scholarly journals Analytic solution for size-dependent behaviors of micro-beam under forced vibration

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Shuai WANG ◽  
Zhiyong WANG ◽  
Feifei WANG ◽  
Bo ZHOU ◽  
Shifeng XUE
Keyword(s):  
Forced Vibration ◽  
Couple Stress Theory ◽  
Separation Of Variables ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Mechanical Behaviors ◽  
Governing Equations ◽  
Euler Beam ◽  
Size Dependent ◽  
Material Length Scale

This paper focuses on the size-dependently mechanical behaviors of a micro-beam under forced vibration. Governing equations of a micro-beam under forced vibration are established by using the modified couple stress theory, Bernoulli-Euler beam theory and D’Alembert’s principle together. A simply supported micro-beam under forced vibration is solved according to the established governing equations and the method of separation of variables. The dimensionless deflection, amplitude mode and period mode are defined to investigate the size-dependently mechanical behaviors of a micro-beam under forced vibration. Results show that the performance of a micro-beams under forced vibration is distinctly size-dependent when the ratio of micro-beam height to material length-scale parameter is small enough. Both frequency ratio and loading location are the important factors that determine the size-dependent performance of a micro-beams under forced vibration.

scholarly journals Modeling size-dependent behaviors of axially functionally graded Bernoulli-Euler micro-beam

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Shuai WANG ◽  
Zetian KANG ◽  
Shichen ZHOU ◽  
Bo ZHOU ◽  
Shifeng XUE
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Axial Direction ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Mechanical Behaviors ◽  
Euler Beam ◽  
Poisson Effect ◽  
Size Dependent

This work focus on the mechanical behaviors, which are related to the size effect, functionally graded (FG) effect and Poisson effect, of an axially functionally graded (AFG) micro-beam whose elastic modulus varies according to sinusoidal law along its axial direction. The displacement field of the AFG micro-beam is set according to the Bernoulli-Euler beam theory. Employing the modified couple stress theory (MCST), the components of strain, curvature, stress and couple stress are expressed by the second derivative of the deflection of the AFG micro-beam. A size-dependent model related to FG effect and Poisson effect, which includes the formulations of bending stiffness, deflection, normal stress and couple stress, is developed to predict the mechanical behaviors of the AFG micro-beam by employing the principle of minimum potential energy. The mechanical behaviors of a simply supported AFG micro-beam are numerically investigated using the developed model for demonstrating the size effects, FG effects and Poisson effects of the AFG micro-beam. Results show that the mechanical behaviors of AFG micro-beams are distinctly size-dependent only when the ratio of micro-beam height to material length-scale parameter is small enough. The FG parameter is an important factor that determines and regulates the size-dependent behaviors of AFG micro-beams. The influences of Poisson’s ratio on the mechanical behaviors of AFG micro-beams are not negligible, and should be also considered in the design and analysis of an AFG micro-beam. This work supplies a theoretical basis and a technical reference for the design and analysis of AFG micro-beams in the related regions.

Size Dependent Pull-in Phenomena in Electro-Statically Actuated Micro-Beam Based on the Modified Couple Stress Theory

2011 ◽  
Vol 335-336 ◽  
pp. 633-640 ◽  
Author(s):  
Jun Feng Zhao ◽  
Shen Jie Zhou ◽  
Bing Lei Wang
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Dynamic Responses ◽  
Euler Beam ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Material Length Scale ◽  
Material Length

A modified continuum model of electro-statically actuated micro-beam is presented based on the modified couple stress theory. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Bernoulli-Euler beam theory. The governing equation of the micro-beam is derived based on the Hamilton’s principle, which accounts for the effects of the moderately large deflection, the residual stress and the fringing electrostatic field. The numerical analysis of mechanical characterization is performed by the Analog Equation Method (AEM). The effects of the couple stress on the static and dynamic responses, pull-in voltage and pull-in time are discussed.

Modeling size-dependent thermal-mechanical behaviors of shape memory polymer Timoshenko micro-beam

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Fei Zhao ◽  
Xueyao Zheng ◽  
Shichen Zhou ◽  
Bo Zhou ◽  
Shifeng Xue
Keyword(s):  
Size Effect ◽  
Shape Memory ◽  
Couple Stress Theory ◽  
Shape Memory Polymer ◽  
Modified Couple Stress Theory ◽  
Mechanical Behaviors ◽  
Equilibrium Equations ◽  
Content Type ◽  
Size Dependent ◽  
Beam Height

PurposeIn this paper, a three-dimensional size-dependent constitutive model of SMP Timoshenko micro-beam is developed to describe the micromechanical properties.Design/methodology/approachAccording to the Hamilton's principle, the equilibrium equations and boundary conditions of the model are established and according to the modified couple stress theory, the model is available to capturing the size effect because of the material length scale parameter. Based on the model, the simply supported beam was taken for example to be solved and simulated.FindingsResults show that the size effect of SMP micro-beam is more obvious when the dimensionless beam height is similar or the larger of the value of loading time. The rigidity and strength of the SMP beam decrease with the increasing of the dimensionless beam height or the loading time. The viscous property of SMP micro-beam plays a more important role with the larger dimensionless beam height. And the smaller the dimensionless beam height is, the more obvious the shape memory effect of the SMP micro-beam is.Originality/valueThis work implies prediction of size-dependent thermo-mechanical behaviors of the SMP micro-beam and will provide a theoretical basis for design SMP microstructures in the field of micro/nanomechanics.

scholarly journals Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage

2019 ◽  
Vol 8 (1) ◽  
pp. 461-469 ◽  
Author(s):  
Xiumei Wang ◽  
Jihai Yuan ◽  
Haorui Zhai
Keyword(s):  
Internal State ◽  
Couple Stress Theory ◽  
Damage Model ◽  
Modified Couple Stress Theory ◽  
Damage Effect ◽  
Bifurcation And Chaos ◽  
Stress Theory ◽  
Size Dependent ◽  
Material Length Scale ◽  
Plate System

