Skip to main content
Log in

Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models

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

Abstract

Hencky (Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Hencky-type model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model.

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

Access this article

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. Hencky, H.: Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann (1921)

  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) (2016)

  3. 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 

  4. 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 

  5. 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 

  6. Challamel N., Kocsis A., Wang C.M.: Discrete and non-local elastica. Int. J. Non-Linear Mech. 77, 128–140 (2015)

    Article  Google Scholar 

  7. Wang, C.M., Zhang, H., Gao, R.P., Duan, W.H., Challamel, N.: Hencky bar-chain model for buckling and vibration of beams with elastic end restraints. Int. J. Struct. Stab. Dyn. 15(7), 1540007 (2014)

  8. Challamel N., Zhang Z., Wang C.M.: Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams. J. Nanomech. Micromech. 5, A4014004-1–A4014004-16 (2014)

    Google Scholar 

  9. 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 

  10. 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 (2016). doi:10.1177/1081286516651324

  11. 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

  12. Boisse, P., Hamila, N., Guzman-Maldonado, E., Madeo, A., Hivet, G., dell’Isola, F.: The bias-extension test for the analysis of in-plane shear properties of textile composite reinforcements and prepregs: a review. Int. J. Mater. Form. (2016). doi:10.1007/s12289-016-1294-7

  13. Barbagallo, G., Madeo, A., Azehaf, I., Giorgio, I., Morestin, F., Boisse, P.: Bias extension test on an unbalanced woven composite reinforcement: experiments and modeling via a second-gradient continuum approach. J. Compos. Mater. (2016). doi:10.1177/0021998316643577

  14. Cao J., Akkerman R., Boisse P. et al.: Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos. Part A Appl. Sci. Manuf. 39(6), 1037–1053 (2008)

    Article  Google Scholar 

  15. Harrison P., Clifford M.J., Long A.C.: Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments. Compos. Sci. Technol. 64(10), 1453–1465 (2004)

    Article  Google Scholar 

  16. Harrison P., Abdiwi F., Guo Z., Potluri P., Yu W.R.: Characterising the shear–tension coupling and wrinkling behaviour of woven engineering fabrics. Compos. Part A Appl. Sci. Manuf. 43(6), 903–914 (2012)

    Article  Google Scholar 

  17. Harrison P.: Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos. Part A Appl. Sci. Manuf. 81, 145–157 (2016)

    Article  Google Scholar 

  18. 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 

  19. 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 

  20. Grillo A., Wittum G., Tomic A., Federico S.: Remodelling in statistically oriented fibre-reinforced materials and biological tissues. Math. Mech. Solids 20(9), 1107–1129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Federico S., Gasser T.C.: Nonlinear elasticity of biological tissues with statistical fibre orientation. J. R. Soc. Interface 7(47), 955–966 (2010)

    Article  Google Scholar 

  22. Grillo A., Federico S., Wittum G.: Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials. Int. J. Non-Linear Mech. 47(2), 388–401 (2012)

    Article  Google Scholar 

  23. Federico S., Grillo A., La Rosa G., Giaquinta G., Herzog W.: A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage. J. Biomech. 38(10), 2008–2018 (2005)

    Article  Google Scholar 

  24. Federico S., Grillo A.: Elasticity and permeability of porous fibre-reinforced materials under large deformations. Mech. Mater. 44, 58–71 (2012)

    Article  Google Scholar 

  25. Engheta N., Ziolkowski R.W.: Metamaterials: Physics and Engineering Explorations. Wiley, New York (2006)

    Book  Google Scholar 

  26. Zouhdi S., Sihvola A., Vinogradov A.P.: Metamaterials and Plasmonics: Fundamentals, Modelling, Applications. Springer, Berlin (2008)

    Google Scholar 

  27. 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 

  28. 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 

  29. Auriault J.-L., Boutin C.: Long wavelength inner-resonance cut-off frequencies in elastic composite materials. Int. J. Solids Struct. 49(23), 3269–3281 (2012)

