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,

[math]\displaystyle{ \varepsilon(u) = \frac{\nabla u + \nabla u^{\top}}{2} }[/math]

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, introduced essentially by P.-M. Suquet in 1978. 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 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 Hⁿ⁻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

[math]\displaystyle{ \varepsilon(u) = e(u) \, \mathrm{d} x + \big( u_{+}(x) - u_{-}(x) \big) \odot \nu_{u} (x) H^{n - 1} | J_{u}, }[/math]

where H ⁿ⁻¹ | J_u denotes H ⁿ⁻¹ on the jump set J_u and [math]\displaystyle{ \odot }[/math] denotes the symmetrized dyadic product:

[math]\displaystyle{ a \odot b = \frac{a \otimes b + b \otimes a}{2}. }[/math]

The collection of all functions of special bounded deformation is denoted SBD(Ω; Rⁿ), or simply SBD.

References

• Suquet, P.-M. (1978). "Existence et régularité des solutions des équations de la plasticité parfaite". Comptes Rendus de l'Académie des Sciences, Série A 286: 1201–1204. 
• Francfort, G. A.; Marigo, J.-J. (1998). "Revisiting brittle fracture as an energy minimization problem". J. Mech. Phys. Solids 46 (8): 1319–1342. doi:10.1016/S0022-5096(98)00034-9. Bibcode: 1998JMPSo..46.1319F. https://hal.archives-ouvertes.fr/hal-03500962/file/Francfort1998.pdf. 
• Francfort, G. A.; Marigo, J.-J. (1999). "Cracks in fracture mechanics: a time indexed family of energy minimizers". in P. Argoul. Dordrecht: Kluwer Academic Publishers. pp. 197–202.