Dynamic Equations of Motion for Inextensible Beams and Plates

Abstract

The large deflections of cantilevered beams and plates are modeled and discussed. Traditional nonlinear elastic models (e.g., that of von Karman) employ elastic restoring forces based on the effect of stretching on bending, and these are less applicable to cantilevers. Recent experimental work indicates that elastic cantilevers are subject to nonlinear inertial and stiffness effects. We review a recently established (quasilinear and nonlocal) cantilevered beam model, and consider some natural extensions to two dimensions -- namely, inextensible plates. Our principal configuration is that of a thin, isotropic, homogeneous rectangular plate, clamped on one edge and free on the remaining three. We proceed through the geometric and elastic modeling to obtain equations of motion via Hamilton's principle for the appropriately specified energies. We enforce {\em effective} inextensibility constraints through Lagrange multipliers. Multiple plate analogs of the established 1D model are obtained, based on various assumptions. For each plate model, we present the modeling hypotheses and the resulting equations of motion. It total, we present three distinct nonlinear partial differential equation models, and, additionally, describe a class of ``higher order" models. Each model has particular advantages and drawbacks for both mathematical and engineering analyses. We conclude with an in depth discussion and comparison of the various systems and some analytical problems.