This paper introduces a novel approach for constructing, at the same scale, a continuum model equivalent to a given nanoscale discrete system, effectively capturing scale effects. Starting from a general formulation of $m$ -particle interaction potentials for discrete particle systems, we propose the use of the Euler–Maclaurin (E–M) summation formula to construct the equivalent continuum model at the same scale. The proposed theory is developed for arbitrary 3D domains. The resulting novel continuum model captures scale effects in both statics and dynamics through additional edge, surface, and volume integrals, which are analytically obtained and driven by the nanostructure. For an arbitrary domain, the proposed approach provides a pathway for its integration into a computational framework. Since nanoscale and microscale systems are inevitably affected by uncertainties, the geometric and constitutive parameters must be modeled as random fields and their identification of must be conducted within a statistical framework. Consequently, we present a discussion on this identification in a probabilistic framework.
