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Identification of Two-Dimensional Pantographic Structures with a Linear D4 Orthotropic Second Gradient Elastic Model Accounting for External Bulk Double Forces

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Mathematical Modelling in Solid Mechanics

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

Abstract

The present paper deals with the identification of the nine constitutive parameters appearing in the strain energy density of a linear elastic second gradient D4 orthotropic two-dimensional continuum model accounting for an external bulk double force \(m^{ext}\). The aim is to specialize the model for the description of pantographic fabrics, which show such a kind of anisotropy. Analytical solutions for model problems, which are here referred to as the heavy sheet, the non-conventional bending and the trapezoidal cases are recalled from a previous paper and further elaborated in order to perform gedanken experiments. We completely characterize the set of nine constitutive parameters in terms of the materials the fibers are made of (i.e. of the Young’s modulus of the fiber materials), of their cross section (i.e. of the area and of the moment of inertia of the fiber cross sections), of the internal rotational spring positioned at each intersection point between the two families of fibers and of the pitch, i.e. the distance between adjacent pivots. Finally, the remarkable form of the strain energy, derived in terms of the displacement field, is shortly discussed.

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Author information

Authors and Affiliations

  1. Faculty of Engineering, International Telematic University Uninettuno, Rome, Italy

    Luca Placidi

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

    Emilio Barchiesi & Alessandro Della Corte

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  1. Luca Placidi
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  2. Emilio Barchiesi
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  3. Alessandro Della Corte
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Correspondence to Luca Placidi .

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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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Placidi, L., Barchiesi, E., Della Corte, A. (2017). Identification of Two-Dimensional Pantographic Structures with a Linear D4 Orthotropic Second Gradient Elastic Model Accounting for External Bulk Double Forces. 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_14

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  • DOI: https://doi.org/10.1007/978-981-10-3764-1_14

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