Vol. 5, No. 2, 2017

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The general form of the relaxation of a purely interfacial energy for structured deformations

Miroslav Šilhavý

Vol. 5 (2017), No. 2, 191–215
DOI: 10.2140/memocs.2017.5.191

This paper deals with the relaxation of energies of media with structured deformations introduced by Del Piero and Owen (1993; 1995). Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth and nonsmooth geometrical changes (disarrangements) at submacroscopic levels. The paper examines the special case of Choksi and Fonseca’s (1997) energetics of structured deformations in which the unrelaxed energy does not contain the bulk contribution. Thus, the energy is purely interfacial but of a general form. New formulas for the relaxed bulk and interfacial energies are proved. The bulk relaxed energy is shown to coincide with the subadditive envelope of the unrelaxed interfacial energy while the relaxed interfacial energy is the restriction of the envelope to rank-1 tensors. Moreover, it is shown that the minimizing sequence required to define the bulk energy in the relaxation scheme of Choksi and Fonseca (1997) can be realized in the more restrictive class required in the relaxation scheme of Baía, Matias and Santos (2012), thus establishing the equality of relaxed energies of the two approaches for general purely interfacial energies. The relaxations of the specific interfacial energies of Owen and Paroni (2015) and Barroso, Matias, Morandotti and Owen (2017) are simple consequences of our general results.

structured deformations, relaxation, subadditive envelope, interfacial energy, bulk energy, functions of measures
Mathematical Subject Classification 2010
Primary: 49J45
Secondary: 74A60, 74G65, 15A99
Received: 25 October 2016
Revised: 7 March 2017
Accepted: 16 May 2017
Published: 7 June 2017

Communicated by Gianpietro Del Piero
Miroslav Šilhavý
Institute of Mathematics
Academy of Sciences of the Czech Republic
Žitná 25
115 67 Pragua 1
Czech Republic