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 stiff- ness 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.
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Theory of Solutions for an Inextensible Cantilever
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.
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- Award ID(s):
- 1907620
- NSF-PAR ID:
- 10285199
- Date Published:
- Journal Name:
- Applied Mathematics & Optimization
- ISSN:
- 0095-4616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00245-021-09798-0
