Strain-gradient theory of hydroelastic travelling waves and young measures of their singular limits

Abstract

This is a mathematical study of steady, two-dimensional, periodic waves which propagate without changing shape on the surface of an infinitely deep fluid that moves under gravity, bounded above by a heavy, frictionless, thin (unshearable) elastic sheet. The sheet stays in contact with the zero streamline of the flow and is deformed by the fluid pressure, internal elastic forces and couples, and inertial forces and gravity, according to the laws of elasticity. The flow, which is irrotational, is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave. Therefore the pressure exerted by the fluid at a steady streamline depends on its height and the fluid velocity. The surface sheet is hyperelastic with a stored energy function that depends on the stretch, strain-gradient, and curvature. This paper studies the balancing of these elastic and hydrodynamic effects to produce a steady hydroelastic wave. It is notable that the stored energy for travelling waves is non-convex in the stretch even when the stored energy of the material at rest is convex. First, the existence of waves is proved by maximizing a Lagrangian in the presence of regularizing strain-gradient effects. Then the limit as the coefficient \({\varepsilon}\) of the strain-gradient term tends to zero is investigated. Finally, for \({\varepsilon = 0}\) we give a general variational principle, for wave profiles and Young measures, that is maximized by limits of maximizers of the regularized problems. The main result is a statement of the free-boundary problem satisfied by generalized maximizers, and a detailed description of the Young measures that arise as limits of a sequences of maximizers of problems regularized by strain-gradient effects.