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Continuum limits of discrete systems without convexity hypotheses. (English) Zbl 1024.74004

Summary: We describe the variational limit of one-dimensional nearest-neighbour systems of interactions, under no structure hypotheses on the discrete energy densities. We show that the continuum limit is characterized by bulk and interfacial energy density, which can be explicitly computed from discrete energies through operations of limit, scaling and regularization that highlight possible bulk oscillations and multiple cracking.

MSC:

74A25 Molecular, statistical, and kinetic theories in solid mechanics
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[1] Truskinovsky L., Contemporary research in the mechanics and mathematics of materials pp 322– (1996)
[2] Braides A., Arch. Rational Mech. Anal. 146 pp 23– (1999) · Zbl 0945.74006
[3] Dal Maso G., An Introduction to {\(\Gamma\)}-convergence (1993)
[4] Braides A., {\(\Gamma\)} -convergence for Beginners
[5] Braides A., Homogenization of Multiple Integrals (1998) · Zbl 0911.49010
[6] Ambrosio L., Functions of Bounded Variation and Free Discontinuity Problems (2000) · Zbl 0957.49001
[7] Pouget J., Phys. Rev. B 43 pp 3575– (1991)
[8] Houchmandzadeh B., J. Phys.: Condens. Matter 4 pp 9779– (1992)
[9] Del Piero, Journal de Physique IV France 8 8 pp 95– (1998)
[10] Braides A., J. Convex Analysis 9 (2002)
[11] Alicandro R., Ann. Scuola Norm. Pisa 29 pp 671– (2000)
[12] De Giorgi E., Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 pp 199– (1988)
[13] Ambrosio L., Homogenization and Applications to Material Sciences pp 1– (1997)
[14] DOI: 10.1007/BFb0097344 · Zbl 0909.49001
[15] Amar M., Nonlinear Anal. TMA 34 pp 953– (1998) · Zbl 0953.49023
[16] Blake A., Visual Reconstruction (1987) · Zbl 0713.93057
[17] Chambolle A., C. R. Acad. Sci., Paris, Ser. I 314 pp 191– (1992)
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