Analytical Approximate Solution for Nonlinear Behavior of Cantilever FGM MEMS Beam with Thermal and Size Dependency
This paper is concerned with the nonlinear static behavior of a cantilever functionally graded material (FGM) microbeam with the influence of thermal stress and the intermolecular force. A significant extension of the recent works of constructing analytical approximate solutions to a clamped-clamped FGM MEMS/NEMS beam is formed in the paper. Based on the modified couple stress theory and an internal material length-scale parameter, the governing equations account for the microbeam size dependency. Note that the thermal force and moment in boundary condition make procedure of solution more complex. The combinations of Galerkin method and the assumption of deflection function are used to establish analytical approximate solutions which have brief expressions. Good agreements of approximate results are found for large range of free-end deflection of cantilever FGM microbeam through comparing them with numerical solutions and other existing results. The influence of various physical parameters, such as the material attributes (Young modulus, Poisson ratio, etc.), the electrostatic gap, and microsize of the beam on mechanical behavior, or Pull-In voltage of electrostatically actuated microbeams, could also be investigated with these analytical expressions.
Tremendous attention in the field of micro- and nanoelectromechanical systems (MEMS/NEMS) has been observed in the last few decades. Sensors and actuators are being actively developed by applying MEMS/NEMS [1–5], because of their advantage of bias networking simplicity and low power consumption. Radio frequency (RF) switches , energy harvesters, inductors , and variable capacitors for high radio frequency circuits  as well as bio-MEMS [9, 10] are typical representatives. The electrostatic force is inversely proportional to the square of the distance between the actuating electrode and the structure. The electrostatic force and large deformations of the structure could usually cause strong nonlinearity. Nathanson  first analyzed Pull-In instability (i.e., equilibrium instability of electrostatically driven MEMS/NEMS). The Pull-In instability including variables of voltages and deflections and limiting the operational procedure of MEMS/NEMS devices is taken as a basic static instability mechanism. Stable equilibrium exists only when the applied voltage does not exceed a critical Pull-In value, and the structure could safely work and not collapse on the actuating electrode, and vice versa. Therefore, analysis of Pull-In instability and mechanical behavior of MEMS/NEMS devices is key work for their designs and application. Scientists and engineering researchers are mainly interested in research of the static and dynamic stability of MEMS/NEMS under various operating conditions as well as Pull-In parameters’ determination.
Investigations on the instability of the single-phase material microbeam are carried out by many researchers [12–21]. For conventional MEMS/NEMS sensors or actuators comprised by a single layer material; however, all material properties and operational requirements posed by MEMS structural layers could not simultaneously be met. Therefore, the advantages of functionally graded materials (FGMs) could be reflected, such as high fracture toughness, improved stress distribution, the superior stress relaxation, capabilities of withstanding high temperatures, and large thermal gradients. Biomaterials are typically used to form the FGM microbeam [22–30]. FGMs have recently been used as micro- and nanostructures, such as MEMS and NEMS [31–33], thin films in the form of shape memory alloys , and atomic force microscopy (AFM) .
Besides residual stress and Pull-In parameters, thermal loading is another basic parameter that can directly affect the system work result. Owing to the electric current, the temperature would be raised, in the operating conditions of the microswitch. As known, thermal actuation could produce large displacements as a result of heat, which is the reason of using FGM as lots of thermal tunable capacitors and microcapacitive thermal sensors [34, 36–43]. Pull-In instabilities in a FGM microplate due to thermal stress caused by the electric current were studied by Hasanyan et al. . The strongly dependent relationship between Pull-In voltage and the temperature and the variation through the two constituents volume fractions thickness could be observed.
FGM microgripper mechanical behavior subjected to thermal load, intermolecular forces, and voltage is studied by Jahangiri et al.  through numerical method. Based on modified couple stress theory, a thin FGM microplate buckling behavior under mechanical and thermal force is investigated by the spline finite strip method . Ashoori and Vanini  observe thermal stability characteristics of FGM plates under the influences of power law index, the material length scale parameter, and inner and outer radii as well as elastic foundation coefficients. The asymmetric buckling patterns are revealed for special boundary conditions. Via applying von Karman’s surface elasticity theory and geometric nonlinearity, Yang and Wang  derive an analytical mechanical model for investigating the Pull-In characteristics of CNTs reinforced nanoactuator. Coupled electrostatic loading, thermal stress, dispersion forces as Casimir force and van der Waals, and surface stress are fully considered in the model. From the careful review of this paper, the effectiveness of electrostatically actuated microstructures caused by temperature changes was discussed in a limited number of papers, especially FGM microbeams caused by nonuniform temperature stress.
