Bounded deformation

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 Ω of R"n", a function "u" : Ω → R"n" is said to be of bounded deformation if the symmetrized 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 of bounded 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 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 "ν""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 "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 symmetrized dyadic 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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