  A mathematical continuum limit of the interaction energy of a random particle
chain is shown to yield new insight into the effect of microscopic
heterogeneities on macroscopic fracture laws in brittle materials. We derive a
formula which yields that either elastic behaviour or a crack is energetically
preferred. The formula explicitly shows the dependence on the boundary
condition and the microstructure of the chain. The mathematical analysis is
based on a variational convergence $\Gamma$-convergence of convex-concave
potentials together with ergodic theorems which are common tools in stochastic
homogenization.
