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Gamma-Convergence for Beginners$
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Andrea Braides

Print publication date: 2002

Print ISBN-13: 9780198507840

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198507840.001.0001

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DISCRETE SYSTEMS AND FREE-DISCONTINUITY PROBLEMS

DISCRETE SYSTEMS AND FREE-DISCONTINUITY PROBLEMS

Chapter:
(p.150) 11 DISCRETE SYSTEMS AND FREE-DISCONTINUITY PROBLEMS
Source:
Gamma-Convergence for Beginners
Author(s):

Andrea Braides

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198507840.003.0012

This chapter shows how general non-convex difference schemes can give rise to functionals defined by piecewise-Sobolev functions with interactions between surface and volume terms. Different identification of discrete schemes with energies on suitable piecewise-Sobolev interpolations are given, leading to a continuous limit energy. As examples, softening and fracture problems with size effects are obtained as limits of convex/concave discrete energies; fracture is described as a phase-transition phenomenon starting from Lennard-Jones potentials; and the Malik-Perona approximation of free-discontinuity problems is considered.

Keywords:   non-convex, piecewise-Sobolev interpolation, softening, fracture mechanics, size effects, Lennard-Jones potential, Malik-Perona functionals

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