Size-dependent Microbeam Research Articles

Active/passive tuning of wave propagation in phononic microbeams via piezoelectric patches

In this paper, size-dependent microbeams made of functionally graded materials (FGMs) with periodic arrays of auxiliary piezoelectric springs are considered small-sized phononic crystals (PCs). The governing equation of motion for the propagation of flexural waves is derived and solved using the transfer matrix method (TMM). Then, the small size effects on the wave propagation in periodic FGM microbeams are investigated. Also, the effects of FGM distribution on the dispersion curves as well as on the frequency ranges and widths of the stop-bands are analyzed. Additionally, by actively modulating the piezoelectric stiffness of the auxiliary springs, tunable wave attenuation is reached via both local resonance (LR) and Bragg scattering mechanisms. The results prove that stop-bands of both LR and Bragg natures can be tuned in microbeams, and tunable wave-attenuation performance can be achieved in different ranges of the frequency spectrum by implementing FGMs and combining it with piezoelectricity. Such findings place this work among the first efforts to provide passive and active control of wave attenuation in size-dependent tiny metamaterials.

Theoretical modeling on the combination resonance of size-dependent microbeams

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

Corrigendum to: Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

Corrigendum to: Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

Abstract In this study, nonlinear couple stress–strain constitutive relationships in the modified couple stress theory (MCST) are derived on the basis of previous classical stress–strain constitutive relationships of nonlinear elasticity materials. Hamilton's principle is employed to obtain higher-order nonlinear governing equations within the framework of the updated MCST, von Kármán geometric nonlinearity and Bernoulli–Euler beam theory. These mathematical formulations are solved numerically by the differential quadrature method together with an iterative algorithm to determine the nonlinear dynamic features of microbeams with four groups of boundary conditions. A detailed parametric study is conducted to analyze the influences of nonlinear elasticity properties on the nonlinear free vibration characteristics of the microbeams. Results show that these microbeams exhibiting nonlinear couple constitutive relationships have lower frequencies than their approximately simplified linear couple constitutive relationships. In addition, the frequencies of microbeams with nonlinear elasticity properties decrease as the vibration amplitude increases.

A microbeam based piezoelectric energy harvester with LASMP

This work developed a piezoelectric micro-beam based energy harvester laminated with light activated shape memory polymer (LASMP). With noncontact mechanism and dynamic Young’s modulus of LASMP, natural frequency of the energy harvester can be tuned during light exposure. Based on modified couple stress theory, the governing equation of the size-dependent micro-beam is derived via Hamilton’s principle. Analytic solution of natural frequency and power output of the micro-beam is presented. Influence of size effect and dynamic Young’s modulus of LASMP on natural frequency and power output is investigated. Present research suggested that application of LASMP in micro-engineering is potential and effective.

Pull‐in analysis of microbeam laminated with LASMP

This work focuses on the pull-in phenomena of electrically actuated size-dependent microbeam laminated with the light-activated shape memory polymer, which can realise the Young's modulus variation under the light exposure. Based on the modified couple stress theory, a size-dependent microbeam model is developed. The dynamic governing equation of the microbeam is derived according to the Hamilton principle and solved by the Galerkin method and Runge–Kutta method numerically. The reliability and accuracy of the present method are validated experimentally. The static and dynamic pull-in behaviours of the microbeam are studied when the microbeam is under a purely direct current voltage and alternating current voltage, respectively. Influences of the size effect and dynamic Young's modulus on pull-in behaviour of the microbeam are discussed comprehensively. This research may be helpful and useful in electrostatically actuated microstructure applications and provide a theoretical foundation for designers and engineers.

WITHDRAWN: Dynamic stability of a size-dependent micro-beam

WITHDRAWN: Dynamic stability of a size-dependent micro-beam

Dynamic stability of a size-dependent micro-beam

The effect of variation of the geometrical dimensions on dynamic instability regions (DIRs) of a rectangular cross-sectioned micro-beam with simply supports is discussed in this study based on modified coupled stress theory. A set of linear equations are derived on basis of the Lagrange method and trial series expansions for vertical displacement and rotation of the Timoshenko micro-beam model while longitudinal displacement is neglected due to the stretching effect of the micro-beams mid-plane. The first approximation of the dynamic stability analysis is done by the application of the Bolotin method besides obtaining the Mathieu-Hill equations. The numerical results demonstrate how the cross section's height and micro-beams length are the only effective parameters on DIRs somehow that when the geometrical dimensions are increased the DIRs are going to coincide.

Bending and vibration analysis of delaminated Bernoulli–Euler micro-beams using the modified couple stress theory

In this paper, we study static bending and free vibration behavior of Bernoulli–Euler micro-beams with a single delamination using the modified couple stress theory. The delaminated beam is modeled by four interconnected sub-beams using the delamination zone as their boundaries. The free and constrained mode theories have been utilized to model the interaction of delamination surfaces in the damaged area. The continuity as well as compatibility conditions are satisfied between the neighboring sub-beams. After verification of the results for some case studies with available solutions, the effect of various parameters such as spanwise and thicknesswise locations of the delamination, material length scale parameter, and boundary conditions on the static bending and free vibration characteristics of the size-dependent micro-beam have been investigated in detail.

