Functionally Graded Microbeams Research Articles

Size dependent pull-in instability of functionally graded microbeams using a finite element formulation

The size dependent static pull-in instability of functionally graded (FG) microbeams in micro-electromechanical systems (MEMS) is studied, considering the influence of the axial force. The material properties of the microbeams are varied in the beam thickness by a power-law function, and they are calculated by the rule of mixture. To account for the microsize effect, the classical Euler-Bernoulli beam theory is employed in combination with the modified couple stress theory to describe the microbeams deformation. Based on Von Kármán nonlinear relationship, a beam element is derived and employed to establish the discretized governing equation for the microbeams. Newton-Raphson iterative procedure is adopted to compute frequencies and pull-in voltages for the microbeam with clamped ends. Numerical result reveals that the pull-in voltage is increased by the increase of the power-law exponent and the microscale parameter. The effects of the material distribution, the axial force as well as the microstructural parameter on the pull-in instability of the FG microbeams are investigated in detail.

Application of Haar wavelet discretization and differential quadrature methods for free vibration of functionally graded micro-beam with porosity using modified couple stress theory

The present investigation is aimed at the implementation of Haar Wavelet Discretization Method (HWDM) and Differential Quadrature Method (DQM) on the free vibration of a Functionally Graded (FG) micro-beam with uniformly distributed porosity along the thickness. As per the power-law exponent model, the material properties such as Young's modulus, and mass density are varied along the thickness of the FG micro-beam and the beam is made up of Aluminum (Al) as metal constituent and Alumina (Al2O3) as ceramic constituent. Modified couple stress theory is employed to capture the small scale effect and pointwise convergence studies for HWDM as well as DQM have also been carried out to exhibit the effectiveness of the methods with respect to the undertaken problem. The results obtained by both methods are compared to demonstrate the accuracy of the present model, revealing excellent accuracy. The effect of power-law exponent, porosity volume fraction index, and thickness to material length scale parameter on the natural frequencies is thoroughly investigated with proper physical explanations for Hinged-Hinged (H-H), Clamped-Hinged (C-H), Clamped-Clamped (C-C), and Clamped-Free (C-F) boundary conditions. Further, mode shapes are also plotted for qualitatively assessing the dynamics of the structural component.

Static and dynamic modeling of functionally graded Euler–Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis

Strain Gradient Theories (SGTs) are often considered to capture the intrinsic microstructural behaviors of the microbeams. In the present work, the recent Reformulated Strain Gradient Theory (RSGT), which accounts the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively, is extended for the static and dynamic modeling of Functionally Graded (FG) Euler–Bernoulli microbeams to account for the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively. The present formulation also considers the shift of neutral axis due to material inhomogeneity in the FG microbeam. The problems of static deflection and free vibrations with different boundary conditions are addressed. Numerical solutions are obtained using isogeometric analysis. Analytical solutions of the present formulation for simply supported FG microbeam for static deflection and free vibration problems are also obtained and employed to validate the numerical results. The influence of the material index, different length scale parameters on the static deflection and free vibration responses of the FG microbeams are discussed. The present FG RSGT microbeam shows the prominent size scale effect in thin beams and predicts stiffer response than the FG microbeam based on modified couple stress theory.

Nonlinear Vibration of an Electrostatically Actuated Functionally Graded Microbeam under Longitudinal Magnetic Field

In this work, we develop a model of an electrostatically actuated functionally graded (FG) microbeam under a longitudinal magnetic field based on the Euler-Bernoulli beam and nonlocal strain gradient theories to investigate the nonlinear vibration problem. The FG microbeam is placed between two electrodes, a DC voltage applied between the two fixed electrodes causes an electrostatic force to be exerted on the FG microbeam. The FG microbeam is composed of metal and ceramic in which the properties of these materials are assumed to change in the thickness direction according to the simple power-law distribution. The Galerkin method and the Hamiltonian Approach are employed to find the approximate frequency of the FG microbeam. The accuracy of the present solution is verified by comparing the obtained results with the numerical results and the published results in the literature. Effects of the power-law index, the material length scale parameter, the nonlocal parameter, the applied voltage and the magnetic force on the nonlinear vibration behaviour of the FG microbeam are studied and discussed.

Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment

A mathematical model is developed, though this article, to investigate a vibrational behaviour of functionally graded (FG) cracked microbeam rested on elastic foundation and exposed to thermal and magnetic fields. The model includes a size scale effect and temperature dependent material properties, for the first time. The crack is modelled as a rotating spring, that is connecting the two parts of the microbeam at the crack’s position. The equation of motion of the FG microbeam is obtained by using the Euler-Bernoulli beam theory for kinematic assumption and nonlocal elasticity theory for size-dependency effects. The transverse Lorentz force induced from the magnetic field is derived using Maxwell's equations. By adding the effects of thermal loading and foundation parameters on the cracked micro beam, the motion equation of the entire system is obtained using the Hamilton’s principle and then solved with a Navier type solution method. Eight constraints equations are used to derived the frequency equation, which are boundary conditions at the end points and the displacement, slope, bending moment and transverse force continuity in the section where the crack is located. The resulting system of equations is solved sequentially, and natural frequencies and vibration modes of the cracked microbeam are obtained. The model is verified with previous published work. Numerical results are presented to illustrate influences of temperature, material composition, foundation parameters and magnetic field on the dynamics of the cracked FG microbeam.

Microbeam Model and Related Differential Quadrature Finite Elements

A size-dependent quasi-3D functionally graded (FG) microbeam model was presented within the combined framework of the modified couple stress theory and a 4-unknown higher-order shear and normal deformation theory. Then the model was applied to analyze the static bending and free vibration of FG microbeams. With the 2nd Lagrange equation, the corresponding motion equations and the appropriate boundary conditions were obtained. A 2-node 16DOF differential quadrature finite element combining the Gauss-Lobatto quadrature rule with the differential quadrature rule was constructed to handle the general static/dynamic boundary value problems of FG microbeams. A comparison study was performed to show the efficacy of the proposed theoretical model and solution method. Finally, the effects of the gradient index, the intrinsic length scale parameter, the geometrical parameters and the boundary conditions on the static and dynamic characteristics of FG microbeams were examined. Numerical results reveal that the developed beam model and element are applicable to the analysis of mechanical behaviors of FG microbeams with various slenderness ratios. Besides, introduction of the couple stress effect can significantly change the static and dynamic characteristics of FG microbeams.

Exact solution for thermal–mechanical post-buckling of functionally graded micro-beams

In this paper, buckling and post-buckling analysis of functionally graded (FG) micro-beams subjected to the thermal gradient are presented. The FG micro-beam is embedded on an elastic substrate medium modeled by Winkler-Pasternak foundation. To capture the size effect, a modified couple stress theory is applied. Based on the minimum potential energy principle, and using Timoshenko beam hypothesis and von Karman strains, governing differential equations are derived. Both Fourier series solution and exact analytical method are presented for solving the system of coupled differential equations. Because Fourier series solution cannot satisfy all boundary conditions, it is not able to predict the post-buckling response of FG micro-beams, correctly. Moreover, in numerical result section, the effects of length-scale parameter, power-law index, transverse and axial load, temperature gradient, and elastic foundation constants on the post-buckling behavior of simply supported FG micro-beams are investigated. Obtained responses demonstrated that the buckling of the FG micro-beams are significantly sensitive to the variation of mentioned parameters.

Nonlinear Thermal Stability of Rotating Pre-twisted Temperature-Dependent FG Microblades

The main objective of this study is to investigate the post-buckling thermal load–deflection path of rotating pre-twisted microblades/beams made of temperature-dependent functionally graded materials (FGMs) via nonlinear non-classical first-order shear deformation theory (FSDT). To this end, modified strain gradient theory (MSGT) as well as the von Karman geometric nonlinearity is used to develop the size-dependent nonlinear beam theory. Thermal loading is exerted by providing nonlinear (non-uniform) temperature rise across the microbeam thickness at steady-state condition. By using the principle of virtual work and Ritz method, the discretized form of nonlinear equations governing the post-buckling behavior of rotating pre-twisted functionally graded (FG) microbeams is achieved. To solve the obtained nonlinear equations numerically, Broyden’s method is employed. The effects of twist angle along the beam axis, angular velocity, index of the material gradient, length-scale parameter and thickness-to-length ratio on the post-buckling path of FG microbeams are carefully investigated. Findings show that making a correct assumption about the temperature dependence of material is an important factor to increase accuracy of results.

