Elsevier

Composite Structures

Volume 93, Issue 6, May 2011, Pages 1516-1525
Composite Structures

On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure

https://doi.org/10.1016/j.compstruct.2010.11.013Get rights and content

Abstract

In the present work, mechanical behavior of a functionally graded cantilever micro-beam subjected to a nonlinear electrostatic pressure and temperature changes has been studied. It has been assumed that the top surface is made of pure metal and the bottom surface from a metal–ceramic mixture. The ceramic constituent percent of the bottom surface varies from 0% to 100%. In addition to the Volume Fractional Rule of material, exponential function has been used for representation of continuous gradation of the material properties through micro-beam thickness. Attention being paid to the ceramic constituent percent of the bottom surface, five different types of FGM micro-beams have been investigated. Nonlinear integro-differential thermo-electro mechanical equation based on Euler–Bernoulli beam theory has been derived and solved using Step-by-Step Linearization Method and Finite Difference Method. The effects of temperature changes and the electrostatic pressure on the deflection and stability of FGM micro-beams having various amounts of the ceramic constituent have been studied and normal stress distributions in the cross section along the beam thickness have been given and compared with a classic metal beam.

Introduction

Functionally graded materials (FGMs) can be described indeed as microscopically inhomogeneous composites which are made from a mixture of two different materials, usually a metal and a ceramic through a powder metallurgy process, with a desired continuous variation of properties as a function of position along certain dimension(s). This type of materials provides the specific benefits of both constituents. The continuously compositional variation of the constituents in FGMs from one surface to the other offers a graceful solution to the problem of appearing high magnitude shear stresses that may be induced in laminated composites, where two materials with great differences in properties are bonded. Today, FGMs have a great practical importance because of their vast applications in many industrial and engineering fields [1], [2].

The concept of FGM was first proposed in 1984 by a group of material scientists, in Sendai, Japan, as novel materials with thermal barrier or heat-shielding properties for aerospace application and fusion reactors. Later on, FGMs were developed for military, automotive, biomedical application, semi-conductor industry and general structural element in high thermal environments [2], [3].

The motivation for using functionally graded materials (FGMs) is their advantages of superior stress relaxation and capabilities of withstanding high temperatures and large temperature gradients. The ceramic constituent of the material provides the high temperature resistance due to its low thermal conductivity. The ductile metal constituent prevents fracture caused by stresses due to high temperature gradient in a very short period of time [4], [5], [6].

Many studies have been performed by researchers on the mechanical behavior of FGM beams. Asghari et al. [1] performed a study on the size-dependent behavior of micro-beams made of functionally graded materials (FGMs) which are analytically investigated on the basis of the modified couple stress theory in the elastic range. Sankar [7] presented an elasticity solution for FGM beams while Aydogdu and Taskin [8] studied on the natural frequencies and mode shapes of a FGM simply supported beam. Two-dimensional elasticity solution of an FGM beam with simply supported edges is investigated by Ying et al. [9] and natural frequencies and mode shapes are presented using state space method. Li [10] presented a new unified approach for analyzing the static behavior of functionally graded beams including rotary inertia and shear deformation. Kapuria et al. [11] validated a third order zigzag theory based model for layered functionally graded beams in conjunction with the modified rule of mixtures (MROM) for effective modulus of elasticity through experiments for static and free vibration response. Simsek and Kocatürk [12], [13] investigated the free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load and solved the governed dynamic equation of motion by using the implicit time integration Newmark-β method. Sina et al. [14] used a new beam theory different from the traditional first-order shear deformation beam theory to analyze free vibration of functionally graded beams. Khalili et al. [15] presented a mixed method to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads. The theoretical formulations are based on Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations. Mahi et al. [16] presented exact solutions to study the free vibration of a beam made of symmetric functionally graded materials. The formulation used is based on a unified higher order shear deformation theory. The beam is assumed to be initially stressed by a temperature rise through the thickness.

