Theory of Solutions for an Inextensible Cantilever

Deliyianni, Maria; Webster, Justin T.

doi:10.1007/s00245-021-09798-0

Cited by 9 publications

(7 citation statements)

References 30 publications

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“…In this section we state recent theoretical results about strong solutions to (1.1). The proofs of these theorems appear in [10], with an effort to have a streamlined presentation of the underlying modeling and theory supporting the numerical simulations below. We begin with a simple well-posedness result for the nonlinear stiffness portion of the model.…”

Section: Well-posedness Resultsmentioning

confidence: 99%

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

Deliyianni¹,

Gudibanda²,

Howell³

et al. 2019

Preprint

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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.

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Section: Well-posedness Resultsmentioning

confidence: 99%

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

Deliyianni¹,

Gudibanda²,

Howell³

et al. 2019

Preprint

Self Cite

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“…• The first point of departure from the beam theory presented in Section 2 is the emergence of nonlinear boundary conditions (Section 3.3.3); this is a necessary byproduct of invoking Hamilton's principle with the chosen potential energy (very similar to what occurs in [23] in an extensible situation). For any future theory of existence and uniqueness of solutions for the inextensible plate mirroring that of the beam [11], nonlinear boundary conditions will be a central issue. This is especially true, as the nonlinear boundary conditions emerge in the notoriously troublesome higher-order plate conditions, and differ markedly from the linear theory.…”

Section: Discussion and Future Workmentioning

confidence: 99%

“…Recently, for the intextensible cantilever as derived in [14], a mathematical theory of solutions has been developed [11,12]. In the present manuscript, we aim to take an analogous first step in this direction, by providing the PDE equations of motion for the "natural" beam extension to a 2D inextensible plate.…”

Section: Modeling and Analysis Of Cantilever Large Deflectionsmentioning

confidence: 99%

“…In future work, we will numerically analyze these models to compare against [42,43]. Moreover, the theory developed in [11] will be adapted, at least in the cases where the 2D equations of motion most closely resemble those of the inextensible cantilevered beam in [11,14]. Indeed, one principal goal in this work is to address inextensibility in the standard coordinate framework-which is to say, to extend standard linear beam theory, rather than using more sophisticated geometric models (such as those of [2]).…”

Section: Modeling and Analysis Of Cantilever Large Deflectionsmentioning

confidence: 99%

“…To provide context for the approach we take to inextensible plates, we first give a summary of the work in [11,14] for the recent inextensible cantilevered beams in Section 2.…”

Section: Outline Of the Remainder Of The Papermentioning

confidence: 99%

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Dynamic Equations of Motion for Inextensible Beams and Plates

Deliyianni,

McHugh,

Webster

et al. 2021

Preprint

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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 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.

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