Calculus of Variations and Geometric Measure Theory
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R. Alicandro - G. Dal Maso - G. Lazzaroni - M. Palombaro

Linearisation of multiwell energies

created by lazzaroni on 16 Jun 2017
modified on 20 Jul 2017

[BibTeX]

Preprint

Inserted: 16 jun 2017
Last Updated: 20 jul 2017

Year: 2017

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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