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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 Ω of Rn, a function u : Ω → Rn is said to be of bounded deformation if the symmetrized gradient ε(u) of u,

\varepsilon(u) = \frac{\nabla u + \nabla u^{\top}}{2}

is a bounded, symmetric n × n matrix-valued Radon measure. The collection of all functions of bounded deformation is denoted BD(Ω; Rn), 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) dx; a jump part, supported on a rectifiable (n − 1)-dimensional set Ju of points where u has two different approximate limits u+ and u, together with a normal vector νu; and a "Cantor part", which vanishes on Borel sets of finite Hn−1-measure (where Hk 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 + \big( u_{+}(x) + u_{-}(x) \big) \odot \nu_{u} (x) H^{n - 1} | J_{u},

where H n−1 | Ju denotes H n−1 on the jump set Ju 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; Rn), or simply SBD.


  • Francfort, G. A. and Marigo, J.-J. (1998). "Revisiting brittle fracture as an energy minimization problem". J. Mech. Phys. Solids 46 (8): 1319–1342.
  • Francfort, G. A. and Marigo, J.-J. (1999). in Variations of domain and free-boundary problems in solid mechanics (Paris, 1997): Cracks in fracture mechanics: a time indexed family of energy minimizers, Solid Mech. Appl.. Kluwer Acad. Publ., 197–202. 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bounded_deformation". A list of authors is available in Wikipedia.
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