The Acoustic Black Hole effect (ABH) is a passive vibration damping technique without added mass based on flexural waves properties in thin structures with variable thickness. The usual implementation on plates is a region where the thickness is reduced with a power law profile, covered with a visco-elastic layer. The inhomogoneity induces a decrease of the wave speed and an increase of the amplitude in the small thickness region, which makes the energy dissipation more efficient due to the absorbing layer. The wave amplitude in the ABH easily reaches the plate thickness and is the origin of geometrical nonlinearities. These nonlin-earities can generate coupling between linear beam eigenmodes of the structure and induce energy transfer between low and high frequency regime. This phenomenon may be used to increase the efficiency of the ABH treatment in the low frequency regime where it is usually inefficient. An experimental investigation shows that the ABH termination displays a nonlinear behaviour and allows for modal coupling. A strongly nonlinear regime can also be observed, which is associated with Wave Turbulence. A model of nonlinear ABH beam as von Kármán plate of variable thickness and a modal resolution of the problem confirm the observed effects and gives more insights on these results.