Non-local energetics of random heterogeneous lattices

TL;DR

This paper develops explicit energetic bounds for two-phase elastic lattices with non-uniform heterogeneity, improving accuracy over first-order methods through second-order statistical analysis and validated by numerical case studies.

Contribution

It introduces second-order statistical bounds for non-uniform elastic lattices, enhancing the accuracy of energetic estimates compared to first-order approaches.

Findings

Explicit bounds are computationally feasible.

Second-order statistics improve accuracy over first-order methods.

Numerical case studies confirm the effectiveness of the approach.

Abstract

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53, 1505-1522, 2005), energetic bounds and estimates of the Hashin-Shtrikman-Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics.