Skip to main content
SpringerLink
Log in

Dynamics of 1D nonlinear pantographic continua

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper a mechanical system consisting of a chain of masses connected by nonlinear springs and a pantographic microstructure is studied. A homogenized form of the energy is justified through a standard passage from finite differences involving the characteristic length to partial derivatives. The corresponding continuous motion equation, which is a nonlinear fourth-order PDE, is investigated. Traveling wave solutions are imposed and quasi-soliton solutions are found and numerically compared with the motion of the system resulting from a generic perturbation.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. It is straightforward to see that the quantity \((u_{i+1}-u_i)-(u_i-u_{i-1})=u_{i+1}-2u_i+u_{i-1}\) is always zero if the beams undergo no deformation.

  2. It is easy to provide an (informal) justification for the passage from the discrete to the continuous case. Indeed, setting \(\varDelta _i[u]=u_{i+1}(t)-u_i(t)\), one has:

    $$\begin{aligned}&-\kappa _3 (\varDelta _i[u]^3-\varDelta _{i-1}[u]^3) = -\kappa _3 (\varDelta _i[u]-\varDelta _{i-1}[u])\\&\quad \times (\varDelta _i[u]^2+\varDelta _{i-1}[u]^2+\varDelta _i[u]\varDelta _{i-1}[u]) \end{aligned}$$

    and then rearranging the terms, in the limit for \(\ell \) going to zero one gets

    $$\begin{aligned}&-\kappa _3 \ell ^3 \left( \frac{\varDelta _i[u]}{\ell ^2}-\frac{\varDelta _{i-1}[u]}{\ell ^2}\right) \\&\quad \times \left( \frac{\varDelta _i[u]^2}{\ell ^2}+\frac{\varDelta _{i-1}[u]^2}{\ell ^2} +\frac{\varDelta _i[u]\varDelta _{i-1}[u]}{\ell ^2}\right) \\&\quad \ell \xrightarrow [\ell \rightarrow 0]{} - 3 k_3 u_{_{XX}} (u_{_X})^2 \text {d}X\end{aligned}$$

  3. A pair L,M with such a property is called the Lax pair for a given PDE.

References

  1. 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 

  2. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  4. Pietraszkiewicz, W., Eremeyev, V.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3), 774–787 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Altenbach, H., Eremeyev, V.: On the linear theory of micropolar plates. ZAMM - Z. Angew. Math. Mech. 89(4), 242–256 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Altenbach, H., Eremeyev, V.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM - Z. Angew. Math. Mech. 88(5), 332–341 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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

  8. Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Band gaps in the relaxed linear micromorphic continuum. ZAMM - Z. Angew. Math. Mech. 95(9), 880–887 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

  11. Aminpour, H., Rizzi, N.: “On the modelling of carbon nano tubes as generalized continua”. In: Generalized Continua as Models for Classical and Advanced Materials, vol. 42, pp. 15–35. Springer (2016)

  12. 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, p. 012018. IOP Publishing (2011)

  13. 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 

  14. Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218, 1239–1262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Alibert, J.-J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–2870 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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. Z. Angew. Math. Phys. 66(6), 3473–3498 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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. Proc. R. Soc. A 472(2185), 20150790 (2016)

    Article  Google Scholar 

  18. Scerrato, D., Zurba 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 - Z. Angew. Math. Mech. 96(11), 1268–1279 (2016). doi:10.1002/zamm201600066

  19. Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys.-ZAMP (2016). doi:10.1007/s00033-016-0650-2

  20. Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: “Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models.”. Z. Angew. Math. Phys. doi:10.1007/s00033-016-0681-8 (2016)

  21. Lee, C.-C., Lee, C.-T., Liu, J.-L., Huang, W.-Y.: Quasi-solitons of the two-mode Korteweg-de Vries equation. Eur. Phys. J. Appl. Phys. 52(1), 11301 (2010)

    Article  Google Scholar 

  22. Munteanu, L., Donescu, S.: Introduction to Soliton Theory: Applications to Mechanics. Springer, Berlin (2006)

    MATH  Google Scholar 

  23. Kudriavtsev, E.M., Abramova, K.B., Scherbakov, I.P. “Soliton-type waves of reflection and conduction in metals at static loading as a possible tool of precatastrophic damage indications”. In: Boulder Damage, pp. 167–172. International Society for Optics and Photonics (2002)

  24. 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

    Google Scholar 

  25. 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 

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

  27. 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 

  28. 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 

  29. Taig, G., Ranzi, G., D’Annibale, F.: An unconstrained dynamic approach for the generalised beam theory. Cont. Mech. Thermodyn. 27(4–5), 879–904 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ruta, G., Varano, V., Pignataro, M., Rizzi, N.: A beam model for the flexural-torsional buckling of thin-walled members with some applications. Thin-Walled Struct. 46(7), 816–822 (2008)

