Keywords: random discrete systems, energy bounds, second-order statistics, Hashin-Shtrikman-Willis variational principles.

Discrete material models, representing a material as a network of particles interacting via inter-particle potentials, have received a steadily increasing attention in the fields of theoretical, computational and applied materials science in the last decade [1]. From the engineering point of view, the interest has been nourished by the possibility of addressing, in a conceptually simple framework, the interplay among the intrinsic material heterogeneity, discreteness and randomness at different levels of resolution.

In this paper, we address, in detail, a specific problem related to mechanics of random discrete media, namely the stored energy estimates for finite two-component lattices with a fixed geometry and the heterogeneity distribution characterized in the sense of the second-order spatial statistics. The major motivation for this study is to establish a well-defined framework for stochastic homogenization of general discrete systems without a need to adopt neither the separation-of-scales hypothesis nor the assumption of statistical homogeneity.

Variational bounds and estimates of the globally stored energy are established following recent extensions of the classical Hashin-Shtrikman-Willis (HSW) variational principles due to Hashin and Shtrikman [2] and Willis [3] to finite-sized random composite bodies due to Luciano and Willis [4,5]. Additional details on the energetic bounds is available in the full-length version of the contribution.

1

M.J. Alava, P.K.V.V. Nukala, S. Zapperi, "Statistical models of fracture", Advances in Physics, 55(3-4), 349-476, 2006. doi:10.1080/00018730300741518

2

Z. Hashin, S. Shtrikman, "On some variational principles in anisotropic and nonhomogeneous elasticity", Journal of the Mechanics and Physics of Solids, 10, 335-342, 1962. doi:10.1016/0022-5096(62)90004-2

3

J.R. Willis, "Bounds and self-consistent estimates for the overall properties of anisotropic composites", Journal of the Mechanics and Physics of Solids, 25(3), 185-202, 1977. doi:10.1016/0022-5096(77)90022-9

4

R. Luciano, J.R. Willis, "FE analysis of stress and strain fields in finite random composite bodies", Journal of the Mechanics and Physics of Solids, 53(7), 1505-1522, 2005. doi:10.1016/j.jmps.2005.02.004

5

R. Luciano, J.R. Willis, "Hashin-Shtrikman based FE analysis of the elastic behaviour of finite random composite bodies", International Journal of Fracture, 137(1-4), 261-273, 2006. doi:10.1007/s10704-005-3067-z

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