    Article  Google Scholar 

  30. Boutin C.: Acoustics of porous media with inner resonators. J. Acoust. Soc. Am. 134(6), 4717–4729 (2013)

    Article  Google Scholar 

  31. Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids (2015). doi:10.1177/1081286515576948

  32. Placidi L., Rosi G., Giorgio I., Madeo A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math. Mech. Solids 19(5), 555–578 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rosi G., Giorgio I., Eremeyev V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. Z. Angew. Math. Mech. 93(12), 914–927 (2013)

    Article  MathSciNet  Google Scholar 

  34. Berezovski A., Giorgio I., Della Corte A.: Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math. Mech. Solids 21(1), 37–51 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Abd-alla A.N., Giorgio I., Galantucci L., Hamdan A.M., Del Vescovo D.: Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Contin. Mech. Thermodyn. 28(1–2), 67–84 (2016)

    Article  MathSciNet  Google Scholar 

  36. 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(3), 1239–1262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Madeo, A., Della Corte, A., Giorgio, I., Scerrato, D.: Modeling and designing micro-and nano-structured metamaterials: towards the application of exotic behaviors. Math. Mech. Solids (2015). doi:10.1177/1081286515616036

  38. Enakoutsa K., Della Corte A., Giorgio I.: A model for elastic flexoelectric materials including strain gradient effects. Math. Mech. Solids 21(2), 242–254 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Giorgio I., Galantucci L., Della Corte A., Del Vescovo D.: Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int. J. Appl. Electromagn. Mech. 47(4), 1051–1084 (2015)

    Article  Google Scholar 

  40. Cuomo M., dell’Isola F., Greco L.: Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres. Z. Angew. Math. Phys. 67(3), 1–23 (2016)

    Article  MathSciNet  Google Scholar 

  41. Steigmann D.J., Pipkin A.C.: Equilibrium of elastic nets. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 335(1639), 419–454 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pipkin A.C.: Some developments in the theory of inextensible networks. Q. Appl. Math. 38(3), 343–355 (1980)

    MathSciNet  MATH  Google Scholar 

  43. Pipkin A.C.: Plane traction problems for inextensible networks. Q. J. Mech. Appl. Math. 34(4), 415–429 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  44. 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: Altenbach, H., Forest, S. (eds) Generalized Continua as Models for Classical and Advanced Materials. Springer, Cham, pp. 77–128 (2016)

  45. 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 

  46. Pideri C., Seppecher P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. Camar-Eddine M., Seppecher P.: Determination of the closure of the set of elasticity functionals. Arch. Ration. Mech. Anal. 170(3), 211–245 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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 

  49. Goda I., Assidi M., Ganghoffer J.-F.: Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization. J. Mech. Phys. Solids 61(12), 2537–2565 (2013)

    Article  Google Scholar 

  50. Cecchi A., Rizzi N.: Heterogeneous elastic solids: a mixed homogenization–rigidification technique. Int. J. Solids Struct. 38(1), 29–36 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. Misra A., Poorsolhjouy P.: Granular micromechanics model of anisotropic elasticity derived from Gibbs potential. Acta Mech. 227(5), 1393–1413 (2016)

    Article  MATH  Google Scholar 

  52. Misra A., Poorsolhjouy P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Misra A., Parthasarathy R., Singh V., Spencer P.: Micro-poromechanics model of fluid-saturated chemically active fibrous media. Z. Angew. Math. Mech. 95(2), 215–234 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. dell’Isola F., Placidi L.: Variational Principles are a Powerful Tool also for Formulating Field Theories, Volume 535 of Variational Models and Methods in Solid and Fluid Mechanics CISM Courses and Lectures. Springer, Berlin (2012)

  55. Federico S., Grillo A., Imatani S., Giaquinta G., Herzog W.: An energetic approach to the analysis of anisotropic hyperelastic materials. Int. J. Eng. Sci. 46, 164–181 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Nadler B., Steigmann D. J.: A model for frictional slip in woven fabrics. Compt. Rendus Mec. 331(12), 797–804 (2003)

