Finite-difference approximation of energies in fracture mechanics
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 29 (2000) no. 3, pp. 671-709.
@article{ASNSP_2000_4_29_3_671_0,
     author = {Alicandro, Roberto and Focardi, Matteo and Gelli, Maria Stella},
     title = {Finite-difference approximation of energies in fracture mechanics},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {671--709},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 29},
     number = {3},
     year = {2000},
     mrnumber = {1817714},
     zbl = {1072.49020},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2000_4_29_3_671_0/}
}
TY  - JOUR
AU  - Alicandro, Roberto
AU  - Focardi, Matteo
AU  - Gelli, Maria Stella
TI  - Finite-difference approximation of energies in fracture mechanics
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2000
SP  - 671
EP  - 709
VL  - 29
IS  - 3
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2000_4_29_3_671_0/
LA  - en
ID  - ASNSP_2000_4_29_3_671_0
ER  - 
%0 Journal Article
%A Alicandro, Roberto
%A Focardi, Matteo
%A Gelli, Maria Stella
%T Finite-difference approximation of energies in fracture mechanics
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2000
%P 671-709
%V 29
%N 3
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2000_4_29_3_671_0/
%G en
%F ASNSP_2000_4_29_3_671_0
Alicandro, Roberto; Focardi, Matteo; Gelli, Maria Stella. Finite-difference approximation of energies in fracture mechanics. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 29 (2000) no. 3, pp. 671-709. http://www.numdam.org/item/ASNSP_2000_4_29_3_671_0/

[1] R. Alicandro - A. Braides - M.S. Gelli, Free-discontinuity problems generated by singular perturbation, Proc. Roy. Soc. Edinburgh 128A (1998), 1115-1129. | MR | Zbl

[2] R. Alicandro - M.S. Gelli, Free discontinuity problems generated by singular perturbation : the n-dimensional case, Proc. Roy. Soc. Edinburgh, to appear. | MR | Zbl

[3] R. Alicandro - A. Braides - J. Shah, Free-discontinuity problems via functionals involving the L1-norm of the gradient and their approximations, Interfaces and Free Boundaries 1 (1999), 17-37. | MR | Zbl

[4] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. 3-B (1989), 857-881. | MR | Zbl

[5] L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), 291-322. | MR | Zbl

[6] L. Ambrosio, A new proof of the SBV compactness theorem, Calc. Var. Partial Differential Equations 3 (1995), 127-137. | MR | Zbl

[7] L. Ambrosio - A. Braides, Energies in SBV and variational models in fracture mechanics, In: "Homogenization and Applications to Material Sciences", D. CIORANESCU - A. DAMLAMIAN - P. DONATO (eds.), GAKUTO, Gakkotosho, Tokio, Japan, 1997, pp. 1-22. | MR | Zbl

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

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

[10] L. Ambrosio - V.M. Tortorelli, Approximation offunctionals depending on jumps by elliptic functionals via r-convergence, Comm. Pure Appl. Math. 43 (1990), 999-1036. | MR | Zbl

[11] L. Ambrosio - V.M. Tortorelli, On the approximation of free-discontinuity problems, Boll. Un. Mat. Ital. 6-B (1992), 105-123. | MR | Zbl

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

[13] M. Born - K. Huang, "Dynamical Theory of Crystal Lattices", Oxford University Press, Oxford, 1954. | Zbl

[14] A. Braides, "Approximation of Free-Discontinuity Problems", Lecture Notes in Mathematics, Springer Verlag, Berlin, 1998. | MR | Zbl

[15] A. Braides, Non-local variational limits of discrete systems, Preprint SISSA, Trieste, 1999. | MR

[16] A. Braides - G. Dal Maso, Non-local approximation of the Mumford-Shah functional, Calc. Var. Partial Differential Equations 5 (1997), 293-322. | MR | Zbl

[17] A. Braides - G. Dal Maso - A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, Arch. Rational Mech. Anal. 146 (1999), 23-58. | MR | Zbl

[18] A. Braides - A. Defranceschi, "Homogenization of Multiple Integral", Oxford University Press, Oxford, 1998. | MR | Zbl

[19] A. Braides - M.S. Gelli, Limits of discrete systems with long-range interactions, Preprint SISSA, Trieste, 1999. | MR

[20] A. Braides - M.S. Gelli, Limits of discrete systems without convexity hypotheses, Math. Mech. Solids, to appear. | MR | Zbl

[21] M. Buliga, Energy minimizing brittle crack propagation, J. Elasticity 52 (1999), 201-238. | MR | Zbl

[22] A. Chambolle, Finite differences discretizations of the Mumford-Shah functional, RAIRO-Model. Math. Anal. Numer., to appear. | Numdam | MR | Zbl

[23] G. Dal Maso, "An Introduction to r-convergence", Birkhäuser, Boston, 1993. | MR | Zbl

[24] E. De Giorgi - L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988), 199-210. | MR | Zbl

[25] E. De Giorgi - T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842-850. | MR | Zbl

[26] L.C. Evans - R.F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC Press, Boca Raton, 1992. | MR | Zbl

[27] H. Federer, "Geometric Measure Theory", Springer Verlag, New York, 1969. | MR | Zbl

[28] M. Focardi, On the variational approximation of free-discontinuity problems in the vectorial case, Math. Models Methods Appl. Sci., to appear. | MR | Zbl

[29] G.A. Francfort - J.J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 No. 8 (1998), 1319-1342. | MR | Zbl

[30] E. Giusti, "Minimal Surfaces and Functions with bounded Variation", Birkhäuser, Basel, 1993. | MR | Zbl

[31] M. Gobbino, Finite difference approximation of the Mumford-Shah functional, Comm. Pure Appl. Math 51 (1998), 197-228. | MR | Zbl

[32] M. Gobbino - M.G. Mora, Finite difference approximation offree discontinuity problems, Preprint SISSA, Trieste, 1999. | MR

[33] A.A. Griffith, The phenomenon of rupture and flow in solids, Phil Trans. Royal Soc. London A 221 (1920), 163-198.

[34] D. Mumford - J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989), 577-685. | MR | Zbl

[35] R. Temam, "Mathematical Problems in Plasticity", Bordas, Paris, 1985.

[36] L. Truskinovsky, Fracture as a phase transition, In: "Contemporary research in the mechanics and mathematics of materials ", R.C. BATRA - M. F. BEATTY (eds.), CIMNE, Barcelona, 1996, pp. 322-332.

[37] W. Ziemer, "Weakly Differentiable Functions", Springer, Berlin, 1989. | MR | Zbl