Trapped modes in curved elastic plates

Abstract

We investigate the existence of trapped modes in elastic plates of constant thickness, which possess bends of arbitrary curvature and flatten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. We present both an asymptotic model and numerical evidence to demonstrate the trapping. In the asymptotic analysis we utilize a dimensionless curvature as a small parameter, whereas the numerical model is based on spectral methods and is free of the small-curvature limitation. The two models agree with each other well in the region where both are applicable. Simple existence conditions depending on Poison's ratio are offered, and finally, the effect of energy build-up in a bend when the structure is excited at a resonant frequency is demonstrated.