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Non-linear Dynamics of Pantographic Fabrics: Modelling and Numerical Study

  • Marco Laudato
  • Emilio BarchiesiEmail author
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 109)

Abstract

In this work, the dynamical behavior of a pantographic sheet undergoing sinusoidal (in time) imposed displacement is numerically investigated. The used model has been largely exploited to analyse the quasi-static behavior of pantographic materials. Here we propose to use a non-linear generalization of such a model for the description of a pantographic material dynamical behavior.

References

  1. 1.
    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)Google Scholar
  2. 2.
    Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials. Springer, pp. 239–258 (2017)Google Scholar
  3. 3.
    Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Mathematics and Mechanics of Solids p. 1081286517735695 (2018)Google Scholar
  4. 4.
    di Cosmo, F., Laudato, M., Spagnuolo, M.: Acoustic metamaterials based on local resonances: Homogenization, optimization and applications. In: Generalized Models and Non-classical Approaches in Complex Materials 1, pp. 247–274. Springer (2018)Google Scholar
  5. 5.
    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):20150,790 (2016)Google Scholar
  6. 6.
    dell’Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D., Giorgio, I., Andreaus, U., Turco, E., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. 1–34 (2018)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Abali, B.E.: Computational Reality: Solving Nonlinear and Coupled Problems in Continuum Mechanics, vol. 55. Springer (2016)Google Scholar
  9. 9.
    Abali, B.E., Müller, W.H., Eremeyev, V.A.: Strain gradient elasticity with geometric nonlinearities and its computational evaluation. Mech. Adv. Mater. Mod. Process. 1(1), 4 (2015)Google Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Abdoul-Anziz, H., Seppecher, P.: Strain gradient and generalized continua obtained by homogenizing frame lattices. Math. Mech. Complex Syst. 6(3), 213–250 (2018)Google Scholar
  12. 12.
    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum. uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys. 133(2):281–345 (2008)Google Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Misra, A., Chang, C.S.: Effective elastic moduli of heterogeneous granular solids. Int. J. Solids Struct. 30(18), 2547–2566 (1993)Google Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Pideri, C., Seppecher, P.: Asymptotics of a non-planar rod in non-linear elasticity. Asymptot. Anal. 48(1,2):33–54 (2006)Google Scholar
  17. 17.
    Seppecher, P.: Second-gradient theory: application to Cahn-Hilliard fluids. In: Continuum thermomechanics, pp. 379–388. Springer (2000)Google Scholar
  18. 18.
    Yang, Y., Misra, A.: Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation. CMES-Comput. Model. Eng. Sci. 64(1), 1–36 (2010)Google Scholar
  19. 19.
    Auffray, N., Dirrenberger, J., Rosi, G.: A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int. J. Solids Struct. 69, 195–206 (2015)Google Scholar
  20. 20.
    Forest, S.: Mechanics of generalized continua: construction by homogenizaton. Le J. Phys. IV 8(PR4):Pr4–39 (1998)Google Scholar
  21. 21.
    Forest, S., Sievert, R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160(1–2), 71–111 (2003)Google Scholar
  22. 22.
    Rosi, G., Giorgio, I., Eremeyev, V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 93(12), 914–927 (2013)Google Scholar
  23. 23.
    Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67(3), 53 (2016)Google Scholar
  24. 24.
    Golaszewski, M., Grygoruk, R., Giorgio, I., Laudato, M., Di Cosmo, F.: Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Contin. Mech. Thermodyn. pp. 1–20 (2018)Google Scholar
  25. 25.
    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 22(10), 1950–1975 (2017)Google Scholar
  26. 26.
    Placidi, L., Barchiesi, E., Della Corte, A.: Identification of two-dimensional pantographic structures with a linear d4 orthotropic second gradient elastic model accounting for external bulk double forces. In: Mathematical Modelling in Solid Mechanics, pp. 211–232. Springer (2017)Google Scholar
  27. 27.
    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 (2016a)Google Scholar
  28. 28.
