Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality

In a bounded domain Ω⊂R3Ω⊂R3 we consider a discrete network of a large number of concentrated masses (particles) connected by elastic springs. We provide sufficient conditions on the geometry of the array of particles, under which the network admits a rigorous continuum limit. Our proof is based on the discrete Korn's inequality. Proof of this inequality is the key point of our consideration. In particular, we derive an explicit upper bound on the Korn's constant. For generic non-periodic arrays of particles we describe the continuum limit in terms of the local energy characteristic on the mesoscale (intermediate scale between the interparticle distances (small scale) and the domain sizes (large scale)), which represents local energy in the neighborhood of a point. For a periodic array of particles we compute coefficients in the limiting continuum problems in terms of the elastic constants of the springs.