×

zbMATH — the first resource for mathematics

An approximation result for special functions with bounded deformation. (English) Zbl 1084.49038
J. Math. Pures Appl. (9) 83, No. 7, 929-954 (2004); addendum ibid. 84, No. 1, 137-145 (2005).
A special displacement with bounded deformation is a function \(u:\Omega \subset \mathbb R^N \to \mathbb R^N\) whose symmetrized gradient is a bounded measure which coincides, outside an \((N-1)\)-dimensional rectifiable jump set \(J_u\), with a summable function \(e(u)\). In the paper the author proves that in dimension \(N=2\) if \(u\) and \(e(u)\) are square integrable and \({\mathcal H}^{N-1}(J_u)\) is finite then \(u\) can be approximated with a sequence \(\{u_n\}\) of piecewise continuous functions whose jump sets \(J_{u_n}\) are relatively closed, with \(u_n\) and \(e(u_n)\) converging strongly in \(L^2\), respectively, to \(u\) and \(e(u)\) and the lengths \({\mathcal H}^{N-1}(J_{u_n})\) converging to \({\mathcal H}^{N-1}(J_u)\). As an application it is given an Ambrosio-Tortorelli type approximation of a functional which appears in the theory of brittle fracture in linearized elasticity. In the addendum the results are extended to dimension \(N\geq 3\) by a careful application of a discretization technique.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
74G05 Explicit solutions of equilibrium problems in solid mechanics
74R10 Brittle fracture
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alberti, G., Variational models for phase transitions, an approach via γ-convergence, (), 95-114 · Zbl 0957.35017
[2] Ambrosio, L., A compactness theorem for a new class of functions with bounded variation, Boll. un. mat. ital. (7), 3-B, 857-881, (1989) · Zbl 0767.49001
[3] Ambrosio, L.; Braides, A., Energies in SBV and variational models in fracture mechanics, (), 1-22 · Zbl 0904.73045
[4] Ambrosio, L.; Coscia, A.; Dal Maso, G., Fine properties of functions with bounded deformation, Arch. rational mech. anal., 139, 3, 201-238, (1997) · Zbl 0890.49019
[5] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variation and free discontinuity problems, (2000), The Clarendon Press, Oxford University Press New York · Zbl 0957.49001
[6] Ambrosio, L.; Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via γ-convergence, Comm. pure appl. math., 43, 8, 999-1036, (1990) · Zbl 0722.49020
[7] Ambrosio, L.; Tortorelli, V.M., On the approximation of free discontinuity problems, Boll. un. mat. ital. (7), 6-B, 1, 105-123, (1992) · Zbl 0776.49029
[8] Bellettini, G.; Coscia, A.; Dal Maso, G., Compactness and lower semicontinuity properties in \(SBD(Ω)\), Math. Z., 228, 2, 337-351, (1998) · Zbl 0914.46007
[9] Bonnetier, E.; Chambolle, A., Computing the equilibrium configuration of epitaxially strained crystalline films, SIAM J. appl. math., 62, 4, 1093-1121, (2002), (electronic) · Zbl 1001.49017
[10] Bourdin, B.; Chambolle, A., Implementation of an adaptive finite-element approximation of the mumford – shah functional, Numer. math., 85, 4, 609-646, (2000) · Zbl 0961.65062
[11] Bourdin, B.; Francfort, G.A.; Marigo, J.-J., Numerical experiments in revisited brittle fracture, J. mech. phys. solids, 48, 4, 797-826, (2000) · Zbl 0995.74057
[12] A. Braides, M. Solci, A remark on the approximation of free discontinuity problems, in preparation · Zbl 1272.49024
[13] Chambolle, A., Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. appl. math., 55, 3, 827-863, (1995) · Zbl 0830.49015
[14] Chambolle, A., Finite-differences discretizations of the mumford – shah functional, M2AN math. model. numer. anal., 33, 2, 261-288, (1999) · Zbl 0947.65076
[15] Chambolle, A., Mathematical problems in image processing, ()
[16] Chambolle, A., A density result in two-dimensional linearized elasticity, and applications, Arch. ration. mech. anal., 167, 3, 211-233, (2003) · Zbl 1030.74007
[17] Dal Maso, G.; Francfort, G.A.; Toader, R., Quasistatic crack growth in finite elasticity, Preprint available on · Zbl 1064.74150
[18] Dal Maso, G.; Toader, R., A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. ration. mech. anal., 162, 2, 101-135, (2002) · Zbl 1042.74002
[19] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, (1992), CRC Press Boca Raton, FL · Zbl 0626.49007
[20] Federer, H., Geometric measure theory, (1969), Springer-Verlag New York · Zbl 0176.00801
[21] Francfort, G.A.; Larsen, C.J., Existence and convergence for quasi-static evolution in brittle fracture, Comm. pure appl. math., 56, 1465-1500, (2003) · Zbl 1068.74056
[22] Francfort, G.A.; Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, J. mech. phys. solids, 46, 8, 1319-1342, (1998) · Zbl 0966.74060
[23] Francfort, G.A.; Marigo, J.-J., Cracks in fracture mechanics: A time indexed family of energy minimizers, (), 197-202
[24] Gobbino, M., Finite difference approximation of the mumford – shah functional, Comm. pure appl. math., 51, 2, 197-228, (1998) · Zbl 0888.49013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.