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Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

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A Correction to this article was published on 24 July 2018

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Abstract

Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

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  • 24 July 2018

    The original version of this article unfortunately contained mistakes.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Elettrica e dell’Informazione, Università di Cassino e del Lazio meridionale, via Di Biasio 43, 03043, Cassino, FR, Italy

    Roberto Alicandro

  2. SISSA, Via Bonomea 265, 34136, Trieste, Italy

    Gianni Dal Maso

  3. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Giuliano Lazzaroni

  4. Department of Mathematics, University of Sussex, Pevensey 2 Building, Falmer Campus, Brighton, BN1 9QH, United Kingdom

    Mariapia Palombaro

Authors
  1. Roberto Alicandro
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  2. Gianni Dal Maso
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  3. Giuliano Lazzaroni
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  4. Mariapia Palombaro
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Corresponding author

Correspondence to Giuliano Lazzaroni.

Additional information

Communicated by I. Fonseca

The original version of this article was revised: Modifications have been made to Table 1 and 2. The end of Theorems 1.7(ii), 1.8(i), 1.8(ii), 1.10, 2.3 and 4.2 are now correctly marked.

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Alicandro, R., Dal Maso, G., Lazzaroni, G. et al. Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals. Arch Rational Mech Anal 230, 1–45 (2018). https://doi.org/10.1007/s00205-018-1240-6

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  • DOI: https://doi.org/10.1007/s00205-018-1240-6

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