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Explicit integral representations of the relaxation of non-local energies for structured deformations

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DNM - The mathematical design of new materials

The theory of structured deformations in the SBV setting developed by Chocki & Fonseca [1]
only takes into account the linear dependance on jumps along the approximating sequences. Following a
model from Del Piero & Owen [2] that captures the non-linear dependence on jumps, the present approach
to relaxation of non-local energies rests on two limiting processes: start from a submacroscopical level
where we have a weighted average of disarrangements within neighborhoods of fixed size r > 0 and pass to
the macrolevel, permitting disarrangements to diffuse through such a neighborhood. This limiting process
determines a structured deformation as well as the non-local dependence of the energy density of such a
structured deformation. Pass to the limit as r ! 0, to obtain purely local bulk and interfacial energy
densities for the structured deformation identified in the first step.
This is a joint work with
Marco Morandotti, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino,
David R. Owen, Department of Mathematical Sciences, Carnegie Mellon University,
Elvira Zappale, Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno.
[1] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational
Mech. Anal. 138 (1997), 37-103.
[2] G. Del Piero and D. R. Owen: Structured Deformations: Part Two. Quaderni

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