    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 (2016b)Google Scholar
  29. 29.
    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 (2016c)Google Scholar
  30. 30.
    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)Google Scholar
  31. 31.
    Laudato, M., Manzari, L., Barchiesi, E., Di Cosmo, F., Göransson, P.: First experimental observation of the dynamical behavior of a pantographic metamaterial. Mech. Res. Commun. 94, 125–127 (2018)Google Scholar
  32. 32.
    Abbas, I.A., Abdalla, A.E.N.N., Alzahrani, F.S., Spagnuolo, M.: Wave propagation in a generalized thermoelastic plate using eigenvalue approach. J. Therm. Stress. 39(11), 1367–1377 (2016).  https://doi.org/10.1080/01495739.2016.1218229Google Scholar
  33. 33.
    Abo-el-nour, N.A.A., Alshaikh, F., Vescovo, D.D., Spagnuolo, M.: Plane waves and eigenfrequency study in a transversely isotropic magneto-thermoelastic medium under the effect of a constant angular velocity. J. Therm. Stress. 40(9), 1079–1092 (2017).  https://doi.org/10.1080/01495739.2017.1334528
  34. 34.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P., Rendón, L.A.: Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech. 80(3), 217–227 (2010)Google Scholar
  35. 35.
    Berezovski, A., Yildizdag, M.E., Scerrato, D.: On the wave dispersion in microstructured solids. Contin. Mech. Thermodyn. pp. 1–20 (2018)Google Scholar
  36. 36.
    Eremeyev, V.A., Lebedev, L.P., Cloud, M.J.: Acceleration waves in the nonlinear micromorphic continuum. Mech. Res. Commun. 93, 70–74 (2018)Google Scholar
  37. 37.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)Google Scholar
  38. 38.
    Eremeyev, V., Zubov, L.: On constitutive inequalities in nonlinear theory of elastic shells. ZAMM 87(2), 94–101 (2007)Google Scholar
  39. 39.
    Eremeyev, V.A., Altenbach, H., Morozov, N.F.: The influence of surface tension on the effective stiffness of nanosize plates. Dokl. Phys. 54(2), 98–100 (2009)Google Scholar
  40. 40.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer Science & Business Media (2012)Google Scholar
  41. 41.
    Di Egidio, A., Luongo, A., Paolone, A.: Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. Contin. Mech. Thermodyn. 42(1), 88–98 (2007)Google Scholar
  42. 42.
    Luongo, A.: Mode localization in dynamics and buckling of linear imperfect continuous structures. In: Normal Modes and Localization in Nonlinear Systems, pp. 133–156. Springer (2001)Google Scholar
  43. 43.
    Luongo, A., Zulli, D., Piccardo, G.: Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables. J. Sound Vib. 315(3), 375–393 (2008)Google Scholar
  44. 44.
    Pagnini, L., Freda, A., Piccardo, G.: Uncertainties in the evaluation of one degree-of-freedom galloping onset. Eur. J. Environ. Civ. Eng. 21(7–8), 1043–1063 (2017)Google Scholar
  45. 45.
    Pagnini, L.C., Piccardo, G.: The three-hinged arch as an example of piezomechanic passive controlled structure. Contin. Mech. Thermodyn. 28(5), 1247–1262 (2016)Google Scholar
  46. 46.
    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)Google Scholar
  47. 47.
    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. pp. 1–21 (2018)Google Scholar
  48. 48.
    Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B: Eng. 43(3), 1315–1328 (2012)Google Scholar
  49. 49.
    Chróścielewski, J., Schmidt, R., Eremeyev, V.A.: Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Contin. Mech. Thermodyn. 31(1), 147–188 (2019)Google Scholar
  50. 50.
    Eremeyev, V.A.: On characterization of an elastic network within the six-parameter shell theory. In: Shell Structures: Theory and Applications Volume 4: Proceedings of the 11th International Conference “Shell Structures: Theory and Applications” (SSTA 2017), October 11–13, 2017, pp. 81–84. Gdansk, Poland, CRC Press, Boca Raton (2018)Google Scholar
  51. 51.
    Spagnuolo, M., Andreaus, U.: A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. Math. Mech. Solids 0(0):1081286517737,000,  https://doi.org/10.1177/1081286517737000 (2018)
  52. 52.
    Franciosi, P.: A decomposition method for obtaining global mean green operators of inclusions patterns. Application to parallel infinite beams in at least transversally isotropic media. Int. J. Solids Struct. (2018)Google Scholar
  53. 53.
    Franciosi, P., Lormand, G.: Using the radon transform to solve inclusion problems in elasticity. Int. J. Solids Struct. 41(3–4), 585–606 (2004)Google Scholar