On the dynamic response of porous functionally graded microbeam under moving load

In this article, the dynamic behavior of functionally graded (FG) microbeams with different porosity distributions and acted by a moving harmonic load is studied. A bi-dimensional FG microbeam with its material properties changing gradually along both the length and thickness directions in a power-law distribution form is presented. Besides, the porosities are distributed evenly and unevenly in the body of microbeams, and the small-scale effect is captured with the help of the modified couple stress theory. The equations describing the motions of the FG microbeam are obtained in the framework of Hamilton's principle in conjunction with the parabolic shear deformation theory. These equations are solved by employing the finite element formulation, which is validated by comparing the fundamental frequency and dynamic response with previous works. The effects of several key factors such as two grading indexes, small scale parameters, porous distribution pattern and volume fraction, as well as moving speed and motivation frequency of the acted load are investigated in detail. It is concluded that the bi-dimensional FG parameters and porous parameter play significant roles in the dynamic response of microbeams subjected to a moving load, which is helpful for the multi-functional and optimal design of microsystems.

Size-dependent behaviour of functionally graded sandwich microbeams based on the modified strain gradient theory

This paper studies the bending, buckling and free vibration analyses of functionally graded (FG) sandwich microbeams using the third-order beam theory. The modified strain gradient theory (MSGT) with three material length scale parameters (MLSPs) is used to capture the size effect. The Mori-Tanaka homogenization scheme is employed to model the material distributions through the thickness. Finite element model is formulated to solve the problems. Verification studies for epoxy and FG microbeams are carried out to validate of the present model. Comparisons of the results of three different models such as MSGT, the modified couple stress theory (MCST) and classical continuum theory (CCT) are presented. Effects of small size, gradient index, shear deformation and boundary conditions on the structural behaviours of microbeams are investigated. Some new results of FG sandwich microbeams for both models (MCST and MSGT) are presented for the first time and can be used as benchmark in future studies.

Size-dependent thermo-mechanical free vibration analysis of functionally graded porous microbeams based on modified strain gradient theory

In present study, vibration analysis of functionally graded (FG) porous microbeam under thermal effects is investigated considering modified strain gradient theory (MSGT) of elasticity. As the main novelty of this study, in order to effectively acquire the effects of size of size-dependent porous structures, MSGT is employed (considers three material length scale parameters) rather than modified couple stress theory (MCST) (considers one material length scale parameter). The nonlinear governing equation of microbeam based on MSGT is derived from Hamilton’s principle and to determine the natural frequency of the system under simply supports, Navier solution is employed. Numerical results presented for different beam models and theories and are compared with those available in previous studies. The obtained results show that frequencies from MSGT are higher than those from MCST and also classical theory (CT), especially when the microbeam thickness is comparable to the microbeam length scale parameter. Also, increases in the temperature of microbeam working environment, lead to decrease of microbeam natural frequency. Finally, parametric study is performed on natural frequency of FG microbeams to indicate the effects of: temperature changes, length scale parameter, slender ratio, and gradient index and porosity volume fraction.

A Comprehensive Review on Vibration Analysis of Functionally Graded Beams

Beams and beam structures are structural components commonly used in mechanical, aerospace, nuclear, and civil engineering. To meet the different engineering design limitations such as operational conditions, weight, and vibrational characteristics, these components may be made of various materials such as functionally-graded materials (FGMs), composites, and homogeneous materials. Functionally-graded (FG) beams play a key role not only in classical structural applications, but also have vast applications in thermal, electric-structural and electric-thermal-structural systems, e.g. in the form of FG beam energy harvesters, sensors and actuators. In all these applications, using new materials like FGMs can greatly improve the efficiency of the structural components and systems. Since FG beams are mostly used as moving components in engineering structures, vibration analysis of these components has been studied by numerous researchers. In order to solve the governing equation and related boundary conditions of the FG beams, powerful numerical methods with a high level of accuracy and fast rate of convergence are often required. The differential quadrature method (DQM) is a powerful and reliable numerical method which has been extensively used by researchers to perform the vibration analyses of FG structures in the last decade. In this paper, firstly various mathematical models which have been used to express the material properties of FGMs are reviewed. Secondly different elasticity theories which have been applied in vibration analysis of FG beams are summarized. In addition, a review on the DQM and its applications is presented. At the next step, a comprehensive review on free vibration analyses of FG beams based on different elasticity theories and in particular those using the DQM is performed. In continue, a brief review on the application of other numerical methods in vibration analysis of FG beams is presented. Moreover, because of the importance of nonlinear vibration analysis of FG beams, a review on the application of various numerical methods and different elasticity theories on nonlinear vibration analysis of FG beams is performed. Finally, a brief review on linear and nonlinear vibration analysis of FG microbeams, as a special type of FG beams, is presented.

