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A review on 2D models for the description of pantographic fabrics

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Abstract

A review on models for pantographic fabrics, a new promising kind of metamaterials, is presented. We treat those models that are able to capture the peculiar effects conferred by their specific microstructure and that can be generalized for the description of more complex metamaterials. For each approach, model formulation and modeling assumptions are discussed along with the presentation of numerical solutions in exemplary cases and no attempt is made to model damage and failure phenomena.

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References

  1. 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 

  2. AminPour, H., Rizzi, N.: On the continuum modelling of carbon nano tubes. In: Proceedings of the 15th International Conference on Civil, Structural and Environmental Engineering Computing, Paper, Vol. 240 (2015)

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. AminPour, H., Rizzi, N.: On the modelling of carbon nano tubes as generalized continua. In: Altenbach H, Forest S (eds.) Generalized Continua as Models for Classical and Advanced Materials, pp. 15–35. Springer, Switzerland (2016)

  5. AminPour, H., Rizzi, N., Salerno, G.: A one-dimensional beam model for single-wall carbon nano tube column buckling. In: Civil-Comp Proceedings (2014)

  6. Andreaus U., Giorgio I., Lekszycki T.: A 2D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. Z. Angew. Math. Mech. 13, 7 (2013)

    MATH  Google Scholar 

  7. Aristodemo M., Turco E.: Boundary element discretization of plane elasticity and plate bending problems. Int. J. Numer. Methods Eng. 37(6), 965–987 (1994)

    Article  MATH  Google Scholar 

  8. Bilotta A., Formica G., Turco E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26(9), 1155–1175 (2010)

    Article  MATH  Google Scholar 

  9. Caggegi C., Pensée N., 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 

  10. Carcaterra A., Akay A., Bernardini C.: Trapping of vibration energy into a set of resonators: theory and application to aerospace structures. Mech. Syst. Sig. Process. 26, 1–14 (2012)

    Article  Google Scholar 

  11. 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 

  12. Cazzani A., Stochino F., Turco E.: On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Z. Angew. Math. Phys. 67(2), 1–30 (2016)

    MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Chiaia B., Kumpyak O., Placidi L., Maksimov V.: Experimental analysis and modeling of two-way reinforced concrete slabs over different kinds of yielding supports under short-term dynamic loading. Eng. Struct. 96, 88–99 (2015)

    Article  Google Scholar 

  15. 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 

  16. 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 

  17. Contrafatto, L., Cuomo, M., Greco, L.: Meso-scale simulation of concrete multiaxial behaviour. Eur. J. Environ. Civ. Eng. 1–16 (2016). doi:10.1080/19648189.2016.1182085

  18. de Felice G., Rizzi N.: Macroscopic modelling of Cosserat media. Trends Appl. Math. Mech. Monogr. Surv. Pure Appl. Math. 106, 58–65 (1999)

    MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Della Corte A., Battista A., dell’Isola F.: Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non Linear Mech. 80, 209–220 (2016)

    Article  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)

  22. 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  MathSciNet  Google Scholar 

  23. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. dell’Isola, F., Bucci, S., Battista, A.: Against the fragmentation of knowledge: the power of multidisciplinary research for the design of metamaterials. In: Naumenko K, Marcus A (eds.) Advanced Methods of Continuum Mechanics for Materials and Structures, vol. 106, pp. 523–545. Springer, Singapore (2016)

  26. dell’Isola F., d’Agostino M., Madeo A., Boisse P., Steigmann D.: Minimization of shear energy in two dimensional continua with two orthogonal families of inextensible fibers: the case of standard bias extension test. J. Elast. 122(2), 131–155 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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 (2016). doi:10.1177/1081286515616034

  28. 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  Google Scholar 

  29. 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  Google Scholar 

  30. Di Carlo A., Rizzi N., Tatone A.: Continuum modelling of a beam-like latticed truss: identification of the constitutive functions for the contact and inertial actions. Meccanica 25(3), 168–174 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Everstine G.C., Pipkin A.C.: Boundary layers in fiber-reinforced materials. J. Appl. Mech. 40, 518–522 (1973)

    Article  MATH  Google Scholar 

  32. Gabriele, S., Rizzi, N., Varano, V.: On the imperfection sensitivity of thin-walled frames. In: Civil-Comp Proceedings, Vol. 99 (2012)

  33. Gabriele, S., Rizzi, N., Varano, V.: On the postbuckling behaviour of thin walled beams with in-plane deformable cross-sections. In: Civil-Comp Proceedings (2013)

  34. Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21, 737–746 (2016)

  35. Gabriele, S., Rizzi, N., Varano, V.: A one-dimensional nonlinear thin walled beam model derived from Koiter shell theory. In: Civil-Comp Proceedings, Vol. 106 (2014)

  36. 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)

    Article  Google Scholar 

  37. 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, Article no. 95 (2016). doi:10.1007/s00033-016-0692-5

  38. Giorgio I., Della Corte A., dell’Isola F., Steigmann D.: Buckling modes in pantographic lattices. C. R. Mec. 344(7), 487–501 (2016)

