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A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results

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Advances in Mechanics of Microstructured Media and Structures

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 87))

Abstract

In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza”, Via Eudossiana 18, 00184, Roma, Italy

    Emilio Barchiesi & Francesco dell’Isola

  2. International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100, L’Aquila, Italy

    Emilio Barchiesi, Francesco dell’Isola, Marco Laudato, Luca Placidi & Pierre Seppecher

  3. Facoltà di Ingegneria, Università Telematica Internazionale UNINETTUNO, Corso Vittorio Emanuele II 39, 00186, Roma, Italy

    Luca Placidi

  4. Dipartimento di Ingegneria Civile, Edile, Ambientale e Architettura, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, 67100, L’Aquila, Italy

    Francesco dell’Isola

  5. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio 1, 67100, L’Aquila, Coppito, Italy

    Marco Laudato

  6. Institut de Mathématiques de Toulon, Universitè de Toulon et du Var, Avenue de l’Université, BP 132, Cedex, 83957, La Garde, France

    Pierre Seppecher

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  1. Emilio Barchiesi
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  2. Francesco dell’Isola
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  4. Luca Placidi
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  5. Pierre Seppecher
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Correspondence to Francesco dell’Isola .

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Editors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, Rome, Italy

    Francesco dell'Isola

  2. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  3. Department of Micromechanics of Materials, Institute of Problems in Mechanical Engineering, Saint Petersburg, Russia

    Alexey Porubov

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Barchiesi, E., dell’Isola, F., Laudato, M., Placidi, L., Seppecher, P. (2018). A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results. In: dell'Isola, F., Eremeyev, V., Porubov, A. (eds) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-73694-5_4

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  • DOI: https://doi.org/10.1007/978-3-319-73694-5_4

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  • Print ISBN: 978-3-319-73693-8

  • Online ISBN: 978-3-319-73694-5

  • eBook Packages: EngineeringEngineering (R0)

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