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A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling. (English) Zbl 1425.74267

Summary: 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 [F. dell’Isola et al., “Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium”, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 472, No. 2185, Article ID 20150790, 23 p. (2016; doi:10.1098/rspa.2015.0790); Z. Angew. Math. Phys. 66, No. 6, 3473–3498 (2015; Zbl 1395.74002); E. Turco et al., ibid. 67, No. 4, Article ID 85, 28 p. (2016; Zbl 1432.74158)].

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G60 Bifurcation and buckling
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
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