Large deflections of inextensible cantilevers: modeling, theory, and simulation

¹ University of Maryland, Baltimore County, MD, USA.
² Carnegie Mellon University, Pittsburgh, PA, USA.

^* Corresponding author: websterj@umbc.edu

Received: 12 October 2019
Accepted: 31 July 2020

Abstract

A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam’s inextensibility – local arc length preservation – rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to: nonlinear stiffness terms, which appear as quasilinear and semilinear effects, as well as nonlinear inertia effects, appearing as nonlocal terms that make the beam implicit in the acceleration. In this paper we discuss the derivation of the equations of motion via Hamilton’s principle with a Lagrange multiplier to enforce the effective inextensibility constraint. We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms present independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight into the features and limitations of the inextensible model.