Functionally Graded Microbeams Research Articles

On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment

This study provides an exact solution for the size dependent buckling and post-buckling behavior of functionally graded (FG) micro-beams with arbitrary boundary conditions which are subjected to combined thermo-mechanical loading. To this end, a theoretical formulation including the effects of size dependency, elastic foundation and uniform temperature distribution is first derived using the modified couple stress theory and through the principle of minimum total potential energy. Next, the nonlinear equations governing bending and stretching behavior of FG micro-beams are uncoupled to a fourth-order ordinary differential equation. Finally, the differential operator method is utilized to exactly solve the decoupled equation. Also, a Fourier series solution is presented for doubly-simply supported FG micro-beams to show the importance of exact solution. In the numerical results section, the effects of the geometric ratios, material distribution, temperature variation, and material length scale parameter on the post-buckling behavior are discussed in detail. Findings show that the Fourier series solution is not able to correctly predict the post-buckling behavior of FG micro-beams, since the effect of flexural-extensional coupling stiffness term appearing in the natural boundary condition is ignored. Also, it is seen that critical values of axial traction obtained from the post-buckling analysis are significantly varied with the transverse force per unit length and temperature variation, while the buckling analysis predicts that buckling values of beam remain constant. Therefore, it can be concluded that the buckling analysis is inadequate for analyzing FG micro-beams under thermo-mechanical loadings.

Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory

A size-dependent three-dimensional dynamic model of rotating functionally graded (FG) microbeams is developed based on the Euler–Bernoulli beam theory. A two-constituent material varying along the thickness is considered following the power law. In addition, Poisson's ratio is assumed constant in the present model. Hamilton's principle is adopted in conjunction with a modified couple stress theory to derive the governing equations with the von Kármán geometric nonlinearity incorporated. The Galerkin method is utilized to solve these equations in which the coupling of the axial, chordwise and flapwise deformations is included. The centrifugal stiffening effect of the rotating FG microbeam is captured by the nonlinear coupling deformations. The convergence, accuracy and validity of the present method are confirmed by several examples. The influence of the size-dependency on modal characteristics is investigated combined with the material gradient index, dimensionless rotational speed and hub radius ratio. Finally, the achieved results of dynamic responses indicate that the deformations of rotating FG microbeams are greatly affected by the size-dependency and material gradient variation.

Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams

Abstract In this paper, thermal and shear deformation effects on the vibrational response of non-homogeneous microbeams made of functionally graded (FG) materials are carried out. It is assumed that the temperature-dependent material properties of FG microbeams change smoothly and gradually throughout the height according to the classical rule of mixture. The governing differential equations and related boundary conditions are derived by implementing Hamilton's principle on the basis of hyperbolic shear deformation beam and modified couple stress theories and they are analytically solved. The results are given together with other beam theories. A detailed parametric study is performed to indicate the influences of slenderness ratio, material length scale parameter, gradient index, shear correction factors and temperature rise on natural frequencies of FG microbeams. It is revealed that the use of modified shear correction factor can provide more accurate and valid results for first-order shear deformable microbeam model.

Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment

The large amplitude free vibration behavior of pre-twisted functionally graded (FG) microbeams in thermal environment is investigated based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory (FSDT). The geometrical nonlinearity effects are taken into account in the sense of von Kármán nonlinear kinematic assumptions. The Chebyshev–Ritz method combined with harmonic balance method is employed to derive the nonlinear algebraic eigenfrequency equations of the microbeams subjected to different boundary conditions. The material properties are assumed to be temperature-dependent and graded in the thickness direction. After numerically demonstrating the fast rate of convergence and accuracy of the method, the effects of different geometrical and material parameters on the nonlinear free vibration behavior of pre-twisted FG microbeams are investigated. It is found that the effect of twist angle on the hardening or softening of the microbeams depend on the boundary conditions, and the largest and smallest values of nonlinear to linear frequency ratios belong to simply supported and fully clamped microbeams, respectively.

Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM

ABSTRACTThermal buckling and vibration of functionally graded (FG) sinusoidal microbeams with temperature-dependent properties and three kinds of temperature distributions are investigated in this article. As one material length scale is introduced, the modified couple stress theory is capable of predicting the small-scale effects. Material properties of FG microbeams are calculated using the Mori–Tanaka method. Furthermore, temperature-dependent properties are taken into account to investigate the mechanical characteristics of FG microbeams in high–thermal-gradient environment. Motion equations and the associated boundary conditions are obtained simultaneously through variational principle. Then Navier procedure and the differential quadrature method incorporating an iterative procedure are used to solve the governing differential equations with temperature-dependent properties and general boundary conditions. Numerical examples are performed for demonstrating the influences of temperature distribution, beam thickness, material length scale, slenderness ratio, shear deformation, functionally graded index, boundary conditions, and temperature-dependent/independent properties on thermal buckling and free vibration behaviors of FG microbeams.

Thermal buckling of rotating pre-twisted functionally graded microbeams with temperature-dependent material properties

As a first endeavor, the thermal buckling of rotating pre-twisted functionally graded (FG) microbeams with temperature-dependent material properties is studied based on the modified strain gradient theory in conjunction with the first-order shear deformation theory of beams. The adjacent equilibrium criterion and Chebyshev–Ritz method are employed to derive the nonlinear algebraic eigenvalue equations governing the thermal buckling behavior of the microbeams, which are solved iteratively. The fast rate of convergence and accuracy of the method are numerically demonstrated. Then, the effects of the twist angle, rate of twist angle (as an important geometrical design parameter), material length scale parameter, material gradient index and angular velocity on the thermal load-bearing capacity of rotating pre-twisted FG microbeams under different boundary conditions are studied. It is shown that by increasing the hub radius, the angular velocity and the length scale parameter, the thermal buckling load increases, but an increase of the material gradient index reduces the critical thermal buckling load. In addition, the formulation can be easily degenerated to those of large-scale rotating pre-twisted FG beams.

Vibration analysis of a rotating functionally graded tapered microbeam based on the modified couple stress theory by DQEM

Due to having difficulty in solving governing nonlinear differential equations of a non-uniform microbeam, a few numbers of authors have studied such fields. In the present study, for the first time, the size-dependent vibration behavior of a rotating functionally graded (FG) tapered microbeam based on the modified couple stress theory is investigated using differential quadrature element method (DQEM). It is assumed that physical and mechanical properties of the FG microbeam are varying along the thickness that will be defined as a power law equation. The governing equations are determined using Hamilton’s principle, and DQEM is presented to obtain the results for cantilever and propped cantilever boundary conditions. The accuracy and validity of the results are shown in several numerical examples. In order to display the influence of size on the first two natural frequencies and consequently changing of some important microbeam parameters such as material length scale, rate of cross section, angular velocity and gradient index of the FG material, several diagrams and tables are represented. The results of this article can be used in designing and optimizing elastic and rotary-type micro-electro-mechanical systems like micro-motors and micro-robots including rotating parts.

Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory

This study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam’s depth and calculated by using classical rule of mixture and Mori–Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton’s principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson’s ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined.

Eigenanalyses of Functionally Graded Micro-Scale Beams Entrapped in an Axially-Directed Magnetic Field with Elastic Restraints

Approximate numerical solutions are obtained for the vibration response of a functionally graded (FG) micro-scale beam entrapped within an axially-directed magnetic field using the differential transformation method (DTM). Idealized as a one-dimensional (1D) continuum with a noticeable microstructural effect and a thickness-directed material gradient, the microbeam’s behavior is studied under a range of nonclassical boundary conditions. The immanent microstructural effect of the micro-scale beam is accounted for through the modified couple stress theory (MCST), while the microscopic inhomogeneity is smoothened with the classical rule of mixture. The study demonstrates the robustness and flexibility of the DTM in providing benchmark results pertaining to the free vibration behavior of the FG microbeams under the following boundary conditions: (a) Clamped-tip mass; (b) clamped-elastic support (transverse spring); (c) pinned-elastic support (transverse spring); (d) clamped-tip mass-elastic support (transverse spring); (e) clamped-elastically supported (rotational and transverse springs); and (f) fully elastically restrained (transverse and rotational springs on both boundaries). The analyses revealed the possibility of using functional gradation to adjust the shrinking of the resonant frequency to zero (rigid-body motion) as the mass ratio tends to infinity. The magnetic field is noted to have a negligibly minimal influence when the gradient index is lower, but a notably dominant effect when it is higher.

