A size-dependent nonlinear Euler–Bernoulli beam is considered in the framework of the nonlocal strain gradient theory. The geometric nonlinearity due to the stretching effect of the mid-plane of the ...size–dependent beam is considered here. The governing equations and boundary conditions are derived by employing the Hamilton principle. The post-buckling deflections and critical buckling forces of simply supported size-dependent beams are analytically derived. The derived results are compared with those of strain gradient theory, nonlocal elasticity theory and classical elasticity theory. It is found that the post-buckling deflections can be increased by increasing the nonlocal parameter or decreasing the material characteristic parameter. The high-order buckling deflections are more sensitive to size-dependent parameters than the low-order buckling deflections. Furthermore, the critical buckling force can be increased by decreasing the nonlocal parameter when the nonlocal parameter is larger than the material characteristic parameter, or increasing the nonlocal parameter when the nonlocal parameter is smaller than the material characteristic parameter.
In the present work, we propose a novel implementation of arc-length method for the computation of equilibrium paths for frame-like structures. We exploit the advantages of velocity-based beam ...formulations and adapt it to quasi-static analysis with path following control. We show that such formulation allows the arc-length control constraint equation to be directly employed in its originally defined differential form, while the arc-length parameter is replaced by time in our adaptation. This simplifies the treatment of the extended system of equations and its solution procedure. Our approach also enables the rotational degrees of freedom to be members of control parameters. The ability of the proposed method to successfully and efficiently compute complex equilibrium paths in post-buckling regime is demonstrated by several numerical examples.
•Our approach is based on temporal derivatives of configuration variables.•Computational procedures are based only on additive quantities.•Path following for velocity-based formultions is proposed.•The arc-length constraint is employed directly in its original differential form.•The rotational degrees of freedom are included in the path-following constraint.
A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic ...structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post-buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.
The conventional harmonic balance method utilizes a numerical approach for solving a set of nonlinear algebraic equations to identify the unknown coefficients. In this research, a modified harmonic ...balance method has been represented to study a free vibration problem with an axial load. The success of this research is that it requires the solution of a single nonlinear algebraic equation and a system of linear algebraic equations to obtain the desired results. As a result, it takes less effort to compute than the traditional harmonic balance technique. To validate and justify the accuracy of the proposed approach, the acquired outcomes are compared to those obtained using the fourth-order Runge–Kutta technique. The results acquired from the proposed process nicely agree with the numerical results.
We propose a novel approach to the linear viscoelastic problem of shear-deformable geometrically exact beams. The generalized Maxwell model for one-dimensional solids is here efficiently extended to ...the case of arbitrarily curved beams undergoing finite displacement and rotations. High efficiency is achieved by combining a series of distinguishing features, that are: (i) the formulation is displacement-based, therefore no additional unknowns, other than incremental displacements and rotations, are needed for the internal variables associated with the rate-dependent material; (ii) the governing equations are discretized in space using the isogeometric collocation method, meaning that elements integration is totally bypassed; (iii) finite rotations are updated using the incremental rotation vector, leading to two main benefits: minimum number of rotation unknowns (the three components of the incremental rotation vector) and no singularity problems; (iv) the same SO(3)-consistent linearization of the governing equations and update procedures as for non-rate-dependent linear elastic material can be used; (v) a standard second-order accurate time integration scheme is made consistent with the underlying geometric structure of the kinematic problem. Moreover, taking full advantage of the isogeometric analysis features, the formulation permits accurately representing beams and beam structures with highly complex initial shape and topology, paving the way for a large number of potential applications in the field of architectured materials, meta-materials, morphing/programmable objects, topological optimizations, etc. Numerical applications are finally presented in order to demonstrate attributes and potentialities of the proposed formulation.
•A new mathematical model is developed to study the contact interactions (structural nonlinearity (SN)) of nano- and micro-electro-mechanical (NEMS/MEMS) beam resonators, both elastic and physically ...nonlinear (FN) materials.•Principal component analysis (PCA) and wavelet analysis are applied to remove noise from the signals.•Various beam interaction scenarios are classified, including static Euler and Rayleigh instabilities, as well as Koning-Taub and Richtmeyer-Meshkov instabilities.•The chaotic behaviour of nonlinear vibrations in the contact interaction of these closely spaced beams is investigated.
