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On Hill’s lemma in continuum mechanics

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Abstract

Hill’s first-order lemma in continuum mechanics has been derived in the literature for quasi-static cases and for dynamic cases in the absence of body forces. In this manuscript, a generalized first-order Hill identity is first derived in an Eulerian dynamic description including body forces. Then, a second-order Hill identity using a Lagrangian description is obtained, involving the variation in time of the associated kinetic energy. Simple examples, that allow for analytical analysis, of one-dimensional elastic deformation of bars are considered that involve dynamics and body forces to illustrate the generalization of Hill’s identities proposed here.

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Authors and Affiliations

  1. IRSTEA, Grenoble, France

    François Nicot

  2. Department of Mechanical Engineering, University of Twente, Enschede, The Netherlands

    Niels P. Kruyt

  3. LaSIE, Université de La Rochelle, La Rochelle, France

    Olivier Millet

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  1. François Nicot
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  2. Niels P. Kruyt
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  3. Olivier Millet
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Correspondence to François Nicot.

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Nicot, F., Kruyt, N.P. & Millet, O. On Hill’s lemma in continuum mechanics. Acta Mech 228, 1581–1596 (2017). https://doi.org/10.1007/s00707-016-1776-1

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  • DOI: https://doi.org/10.1007/s00707-016-1776-1

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