Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas

Abstract

In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler–Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

Nonlinear dynamics of origami metamaterials: energetic discrete approach accounting for bending and in-plane deformation of facets

“…However, the approach presented in [41,55,56] is affected by some singularities that pose difficulties in computing the structural reaction and the stiffness matrix for some configurations using automatic symbolic computation. In this regard, it is worth to mention in this concluding section that, similarly to what has been done in [57], a tensorial definition of such a relative rotation has been introduced in this paper, which is not biased by in-plane affine deformation of triangles and can get around the singularities affecting the approach presented in [41,55,56].…”

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff’s law

“… 2021 ; Torabi and Niiranen 2023 ) or an ad hoc discretization (Turco et al. 2016 ; Turco 2018 , 2022 ) in a subsequent phase of the development of the model.…”

Nonlinear waves in pantographic beams induced by transverse impulses

“…These materials assume special properties depending on their microstructure. Complex microstructures have been proven to confer remarkable macro-scale properties to metamaterials, like extension-bending/extension-torsion coupling [24][25][26][27], band gap properties for vibration attenuation [28][29][30], extreme stiffness [31], fluid-like behavior [32], extreme elastic range [33,34], damage-tolerant behavior [35], enhanced and/or tailored stability properties [36,37], unidirectional response [38,39], and many others [40]. In particular, it is very interesting the extension of the model and methodology presented in this work to the study of MEMS (Micro Electro-Mechanical Systems) or NEMS (Nano Electro-Mechanical Systems), following, e.g., the suggestions on deformations of flexoelectric pantographic structures reported in [41].…”