Recently, FGMs are widely used in micro- and nano-structures such as thin films in the form of shape memory alloys [17], [18], micro- and nano-electromechanical systems (MEMS and NEMS) [19], [20], [21] and also atomic force microscopes (AFMs) [22].

Study of electrostatically actuated MEMS is a branch of micromechanics whose principle is a very commonly used one on sensing or actuating devices in MEMS. The foregoing systems have found wide applications in switches [23], micro-mirrors [24], micro-resonators [25], micro-actuators [26], micro thermo-meters [27], and tunable capacitors [28].

Today because of advantages of electrostatic MEM actuators, such as, favorable scaling property, low energy consumption, low cost, low driving power, large deflection capacity, relative ease of fabrication and others, electrostatic MEM actuators are widely used in micro-structure. Therefore, electrostatic MEM actuators are one of the most common and important actuators in MEMS [26].

MEMS are usually comprised of a conductive deformable body suspended above a rigid grounded body [29]. An applied direct current potential difference between the two bodies induces the Coulomb force that deflects the deformable body, and consequently changes the system capacitance. When an additional alternating current is applied to excite harmonic motions of the deformable electrode, resonant devices are obtained. These devices are used in signal filtering, and chemical and mass sensing, see e.g., [30], [31], [32], [33]. The applied direct current voltage has an upper limit beyond which the electrostatic force is not balanced by the elastic restoring force in the deformable conductor, and the MEMS eventually collapses. This phenomenon, called pull-in instability, has been observed experimentally [34], [35]. The critical voltage associated with this instability is called the pull-in voltage. In micro-mirrors [36] and micro-resonators [37], the designer avoids this instability to achieve stable motions; while in switching applications [38], the designer exploits this effect to optimize device’s performance. In fact as a result of the nonlinearity, stable force balance and deflection can only be achieved to one third (in lamped modeling) of the initial gap distance [39].

Tilmans and Legtenberg [31] studied the static and dynamic characteristics of electrostatically driven vacuum-encapsulated polysilicon resonators using Rayleigh–Ritz method. Abdel-Rahman et al. [40] and Kuang and Chen [32] presented the nonlinear model for electrostatically actuated micro-beams with both ends fixed. A distributed parameter model was used by Ramezani et al. [41] to study the pull-in instability of cantilever nano-mechanical switches subjected to intermolecular and electrostatic forces.

Since it is sometimes difficult for a single layer to meet all material and economical requirements posed to an MEMS structural layer, Witvrouw and Mehta [20] proposed the use of a non-homogenous functionally graded material (FGM) layer to achieve the desired electrical and mechanical properties and suggested that a polycrystalline-SiGe (poly-SiGe) layer can be an appropriate choice. Hasanyan et al. [42] studied the pull-in instabilities in a functionally graded MEMS due to the heat produced by the electric current. The material properties of the two-phase MEMS are assumed to vary continuously in the thickness direction. Recently, Jia et al. [43] investigated the nonlinear pull-in characteristics of the microswitches made of either homogeneous material or non-homogeneous functionally graded material (FGM) with two material phases under the combined electrostatic and intermolecular force. The effects of material composition, gap ratio, slenderness ratio, intermolecular force, axial residual stress on the pull-in instability was shown.

Temperature change is one of the basic actuation parameters that can tune the system directly. Thermal actuation is known for its capability in producing a large and linear displacement with respect to a heating power. This mechanism is derived using a FGM micro-beam including variable thermal expansion coefficient along its thickness.

In micro capacitive thermal sensors and thermal tunable capacitors, the used FGM micro-beam is not only subjected to an electrostatic force but also to a thermal bending moment. In addition, all of the micro-actuators and micro sensors are exposed to temperature changes.

By applying simultaneously effect of electrostatic force and thermal moment, the pull-in parameters can change. But there are no enough studies in the effect of temperature changes on the stability of electrostatically actuated micro-structures specially FGM micro-beams. In the present work, a FGM micro-beam, which is loaded by a nonlinear electrostatic pressure and a thermal bending moment, is studied. The static behavior of the micro-beam and its static instability are investigated.

