Nonlinear Problems of Elasticity

Antman, Stuart S.

doi:10.1007/978-1-4757-4147-6

Cited by 889 publications

(923 citation statements)

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“…In fact, the cables under only one vertical load chosen in one case for illustrating the concept of tangent stiffness matrix is incapable of exhibiting the configurational displacements under (only proportional) load increments (Altin, et al, 2012). Of course, the kinematic description of the deformed state of the cable in terms of position vector as a function of the Lagrangian coordinate (Luongo, et al, 1984;Antman, 2005;Locarbonara, 2013;Greco et al, 2014) includes, perhaps unknowingly so, the configurational response in addition to the effect of finite elastic displacements.…”

Section: Discussionmentioning

confidence: 99%

“…Kinematical description is based upon the position vector as a function of the Lagrangian coordinate along the cable length and the assumed constitutive equation relates the local axial tension with the local stretch (Luongo et al, 1984, Antman, 2005Lacarbonara, 2013). In the incremental or updated Lagrangian formulations, there exists a diversity of approaches for establishing the tangent stiffness of the cable.…”

Section: Introductionmentioning

confidence: 99%

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Mechanics of Cable-Suspended Beams

Kumar

Ganguli

Benipal

2017

Lat. Am. j. solids struct.

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The cable-suspended bridges differ from the elastic structures because of the inherent nonlinearity of the suspension cables. The primary focus of the available theories is to investigate the effect of nonlinearities associated with the distributed self-weight of the cable and its finite elastic displacements. The main point of departure of this paper is to study the effect of the configurational nonlinearity of the weightless linear elastic suspension cables undergoing small elastic displacements. Based upon authors' theory of weightless elasto-flexible planar sagging cables, a new Beam with Elasto-Flexible Support (BEFS) model of the cable-deck interaction is proposed here. Rate-type constitutive equations and third order differential equations of motion are derived for a simple four-node cable-suspended beam structure undergoing small elastic vertical displacements. Static and dynamic analysis of the cable-suspended structures is carried out to reveal their characteristic configurational response. The proposed theory is critically evaluated in the context of existing theories of suspension bridges.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mechanics of Cable-Suspended Beams

Kumar

Ganguli

Benipal

2017

Lat. Am. j. solids struct.

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“…Using this known frame R(s; R 0 ) as well as the given material shear strains v(s) we obtain the centerline curve by means of integrating (8) with the final result r(s) = r 0 + s sc R(s; R 0 ) · v(s) ds, which indicates how the solution s → (r(s), R(s)) depends parametrically on rigid body motions (r 0 , R 0 ) as integration constants. More specifically one can show that for given (u, v) any two solutions differ by at most a rigid body motion (see [11], Chapter 8.6 for a detailed proof). Remark 1.…”

Section: Strain Measuresmentioning

confidence: 99%

A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

Jung

Leyendecker

Linn

et al. 2010

Numerical Meth Engineering

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A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null‐space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so‐called symmetries. The symmetries relevant for rod mechanics are frame‐indifference, isotropy, and uniformity. We show that a completely analogous and self‐contained theory of discrete rods can be formulated in which the arc‐length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc‐length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright © 2010 John Wiley & Sons, Ltd.

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“…Antman (1995aAntman ( , 1995b presents an application of the Cosserat theory to describe the behavior of rods. This author states that the Cosserat theory presents several advantages with respect to others used in structural mechanics, as it is not based on specific geometrical approximations or mechanical assumptions.…”

Section: Introductionmentioning

confidence: 99%

Time Domain Modeling and Simulation of Nonlinear Slender Viscoelastic Beams Associating Cosserat Theory and a Fractional Derivative Model

Borges

Faria

et al. 2017

Lat. Am. j. solids struct.

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A broad class of engineering systems can be satisfactory modeled under the assumptions of small deformations and linear material properties. However, many mechanical systems used in modern applications, like structural elements typical of aerospace and petroleum industries, have been characterized by increased slenderness and high static and dynamic loads. In such situations, it becomes indispensable to consider the nonlinear geometric effects and/or material nonlinear behavior. At the same time, in many cases involving dynamic loads, there comes the need for attenuation of vibration levels. In this context, this paper describes the development and validation of numerical models of viscoelastic slender beam-like structures undergoing large displacements. The numerical approach is based on the combination of the nonlinear Cosserat beam theory and a viscoelastic model based on Fractional Derivatives. Such combination enables to derive nonlinear equations of motion that, upon finite element discretization, can be used for predicting the dynamic behavior of the structure in the time domain, accounting for geometric nonlinearity and viscoelastic damping. The modeling methodology is illustrated and validated by numerical simulations, the results of which are compared to others available in the literature.

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Nonlinear Problems of Elasticity

Abstract: Library of Congress Cataloging-in-Publication Data Antman, S.S. (Stuart S.) Nonlinear problems of elasticity / Stuart S. Antman. p. cm. -(Applied mathematical sciences ; v. 107) IncIudes bibliographical references and index.

Cited by 889 publications

References 0 publications

Mechanics of Cable-Suspended Beams

Mechanics of Cable-Suspended Beams

A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

Time Domain Modeling and Simulation of Nonlinear Slender Viscoelastic Beams Associating Cosserat Theory and a Fractional Derivative Model

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