Predicting the mechanical properties of spring networks

The elastic response of mechanical, chemical, and biological systems is often modeled using a discrete arrangement of Hookean springs, representing either 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, with arbitrary geometry, and its 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 "nonaffine"displacements. Explicit comparison of our calculations to simulations of different crystalline and disordered configurations shows that we successfully capture the mechanics even of auxetic materials. Our method is valid for residually stressed systems with nontrivial geometries, and is an important 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.