Section snippets

Mathematical modeling

The proposed model consists of a deformable FGM cantilever micro-beam separated from a fixed ground plate as a substrate by an air gap g0 (Fig. 1).

A DC voltage source, V, between the capacitor electrodes, is used to create an electrostatic force along the micro-beam.

The following reasons can cause the deflection of the micro-beam:

  • Temperature changes (variable thermal expansion coefficient of FGM micro-beam along its thickness cause the deflection of the micro-beam as a result of temperature

Analytical solution (only subjected to a thermal moment)

When the applied voltage is zero, there are no external force (no Electrostatic Pressure) and the bending moment at a cross section is zero: (M(x) = 0). Therefore, the equation of FGM micro-beam deflection can be obtained as:-(EI)eqd2wdx2=MT

Then by integration, deflection of the micro-beam at a given x, along its length can be calculated as below in terms of temperature changes:w=-Ωθ2(EI)eqx2

The constants of integration, from below boundary conditions at fixed end, are evaluated.w(x=0)=0,dwdx(x=0)

FGM Micro-Beam subjected to only thermal moment

The investigated micro-beam in this paper has the material and geometrical properties given in Table 1, Table 2.

They are reference values for any case where no values are given. High difference of thermal expansion coefficients of two materials (Silicon nitride and Nickel) is suitable to have a large and sensible deflection.

In order to verify the model, obtained results were compared with those published in Refs. [39], [48], [49].

It is assumed that the top surface is made of pure metal but the

Conclusion

In the present work, a cantilever FGM micro-beam subjected to nonlinear electrostatic pressures and temperature changes was studied.

It was assumed that the top surface was made of pure metal but the bottom surface from a mixture of metal and ceramic. The ceramic constituent percent of bottom surface varies from 0% to 100%. Considering an exponential function for representation of continuous gradation of the material properties through micro-beam thickness, nonlinear integro-differential

References (54)

  • M. Simsek et al.

    Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load

    J Compos Struct

    (2009)
  • M. Simsek

    Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load

    J Compos Struct

    (2010)
  • S.A. Sina et al.

    An analytical method for free vibration analysis of functionally graded beams

    J Mater Des

    (2009)
  • S.M.R. Khalili et al.

    A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads

    Compos Struct

    (2010)
  • A. Mahi et al.

    An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions

    J Compos Struct

    (2010)
  • Y.Q. Fu et al.

    Functionally graded TiN/TiNi shape memory alloy films

    J Mater Lett

    (2003)
  • Y.Q. Fu et al.

    TiNi-based thin films in MEMS applications: a review

    J Sens Actuat A

    (2004)
  • H.A. Tilmans et al.

    Electrostatically driven vacuum-encapsulated polysilicon resonators: part II. Theory and performance

    Sens Actuat A – Phys

    (1994)
  • A. Ramezani et al.

    Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces

    Int J Solids Struct

    (2007)
  • M. Simsek

    Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories

    Nucl Eng Des

    (2010)
  • J. Yang et al.

    Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties

    Int J Solids Struct

    (2006)
  • L.L. Ke et al.

    Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

    J Compos Struct

    (2010)
  • A. Alibeigloo

    Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers

    J Compos Struct

    (2010)
  • G. Anandakumar et al.

    On the modal behavior of a three-dimensional functionally graded cantilever beam: Poisson’s ratio and material sampling effects

    J Compos Struct

    (2010)
  • Y.C. Hu et al.

    Some design consideration on the electrostatically actuated microstructures

    J Sens Actuat

    (2004)
  • R. Javaheri et al.

    Thermal buckling of functionally graded plates based on higher order theory

    J Therm Stress

    (2002)
  • C.M. Craciunescu et al.

    New ferromagnetic and functionally grade shape memory alloys

    J Optoelectron Adv Mater

    (2003)
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