  54. 54.
    Franciosi, P., Brenner, R., El Omri, A.: Effective property estimates for heterogeneous materials with cocontinuous phases. J. Mech. Mater. Struct. 6(5), 729–763 (2011)Google Scholar
  55. 55.
    Franciosi, P., Spagnuolo, M., Salman, O.U.: Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Contin. Mech. Thermodyn. pp. 1–32 (2018)Google Scholar
  56. 56.
    Marmo, F., Rosati, L.: The fiber-free approach in the evaluation of the tangent stiffness matrix for elastoplastic uniaxial constitutive laws. Int. J. Numer. Methods Eng. 94(9), 868–894 (2013)Google Scholar
  57. 57.
    Romano, G., Rosati, L., Ferro, G.: Shear deformability of thin-walled beams with arbitrary cross sections. Int. J. Numer. Methods Eng. 35(2), 283–306 (1992)Google Scholar
  58. 58.
    Rosati, L., Marmo, F., Serpieri, R.: Enhanced solution strategies for the ultimate strength analysis of composite steel-concrete sections subject to axial force and biaxial bending. Comput. Methods Appl. Mech. Eng. 197(9–12), 1033–1055 (2008)Google Scholar
  59. 59.
    Contrafatto, L., Cuomo, M.: A framework of elastic-plastic damaging model for concrete under multiaxial stress states. Int. J. Plast. 22(12), 2272–2300 (2006)Google Scholar
  60. 60.
    Cuomo, M.: Continuum model of microstructure induced softening for strain gradient materials. Math. Mech. Solids p. 1081286518755845 (2018)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 (2018a)Google Scholar
  62. 62.
    Placidi, L., Misra, A., Barchiesi, E.: Simulation results for damage with evolving microstructure and growing strain gradient moduli. Contin. Mech. Thermodyn. pp. 1–21 (2018b)Google Scholar
  63. 63.
    Placidi, L., Misra, A., Barchiesi, E.: Wo-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3):56 (2018c)Google Scholar
  64. 64.
    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)Google Scholar
  65. 65.
    Yang, Y., Misra, A.: Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49(18), 2500–2514 (2012)Google Scholar
  66. 66.
    Beirão Da Veiga, L., Hughes, T., Kiendl, J., Lovadina, C., Niiranen, J., Reali, A., Speleers, H.: A locking-free model for reissner-mindlin plates: analysis and isogeometric implementation via nurbs and triangular nurps. Math. Model. Methods Appl. Sci. 25(08), 1519–1551 (2015)Google Scholar
  67. 67.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21(5):562–577 (2016a)Google Scholar
  68. 68.
    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 (2016b)Google Scholar
  69. 69.
    Greco, L., Cuomo, M.: B-spline interpolation of kirchhoff-love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)Google Scholar
  70. 70.
    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)Google Scholar
  71. 71.
    Khakalo, S., Niiranen, J.: Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software. Comput.-Aided Des. 82, 154–169 (2017)Google Scholar
  72. 72.
    Niiranen, J., Kiendl, J., Niemi, A.H., Reali, A.: Isogeometric analysis for sixth-order boundary value problems of gradient-elastic kirchhoff plates. Comput. Methods Appl. Mech. Eng. 316, 328–348 (2017)Google Scholar
  73. 73.
    Yildizdag, M.E., Demirtas, M., Ergin, A.: Multipatch discontinuous galerkin isogeometric analysis of composite laminates. Contin. Mech. Thermodyn. 1–14 (2018)Google Scholar
  74. 74.
    Cazzani, A., Atluri, S.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11(4), 229–251 (1993)Google Scholar
  75. 75.
    Javili, A., McBride, A., Steinmann, P., Reddy, B.: A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology. Comput. Mech. 54(3), 745–762 (2014)Google Scholar
  76. 76.
    McBride, A., Mergheim, J., Javili, A., Steinmann, P., Bargmann, S.: Micro-to-macro transitions for heterogeneous material layers accounting for in-plane stretch. J. Mech. Phys. Solids 60(6), 1221–1239 (2012)Google Scholar
  77. 77.
    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):050,801 (2016)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità degli Studi dell’AquilaL’AquilaItaly
  2. 2.International Center M&MOCS Mathematics and Mechanics of Complex SystemsDICEAA, Universitá degli Studi dell’AquilaL’AquilaItaly
  3. 3.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversità degli Studi di Roma “La Sapienza” RomeRomeItaly

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