Abstract

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.

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