The purpose of this paper is to propose alternative methods to solve nonlinear response of a cantilever FGM MEMS/NEMS beam actuator, including the effect of fringing field, the length scale, the nonuniform temperature field, van der Waals force, and Casimir force. This paper forms an important extension of the recent research of establishing analytical approximate solutions to a clamped-clamped FGM MEMS/NEMS beam . The present approaches include numerical method as well as analytical approximate method. The brief analytical approximate expressions are constructed via coupling Galerkin formulation with the selection of the shape function. The numerical solutions of this problem are then obtained by establishing an extended system and using shooting method . Compared with numerical shooting solutions, analytical approximate results could give excellent agreements for a wide range of microbeam-center displacement. Furthermore, these brief analytical solutions can be applied to investigate the dependent relationship between mechanical behavior of electrostatically actuated microbeams and various physical parameters.
2. Governing Equation
Figure 1 depicts a FGM microbeam electrostatically actuated by a fixed electrode. A voltage is used across the microbeam and the electrode fixed below it. Let be the horizontal coordinate along the microbeam length and be the vertical coordinate along the cross section. Furthermore, are the axial and vertical translation of a point on the midplane (i.e., ). In this model, the fringing field effect denoted by its first-order correction and the influence of the intermolecular force are taken into account. The nonlinear differential governing equations for the equilibrium of the electromechanical system can be written in the following form [1, 3, 31, 46]:The cantilever boundary conditions arewhere
Here, present Young’s modulus, coefficient of thermal expansion, the Poisson ratio, a length-scale parameter of material, the width, the thickness , the length of the beam, and the nominal gap between microbeam and the ground, respectively. Furthermore, is the temperature transformation. The parameters and present thermal force and moment, respectively. The other physical parameters are as follows: the permittivity of free space , Planck’s constant (divided by ) , the Hamaker constant , the speed of light , and the potential of the fixed electrode . A planar stress statement is considered. For a wide beam , planar strain statement prevails and must be substituted by and by .
The material attributes of FGM microbeam are assumed to vary along its thickness , and the top surface is comprised with pure metal, the bottom surface from a mixture of ceramic with the metal. The material attributes about could be written aswhere is the material attributes, such as , , , and thermal conductivity . The constant is , where , are the material attributes at the top and the bottom surface of the microbeam, respectively.
In this paper, thermal effects in two conditions are considered: temperature is raised from the operating conditions of the microswitch; thermal stress is used to the microswitch. For saving space, about the details of (1) and (2) derivation, please see [44, 46]. Note that, compared with Sun et al. , the thermal force and moment in boundary condition (3) make procedure of solution more complex.
Next, (1) and (2) will be converted into dimensionless form. Substituting (1) into (2), and using (3), the controlling equations of the system could be rewritten as the following dimensionless forms:And the boundary conditions arewhere
The exact solution for the above mathematical model of electrostatic-actuated beam problem is hard to obtain, since the nonlinear terms in the model, such as nonlinear electrostatic force, van der Waals force, and the Casimir force, are coupled with the structural deflection. Moreover, the thermal force and moment in boundary condition (7) make solution procedure more complex. This paper is focused on establishing analytical approximate expressions. These analytical approximate solutions are presented via applying Galerkin formulation and the assumed deflection function of the micro-/nanobeam.
3. Analytical Approximate Solution
In this part, Galerkin formulation is used to construct the analytical approximate expressions. An assumed deflection function which satisfies the conditions in (7) and (i.e., ) can be expressed as
Substituting (9) into (6), multiplying (6) by and , and then calculating the integral subjected to , from to , yieldApplying (10), the approximate voltage expression is obtained:where the expressions for are presented in Appendix.
In the next part, (12) is explained to excellently calculate the actuating voltage for a classical example of MEMS/NEMS. It is noted that if all the three factors of Casimir force, the fringing field effect, and van der Waals force are included, the actuating voltage expression is more complex than the statement neglecting them.
The dimensionless Pull-In parameters can be expressed in terms of , through equation . In the same way, the actuating Pull-In voltages and Pull-In free-end deflections could be expressed in terms of , by calculating from equation . However, equation has multiple roots; the real solution relating to the Pull-In instability should be determined.
4. Results and Discussion
4.1. Validation of the Present Solution
In this part, the classical example will be given to illustrate accuracy of the proposed analytical approximate expressions via comparing it with numerical solution and other existing results [48–52]. The corresponding numerical solutions are calculated by applying the shooting method . To apply the shooting method to solve the model in (6) and (7), differential equations with initial conditions are expressed asand the shooting conditions are
Using the shooting method, the depending relation of the normalized applied voltage and the coefficient “” will be obtained. Furthermore, adding into the shooting condition (16), the Pull-In voltage would be presented. For saving space, the detail procedure of the shooting method would not be presented here, while the readers are kindly referred to Yu et al. . It is noted that the above numerical method can be conveniently carried out for solving nonlinear ordinary integral-differential equations describing electrostatically actuated microstructures .
