- Bounded deformation
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",
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 continuouswith 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
where "H" "n"−1 | "J""u" denotes "H" "n"−1 on the jump set "J""u" and denotes the symmetrized
The collection of all functions of bounded deformation is denoted SBD(Ω; R"n"), or simply SBD.
* 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
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… … Wikipedia
Superior craton — in the west. The western margin extends from northern Minnesota through eastern Manitoba to northwestern Ontario.The formation of the Superior craton is best explained within the context of 2.72 2.68 Ga accretion of small continental plates and… … Wikipedia
mountain — mountainless, adj. /mown tn/, n. 1. a natural elevation of the earth s surface rising more or less abruptly to a summit, and attaining an altitude greater than that of a hill, usually greater than 2000 ft. (610 m). 2. a large mass of something… … Universalium
Asia — /ay zheuh, ay sheuh/, n. a continent bounded by Europe and the Arctic, Pacific, and Indian oceans. 2,896,700,000; ab. 16,000,000 sq. mi. (41,440,000 sq. km). * * * I Largest continent on Earth. It is bounded by the Arctic Ocean, the Pacific Ocean … Universalium
Yilgarn Craton — The Yilgarn Craton is a large craton which constitutes the bulk of the Western Australian land mass. It is bounded by a mixture of sedimentary basins and Proterozoic fold and thrust belts. Zircon grains in the Jack Hills, Narryer Gneiss Terrane… … Wikipedia
North America — North American. the northern continent of the Western Hemisphere, extending from Central America to the Arctic Ocean. Highest point, Mt. McKinley, 20,300 ft. (6187 m); lowest, Death Valley, 276 ft. (84 m) below sea level. 400,000,000 including… … Universalium
Australia — /aw strayl yeuh/, n. 1. a continent SE of Asia, between the Indian and the Pacific oceans. 18,438,824; 2,948,366 sq. mi. (7,636,270 sq. km). 2. Commonwealth of, a member of the Commonwealth of Nations, consisting of the federated states and… … Universalium
Africa — /af ri keuh/, n. 1. a continent S of Europe and between the Atlantic and Indian oceans. 551,000,000; ab. 11,700,000 sq. mi. (30,303,000 sq. km). adj. 2. African. * * * I Second largest continent on Earth. It is bounded by the Mediterranean Sea,… … Universalium
Colorado Plateau — A map of the Colorado Plateau. The … Wikipedia
BDDC — In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arize from the finite element method. BDDC is used as a… … Wikipedia