A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling

Abstract

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell’Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)].

Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators

“…where b i = Y i j i /η i is the lumped bending stiffness of the introduced torsional springs, Y i is the Young modulus of constituting material, and j i is the second moment of area of the link's cross-section. We remark that this model is able to take into account also nonlinear elastic behaviors of the system; indeed, it has been shown ] that such a model, in the homogenized limit, converges to that of a nonlinear beam [Turco 2018;Pietraszkiewicz and Eremeyev 2009;Spagnuolo and Andreaus 2019] being shear undeformable, suitable for the description of problems involving large displacements and large deformations (see [Rosi et al 2018;Placidi et al 2017;Baroudi et al 2019] for more details on methods for obtaining material parameters), and whose deformation energy density depends only upon the exact curvature. Note that the linearized form of (9) is simply a quadratic form in the relative angle ϕ i j .…”

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“…where b i = Y i j i /η i is the lumped bending stiffness of the introduced torsional springs, Y i is the Young modulus of constituting material, and j i is the second moment of area of the link's cross-section. We remark that this model is able to take into account also nonlinear elastic behaviors of the system; indeed, it has been shown ] that such a model, in the homogenized limit, converges to that of a nonlinear beam [Turco 2018;Pietraszkiewicz and Eremeyev 2009;Spagnuolo and Andreaus 2019] being shear undeformable, suitable for the description of problems involving large displacements and large deformations (see [Rosi et al 2018;Placidi et al 2017;Baroudi et al 2019] for more details on methods for obtaining material parameters), and whose deformation energy density depends only upon the exact curvature. Note that the linearized form of (9) is simply a quadratic form in the relative angle ϕ i j .…”

“…However, from a computational point of view, this method requires fewer mathematical operations; therefore, it is particularly suited for dynamic model-based online controller implementations [Theodore and Ghosal 1995]. Although the finite element method can be identified as a different version of the assumed modes method, it can be generalized to be used in a wider context, in particular, when nonlinear effects arise as for a multilink manipulator [Sharf 1996;Eugster et al 2014;Luongo and D'Annibale 2013]. To address some issues related to failures in convergence that are occasionally experienced, some authors have proposed a mixed formulation, based on both stress and displacement degrees of freedom, which appears very promising in this respect [Hodges 1990;Garcea et al 1998].…”

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

“…The literature in which the theory of Euler–Bernoulli beam theory and/or its applications are studied is so huge that it is probably impossible to give a reasonable account for it. Among the many relevant problems which have been studied with great details and care we, simply as they are instrumental to our investigations, cite [].…”

“…The literature in which the theory of Euler-Bernoulli beam theory and/or its applications are studied is so huge that it is probably impossible to give a reasonable account for it. Among the many relevant problems which have been studied with great details and care we, simply as they are instrumental to our investigations, cite [8,13,62,71,94,102,105]. However, the importance of the model is so great that there are still left interesting and practically significant problems to be studied anew or whose study needs to be completed.…”