Large Deflections of Inextensible Cantilevers: Modeling, Theory, and Simulation

TL;DR

This paper develops a comprehensive model for large deflections of inextensible cantilevers, incorporating nonlinear effects due to inextensibility, and provides theoretical analysis and numerical simulations to understand their behavior.

Contribution

It introduces a novel derivation of the equations of motion using Hamilton's principle with inextensibility constraints and analyzes existence and uniqueness of solutions.

Findings

Nonlinear stiffness and inertia effects are characterized.

Existence and uniqueness of strong solutions are established.

Numerical simulations reveal features and limitations of the model.

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 prevent 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 to the features and limitations of the inextensible model.