Predicting the mechanical properties of spring networks

TL;DR

This paper introduces a method to derive the exact elastic continuum model from any discrete spring network using only its geometry and topology, enabling efficient analysis of complex systems.

Contribution

The authors present a novel approach to directly connect discrete spring networks with their continuum elastic models, including non-affine displacements, applicable to complex and stressed geometries.

Findings

Successfully captures mechanics of crystalline and disordered networks

Valid for systems with residual stress and complex geometries

Enables rational design of elastic materials

Abstract

The elastic response of mechanical, chemical, and biological systems is often modeled using a discrete arrangement of Hookean springs, either representing finite material elements or even the molecular bonds of a system. However, to date, there is no direct derivation of the relation between a general discrete spring network\blu{, with arbitrary geometry,} and it's corresponding elastic continuum. Furthermore, understanding the network's mechanical response requires simulations that may be expensive computationally. Here we report a method to derive the exact elastic continuum model of any discrete network of springs, requiring network geometry and topology only. We identify and calculate the so-called "non-affine" displacements. Explicit comparison of our calculations to simulations of different crystalline and disordered configurations, shows we successfully capture the mechanics even of auxetic materials. Our method is valid for residually stressed systems with non-trivial geometries, and is an essential step in generalizing active stresses on such discrete systems. It is easily generalizable to other discrete models, and opens the possibility of a rational design of elastic systems.