Invariants of nonlinearity in the phononic characteristics of granular chains

In this work, we analyze the phononic characteristics of wave motion in precompressed monoatomic and diatomic granular chains, with emphasis on the evolving spatial features of wave packets. A Taylor series expansion of the governing equations is considered to approximate the granular chain with the Fermi-Pasta-Ulam chain. Within this approximation, the envelope modulation of the first-order features of the wave profile is monitored and the characteristics of this modulation are determined by studying the evolution of one of the distinctive features of the spatial profile. A set of constants that describe the quantitative effects of nonlinearity on the response are determined for monoatomic and diatomic chains and interpreted as invariants of quadratic nonlinearity. The universality of these invariants is verified by constructing inverse problems to estimate the contact power law from the wave response of granular chains with arbitrary nonlinear force interaction. The imposed power law is recovered exactly from numerical simulations for a number of considered scenarios, paving the way for inverse characterization of nonlinearity from experimental data.