A novel functional and mixed finite element analysis of functionally graded micro-beams based on modified couple stress theory

Static bending, free vibration and buckling analyses of functionally graded (FG) Timoshenko micro-beams based on modified couple stress theory are presented under the framework of a new functional and mixed finite element method (MFEM). The new functional has been constituted for FG Timoshenko micro-beams through a scientific procedure based on the Gâteaux differential. Detailed derivation of mixed finite element formulation which allows C0 type continuous shape functions is given. The results of MFEM formulation for static bending, fundamental vibration frequency and critical buckling load of a straight FG micro-beam under various boundary conditions are obtained and compared with the existing literature. It is shown that, with simple C0 type shape functions it is possible to effectively capture variations of dimensionless fundamental frequencies and critical buckling load of moderately thick FG micro-beams for various boundary conditions and material parameters by completely avoiding the shear locking. Also FG tapered micro-beams are analyzed as a demonstration of versatility. Another distinctive feature of the new formulation is its capability of identification of the separate contributions of the coupled and the classical parts of the section moment. The new functional and the mixed finite element formulation are the original contributions of this study.

Nonlinear forced vibration of size-dependent functionally graded microbeams with damping effects

• A coupled nonlinear model of FG microbeams with damping effects is presented. • The periodic, period-n, chaotic oscillations and three-to-one internal resonance are detected. • The effects of key parameters on the nonlinear vibrations of FG microbeams are analyzed. • The comparisons are performed and the investigations demonstrate good reliability. The nonlinear dynamics of functionally graded (FG) microbeams have been studied based on the von Kármán nonlinear theory, the modified couple stress and Euler–Bernoulli beam theories. The internal damping of materials is taken into account using the Kelvin–Voigt model. The coupled nonlinear mode equations of FG microbeams are obtained using Hamilton's principle and Galerkin's method. The primary resonance and internal resonance are investigated by means of the method of multiple scales and the Runge–Kutta numerical method. The effects of the length scale parameter, volume fraction exponent and internal damping constant on the nonlinear vibration are discussed using the numerical simulation. The numerical results show that periodic, period-n, and chaotic oscillations can be displayed by changing the length scale parameter, volume fraction exponent and internal damping constant. Boundary conditions can also change the nonlinear vibration behavior of FG microbeams. In the present study, the second natural frequency is approximately equal to three times the first one for the FG clamped–clamped microbeam, and a three-to-one internal resonance is detected using the time response curves. The present analysis is validated by comparing the numerical results with existing results and very good agreement is obtained.

Post-buckling and vibration of post-buckled rotating pre-twisted FG microbeams in thermal environment

In this paper, comprehensive studies on the size-dependent thermal post-buckling and coupled bi-directional transverse-longitudinal free vibration behavior of post-buckled rotating pre-twisted functionally graded (FG) microbeams in thermal environment are presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction with mid-plane symmetric distribution, i.e., symmetric functionally graded (S-FG). At first, the discretized nonlinear post-buckling governing equations are derived based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory (FSDT) of beams under the von Kármán geometric nonlinearity assumptions using Hamilton's principle and Chebyshev-Ritz method. The resulting nonlinear equations under different boundary conditions are solved iteratively by employing Newton-Raphson's method. Then, using the same theories and solution procedure, the free vibration equations around the post-buckled configuration are derived and solved. After validating the approach, the effects of angular velocity, twist angle, length scale parameters, thickness-to-length ratio, hub radius, material gradient index and boundary conditions on the post-buckling equilibrium path and linear free vibration behavior of rotating pre-twisted S-FG microbeams are investigated. The results are prepared for both linear and nonlinear variations of the twist angle along the microbeam axis.

Computational hygro-thermal vibration and buckling analysis of functionally graded sandwich microbeams

In this study, for the first time, hygro-thermal behaviour of functionally graded (FG) sandwich microbeams based on nonlocal elasticity theory is investigated. Temperature-dependent material properties are considered for the FG microbeam, which are assumed to change continuously through the thickness based on the power-law form. The equations of motion are obtained on the basis of first-order shear deformation beam theory via Hamilton's principle. The size effects are considered in the framework of the nonlocal elasticity theory of Eringen. The detailed variational and finite element procedure for FG sandwich microbeams are presented with a five-noded beam element and numerical examinations are performed. The influence of several parameters such as temperature and moisture gradients, material graduation, nonlocal parameter, face-core-face and span to depth ratios on the critical buckling temperature and the nondimensional fundamental frequencies of the FG sandwich microbeams are analysed. Based on the results of this study, temperature and moisture rise soften the FG sandwich microbeam and result in the reduction of the critical buckling load and vibration frequency. In addition, the FG sandwich microbeam with a thicker ceramic core can resist higher temperature and moisture gradients.

Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory

A novel and effective computational approach within the context of isogeometric analysis (IGA) is developed for analyzing size-dependent mechanical behaviors of functionally graded (FG) microbeams. To capture the size effects, an extension of quasi-3D theory is established to integrate with the modified couple stress theory. The nonuniform rational B-spline (NURBS) basis functions are employed and can directly meet the first-order derivative demand of the quasi-3D theory, where four variables are involved at each node. In this new setting, both normal and shear deformations are considered, while the shear correction factor is avoided. Numerical examples are studied, in which the effects of several factors, including material gradient factor, boundary conditions, parameter of material length scale, and aspect ratio, on deflections, stresses, and fundamental frequencies of FG microbeams are explored.

Viscoelastic dynamics of axially FG microbeams

A viscoelastic axially functionally graded (FG) imperfect microbeam is considered and the complex nonlinear dynamics is analysed for the first time. The distribution of the material properties from beam's left end to the right one is of an exponential form. The small nature of the system is incorporated using the couple stress theory (CST) in a modified form. The viscosity between microbeam's infinitesimal elements is incorporated through the Kelvin-Voigt scheme. The elastic part of the energy is formulated via the elastic component of the stress and its couple; the viscous parts are incorporated via a negative work. The energy of kinetic type is also formulated. Without ignoring the axial displacement/inertia, Hamilton's principle of energy is used to formulate the viscoelastic mechanical model of the microsystem. A high-dimensional truncation/discretisation is conducted and numerical simulations are performed based upon it. The complex nonlinear dynamics of the axially FG imperfect beam is analysed in the presence/absence of viscosity.

Size-Dependent Vibration Analysis of FG Microbeams in Thermal Environment Based on Modified Couple Stress Theory

In this paper, free vibration of functionally graded (FG) microbeams under thermal environment based on modified couple stress theory is investigated. The internal length-scale parameter is considered to account for material size effect. The Rayleigh–Ritz method is employed, and the eigenvalue problem for obtaining the natural frequencies of FG microbeam is solved for different boundary conditions. Trial functions are expressed in simple algebraic polynomial forms, and application of different boundary conditions in Rayleigh–Ritz method is explained in detail. The convergence of the method is investigated, and the results are compared with the available results in the literature. The material properties of the microbeam are assumed to vary in the thickness direction with a power-law function. It is shown that by increasing the temperature change in the microbeam, the natural frequencies decrease. In addition, it is observed that natural frequencies of microbeam based on modified couple stress theory differ significantly from natural frequencies of classical beam. It is also concluded that increase in FG index results in increase in natural frequencies, but for very high FG indexes, natural frequencies do not change considerably.

Static Pull-in Analysis of Electrostatically Actuated Functionally Graded Micro-Beams Based on the Modified Strain Gradient Theory

In this paper, the static pull-in behavior of electrostatically actuated functionally graded (FG) micro-beams resting on an elastic medium is studied using the modified strain gradient (MSG) theory. To this end, the equilibrium equation along with classical and non-classical boundary conditions is obtained by considering the fringing field and elastic foundations effects within the principle of minimum total potential energy. Also, the elastic medium is composed of a shear layer (Pasternak foundation) and a linear normal layer (Winkler foundation). The governing differential equation is solved for cantilever and doubly fixed FG beams using an iterative numerical method. This method is a combination of the reduced-order technique, the fourth-order Runge–Kutta and shooting methods. The pull-in voltages of silicon beams obtained from the present model are compared with the experimental and theoretical results reported in the literature. It is seen that the MSG theory is able to reduce significantly the difference between pull-in voltages predicted by theoretical approaches based on the classical continuum theory and the experimental observations. Finally, a parametric study is carried out to analyze in detail the influences of power index, length scale parameters, coefficients of elastic foundations and boundary conditions on the pull-in behavior of FG micro-beams. Findings indicate that the size effect on the pull-in instability of FG micro-beams with both boundary conditions is significant; however, it is seen that the variation with the normalized length scale parameter of static pull-in voltages for doubly fixed beams is larger than cantilever beams. Also, it is shown that the increase of ceramic volume fraction can improve the pull-in resistance of beams. However, this influence becomes smaller with rise of power index for both cases.