Nonlinear generation of a zero group velocity mode in an elastic plate by non-collinear mixing

Zero Group Velocity (ZGV) modes are peculiar guided waves that can exist in elastic plates or cylinders, and have proved to be very sensitive tools in characterizing materials or thickness variations with sub-percent accuracy at space resolutions of about the plate thickness. In this article we show theoretically and experimentally how such a mode can be generated as the sum-frequency interaction of two high amplitude primary waves, and then serve as a local probe of material non-linearity. The solutions to the phase matching condition, i.e. condition for a constructive non-linear effect, are obtained numerically in the mark of classical, quadratic non-linearity. The coupling coefficients that measure the transfer rate of energy as a function of time from primary to secondary modes are derived. Experiments are conducted on an aluminum plate using piezo-electric transducers and a laser interferometer, and explore the interaction for incident symmetric and anti-symmetric fundamental Lamb modes. In an experiment operated without voltage amplifier we demonstrate that the resonant nature of these ZGV modes can be leveraged to accumulate energy from long excitations and produce detectable effects at extraordinarily low input power even in such weakly non-linear material.