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Material characterization and computations of a polymeric metamaterial with a pantographic substructure

  • Hua Yang
  • Gregor Ganzosch
  • Ivan GiorgioEmail author
  • B. Emek Abali
Article
  • 111 Downloads

Abstract

The development of additive manufacturing methods, such as 3D printing, allows the design of more complex architectured materials. Indeed, the main structure can be obtained by means of periodically (or quasi-periodically) arranged substructures which are properly conceived to provide unconventional deformation patterns. These kinds of materials which are ‘substructure depending’ are called metamaterials. Detailed simulations of a metamaterial is challenging but accurately possible by means of the elasticity theory. In this study, we present the steps taken for analyzing and simulating a particular type of metamaterial composed of a pantographic substructure which is periodic in space—it is simply a grid. Nevertheless, it shows an unexpected type of deformation under a uniaxial shear test. This particular behavior is investigated in this work with the aid of direct numerical simulations by using the finite element method. In other words, a detailed mesh is generated to properly describe the substructure. The metamaterial is additively manufactured using a common polymer showing nonlinear elastic deformation. Experiments are undertaken, and several hyperelastic material models are examined by using an inverse analysis. Moreover, a direct numerical simulation is repeated for all studied material models. We show that a good agreement between numerical simulations and experimental data can be attained.

Keywords

Nonlinear elasticity Metamaterials Material identification Hyperelastic models 

Mathematics Subject Classification

74A30 74B20 74S05 

Notes

Acknowledgements

IG is supported by a grant from the Government of the Russian Federation (No. 14.Y26.31.0031). We thank Prof. Wolfgang H. Müller for fruitful discussions.

