Elastic properties of central-force networks with bond-length mismatch

We study a triangular network containing two kinds of Hooke springs with different natural lengths. If the two spring constants are the same, we can solve the model exactly and show that Vegard’s law is obeyed, irrespective of whether the bonds are arranged randomly or in a correlated way. A more complete description of these networks is obtained through the mean lengths, the length fluctuations, and the strain energy. The complete distribution of bond lengths is obtained numerically and shows an interesting and unexpected symmetry for the random case. Finally we show that numerical results for a similar system, but with different force constants as well as different natural lengths, can be well accounted for by using an effective-medium theory that reduces to the exact results when the two spring constants are made equal. These lattices can be described very accurately up to about 50% length mismatches, when ‘‘pleating’’ occurs and the lattices develop local instabilities.