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How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert”

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Abstract

Navier–Cauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how—by means of the Principle of Virtual Powers—it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green–Saint-Venant tensor and all its N − 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.

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Authors and Affiliations

  1. DISG, Università di Roma “La Sapienza”, via Eudossiana 18, 00184, Rome, Italy

    Francesco dell’Isola

  2. International Research Center M&MoCS, Palazzo Caetani, Cisterna di Latina, Latina, Italy

    Francesco dell’Isola, Pierre Seppecher & Angela Madeo

  3. IMath, Université du Sud-Toulon-Var, Avenue de l’Université, BP 132, 83957, La Garde Cedex, Toulon, France

    Pierre Seppecher

  4. LGCIE, Université de Lyon INSA, 20 Avenue Albert Einstein, 69621, Villeurbanne Cedex, France

    Angela Madeo

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  1. Francesco dell’Isola
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  2. Pierre Seppecher
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  3. Angela Madeo
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Correspondence to Francesco dell’Isola.

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dell’Isola, F., Seppecher, P. & Madeo, A. How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert” . Z. Angew. Math. Phys. 63, 1119–1141 (2012). https://doi.org/10.1007/s00033-012-0197-9

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  • DOI: https://doi.org/10.1007/s00033-012-0197-9

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