Stability analysis of gradient elastic microbeams with arbitrary boundary conditions
09 Aug 2015-Journal of Mechanical Science and Technology (Korean Society of Mechanical Engineers)-Vol. 29, Iss: 8, pp 3373-3380
TL;DR: Based on gradient elasticity theory with surface energy, a simple and unified method is presented for the stability analysis of a generally supported microbeam in this paper, which conveniently computes an accurate buckling parameter for microbeams using both classical and non-classical boundary conditions restrained by translational and rotational springs.
Abstract: Based on gradient elasticity theory with surface energy, a simple and unified method is presented for the stability analysis of a generally supported microbeam. The proposed method conveniently computes an accurate buckling parameter for microbeams using both classical and non-classical boundary conditions restrained by translational and rotational springs. The Fourier coefficient and fundamental relations of strain gradient beams are obtained first. Stokes’ transformation is applied to transform these equations into a set of algebraic equations with buckling parameter. The derived expressions can be useful for theoretical investigation that leads to a determinant calculation of a 4 × 4 matrix. The critical buckling loads of microbeams for variant scale parameters under different boundary conditions are computed using the proposed method. Comparing results with those in the literature validates the present analysis.
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TL;DR: In this paper, the buckling of elastically restrained embedded microbeams under axial compression load is investigated and a coefficient matrix is obtained with the aid of applying Stokes' transformation to corresponding boundary conditions.
Abstract: The buckling of elastically restrained embedded microbeam under axial compression load is researched. The effects of small size, axial compression load and surrounding elastic medium are taken into account at the same time. Winkler elastic foundation approach is used to simulate the interaction between microbeam and elastic medium. Fourier sine series is employed for the simulation of microbeam deflections. A coefficient matrix is obtained with the aid of applying Stokes' transformation to corresponding boundary conditions. The buckling characteristics of elastically restrained embedded microbeams are investigated in some numerical examples. There are very good agreements between this study and the previous results indicating the validity of the presented method.
43 citations
TL;DR: In this paper, a mathematical model is developed to investigate a vibrational behavior of functionally graded (FG) cracked microbeam rested on an elastic foundation and exposed to thermal and magnetic fields.
41 citations
TL;DR: In this article, buckling analysis of silicon carbide nanowires has been investigated including size effect by using different size-dependent continuum theories including modified couple stress theory, modified strain gradient theory, nonlocal elasticity theory, surface elasticity theories, and nonlocal surface linearity theory.
Abstract: In the present paper, buckling analysis of silicon carbide nanowires has been investigated including size effect. The size effect has been taken into consideration by using different size-dependent continuum theories. These theories are modified couple stress theory, modified strain gradient theory, nonlocal elasticity theory, surface elasticity theory, and nonlocal surface elasticity theory. Analyses have been made for a continuum model which is embedded in double-parameter elastic foundation. The foundation has been modeled by using both Winkler- and Pasternak-type elastic foundation models. Simply supported boundary conditions have been used. Buckling equations have been obtained by using energy principle and solved via Navier’s solution procedure. Results are given and compared in figures and tables.
32 citations
TL;DR: In this article, the free vibration and buckling behaviors of a Timoshenko functionally graded nanobeam under to thermal and magnetic environment were investigated using nonlocal strain gradient theory, where the gradation of material properties throughout the beam thickness is described by power-law function.
Abstract: Due to the increasing usage of nanostructures in nanotechnology and nanodevice, the following article aims to investigate the free vibration and buckling behaviours of a Timoshenko functionally graded nanobeam under to thermal and magnetic environment. The nanoscale and microstructure effects of FG nanobeam are included to classical mechanics using nonlocal strain gradient theory. The gradation of material properties throughout the beam thickness is described by power-law function. The material properties are assumed to be temperature dependent. Considering the thermal and Lorentz forces, the equations of motion of the functionally graded Timoshenko nanobeam are obtained using the strain gradient and nonlocal elasticity theories. The transverse Lorentz force induced by the horizontal magnetic field vector is derived using Maxwell’s equations. External compressive axial and transverse point loads are included in the formulation and the motion equations are solved using a Navier-type approach. The effects nonlocal size scale, and strain gradient microstructure influence, thermal loadings and magnetic field intensities on the free vibration, transverse bending and buckling behaviours of the functionally graded nanobeam are presented. The following model can be used as benchmark to analyse the nanobeam structure under thermomagnetic field using a finite element or any other numerical method.
32 citations
01 Aug 2020-Microsystem Technologies-micro-and Nanosystems-information Storage and Processing Systems
TL;DR: In this paper, the free axial vibration of Rayleigh nanorods with axial restraints is studied via Eringens' nonlocal elasticity theory, which takes into account the size effect into the formulation due to dealing with micro and nanostructures.
Abstract: In this study, the free axial vibration of Rayleigh nanorods with axial restraints is studied via Eringens’ nonlocal elasticity theory. This higher order elasticity theory takes into account the size effect into the formulation due to dealing with micro and nanostructures. The boundary conditions and equation of motion are obtained using Hamilton’s principle. Two symmetrical axial elastic springs are attached to a nanorod at both ends. The novelty of the present study is that it seeks to obtain a general eigen value algorithm for the angular frequencies subjected to the rigid or restrained boundary conditions in a nanorod for the first time. A Fourier sine series is used to work Stokes’ transformation for the Rayleigh nanorods with elastic springs at the ends. Afterward, the effect of the spring coefficient on the the eigen-frequency is investigated. Also, the effects of the nonlocal parameter and the elastic springs on the eigen-frequency is reported.
20 citations
References
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TL;DR: In this paper, an equilibrium relation is developed to govern the behavior of the couples, which constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system.
2,725 citations
TL;DR: HAL as discussed by the authors is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Elastic materials with couple-stresses R. Toupin
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2,228 citations
TL;DR: In this article, a continuum theory of non-local polar bodies is developed for nonlinear micromorphic elastic solids, and the balance laws and jump conditions are given.
1,788 citations
TL;DR: In this paper, a microstructure-dependent Timoshenko beam model is developed using a variational formulation, which is based on a modified couple stress theory and Hamilton's principle.
Abstract: A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.
995 citations