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On the effect of interactions beyond nearest neighbours on non-convex lattice systems

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Abstract

We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.

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Acknowledgements

The authors thank Sergio Conti for useful discussions. This work has been partially supported by the European Research Council through the Advanced Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture” and by the INdAM-GNAMPA Project 2016 “Multiscale analysis of complex systems with variational methods”.

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Authors and Affiliations

  1. DIEI, 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

    Giuliano Lazzaroni

  3. Department of Mathematics, University of Sussex, Pevensey 2 Building, Falmer Campus, Brighton, BN1 9QH, UK

    Mariapia Palombaro

Authors
  1. Roberto Alicandro
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  2. Giuliano Lazzaroni
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  3. Mariapia Palombaro
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Corresponding author

Correspondence to Roberto Alicandro.

Additional information

Communicated by L. Ambrosio.

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Alicandro, R., Lazzaroni, G. & Palombaro, M. On the effect of interactions beyond nearest neighbours on non-convex lattice systems. Calc. Var. 56, 42 (2017). https://doi.org/10.1007/s00526-017-1129-5

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  • DOI: https://doi.org/10.1007/s00526-017-1129-5

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