סמינר בחומר מעובה: A first first-principle derivation of elasticity

The elastic response of mechanical, chemical, and biological systems is often modeled using a discrete arrangement of Hookean springs, either representing finite aterial 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 residual stresses, 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 and to any possible loading scheme, and 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 nontrivial geometries, and is an essential step in any accurate continuous description of multi-element elastic systems, including active ones. Since the use of spring networks is ubiquitous in many fields, this works is highly relevant to engineers, physicists, chemists, and biologist. The formulation can be easily generalized to other discrete models, and opens the possibility of a rational design of elastic systems.