Relaxation of elastic energies with free discontinuities and constraint on the strain
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 275-317.

As a model for the energy of a brittle elastic body we consider an integral functional consisting of two parts: a volume one (the usual linearly elastic energy) which is quadratic in the strain, and a surface part, which is concentrated along the fractures (i.e. on the discontinuities of the displacement function) and whose density depends on the jump part of the strain. We study the problem of the lower semicontinuous envelope of such a functional under the assumptions that the surface energy density is positively homogeneous of degree one and that additional geometrical constraints, such as a shearing condition or a normal detachement condition, are imposed on the fractures.

Classification : 49J45, 74R10
@article{ASNSP_2002_5_1_2_275_0,
     author = {Braides, Andrea and Defranceschi, Anneliese and Vitali, Enrico},
     title = {Relaxation of elastic energies with free discontinuities and constraint on the strain},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {275--317},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {2},
     year = {2002},
     mrnumber = {1991141},
     zbl = {1170.49306},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2002_5_1_2_275_0/}
}
TY  - JOUR
AU  - Braides, Andrea
AU  - Defranceschi, Anneliese
AU  - Vitali, Enrico
TI  - Relaxation of elastic energies with free discontinuities and constraint on the strain
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 275
EP  - 317
VL  - 1
IS  - 2
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_2002_5_1_2_275_0/
LA  - en
ID  - ASNSP_2002_5_1_2_275_0
ER  - 
%0 Journal Article
%A Braides, Andrea
%A Defranceschi, Anneliese
%A Vitali, Enrico
%T Relaxation of elastic energies with free discontinuities and constraint on the strain
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 275-317
%V 1
%N 2
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_2002_5_1_2_275_0/
%G en
%F ASNSP_2002_5_1_2_275_0
Braides, Andrea; Defranceschi, Anneliese; Vitali, Enrico. Relaxation of elastic energies with free discontinuities and constraint on the strain. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 275-317. http://archive.numdam.org/item/ASNSP_2002_5_1_2_275_0/

[1] L. Ambrosio - A. Braides, Energies in SBV and variational models in fracture mechanics, In: “Homogenization and applications to material sciences”, (Nice, 1995) D. Cioranescu - A. Damlamian - P. Donato (eds.), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, 1995, pp. 1-22. | MR | Zbl

[2] L. Ambrosio - A. Coscia - G. Dal Maso, Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal. 139 (1997), 201-238. | MR | Zbl

[3] L. Ambrosio - N. Fusco - D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford University Press, 2000. | MR | Zbl

[4] G. Anzellotti, A class of convex non-coercive functionals and masonry-like materials, Ann. Inst. H. Poincaré Anal. Non Linéaire 2(4) (1985), 261-307. | Numdam | MR | Zbl

[5] A. C. Barroso - I. Fonseca - R. Toader, A relaxation theorem in the space of functions with bounded deformation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), XXIX (2000), 19-49. | Numdam | MR | Zbl

[6] G. Bellettini - A. Coscia - G. Dal Maso, Compactness and lower semicontinuity in SBD(Ω), Math. Z. 228 (1998), 337-351. | MR | Zbl

[7] G. Bouchitté - I. Fonseca - L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145(1) (1998), 51-98. | MR | Zbl

[8] A. Braides - V. Chiadò Piat, Integral representation results for functionals defined on SBV (Ω; m ), J. Math. Pures Appl.(9) 75(6) (1996), 595-626. | MR | Zbl

[9] A. Braides - A. Defranceschi, “Homogenization of multiple integrals”, Oxford Lecture Series in Mathematics and its Applications, 12, Clarendon Press, Oxford University Press, New York, 1998. | MR | Zbl

[10] A. Braides - A. Defranceschi - E. Vitali, A relaxation approach to Hencky's plasticity, Appl. Math. Optim. 35 (1997), 45-68. | MR | Zbl

[11] M. Buliga, Energy minimizing brittle crack propagation, J. Elasticity 52(3) (1998/99), 201-238. | MR | Zbl

[12] G. Buttazzo, “Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations”, Pitman Research Notes in Mathematics Ser. 207, Longman, Harlow, 1989. | MR | Zbl

[13] G. Buttazzo - G. Dal Maso, Integral representation and relaxation of local functionals, Nonlinear Anal. 9(6) (1985), 515-532. | MR | Zbl

[14] G. Dal Maso, “An Introduction to Γ-Convergence”, Birkhäuser, Boston, 1993. | MR | Zbl

[15] E. De Giorgi - M. Carriero - A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Arch. Ration. Mech. Anal. 108 (1989), 195-218. | MR | Zbl

[16] J. A. Francfort - J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46(8) (1998), 1319-1342. | MR | Zbl

[17] M. Giaquinta - E. Giusti, Researches on the statics of masonry structures, Arch. Ration. Mech. Anal. 88 (1985), 359-392. | MR | Zbl

[18] C. Goffman - J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159-178. | MR | Zbl

[19] Yu. G. Reshetnyak, Weak convergence of completely additive vector functions on a set, Siberian Math. J. 9 (1968), 1039-1045. | Zbl

[20] R. T. Rockafellar, “Convex Analysis”, Princeton University Press, Princeton, 1970. | MR | Zbl

[21] R. Temam, “Problèmes mathématiques en plasticité”, Gauthier-Villars, Paris, 1983. | Zbl

[22] R. Temam - G. Strang, Functions of bounded deformation, Arch. Ration. Mech. Anal. 75 (1980), 7-21. | MR | Zbl