Dispersion in Nonlinear Plates: Amplitude Dependent Lamb Waves

This paper presents results from a perturbation-based analysis approach, and accompanying numerical validation, for calculating amplitude-dependent Lamb wave dispersion in nonlinear plates. Nonlinearities considered include those arising from geometric and material nonlinearities. Using a Lindstedt-Poincaré perturbation analysis, nonlinear dispersion relationships are presented in closed form using the partial wave technique. Solvability conditions, based on an operator formalism accompanied by inner product projections against adjoint solutions, yield higher-order dispersion approximations capturing amplitude-dependent Lamb wave propagation. Numerical simulations using a cellular automata approach verify the predicted dispersion shifts for an example nonlinear plate. The analysis and identification of amplitude-dependent, nonlinear Lamb wave dispersion complements recent research focusing on higher harmonic generation and internally resonant waves, which require precise frequency-wavenumber matching, including at large amplitudes.