    Article  MATH  Google Scholar 

  57. Scerrato D., Giorgio I., Madeo A., Liman A., Darve F.: A simple non-linear model for internal friction in modified concrete. Int. J. Eng. Sci. 80, 136–152 (2014)

    Article  MathSciNet  Google Scholar 

  58. Scerrato D., Giorgio I., Della Corte A., Madeo A., Limam A.: A micro-structural model for dissipation phenomena in the concrete. Int. J. Numer. Anal. Methods Geomech. 39(18), 2037–2052 (2015)

    Article  Google Scholar 

  59. Germain P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  60. Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)

    Article  Google Scholar 

  61. dell’Isola F., Seppecher P., Madeo A.: How contact interactions may depend on the shape of cauchy cuts in n-th gradient continua: approach á la D’Alembert. Z. Angew. Math. Phys. 63(6), 1119–1141 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. 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. (2015). doi: 10.1007/s00205-015-0879-5

  63. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. Gabriele, S., Rizzi, N., Varano V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 6 (2014). doi:10.1177/1081286514536721

  66. dell’Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S.: The complete works of Gabrio Piola: volume I—Commented English translation. Adv. Struct. Mater. (2014). doi:10.1007/978-3-319-00263-7

  67. Navier, L.: Mémoire sur les lois de l’équilibre et du mouvement des corps solides élastiques. Mem. Inst. Nat. 3, 375–393 (1827)

  68. Cercignani C: Ludwig Boltzmann: The Man who Trusted Atoms. OUP Oxford, Oxford (2006)

    Book  MATH  Google Scholar 

  69. Truesdell C., Noll W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  70. Kuhn T.S.: The Structure of Scientific Revolutions. University of Chicago Press, Chicago (2012)

    Book  Google Scholar 

  71. Russo L.: Forgotten Revolution: How Science was Born in 300 BC and Why it Had to be Reborn. Springer, Berlin (2013)

    Google Scholar 

  72. de Solla Price D.: Gears from the Greek. The Antikythera mechanism: a calendar computer from ca. 80 BC. Trans. Am. Philos. Soc. 64(7), 1–70 (1974)

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

    Article  MATH  Google Scholar 

  74. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams. Z. Angew. Math. Mech. 1–25 (2016). doi:10.1002/zamm.201500280

  75. 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 

  76. 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). doi:10.1177/1081286515577043

    Article  MathSciNet  MATH  Google Scholar 

  77. 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 

  78. Cuomo M., Contraffatto 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  80. 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 

  81. 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)

  82. Andreaus U., Casini P.: Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle. Int. J. Non-Linear Mech. 37(1), 117–133 (2002)

    Article  MATH  Google Scholar 

  83. 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 

  84. Cuomo M., Ventura G.: Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation. Math. Comput. Model. 28(4), 185–204 (1998)

    Article  MATH  Google Scholar 

  85. D’Annibale F., Rosi G., Luongo A.: On the failure of the ‘similar piezoelectric control’ in preventing loss of stability by nonconservative positional forces. Z. Angew. Math. Phys. 66(4), 1949–1968 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  86. 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 

  87. Piccardo G., Pagnini L.C., Tubino F.: Some research perspectives in galloping phenomena: critical conditions and post-critical behavior. Contin. Mech. Thermodyn. 27(1–2), 261–285 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  88. Luongo A., Paolone A., Piccardo G.: Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica 33(3), 229–242 (1998)

    Article  MATH  Google Scholar 

  89. 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 

  90. AminPour, H., Rizzi, N.: A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis. Math. Mech. Solids 4, 168–181 (2015)

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

  92. Eremeyev V.A., Pietraszkiewicz W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49(14), 1993–2005 (2012)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

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., dell’Isola, F., Cazzani, A. et al. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67, 85 (2016). https://doi.org/10.1007/s00033-016-0681-8

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-016-0681-8

Mathematics Subject Classification

Keywords

Navigation