    Article  Google Scholar 

  39. Goda I., Assidi M., Ganghoffer J.F.: A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure. Biomech. Model. Mechanobiol. 13, 53–83 (2014)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Hilgers M.G., Pipkin A.C.: Elastic sheets with bending stiffness. Q. J. Mech. Appl. Math. 45, 57–75 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hilgers M.G., Pipkin A.C.: Energy-minimizing deformations of elastic sheets with bending stiffness. J. Elast. 31, 125–139 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hilgers M.G., Pipkin A.C.: Bending energy of highly elastic membranes II. Q. Appl. Math. 54, 307–316 (1996)

    MathSciNet  MATH  Google Scholar 

  45. Hu M.Z., Kolsky H., Pipkin A.C.: Bending theory for fiber-reinforced beams. J. Compos. Mater. 19, 235–249 (1985)

    Article  Google Scholar 

  46. Luongo A., Zulli D., Piccardo G.: A linear curved-beam model for the analysis of galloping in suspended cables. J. Mech. Mater. Struct. 2(4), 675–694 (2007)

    Article  Google Scholar 

  47. Luongo A., Zulli D.: A non-linear one-dimensional model of cross-deformable tubular beam. Int. J. Non Linear Mech. 66, 33–42 (2014)

    Article  Google Scholar 

  48. 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, 900–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Piccardo G., Ranzi G., Luongo A.: A direct approach for the evaluation of the conventional modes within the gbt formulation. Thin Wall. Struct. 74, 133–145 (2014)

    Article  Google Scholar 

  50. Pignataro M., Ruta G., Rizzi N., Varano V.: Effects of warping constraints and lateral restraint on the buckling of thin-walled frames. ASME Int. Mech. Eng. Congr. Expos. 10(B), 803–810 (2010)

    Google Scholar 

  51. Pignataro M., Rizzi N., Luongo A.: Stability, Bifurcation and Postcritical Behaviour of Elastic Structures, Vol. 39. Elsevier, Amsterdam (2013)

    Google Scholar 

  52. Pipkin A.C.: Generalized plane deformations of ideal fiber-reinforced materials. Q. Appl. Math. 32, 253–263 (1974)

    MATH  Google Scholar 

  53. Pipkin A.C.: Energy changes in ideal fiber-reinforced composites. Q. Appl. Math. 35, 455–463 (1978)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  55. 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 

  56. Placidi, L., El Dhaba, A.: Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids (2015). doi:10.1177/1081286515616043

  57. Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear 4D orthotropic second gradient elastic model. J. Eng. Math. 1–21 (2016). doi:10.1007/s10665-016-9856-8

  58. Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angew. Math. Phys. 67, Article no. 114 (2016)

  59. Ranzi G., Luongo A.: A new approach for thin-walled member analysis in the framework of gbt. Thin Wall. Struct. 49, 1404–1414 (2011)

    Article  Google Scholar 

  60. Rivlin R.S.: Plane strain of a net formed by inextensible cords. J. Ration. Mech. Anal. 4(6), 951–974 (1955)

    MathSciNet  MATH  Google Scholar 

  61. 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 

  62. Rizzi, N., Varano, V.: On the postbuckling analysis of thin-walled frames. In: 13th International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press (2011)

  63. 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 

  64. Roveri N., Carcaterra A., Akay A.: Energy equipartition and frequency distribution in complex attachments. J. Acoust. Soc. Am. 126, 122–128 (2009)

    Article  Google Scholar 

  65. Roveri N., Carcaterra A., Akay A.: Vibration absorption using non-dissipative complex attachments with impacts and parametric stiffness. J. Acoust. Soc. Am. 126, 2306–2314 (2009)

    Article  Google Scholar 

  66. Scerrato D., Giorgio I., Della Corte A., Madeo A., Dowling N., Darve F.: Towards the design of an enriched concrete with enhanced dissipation performances. Cem. Concr. Res. 84, 48–61 (2016)

    Article  Google Scholar 

  67. 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 

  68. 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. Mech. (2016). doi:10.1002/zamm.201600066

  69. 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. 67 (2016)

  70. Zhu B., Yu T.X., Tao X.M.: Large deformation and slippage mechanism of plain woven composite in bias extension. Compos. A Appl. Sci. Manuf. 38(8), 1821–1828 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. International Telematic University Uninettuno, C.so Vittorio Emanuele II, 39 00186, Rome, Italy

    Luca Placidi

  2. Department of Structural and Geotechnical Engineering, Universitã di Roma La Sapienza, Rome, Italy

    Emilio Barchiesi

  3. Universitã degli studi di Sassari, Alghero, Italy

    Emilio Turco

  4. Dipartimento di Architettura, Universitã degli studi Roma Tre, Rome, Italy

    Nicola Luigi Rizzi

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  1. Luca Placidi
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  2. Emilio Barchiesi
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  3. Emilio Turco
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  4. Nicola Luigi Rizzi
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Correspondence to Luca Placidi.

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Placidi, L., Barchiesi, E., Turco, E. et al. A review on 2D models for the description of pantographic fabrics. Z. Angew. Math. Phys. 67, 121 (2016). https://doi.org/10.1007/s00033-016-0716-1

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