- Bounded deformation
In

mathematics , a function of**bounded deformation**is a function whosedistributional derivative s are not quitewell-behaved -enough to qualify as functions ofbounded 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 Ω of

**R**^{"n"}, a function "u" : Ω →**R**^{"n"}is said to be of**bounded deformation**if thesymmetrized gradient "ε"("u") of "u",:$varepsilon(u)\; =\; frac\{\; abla\; u\; +\; abla\; u^\{\; op\{2\}$

is a bounded, symmetric "n" × "n" matrix-valued

Radon measure . The collection of all functions of bounded deformation is denoted BD(Ω;**R**^{"n"}), or simply BD. BD is a strictly larger space than the space BV of functions ofbounded variation .One can show that if "u" is of bounded deformation then the measure "ε"("u") can be decomposed into three parts: one

absolutely continuous with respect toLebesgue 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 anormal vector "ν"_{"u"}; and a "Cantor part", which vanishes on Borel sets of finite "H"^{"n"−1}-measure (where "H"^{"k"}denotes "k"-dimensionalHausdorff measure ).A function "u" is said to be of

**special bounded deformation**if the Cantor part of "ε"("u") vanishes, so that the measure can be written as:$varepsilon\; u\; =\; e(u)\; ,\; mathrm\{d\}\; x\; +\; ig(\; u\_\{+\}(x)\; +\; u\_\{-\}(x)\; ig)\; odot\; u\_\{u\}\; (x)\; H^\{n\; -\; 1\}\; |\; J\_\{u\},$

where "H"

^{ "n"−1}| "J"_{"u"}denotes "H"^{ "n"−1}on the jump set "J"_{"u"}and $odot$ denotes the symmetrizeddyadic product ::$a\; odot\; b\; =\; frac\{a\; otimes\; b\; +\; b\; otimes\; a\}\{2\}.$

The collection of all functions of bounded deformation is denoted SBD(Ω;

**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 = 1319–1342

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 = 197–202

publisher = Kluwer Acad. Publ.

address = Dordrecht

year = 1999

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