Consider a cantilever microbeam without effect of temperature stress. The Pull-In voltage results of the cantilever microbeam are compared with the existing results in this case . The microbeam consisted of single material in this validation example, with the relevant material and geometric parameters listed in Table 1. Comparisons of the Pull-In voltages: the numerical result obtained with the shooting method , the analytical approximate voltage , and other results are listed in Table 2. The relative errors of Pull-In voltage between the present analytical results and  are 1.626% and 1.322%, for narrow and wide beams, respectively. The results from Table 2 reveal good agreements. Moreover, Figure 2 shows that Pull-In voltages of the present analytical approximate results could give excellent agreements with the numerical ones obtained by shooting method, respected to the length of the beam. In particular, the maximum relative errors of Pull-In voltage between the present analytical and numerical results with respect to the length of the beam are 0.954% and 1.546%, for narrow and wide beams, respectively.
4.2. Pull-In Voltage of FGM Microbeam
In this part, the dependent relationship between the Pull-In voltage and uniform/nonuniform temperature field stresses, the microbeam’s material attributes, Casimir and van der Waals forces, and length-scale parameter of material are proposed.
Consider a FGM microbeam with the material and geometric parameters given in Table 3. It consisted of two materials: silicon nitride () and nickel (Ni). The FGM microbeam is diffused from five different silicon nitrides to nickel. According to the various percentage of diffused in FGM microbeam, Cases 1–5 are determined and relative to 0–100%, in increments of 25%. The characteristics of Cases 1–5 are given in Table 4. The coefficient of thermal expansion in the thickness direction and variation in Young’s modulus are calculated by (5).
Next, some good features of the FGM microbeam over single-phase microbeam with respect to both thermal and electrostatic stress will be illustrated. The microbeam subjected to , where the initial temperature is with being the temperature change, will initially be considered. The nonuniform temperature field is with the following boundary conditions: at the bottom, and at the top of the microbeam.
Firstly, consider the statement without thermal effects. For the cantilever boundary condition, the dependent relationship between the actuating voltage and the free-end deflection of the FGM microbeam is shown in Figure 3. It can be observed that the stable and unstable solutions are shown by thick solid lines and dot lines in Figure 3, respectively. The increased stiffness of the microbeam (which is due to the inclusion of ) gives rise to the increase in Pull-In voltage.
The influence of a uniform temperature field stress on the actuating voltage for a certain free-end deflection of the microbeam for Case 2 is presented in Figure 4. From Figure 4, it is concluded that a decrease in the FGM (Case 2) microbeam Pull-In voltage will be caused by uniform thermal loading. Also the stable and unstable solutions are shown by thick solid lines and dot lines in Figure 4, respectively.
In the following section, the case of a thermally actuated microbeam () will be investigated. Figure 5 shows the dependent relationship between FGM microbeam free-end deflection and the temperature field change () under (a) uniform temperature field and (b) nonuniform temperature field ( and ). In Figure 5(a), the free-end deflection of the microbeam severely increases, when is large. However, the values of the free-end deflection vary linearly with increase, when is small. Moreover, from Figure 5(a), one could find that in the case of uniform temperature field (the temperature field boundary condition: and ) the pure nickel microbeam deflection is not affected by the temperature change. In short, nonuniform thermal stress is less effective than uniform temperature loading for microbeam. The dependent relationship between the Pull-In parameters of the FGM microbeam and other different nonuniform temperature distributions, such as and , can also be studied with the present analytical approximate expressions.
In the end, the dependent relationship between Pull-In parameters (voltage and deflection) and the microbeam length scale parameter will be investigated in Figure 6. The length scale parameter plays a significant role in the Pull-In parameters. If the length scale parameter is included into the model, the Pull-In voltage and deflection are larger than the case of neglecting it. In addition, with the increase of material length parameters, the results based on modified coupled stress theory are significantly different from those based on classical theory.
5. Conclusions and Future Work
In this paper, numerical and analytical approximate expressions to the nonlinear response of a cantilever actuator modeled as a FGM microbeam have been developed. The thermal force and moment in boundary condition make procedure of solution more complex. Based on combination of Galerkin formulation and an assumed deflection shape function, the analytical approximate solution expression to the static behavior of a cantilever FGM microbeam has been constructed. Good agreements of approximate results have been demonstrated for wide range of free-end deflection of cantilever FGM microbeam through comparing them with numerical solutions and other existing results. The effect of thermal stress to the critical parameters of the FGM microbeam, static deformation and the applied voltage, is explicitly analyzed via the analytical solution. Theoretical results are very useful and convenient for implementation in MEMS/NEMS designs and tests, because the approximate solution expressions are explicit and brief functions of free-end deflection and other geometric and material parameters. In future, advanced approximate solution method, such as [54, 55], will be applied and improved for solving nonlinear vibration of MEMS and some experiments in lab will be done to validate the theoretical results.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
The authors gratefully acknowledge the financial support from the Science and Technology Developing Plan Project of Jilin Province (Grant no. 20160520021JH), Science and Technology Project of the 13th Five-Year Plan of Jilin Provincial Department of Education (Grant no. JJKH20190126KJ), and project from Guangzhou Marine Geological Survey (Grant no. 3S2180254424).
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