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Variational limit of a one-dimensional discrete and statistically homogeneous system of material points. (English) Zbl 1031.74041
The authors show that the energy of a discrete system of material points on a line and subjected to random nearest-neighbour interactions, almost surely converges in a variational sense to a deterministic energy defined on spaces of functions with bounded variation. These material points, in the reference configuration, occupy the points of the lattice \( \varepsilon Z, \varepsilon = {1 \over n} \), contained in the interval \([0,1]\).Their first result extends, in the context of statistical physics but for more specific energy density functions, the result of A. Braides, G. Dal Maso and A. Garroni [Arch. Rational Mech. Anal. 146, 23-58 (1999; Zbl 0945.74006)]. Next, following the same procedure, the authors study a new discrete model for which the interaction between each pair of contigous points is described by a random energy density which is no longer assumed to be convex but which fulfils the same condition in the neighborhood of \( 0^{+}\). In this case, their result extends, for more specific density functions and in a stochastic setting, that of A. Brades and M. S. Gelli [Limits of discrete system without convexity hypotheses, Preprint SISSA, Triest (1999)].
Reviewer: T.N.Pham (Hanoi)

74Q05 Homogenization in equilibrium problems of solid mechanics
74R99 Fracture and damage
74G65 Energy minimization in equilibrium problems in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
49J55 Existence of optimal solutions to problems involving randomness