Braides, Andrea; Gelli, Maria Stella Continuum limits of discrete systems without convexity hypotheses. (English) Zbl 1024.74004 Math. Mech. Solids 7, No. 1, 41-66 (2002). 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. Cited in 36 Documents MSC: 74A25 Molecular, statistical, and kinetic theories in solid mechanics Keywords:lattice systems; gamma-convergence; microcracking; one-dimensional nearest-neighbour interactions; variational limit; discrete energy densities; continuum limit; scaling; regularization PDFBibTeX XMLCite \textit{A. Braides} and \textit{M. S. Gelli}, Math. Mech. Solids 7, No. 1, 41--66 (2002; Zbl 1024.74004) Full Text: DOI References: [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) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.