    Article  Google Scholar 

  31. Greco, L., Cuomo, M.: Consistent tangent operator for an exact Kirchhoff rod model. Cont. Mech. Thermodyn. 27(4–5), 861–877 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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 

  33. 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 

  34. Cazzani A., Stochino F., Turco E. (2016) An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. ZAMM-Z. Angew. Math. Mech. doi:10.1002/zamm.201500280

  35. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Cont. Mech. Thermodyn. 28(1–2), 139–156 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Cuomo, M. Greco, L.: Isogeometric analysis of space rods: considerations on stress locking. In: Computational Methods in Applied Sciences and Engineering, pp. 5094–5112. (2012)

  37. Greco, L., Cuomo, M.: Multi-patch isogeometric analysis of space rods. In YIC2012-ECCOMAS Young Investigators Conference, pp 24–27. (2012)

  38. Greco, L., Cuomo, M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  MATH  Google Scholar 

  39. 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  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Pignataro, M., Ruta, G., Rizzi, N., Varano, V.: “Effects of warping constraints and lateral restraint on the buckling of thin-walled frames”. In: ASME 2009 International Mechanical Engineering Congress and Exposition, pp. 803–810. American Society of Mechanical Engineers (2009)

  42. Gabriele, S., Rizzi, N., Varano, V.: On the imperfection sensitivity of thin-walled frames. In: Topping, B.H.V. (ed.) Proceedings of the 11th International Conference on Computational Structures Technology. Civil-Comp Press, Stirlingshire, UK, Paper 15 (2012). doi:10.4203/ccp.99.15

  43. Amin Pour, H., Rizzi, N., 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. Civil-Comp Press, Stirlingshire, UK, Paper 155 (2014). doi:10.4203/ccp.106.155

  44. Rizzi, N., Varano, V.: On the postbuckling analysis of thin-walled frames. In: Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press (2011)

  45. 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 

  46. Rizzi, N., Varano, V.: The effects of warping on the postbuckling behaviour of thin-walled structures. Thin-Walled Struct. 49(9), 1091–1097 (2011)

    Article  Google Scholar 

  47. Pignataro, M., Rizzi, N., Ruta, G., Varano, V.: The effects of warping constraints on the buckling of thin-walled structures. J. Mech. Mater. Struct. 4(10), 1711–1727 (2010)

    Article  Google Scholar 

  48. Luongo, A., D’Annibale, F.: Linear stability analysis of multiparameter dynamical systems via a numerical-perturbation approach. AIAA J. 49(9), 2047–2056 (2011)

    Article  Google Scholar 

  49. Luongo, A., D’Annibale, F.: Bifurcation analysis of damped visco-elastic planar beams under simultaneous gravitational and follower forces. Int. J. Mod. Phys. B 26(25), 1246015 (2012)

    Article  Google Scholar 

  50. Calogero, F., Degasperis, A.: Spectral Transform and Solitons. Elsevier, Amsterdam (2011)

    MATH  Google Scholar 

  51. Zakharov, V.E.: On stochastization of one-dimensional chains of nonlinear oscillators. Sov. Phys.-JETP 38, 108–110 (1974)

    Google Scholar 

  52. Placidi, L.: A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Cont. Mech. Thermodyn. 28(1), 119–137 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  53. Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Cont. Mech. Thermodyn. 27(4), 623–638 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yang, Y., Ching, W.Y., Misra, A.: Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. J. Nanomech. Micromech. 1(2), 60–71 (2011)

    Article  Google Scholar 

  55. Cuomo, M., Nicolosi, A.: A poroplastic model for hygro-chemo-mechanical damage of concrete. In: EURO-C; Computational Modelling of Concrete Structures Conference, EURO-C; Computational Modelling of Concrete Structures

  56. Contrafatto, L., Cuomo, M., Fazio, F.: An enriched finite element for crack opening and rebar slip in reinforced concrete members. Int. J. Fract. 178(1–2), 33–50 (2012)

    Article  Google Scholar 

  57. 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)

    Article  MathSciNet  Google Scholar 

  58. Contrafatto, L., Cuomo, M., Gazzo, S.: A concrete homogenisation technique at meso-scale level accounting for damaging behaviour of cement paste and aggregates. Comput. Struct. 173, 1–18 (2016)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Università di Roma La Sapienza, Rome, Italy

    Ivan Giorgio & Francesco dell’Isola

  2. International Research Center for the Mathematics and Mechanics of Complex Systems - MeMoCS, Università dell’Aquila, Cisterna di Latina, Italy

    Ivan Giorgio, Alessandro Della Corte & Francesco dell’Isola

  3. Department of Mechanical and Aerospace Engineering, Università di Roma La Sapienza, Rome, Italy

    Alessandro Della Corte

Authors
  1. Ivan Giorgio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Della Corte
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesco dell’Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Giorgio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Giorgio, I., Della Corte, A. & dell’Isola, F. Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn 88, 21–31 (2017). https://doi.org/10.1007/s11071-016-3228-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3228-9

Keywords

Search

Navigation