Trapped modes in bent elastic rods

We investigate the existence of trapped modes in elastic rods of constant circular cross-section that possess bends of arbitrary curvature and straighten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. An asymptotic model assuming smallness of dimensionless curvature is developed to describe the trapping. Existence conditions depending on Poisson's ratio are offered, and the equations from which they derive are numerically validated. A physical explanation of why trapped modes should be expected is also given.