# Elastic properties of isotropic discrete systems: Connections between geometric structure and Poisson's ratio

## TL;DR

This paper analytically investigates the Poisson's ratio in isotropic discrete systems, establishing theoretical limits based on geometric configurations and comparing them with numerical results, highlighting deviations due to complex geometries.

## Contribution

It provides the first analytical derivation of Poisson's ratio limits for isotropic discrete models considering arbitrary geometries.

## Key findings

- Theoretical limits depend on contact normal directions and their alignment with connecting vectors.
- Deviations from parallelism reduce the Poisson's ratio limits.
- Numerical results show smaller actual restrictions than theoretical predictions.

## Abstract

The use of discrete material representation in numerical models is advantageous due to the straightforward way it takes into account material heterogeneity and randomness, and the discrete and orientated nature of cracks. Unfortunately, it also restricts the macroscopic Poisson's ratio and therefore narrows its applicability. The paper studies the Poisson's ratio of a discrete model analytically. It derives theoretical limits for cases where the geometry of the model is completely arbitrary, but isotropic in the statistical sense. It is shown that the widest limits are obtained for models where normal directions of contacts between discrete units are parallel with the vectors connecting these units. Any deviation from parallelism causes the limits to shrink. A comparison of the derived equations to the results of the actual numerical model is presented. It shows relatively large deviations from the theory because the fundamental assumptions behind the theoretical derivations are largely violated in systems with complex geometry. The real shrinking of the Poisson's ratio limit is less severe compared to that which is theoretically derived.