Dynamic analysis of size-dependent micro-beams with nonlinear elasticity under electrical actuation

This paper presents a nonlinear electro-dynamic analysis for a size dependent micro-beam made of materials with nonlinear elasticity by employing the modified couple stress theory based on Euler–Bernoulli beam model. By employing Hamilton's principle, the nonlinear partial differential governing equation is derived then solved by using Galerkin method in the space domain and backward differential method in the time domain to obtain the nonlinear electro-dynamic response and phase portrait of the micro-beam. A parametric study is conducted, with a particular focus on the influences of nonlinear elasticity and size-dependency of the micro-beam on its nonlinear dynamic behavior. Numerical results show that a micro-beam exhibiting nonlinear elastic stress–strain relationship has reduced effective stiffness while the size effect has the opposite effect.

A size-dependent nonlinear microbeam model based on the micropolar elasticity theory

A size-dependent microbeam model including von Karman geometric nonlinearity is developed based on the micropolar elasticity theory. The governing equations and the boundary conditions are obtained using Hamilton’s principle. In the present model, the microrotation quantity and two new material parameters (characteristic length of material and second shear modulus) are introduced. The size effects, nonlinear static and dynamic behaviors are investigated by directly applying the formulas derived. Special attention is paid on the operating mechanism of microrotation and effects of the second shear modulus on the system. The numerical results for the static bending problem reveal that (1) geometric nonlinearity and the size effect enhance the stiffness of the beam; (2) the microrotation has an opposite motion trend to those of deflection and macroangle of the centroidal axis with changing beam thickness or second shear modulus. A multiple-scale method is employed to obtain an approximate analytical solution for the natural frequency and time response of the beam. It shows that (1) the second shear modulus has significant effects on the lower-order frequency, while the effect on the higher-order frequency is negligible; (2) for increasing the natural frequency of the beam, geometric nonlinearity plays the essential role for the beam with larger thickness, while the size effect plays the essential role for the beam with smaller thickness.

A novel size-dependent microbeam element based on Mindlin’s strain gradient theory

Based on Mindlin's strain gradient elasticity and Euler---Bernoulli beam theory, a non-classical beam element capable of considering micro-structure effects is developed. To accomplish this aim, the higher-order tensors of energy pairs in the energy functional are vectorized and written in the quadratic representation, from which the stiffness and mass matrices of the element are obtained. In comparison with the classical Euler---Bernoulli beam element, the new element needs one additional nodal degree of freedom (DOF) which results in a total of three DOFs per node. The formulation of the paper is general so that it can be reduced to that based on the modified couple stress theory, the modified strain gradient theory, and the classical elasticity theory. To show the reliability of the proposed element, the bending and free vibration problems of microbeams under different kinds of end conditions are addressed. It is revealed that the present finite element results are in excellent agreement with the ones achieved through analytical solutions.

Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect

Presented herein is the prediction of buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) including thermal environment effect. To this purpose, strain gradient elasticity theory is incorporated into the classical third-order shear deformation beam theory to develop a non-classical beam model which contains three additional internal material length scale parameters to consider the effects of size dependencies. The higher-order governing differential equations are derived on the basis of Hamilton’s principle. Afterward, the size-dependent differential equations and related boundary conditions are discretized along with commonly used end supports by employing generalized differential quadrature (GDQ) method. A parametric study is carried out to demonstrate the influences of the dimensionless length scale parameter, material property gradient index, temperature change, length-to-thickness aspect ratio and end supports on the buckling characteristics of FGM microbeams. It is revealed that temperature change plays more important role in the buckling behavior of FGM microbeams with higher values of dimensionless length scale parameter.

Buckling analysis of functionally graded microbeams based on the strain gradient theory

The buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) for different boundary conditions is investigated on the basis of Bernoulli–Euler beam and modified strain gradient theory. The higher-order governing differential equation for buckling with all possible classical and non-classical boundary conditions is obtained by a variational statement. The effects of the power of the material property variation function, boundary conditions, slenderness ratio, ratio of additional material length scale parameters for two constituents, beam thickness-to-additional material length scale parameter ratio on the buckling response of FGM microbeams are investigated. Some comparative results are presented in tabular and graphical form in order to show the differences between the results obtained by the present model and those predicted by modified couple stress and classical continuum models.

Thermal effect on free vibration and buckling of size-dependent microbeams

Based on the modified couple stress theory, free vibration and buckling of the microbeams with the effect of the temperature change are investigated. The non-classical Timoshenko beam model, which contains a material length scale parameter, is developed to interpret the size effect in microscale structures. The higher-order governing equations and boundary conditions are derived by using the Hamilton principle. The differential quadrature method is employed to determine the natural frequency and the critical buckling load of the microbeams with different boundary conditions. The effects of the temperature change, length scale parameter, slenderness ratio and end supported conditions on the free vibration and buckling of the microbeams are discussed in detail. Results show that the thermal effect on the fundamental frequency and critical buckling load is slight when the thickness of the microbeam has a similar value to the material length scale parameter, but it becomes significant when the thickness of the microbeams becomes larger.