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Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions

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Abstract

3D printing technology has opened application perspectives which were difficult to imagine only few years ago. In this paper, we show how it is possible to design and print some microstructures in which relative displacements are allowed at micro-level. Some structural elements have been built in the aforementioned structures which can be confidently modeled as perfect pivots or as soft elastic connections. The obtained specimens can be regarded as constituted by very exotic materials, as forecast theoretically. We numerically study the behavior of pantographic structures including soft or nearly perfect pivots in large deformations, and we experimentally observe an enlarged elastic range and peculiar buckling mechanisms. The presented results are extremely promising: We consider now as proven that higher gradient metamaterials can be realized by using microstructures having micro-characteristic lengths of the order on tenth of millimeters.

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References

  1. Abali, B., Müller, W., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87, 1–16 (2017)

    Article  Google Scholar 

  2. Alibert, 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 

  3. Alibert, J.J., Della Corte, A., Giorgio, I., Battista, A.: Extensional elastica in large deformation as\(\backslash \)gamma-limit of a discrete 1D mechanical system. Z. Angew. Math. Phys. 68(2), 42 (2017)

    Article  MathSciNet  MATH  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Sumbatyan, M.A. (ed.) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, pp. 239–258. Springer, Berlin (2017)

    Chapter  Google Scholar 

  8. Berrehili, Y., Marigo, J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181–207 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bertram, A., Rainer, G.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bevill, G., Eswaran, S.K., Gupta, A., Papadopoulos, P., Keaveny, T.M.: Influence of bone volume fraction and architecture on computed large-deformation failure mechanisms in human trabecular bone. Bone 39(6), 1218–1225 (2006)

    Article  Google Scholar 

  11. Boutin, C., Giorgio, I., Placidi, L., et al.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caprino, S., Esposito, R., Marra, R., Pulvirenti, M.: Hydrodynamic limits of the Vlasov equation. Commun. Partial Differ. Equ. 18(5), 805–820 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogeneous 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 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Super-hydrodynamic limit in interacting particle systems. J. Stat. Phys. 155(5), 867–887 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. 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. Part B Eng. 115, 423–448 (2016)

    Article  Google Scholar 

  17. 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 

  18. De Masi, A., Olla, S.: Quasi-static hydrodynamic limits. J. Stat. Phys. 161(5), 1037–1058 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. 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, 852–872 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  22. 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, 219–225 (2015)

    Article  Google Scholar 

  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, 20150790 (2016)

    Article  ADS  Google Scholar 

  24. Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory. Z. Angew. Math. Phys. 65(3), 587–612 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21, 737–746 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ganghoffer, J.: Spatial and material stress tensors in continuum mechanics of growing solid bodies. Math. Mech. Complex Syst. 3(4), 341–363 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Giorgio, I.: Numerical identification procedure between a micro-cauchy model and a macro-second gradient model for planar pantographic structures. Z. Angew. Math. Phys. 67(4), 95 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)

    Article  Google Scholar 

  29. Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.: Buckling modes in pantographic lattices. C. R. Mec. 344, 487–501 (2016)

    Article  ADS  Google Scholar 

  30. Dell Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471, 20150415 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Javili, A., McBride, A., Mergheim, J., Steinmann, P., Schmidt, U.: Micro-to-macro transitions for continua with surface structure at the microscale. Int. J. Solids Struct. 50(16), 2561–2572 (2013)

    Article  Google Scholar 

  32. Li, J., Slesarenko, V., Galich, P., Rudykh, S.: Oblique shear wave propagation in finitely deformed layered composites. Mech. Res. Commun. 87, 21–28 (2018)

    Article  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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 

  35. Nadler, B., Papadopoulos, P., Steigmann, D.: Multiscale constitutive modeling and numerical simulation of fabric material. Int. J. Solids Struct. 43(2), 206–221 (2006)

    Article  MATH  Google Scholar 

  36. Pideri, C., Seppecher, P.: Asymptotics of a non-planar rod in non-linear elasticity. Asymptot. Anal. 48(1, 2), 33–54 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Pignataro, M., Rizzi, N., Luongo, A.: Stability, Bifurcation and Postcritical Behaviour of Elastic Structures, vol. 39. Elsevier, New York (2013)

    Google Scholar 

  38. Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z. Angew. Math. Phys. 66(6), 3699–3725 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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, 1–21 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.: A review on 2D models for the description of pantographic fabrics. Z. Angew. Math. Phys. 67(5), 121 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Placidi, L., El Dhaba, A.: Semi-inverse method à la saint-venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids 22, 919–937 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angew. Math. Phys. 67(5), 114 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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 

  44. Saeb, S., Steinmann, P., Javili, A.: Aspects of computational homogenization at finite deformations: a unifying review from reuss’ to Voigt’s bound. Appl. Mech. Rev. 68(5), 050801 (2016)

    Article  ADS  Google Scholar 

  45. Scerrato, D., Giorgio, I., Rizzi, N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys. 67(3), 1–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Scerrato, D., Zhurba Eremeeva, I., Lekszycki, T., Rizzi, N.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM J. Appl. Math. Mech. Z. Angew. Math. Phys. 96, 1268–1279 (2016)

    Article  MathSciNet  Google Scholar 

  47. Sciarra, G., dell’Isola, F., Coussy, O.: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

  49. Steigmann, D.: Two-dimensional models for the combined bending and stretching of plates and shells based on three-dimensional linear elasticity. Int. J. Eng. Sci. 46(7), 654–676 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Steigmann, D., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta. Mech. Sin. 31(3), 372–382 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  51. Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.: 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

    Article  MathSciNet  MATH  Google Scholar 

  52. 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)

    Article  Google Scholar 

  53. Turco, E., Rizzi, N.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. 77, 65–69 (2016)

    Article  Google Scholar 

  54. Shirani, M., Luo, C., Steigmann, D.J.: Cosserat elasticity of lattice shells with kinematically independent flexure and twist. Continuum Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0679-x

    Article  Google Scholar 

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Acknowledgments

3D measurements were carried out using GOMs company system-ARAMIS. Measurements were made possible thanks to the cooperation of the Institute of Mechanics and Printing Technology with LENSO partner and supplier in the field of 3D scanning and 3D measurements. This work was supported by a grant from the Government of the Russian Federation (contract No. 14.Y26.31.0031). The authors express their gratitude to the work made by the referees. Their useful comments have deeply contributed to current version of the paper.

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Authors and Affiliations

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

    Maciej Golaszewski & Roman Grygoruk

  2. International Research Centre on Mathematics and Mechanics of Complex Systems M&MOCS, L’Aquila, Italy

    Maciej Golaszewski, Roman Grygoruk, Ivan Giorgio, Marco Laudato & Fabio Di Cosmo

  3. DISG, University of Roma “La Sapienza”, Rome, Italy

    Ivan Giorgio

  4. Research Institute for Mechanics, National Research Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia

    Ivan Giorgio, Marco Laudato & Fabio Di Cosmo

  5. Department of Information Engineering Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    Marco Laudato

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  1. Maciej Golaszewski
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  2. Roman Grygoruk
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  3. Ivan Giorgio
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  4. Marco Laudato
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  5. Fabio Di Cosmo
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Correspondence to Fabio Di Cosmo.

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Communicated by Francesco dell’Isola.

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Golaszewski, M., Grygoruk, R., Giorgio, I. et al. Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Continuum Mech. Thermodyn. 31, 1015–1034 (2019). https://doi.org/10.1007/s00161-018-0692-0

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