Finite element analysis of functionally graded hyperelastic beams under plane stress
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ORIGINAL ARTICLE
Finite element analysis of functionally graded hyperelastic beams under plane stress João Paulo Pascon1 Received: 9 February 2018 / Accepted: 25 April 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract An alternative finite element formulation to analyze highly deformable beams composed of functionally graded (FG) hyperelastic material is presented. The 2D beam element adopted has a general order and seven degrees of freedom per node, allowing both axial and shear effects, as well as cubic variation of transverse strains. The constitutive law employed is the hyperelastic compressible neo-Hookean model for plane stress conditions. The material coefficients vary along the beam thickness according to the power law. The nonlinear system of equilibrium equations is solved numerically by the Newton– Raphson iterative technique. Full integration scheme and division in load steps are employed to obtain accuracy and stability, respectively. Four illustrative examples involving highly deformable elastic beams under plane stress and static conditions are analyzed: cantilever beam under free-end shear force, semicircular cantilever beam, column under buckling and shallow thin arch. The effects of the mesh refinement, the FG power coefficient and the transverse enrichment scheme on the beam behavior are investigated. In general, mesh refinement provides more accurate results, the power coefficient has a more significant effect on displacements and the second enrichment rate is needed to correctly predict the nonlinear variation of transverse strains across the thickness. This nonlinear variation, together with the moderate values of strains and stresses achieved, reinforces the need of adopting a nonlinear hyperelastic model. Finally, the determination of the out-of-plane strain from the plane components is solved numerically by a proposed Newton’s method. Keywords Highly deformable beams · 2D Finite elements with transverse enrichment · Plane stress neo-Hookean hyperelastic model · Functionally graded materials
1 Introduction Highly deformable beams have been widely used in engineering industries, such as aerospace, energy and robotics. Some applications can be cited: transmission cables, helicopter rotor blades, wind turbine blades, aircraft structures, robot manipulators, among many others [1, 2]. The proper mechanical modeling of these flexible beams is essential along their design process. A common restriction adopted at this stage is that the beam remains in elastic regime, that is, the deformation process is reversible. Small displacements and strains may be assumed in some cases, in which the classical Hooke’s law from solid mechanics suffices. In other * João Paulo Pascon [email protected] 1
Materials Engineering Department, Lorena School of Engineering, University of São Paulo, Pólo‑Urbo Industrial, Gleba AI‑6, s/no, Lorena, SP 12602‑810, Brazil
situations, such as a flexible thin beam, the structure may present finite displacements and rotations, together with relativel
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