Theory of Solutions for An Inextensible Cantilever

TL;DR

This paper develops a mathematical theory for the large deflections of an inextensible cantilever beam, incorporating nonlinear effects, damping, and proving well-posedness and exponential decay of solutions.

Contribution

It introduces a novel analysis of inextensible cantilever dynamics, including existence, uniqueness, and decay properties of solutions with nonlinear and nonlocal effects.

Findings

Smooth solutions constructed via spectral Galerkin method

Global well-posedness with damping and small data

Exponential decay of solutions over time

Abstract

Recent equations of motion for the large deflections of a cantilevered elastic beam are analyzed. In the traditional theory of beam (and plate) large deflections, nonlinear restoring forces are due to the effect of stretching on bending; for an inextensible cantilever, the enforcement of arc-length preservation leads to quasilinear stiffness effects and inertial effects that are both nonlinear and nonlocal. For this model, smooth solutions are constructed via a spectral Galerkin approach. Additional compactness is needed to pass to the limit, and this is obtained through a complex procession of higher energy estimates. Uniqueness is obtained through a non-trivial decomposition of the nonlinearity. The confounding effects of nonlinear inertia are overcome via the addition of structural (Kelvin-Voigt) damping to the equations of motion. Local well-posedness of smooth solutions is shown first in the absence of nonlinear inertial effects, and then shown with these inertial effects present, taking into account structural damping. With damping in force, global-in-time, strong well-posedness result is obtained by achieving exponential decay for small data.