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Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

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Abstract

In the present paper, a two-dimensional solid consisting of a linear elastic isotropic material, for which the deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lamé constants (\({\lambda}\) and \({\mu}\)) and 4 more parameters (instead of 5 as it is in the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and flexure are provided. The idea is very simple: The solutions of the corresponding problem of first gradient classical case are imposed, and the corresponding forces, double forces and wedge forces are found. On the basis of such solutions, a method is outlined, which is able to identify the six constitutive parameters. Ideal (or Gedanken) experiments are designed in order to write equations having as unknowns the six constants and as known terms the values of suitable experimental measurements.

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Authors and Affiliations

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

    Luca Placidi

  2. Universitá di Roma La Sapienza, Rome, Italy

    Ugo Andreaus & Alessandro Della Corte

  3. Faculty of Engineering Production, Warsaw University of Technology, Warsaw, Poland

    Tomasz Lekszycki

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  1. Luca Placidi
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  2. Ugo Andreaus
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  3. Alessandro Della Corte
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  4. Tomasz Lekszycki
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Correspondence to Luca Placidi.

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Placidi, L., Andreaus, U., Corte, A.D. et al. Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z. Angew. Math. Phys. 66, 3699–3725 (2015). https://doi.org/10.1007/s00033-015-0588-9

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  • DOI: https://doi.org/10.1007/s00033-015-0588-9

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