In this study, a new mathematical model is developed to study the contact interactions of nano- and micro-electro-mechanical (NEMS/MEMS) beam resonators. The structural elements are considered as porous, size-dependent Euler-Bernoulli beams subjected to a variable transverse load. The beams resonators are located one above the other with minimal clearance. The analysis includes interactions between beams of both linear elastic and physically nonlinear materials, following the approach of Professor B. Ya. Kantor. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. Hamilton's principle is applied to derive new size-dependent equations of motion together with the corresponding boundary and initial conditions. This study presents a new approach to the analysis of the chaotic dynamics of the contact interaction of porous, size-dependent Euler-Bernoulli beams. They are considered as systems with an "almost" infinite number of degrees of freedom. This innovative approach uses Principal Component Analysis (PCA) and Wavelet Analysis to filter out noise from signals. In addition, different beam interaction scenarios are classified including static Euler and Rayleigh instabilities as well as interactions between Koning-Taub and Richtmeyer-Meshkov instabilities. A significant discovery of this investigation is the chaotic behaviour of nonlinear vibrations during the contact interaction of these closely positioned beams. This demonstrates the complex dynamics involved in these high technology mechanical systems.
We extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any ...restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.
•Isogeometric collocation is extended to geometrically exact beams.•Consistent linearization of the strong form of the governing equations is derived.•Incremental rotations are parametrized through Eulerian rotation vectors.•Numerical tests show efficiency and high accuracy.
The general form of Reissner stationary variational principle is established in the framework of the nonlocal strain gradient theory of elasticity. Including two size-dependent characteristic ...parameters, the nonlocal strain gradient elasticity theory can demonstrate the significance of the strain gradient as well as the nonlocal elastic stress field. Based on the Reissner functional, the governing differential and boundary conditions of dynamic equilibrium and differential constitutive equations of the classical and first-order nonlocal stress tensor are derived in the most general form. Additionally, the boundary congruence conditions are formulated and discussed for the nonlocal strain gradient theory. To exhibit the application value of Reissner variational principle, it is employed to examine the nonlinear vibrations of size-dependent Bernoulli-Euler and Timoshenko beams. In the case of immovable boundary conditions, employing the weighted residual Galerkin method, the homotopy analysis method is also utilized to determine the closed form analytical solutions of the geometrically nonlinear vibration equations. Consequently, the analytical expressions for the nonlinear natural frequencies of Bernoulli-Euler and Timoshenko nonlocal strain gradient beams are derived.
•Reissner variational principle is established for nonlocal strain gradient theory.•Governing equations are derived in general form based on the Reissner principle.•Boundary congruence conditions are formulated and comprehensively discussed.•Nonlinear vibrations of Bernoulli-Euler and Timoshenko beams are examined.•Analytical solutions for the nonlinear fundamental frequencies are presented.
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).
Reissner mixed variational principle is employed for establishment of the nonlinear differential and boundary conditions of dynamic equilibrium governing the flexure of beams when the effects of true ...shear stresses are included. Based on the Reissner mixed variational principle, the nonlinear size-dependent model of the Reissner nano-beam is derived in the framework of the nonlocal strain gradient elasticity theory. Furthermore, the closed form analytical solutions for the geometrically nonlinear flexural equations are derived and compared to the nonlinear flexural results of the Timoshenko size-dependent beam theory. The profound differences in the assumptions and formulations between the Timoshenko and the Reissner beam theory are also comprehensively discussed. The Reissner beam model is shown not to be a first-order shear deformation theory while comprising the influences of the true transverse shearing stress and the applied normal stress. Moreover, it is exhibited that the linear and nonlinear deflections obtained based on the Reissner beam theory are consistently lower than their Timoshenko counterparts for various gradient theories of elasticity.
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