Abstract In this research, nonlinear dynamics and characteristics of a micro–plate system under electrostatic forces on both sides are studied. A novel model, which takes micro-scale effect and damage effect into account, is established on the basis of the Talreja’s tensor valued internal state damage model and modified couple stress theory. According to Hamilton principle, the dynamic governing equations of the size-dependent micro–plate are derived by variational method and solved via Galerkin method and the fourth order Runge-Kutta method. The effects of damage variable and material length scale parameter on bifurcation and chaos of the micro–plate system are presented with numerical simulations using the bifurcation diagram, Poincare map. Results provide a theoretical basis for the design of dynamic stability of electrically actuated micro- structures.

Resonance Responses of Geometrically Imperfect Functionally Graded Extensible Microbeams

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Hamed Farokhi ◽  
Alireza Gholipour ◽  
Shahid Hussain ◽  
Maziar Arjomandi
Keyword(s):  
Couple Stress Theory ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Dynamical Response ◽  
Weighted Residual Method ◽  
Size Dependent ◽  
Rotational Motions

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

Thermal Postbuckling and Free Vibration of Extensible Microscale Beams Based on Modified Couple Stress Theory

2014 ◽  
Vol 31 (1) ◽  
pp. 37-46 ◽  
Author(s):  
Y.-G. Wang ◽  
W.-H. Lin ◽  
C.-L. Zhou ◽  
R.-X. Liu
Keyword(s):  
Free Vibration ◽  
Temperature Rise ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Modified Couple Stress ◽  
Thermal Postbuckling ◽  
Material Length Scale

AbstractThis paper presents a mathematical model and a computational approach for the thermal post-buckling and free vibration in the vicinity of the buckled equilibrium position of microbeams based on the modified couple stress Euler-Bernoulli beam theory and geometrically accurate relation. The governing equations for the whole analysis are established with a intrinsic material length scale parameter to capture the size effect. These equations, in conjunction with the corresponding boundary conditions, are decomposed into two two-point boundary value problems, which are solved using the shooting method. For static deformation, the geometric nonlinearity is involved and the size dependent postbuckling configuration is obtained as a function of temperature rise. For dynamic one, the small amplitude free vibration about the prebuckled position is developed following an assumed harmonic time mode, and the length scale and temperature rise dependent fundamental natural frequency is presented. Numerical computations are executed to illustrate the size dependency in the thermal postbuckling behaviors and fundamental frequency of the vibration around the buckled configuration of microbeams.

scholarly journals Snap-through buckling of initially curved microbeam subject to an electrostatic force

2015 ◽  
Vol 471 (2177) ◽  
pp. 20150072 ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid
Keyword(s):  
Electrostatic Force ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Intermolecular Forces ◽  
Modified Couple Stress Theory ◽  
Casimir Forces ◽  
Governing Equations ◽  
Stress Theory ◽  
Fringing Field ◽  
Curved Microbeam

In this paper, the snap-through buckling of an initially curved microbeam subject to an electrostatic force, accounting for fringing field effect, is investigated. The general governing equations of the curved microbeam are developed using Euler–Bernoulli beam theory and used to develop a new criterion for the snap-through buckling of that beam. The size effect of the microbeam is accounted for using the modified couple stress theory, and intermolecular effects, such as van der Waals and Casimir forces, are also included in our snap-through formulations. The snap-through governing equations are solved using Galerkin decomposition of the deflection. The results of our work enable us to carefully characterize the snap-through behaviour of the initially curved microbeam. They further reveal the significant effect of the beam size, and to a much lesser extent, the effect of fringing field and intermolecular forces, upon the snap-through criterion for the curved beam.

scholarly journals Size-Dependent Dynamic Behavior of a Microcantilever Plate

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaoming Wang ◽  
Fei Wang
Keyword(s):  
Size Effect ◽  
Dynamic Behavior ◽  
Size Effects ◽  
Numerical Solutions ◽  
Couple Stress Theory ◽  
Differential Quadrature Method ◽  
Modified Couple Stress Theory ◽  
Cantilever Plate ◽  
Size Dependent ◽  
Material Length Scale

Material length scale considerably affects the mechanical properties of microcantilever components. Recently, cantilever-plate-like structures have been commonly used, whereas the lack of studies on their size effects constrains the design, testing, and application of these structures. We have studied the size-dependent dynamic behavior of a cantilever plate based on a modified couple stress theory and the differential quadrature method in this note. The numerical solutions of microcantilever plate equation involving the size effect have been presented. We have also analyzed the bending and vibration of the microcantilever plates considering the size effect and discussed the dependence of the size effect on their geometric dimensions. The results have shown that (1) the mechanical characteristics of the cantilever plate show obvious size effects; as a result, the bending deflection of a microcantilever plate reduces whereas the natural frequency increases effectively and (2) for the plates with the same material, the size effect becomes more obvious when the plates are thinner.

Size Effect on Buckling Behaviors of Gradient Elastic Slender Columns

2013 ◽  
Vol 300-301 ◽  
pp. 1158-1161
Author(s):  
Sheng Li Kong
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Governing Equations ◽  
Buckling Loads ◽  
Stress Theory ◽  
Size Dependent ◽  
Modified Couple Stress ◽  
Slender Columns ◽  
Variational Statement

For the buckling problems of slender columns, size effects on buckling behaviors have been studied based on the modified couple stress theory. The governing equations are obtained by using variational statement and the buckling loads of slender columns are assessed. The results show that the buckling loads predicted by the new model are size-dependent.

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