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Convergence of Hencky-Type Discrete Beam Model to Euler Inextensible Elastica in Large Deformation: Rigorous Proof

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Mathematical Modelling in Solid Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 69))

Abstract

The present chapter concerns rigorous homogenization of a Hencky-type discrete beam model, which is useful for the numerical study of complex fibrous systems as pantographic sheets as well as woven fabrics. \(\varGamma \)-convergence of the discrete model towards the inextensible Euler’s beam model is proven and the result is established for placements in \(\mathbb {R}^d\) in large deformation regime.

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References

  1. 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 

  2. 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 (2016)

    Google Scholar 

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

    Google Scholar 

  4. 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 

  5. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2012)

    MATH  Google Scholar 

  6. Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3), 774–787 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yeremeyev, V.A., Zubov, L.M.: The theory of elastic and viscoelastic micropolar liquids. J. Appl. Math. Mech. 63(5), 755–767 (1999)

    Article  MathSciNet  Google Scholar 

  8. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 89(4), 242–256 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Scerrato, D., Giorgio, I., Madeo, A., Limam, 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 

  10. Giorgio, I., Scerrato, D.: Multi-scale concrete model with rate-dependent internal friction. Eur. J. Environ. Civ. Eng. 1–19 (2016)

    Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. Misra, A., Yang, Y.: Micromechanical model for cohesive materials based upon pseudo-granular structure. Int. J. Solids Struct. 47(21), 2970–2981 (2010)

    Article  MATH  Google Scholar 

  13. Chang, C.S., Misra, A.: Packing structure and mechanical properties of granulates. J. Eng. Mech. 116(5), 1077–1093 (1990)

    Article  Google Scholar 

  14. Seddik, H., Greve, R., Placidi, L., Hamann, I., Gagliardini, O.: Application of a continuum-mechanical model for the flow of anisotropic polar ice to the edml core, antarctica. J. Glaciol. 54(187), 631–642 (2008)

    Article  Google Scholar 

  15. Placidi, L., Faria, S.H., Hutter, K.: On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets. Ann. Glaciol. 39(1), 49–52 (2004)

    Article  Google Scholar 

  16. Placidi, L., Greve, R., Seddik, H., Faria, S.H.: Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor. Contin. Mech. Thermodyn. 22(3), 221–237 (2010)

    Article  MATH  Google Scholar 

  17. F dell’Isola, G Sciarra, and S Vidoli. Generalized hooke’s law for isotropic second gradient materials. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. doi:10.1098/rspa.2008.0530, (2009)

  18. Andreaus, U., dell’Isola, F., Giorgio, I., Placidi, L., Lekszycki, T., Rizzi, N.L.: Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int. J. Eng. Sci. 108, 34–50 (2016)

    Article  MathSciNet  Google Scholar 

  19. dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, physique, astronomie, p. 7 (1995)

    Google Scholar 

  20. Sciarra, G., Ianiro, N., Madeo, A., et al.: A variational deduction of second gradient poroelasticity i: general theory. J. Mech. Mater. Struct. 3(3), 507–526 (2008)

    Article  Google Scholar 

  21. dell’Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Placidi, L.: A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Contin. Mech. Thermodyn. 28(1–2), 119–137 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Alibert, J.J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für angewandte Mathematik und Physik 66(5), 2855–2870 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 6 (2016). doi:10.1007/s10665-016-9856-8

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

    Google Scholar 

  27. Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)

    Article  MathSciNet  Google Scholar 

  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 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2d lattices: a numerical analysis on the static response and wave propagation. Proc. Est. Acad. Sci. 64(3), 219 (2015)

    Article  Google Scholar 

  32. 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 (2016)

    Article  Google Scholar 

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

    Google Scholar 

  34. Fertis, D.J.: Nonlinear Structural Engineering. Springer, Berlin (2006)

    MATH  Google Scholar 

  35. Forest, S., Sievert, R.: Nonlinear microstrain theories. Int. J. Solids Struct. 43(24), 7224–7245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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), 1 (2015)

    Article  Google Scholar 

  37. Milton, G.W.: Adaptable nonlinear bimode metamaterials using rigid bars, pivots, and actuators. J. Mech. Phys. Solids 61(7), 1561–1568 (2013)

    Article  MathSciNet  Google Scholar 

  38. Braides, A.: Gamma-Convergence for Beginners, vol. 22. Clarendon Press, Oxford (2002)

    Book  MATH  Google Scholar 

  39. 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, 20150790 (2016)

    Google Scholar 

  40. 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. für angew. Math. und Phys. 67(4), 1–28 (2016)

    MathSciNet  Google Scholar 

  41. 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 

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

    Article  Google Scholar 

  43. 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(8), 900–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Piccardo, G., Ferrarotti, A., Luongo, A.: Nonlinear generalized beam theory for open thin-walled members. Math. Mech. Solids 1081286516649990 (2016)

    Google Scholar 

  45. 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 

  46. Taig, G., Ranzi, G., Luongo, A.: Gbt pre-buckling and buckling analyses of thin-walled members under axial and transverse loads. Contin. Mech. Thermodyn. 28(1–2), 41–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Luongo, A., D’Annibale, F.: A paradigmatic minimal system to explain the ziegler paradox. Contin. Mech. Thermodyn. 27(1–2), 211–222 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Luongo, A., D’Annibale, F.: Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads. Int. J. Non-Linear Mech. 55, 128–139 (2013)

    Article  Google Scholar 

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Authors and Affiliations

  1. IMATH, Université de Toulon, Toulon, France

    Jean-Jacques Alibert & Pierre Seppecher

  2. L.M.A CNRS AMU Marseille, Marseille, France

    Alessandro Della Corte

  3. Dipartimento di Ingegneria Meccanica e Aerospaziale, Università di Roma La Sapienza, Via Eudossiana 18, 00184, Rome, Italy

    Pierre Seppecher

Authors
  1. Jean-Jacques Alibert
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  2. Alessandro Della Corte
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  3. Pierre Seppecher
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Correspondence to Pierre Seppecher .

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Editors and Affiliations

  1. Università di Roma "La Sapienza" , Roma, Italy

    Francesco dell'Isola

  2. Université de Perpignan Via Domitia, LAboratoire de Mathématiques et PhySique Université de Perpignan Via Domitia, Perpignan, France

    Mircea Sofonea

  3. Dept Mechanical Engineering, UC Berkeley Dept Mechanical Engineering, Berkeley, California, USA

    David Steigmann

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Alibert, JJ., Della Corte, A., Seppecher, P. (2017). Convergence of Hencky-Type Discrete Beam Model to Euler Inextensible Elastica in Large Deformation: Rigorous Proof. In: dell'Isola, F., Sofonea, M., Steigmann, D. (eds) Mathematical Modelling in Solid Mechanics. Advanced Structured Materials, vol 69. Springer, Singapore. https://doi.org/10.1007/978-981-10-3764-1_1

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  • DOI: https://doi.org/10.1007/978-981-10-3764-1_1

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