References

  1. 1.
    Abali, B.E.: Thermodynamically Compatible Modeling, Determination of Material Parameters, and Numerical Analysis of Nonlinear Rheological Materials. PhD thesis, Technische Universität Berlin, Institute of Mechanics (2014)Google Scholar
  2. 2.
    Abali, B.E.: Computational Reality, Solving Nonlinear and Coupled Problems in Continuum Mechanics. Advanced Structured Materials. Springer, Berlin (2017)zbMATHGoogle Scholar
  3. 3.
    Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)CrossRefGoogle Scholar
  4. 4.
    Abali, B.E., Wu, C.-C., Müller, W.H.: An energy-based method to determine material constants in nonlinear rheology with applications. Contin. Mech. Thermodyn. 28(5), 1221–1246 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Altenbach, H., Eremeyev, V.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 88(5), 332–341 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Altenbach, H., Eremeyev, V.A. (eds.): Generalized Continua—from the Theory to Engineering Applications, Volume 541 of CISM International Centre for Mechanical Sciences. Springer, Wien (2013)Google Scholar
  8. 8.
    Altenbach, H., Eremeyev, V.A.: Surface viscoelasticity and effective properties of materials and structures. In: Altenbach, H., Kruch, S. (eds.) Advanced Materials Modelling for Structures, pp. 9–16. Springer, Berlin (2013)Google Scholar
  9. 9.
    Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00161-018-0665-3
  10. 10.
    Arruda, E.M., Boyce, M.C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41(2), 389–412 (1993)CrossRefzbMATHGoogle Scholar
  11. 11.
    Attard, M.M., Hunt, G.W.: Hyperelastic constitutive modeling under finite strain. Int. J. Solids Struct. 41(18–19), 5327–5350 (2004)CrossRefzbMATHGoogle Scholar
  12. 12.
    Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00161-018-0626-x
  13. 13.
    Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids (2018).  https://doi.org/10.1177/1081286517735695
  14. 14.
    Battista, A., Cardillo, C., Del Vescovo, D., Rizzi, N.L., Turco, E.: Frequency shifts induced by large deformations in planar pantographic continua. Nanosci. Technol. Int. J. 6(2), 161–178 (2015)Google Scholar
  15. 15.
    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
  16. 16.
    Biderman, V.L.: Calculation of rubber parts. Rascheti na prochnost, Moscow (1958)Google Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cuomo, M.: Forms of the dissipation function for a class of viscoplastic models. Math. Mech. Complex Syst. 5(3), 217–237 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    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, 887–928 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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), 1–23 (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(6), 3473–3498 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    dell’Isola, F., Seppecher, P., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00161-018-0689-8
  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)CrossRefGoogle Scholar
  27. 27.
    Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152–168 (2015)CrossRefGoogle Scholar
  28. 28.
    Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Syst. 3(1), 43–82 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Eringen, A.C.: Theory of Micropolar Elasticity. Technical report, DTIC Document (1967)Google Scholar
  31. 31.
    Eugster, S.R., Hesch, C., Betsch, P., Glocker, C.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng. 97(2), 111–129 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Flory, P.J., Rehner Jr., J.: Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity. J. Chem. Phys. 11(11), 512–520 (1943)CrossRefGoogle Scholar
  33. 33.
    Ganzosch, G., dell’Isola, F., Turco, E., Lekszycki, T., Müller, W.H.: Shearing tests applied to pantographic structures. Acta Polytech. CTU Proc. 7, 1–6 (2016)CrossRefGoogle Scholar
  34. 34.
    Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67(4), 95 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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), 1–21 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Harrison, P., Clifford, M.J., Long, A.C., Rudd, C.D.: A constituent-based predictive approach to modelling the rheology of viscous textile composites. Compos. A Appl. Sci. Manuf. 35(7–8), 915–931 (2004)CrossRefGoogle Scholar
  37. 37.
    Hoffman, J., Jansson, J., Johnson, C., Knepley, M., Kirby, R.C., Logg, A., Scott, L.R., Wells, G.N.: Fenics (2005). http://www.fenicsproject.org/
  38. 38.
    Holzapfel, A.G.: Nonlinear Solid Mechanics II. Wiley, New York (2000)zbMATHGoogle Scholar
  39. 39.
    Isihara, A., Hashitsume, N., Tatibana, M.: Statistical theory of rubber-like elasticity. IV (two-dimensional stretching). J. Chem. Phys. 19(12), 1508–1512 (1951)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Itskov, M., Aksel, N.: A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. Int. J. Solids Struct. 41(14), 3833–3848 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    James, A.G., Green, A., Simpson, G.M.: Strain energy functions of rubber. I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19(7), 2033–2058 (1975)CrossRefGoogle Scholar
  42. 42.
    Julio García Ruíz, M., Yarime Suárez González, L.: Comparison of hyperelastic material models in the analysis of fabrics. Int. J. Cloth. Sci. Technol. 18(5), 314–325 (2006)CrossRefGoogle Scholar
  43. 43.
    Khakalo, S., Balobanov, V., Niiranen, J.: Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics. Int. J. Eng. Sci. 127, 33–52 (2018)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)CrossRefzbMATHGoogle Scholar
  45. 45.
    Logg, A., Mardal, K.A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method, the FEniCS Book, Volume 84 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2011)Google Scholar
  46. 46.
    Marckmann, G., Verron, E.: Comparison of hyperelastic models for rubber-like materials. Rubber Chem. Technol. 79(5), 835–858 (2006)CrossRefGoogle Scholar
  47. 47.
    Martins, P., Natal Jorge, R.M., Ferreira, A.J.M.: A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues. Strain 42(3), 135–147 (2006)CrossRefGoogle Scholar
  48. 48.
    Milton, G.W., 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)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Mindlin, R.D., Eshel, N.N.: On first strain–gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)CrossRefzbMATHGoogle Scholar
  50. 50.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Müller, W.H., dell’Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun. 89, 6–10 (2018)CrossRefGoogle Scholar
  52. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1–2), 215–234 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11(9), 582–592 (1940)CrossRefzbMATHGoogle Scholar
  55. 55.
    Nadler, B., Papadopoulos, P., Steigmann, D.J.: Multiscale constitutive modeling and numerical simulation of fabric material. Int. J. Solids Struct. 43(2), 206–221 (2006)CrossRefzbMATHGoogle Scholar
  56. 56.
    Niiranen, J., Balobanov, V., Kiendl, J., Hosseini, S.B.: Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Math. Mech. Solids (2017).  https://doi.org/10.1177/1081286517739669
  57. 57.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474(2210), 1–19 (2018)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Placidi, L., Barchiesi, E., Battista, A.: An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations. In: dell’Isola, F., Sofonea, M., Steigmann, D. (eds.) Mathematical Modelling in Solid Mechanics, pp. 193–210. Springer, Berlin(2017)Google Scholar
  61. 61.
    Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6(2), 77–100 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3), 1–19 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Rivlin, R.S.: Large elastic deformations of isotropic materials iv. further developments of the general theory. Philos. Trans. R. Soc. Lond. A 241(835), 379–397 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109(1), 1–37 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Shirani, M., Luo, C., Steigmann, D.J.: Cosserat elasticity of lattice shells with kinematically independent flexure and twist. Contin. Mech. Thermodyn. (2018).  https://doi.org/10.1007/s00161-018-0679-x
  67. 67.
    Soe, S.P., Martindale, N., Constantinou, C., Robinson, M.: Mechanical characterisation of Duraform\(^{\textregistered }\) Flex for FEA hyperelastic material modelling. Polym. Test. 34, 103–112 (2014)CrossRefGoogle Scholar
  68. 68.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  69. 69.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Thai, H.-T., Vo, T.P., Nguyen, T.-K., Kim, S.-E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017)CrossRefGoogle Scholar
  71. 71.
    Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Treloar, L.R.G.: The elasticity of a network of long-chain molecules—II. Trans. Faraday Soc. 39, 241–246 (1943)CrossRefGoogle Scholar
  73. 73.
    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(4), 85 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    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)CrossRefGoogle Scholar
  75. 75.
    Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R. Soc. Open Sci. 4(10), 171153 (2017)CrossRefGoogle Scholar
  76. 76.
    Weber, G., Anand, L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic–viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79(2), 173–202 (1990)CrossRefzbMATHGoogle Scholar
  77. 77.
    Yeoh, O.H.: Some forms of the strain energy function for rubber. Rubber Chem. Technol. 66(5), 754–771 (1993)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Chair of Continuum Mechanics and Constitutive Theory, Institute of MechanicsTechnische Universität BerlinBerlinGermany
  2. 2.International Research Center for the Mathematics and Mechanics of Complex SystemsUniversità degli studi dell’AquilaL’AquilaItaly
  3. 3.Department of Mechanical and Aerospace EngineeringSapienza Università di RomaRomeItaly
  4. 4.Research Institute for Mechanics, National Research LobachevskyState University of Nizhni NovgorodNizhni NovgorodRussian Federation

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