  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 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, is easily
generalizable to other discrete models, and opens the possibility of a rational
design of elastic systems.
