# Bounded deformation

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Bounded deformation

In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition. Thought of as deformations of elasto-plastic bodies, functions of bounded deformation play a major role in the mathematical study of materials, e.g. the Francfort-Marigo model of brittle crack evolution.

More precisely, given an open subset &Omega; of R"n", a function "u" : &Omega; &rarr; R"n" is said to be of bounded deformation if the symmetrized gradient "&epsilon;"("u") of "u",

:$varepsilon\left(u\right) = frac\left\{ abla u + abla u^\left\{ op\left\{2\right\}$

is a bounded, symmetric "n" &times; "n" matrix-valued Radon measure. The collection of all functions of bounded deformation is denoted BD(&Omega;; R"n"), or simply BD. BD is a strictly larger space than the space BV of functions of bounded variation.

One can show that if "u" is of bounded deformation then the measure "&epsilon;"("u") can be decomposed into three parts: one absolutely continuous with respect to Lebesgue measure, denoted "e"("u") d"x"; a jump part, supported on a rectifiable ("n" − 1)-dimensional set "J""u" of points where "u" has two different approximate limits "u"+ and "u", together with a normal vector "&nu;""u"; and a "Cantor part", which vanishes on Borel sets of finite "H""n"−1-measure (where "H""k" denotes "k"-dimensional Hausdorff measure).

A function "u" is said to be of special bounded deformation if the Cantor part of "&epsilon;"("u") vanishes, so that the measure can be written as

:

where "H" "n"−1 | "J""u" denotes "H" "n"−1 on the jump set "J""u" and $odot$ denotes the symmetrized dyadic product:

:$a odot b = frac\left\{a otimes b + b otimes a\right\}\left\{2\right\}.$

The collection of all functions of bounded deformation is denoted SBD(&Omega;; R"n"), or simply SBD.

References

* cite journal
author = Francfort, G. A. and Marigo, J.-J.
title = Revisiting brittle fracture as an energy minimization problem
journal = J. Mech. Phys. Solids
volume = 46
year = 1998
issue = 8
pages = 1319&ndash;1342
doi = 10.1016/S0022-5096(98)00034-9

* cite book
author = Francfort, G. A. and Marigo, J.-J.
title = Cracks in fracture mechanics: a time indexed family of energy minimizers
editor = Variations of domain and free-boundary problems in solid mechanics (Paris, 1997)
series = Solid Mech. Appl.
volume = 66
pages = 197&ndash;202
year = 1999

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