Free flexural vibration of geometrically imperfect functionally graded microbeams

Free flexural vibration characteristics of functionally graded (FG) microbeams with geometric imperfection are explored numerically, taking into account the size effect phenomenon based on modified couple stress theory. This theory employs only one material length scale parameter to interpret the size-dependent mechanical behavior of microstructures. The mechanical and physical properties of FG microbeam are assumed to vary smoothly and continuously through the thickness direction according to a power-law distribution. Hamilton's principle in conjunction with Euler-Bernoulli beam theory is used to establish the coupled longitudinal-transverse equations of motion and associated boundary conditions. A weighted-residual method is utilized to evaluate the size-dependent free flexural vibration behavior of FG microbeams with clamped-clamped, clamped-pinned, and pinned-pinned boundary conditions. The influences of different dimensionless parameters i.e., maximum imperfection amplitude-to-length ratio, length-to-thickness ratio, flexural rigidity ratio, and power-law index on the flexural frequencies and mode shapes of FG microbeams are investigated. The mode veering phenomenon is also explored. Finally, the role of longitudinal displacement in the free flexural vibration of the geometrically imperfect FG microbeams is examined.

Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM

In this paper, for the first time, the size dependent vibration behavior of a rotating non-uniform functionally graded (FG) Timoshenko and Euler–Bernoulli microbeam based on the modified couple stress theory is presented. Also, the impact of the shear deformation on natural frequencies of the microbeam considering different values of the material length scale parameter, the angular velocity and the rate of cross section change is investigated. The mechanical and physical properties of the FG microbeam are varying along the thickness according to a power law equation. To determine governing equations, Hamilton’s principle and a generalized differential quadrature element method (GDQEM) are applied. The natural frequencies of the microbeam for cantilever and propped cantilever boundary conditions are calculated. The accuracy and validity of the results are shown by several numerical examples. The influences of some parameters such as the small scale, the rate of cross section change, the angular velocity and the gradient index of the FG material on the first two natural frequencies, are plotted in several diagrams and are shown in tables. The results of this study can be used in designing and optimizing elastic and rotary type micro-electro-mechanical systems (MEMS) like micro-motors and micro-robots included rotating parts.

Exact solution for thermal damping of functionally graded Timoshenko microbeams

ABSTRACTThis article deals with the thermoelastic damping problem in a functionally graded (FG) Timoshenko microbeam. Thermal and mechanical properties of the microbeam vary in the thickness direction according to the power law relation. Employing Timoshenko beam theory, the governing dynamic equation coupled with thermal effects of the FG microbeam is developed. Afterwards, Using the Taylor series expansion for material properties, the heat conduction equation is solved analytically for temperature in the form of a power series. The free vibration of the FG microbeam is analyzed to achieve the natural frequencies and thermal damping ratio of the FG microbeam. The effect of FG index on the thermoelastic damping ratio is investigated in different aspect ratios. Also comparison studies are made between the results obtained from the models based on the Euler–Bernoulli and Timoshenko beam theories.

THERMALLY INDUCED VIBRATION OF A FUNCTIONALLY GRADED MICRO-BEAM SUBJECTED TO A MOVING LASER BEAM

In this paper, the thermally induced vibration of a functionally graded (FG) cantilever micro-beam subjected to a moving laser beam is investigated via simulating the equivalent third-order dynamic system. The material properties of the FG micro-beam are defined by an exponential function through the thickness. The coupled thermo-elastic equations are obtained utilizing the first law of thermodynamics under the assumption of the classical Fourier heat conduction model and Euler–Bernoulli beam theory. To evaluate the dynamic response of the micro-beam, the coupled equations are first discretized by employing a Galerkin based reduced-order model and then decoupled by applying the Cramer's rule. Solving the decoupled equations analytically, effects of key parameters are illustrated by means of time histories and phase portraits. Furthermore, by investigating the case of resonant excitation, the critical velocity corresponding to mobile heat source is obtained.

Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium

Thermo-mechanical size-dependent buckling analysis of embedded functionally graded (FG) microbeams is performed based on sinusoidal shear deformation beam and modified couple stress theories. It is assumed that material properties vary smoothly and continuously throughout the thickness. Winkler elastic foundation model is used to simulate the interaction between FG microbeam and elastic medium. The governing equations and corresponding boundary conditions are obtained with the aid of minimum total potential energy principle. The buckling characteristics of simply supported embedded FG microbeams in thermal environment are investigated. The obtained results are compared with the results of simple beam theory with no shear deformation effects and classical theory. Influences of thickness-to-material length scale parameter ratio, material property gradient index, slenderness ratio, temperature change and Winkler parameter on critical buckling loads of embedded FG microbeams are discussed in detail.

Simulation of MEMS Capacitive Thermal Sensor Based on Tip Deflection of a Functionally Graded Micro-Beam

This work presents a simulation study on a novel micro thermal sensor to evaluate a different range of temperature. Micro Electro Mechanical Systems (MEMS) technology is an interesting field in mechanics andmetrology. In this work, a capacitive micro thermal sensor based on tip deflection of Functionally Graded Micro-beam (FGM) was designed. The thermo-electric mechanical equations based on Euler-Bernoulli beam theory were derived and solved using step-by-step linearization method. The increase in temperature was expressed with respect to the changes in the capacitance. The beam deflections were compared with the existing results showing good conformity with the highest error obtained as 4.3% at 65°C.

Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

ABSTRACTBased on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory

In this paper, a novel size-dependent beam model made of functionally graded materials (FGMs) is developed based on the strain gradient elasticity theory and sinusoidal shear deformation theory. The material properties of the functionally graded (FG) microbeams are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. Governing equations and boundary conditions are derived simultaneously by using Hamilton’s principle. The new model contains three material length scale parameters and can consequently capture the size effect. In addition, the newly developed model degenerates to the modified couple stress sinusoidal beam model or the classical sinusoidal beam model by setting two or all material length scale parameters to zero. The Navier-type solution is developed for simply-supported boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, different material compositions, and shear deformation on the bending and free vibration behavior of FG microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. It is established that the present FG microbeams exhibit significant size-dependence when the thickness of the microbeam approaches to the material length scale parameter.

Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes

Stability of a functionally graded (FG) micro-beam, based on modified couple stress theory (MCST), subjected to nonlinear electrostatic pressure and thermal changes regarding convection and radiation, is the main purpose of this paper. It is assumed that the functionally graded beam, made of metal and ceramic, follows the volume fraction definition and law of mixtures, and its properties change as an exponential function through its thickness. By changing the ceramic constituent percent of the bottom surface, five different types of the micro-beams are investigated. The static pull-in voltages in presence of temperature changes are obtained by using step-by-step linearization method (SSLM) and, by adapting Runge–Kutta approach, the dynamic pull-in voltages are obtained numerically. Though the temperature distribution through the thickness of FG micro-beam (due to its too small measurement) is considered uniform, owing to the different thermal expansions of layers, temperature changes cause deflection in the micro-beam, and consequently affect pull-in values. Hence the profound effects of different material constituent over the pull-in voltages are illustrated and it is graphically displayed that how in some cases neglecting components of the couple stress leads to inaccurate results.

A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory

Based on the modified couple stress theory (MCST), a unified higher order beam theory which contains various beam theories as special cases is proposed for buckling of a functionally graded (FG) microbeam embedded in elastic Pasternak medium. This non-classical microbeam model incorporates the material length scale parameter which can capture the size effect. The non-classical beam model reduces to the classical beam model when the material length scale parameter is set to zero. The material properties of the FG microbeam are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique and the classical rule of mixture. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and explicit expressions related to each type of beam theory are proposed for the critical buckling load. Numerical results are presented to investigate the influences the material length scale parameter, aspect ratio, different estimation method of material properties, various material compositions, and the parameters of the elastic medium on the critical buckling load. Comparison study is also performed to verify the present formulation.

Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory

Static bending and free vibration of functionally graded (FG) microbeams are examined in this paper based on the modified couple stress theory (MCST) and various higher order beam theories (HOBTs). This non-classical microbeam model incorporates the material length scale parameter which can capture the size effect. The material properties of the FG microbeams are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The governing equations and the related boundary conditions are derived using Hamilton’s principle. The Navier-type solution is developed for simply-supported boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, different material compositions, and shear deformation on the bending and free vibration behavior of FG microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. It is established that the present FG microbeams exhibit significant size-dependence when the thickness of the microbeam approaches to the material length scale parameter.