V.(2012) On the imperfection sensitivity of thin-walled frames. In: Topping, B.H.V. (ed.) Proceedings of the Eleventh International Conference on Computational Structures Technology, vol 99. doi:10.4203/ccp.99.15, Stirlingshire, UK, 2012. Civil-Comp Press.

  62. Pignataro M., Ruta G., Rizzi N.L., Varano V.: Effects of warping constraints and lateral restraint on the buckling of thin-walled frames. ASME Int. Mech. Eng. Congr. Expo. 10, 803–810 (2010)

    Google Scholar 

  63. Rizzi N., Varano V., Gabriele S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin-Walled Struct. 68, 124–134 (2013)

    Article  Google Scholar 

  64. Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21(6), 737–746 (2016)

  65. AminPour H., Rizzi N.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 21(2), 168–181 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  66. Gabriele S., Rizzi N.L., Varano V.: A 1D nonlinear TWB model accounting for in plane cross-section deformation. Int. J. Solids Struct. 94–95, 170–178 (2016)

    Article  Google Scholar 

  67. AminPour, H., Rizzi, N.L.: On the continuum modelling of carbon nano tubes. In: Kruis, J., Tsompanakis, Y., Topping, B.H.V. (eds.) Proceedings of the 15th International Conference on Civil, Structural and Environmental Engineering Computing, vol. 108. Civil-Comp Press, Stirlingshire (2015)

  68. Gabriele, S., Rizzi, N.L., Varano, V.: A one-dimensional nonlinear thin walled beam model derived from Koiter shell theory. In: Topping, B.H.V., Iványi, P. (eds.) Proceedings of the 12th International Conference on Computational Structures Technology. Civil-Comp Press, Stirlingshire (2014)

  69. AminPour, H., Rizzi, N.L., Salerno, G.: A one-dimensional beam model for single-wall carbon nano tube column buckling. In: Topping, B.H.V., Iványi, P. (eds.) Proceedings of the 12th International Conference on Computational Structures Technology, vol. 106. Civil-Comp Press, Stirlingshire (2014)

  70. Rizzi, N.L., Varano, V.: On the postbuckling analysis of thin-walled frames. In: Topping, B.H.V., Tsompanakis, Y. (eds.) Proceedings of the 13th International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press, Stirlingshire (2011)

Download references

Author information

Authors and Affiliations

  1. Department of Architecture, Design and Urban planning (DADU), University of Sassari, Alghero, Sassari, Italy

    Emilio Turco

  2. Institute of Mechanics and Printing, Warsaw University of Technology, Warsaw, Poland

    Katarzyna Barcz & Marek Pawlikowski

  3. Department of Architecture, Roma Tre University, Rome, Italy

    Nicola Luigi Rizzi

Authors
  1. Emilio Turco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katarzyna Barcz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marek Pawlikowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nicola Luigi Rizzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Turco.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Turco, E., Barcz, K., Pawlikowski, M. et al. Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Z. Angew. Math. Phys. 67, 122 (2016). https://doi.org/10.1007/s00033-016-0713-4

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-016-0713-4

Mathematics Subject Classification

Keywords

Search

Navigation