Free Vibration Behavior of a Gradient Elastic Beam with Varying Cross Section

Based on strain gradient elasticity theory, a finite element procedure is proposed for computation of natural frequencies for the microbeams of constant width and linear varying depth. Weak form formulation of the equation of motion is obtained first as in common classical finite element procedure in terms of various kinds of boundary conditions. Gradient elastic shape functions are used for interpolating deflection inside a finite element. Stiffness and mass matrices are then calculated to solve the microbeam eigen value problem. A solution for natural frequencies is obtained using characteristic equation ofmicrobeam in gradient elasticity. The results are given in a series of figures and compared with their classical counterparts. The effect of various slope values on the natural frequencies are examined in some numerical examples. Comparison with the classical elasticity theory is also performed to verify the present study.


Introduction
Microbeams have held wide applications in micro-electronicmechanical systems (MEMS) such as those in actuators [1], microswitches [2], microresonators [3], Atomic Force Microscopes [4], and sensors [5] in which thicknesses and lengths of microbeams are typical on the order of microns and submicrons; therefore the small scale effects in their static and dynamic behavior are considerable.
In the past decades, some researchers have tried to establish experimental and theoretical models for vibration of microbeams.However, most of these studies are based on the classical continuum mechanics.The classical model is doubtful whether it can describe the static or dynamic behaviour of elastic materials with microstructure, since it is associated with the concepts of locality of the stress.The experimental studies have also revealed that the classical continuum mechanics is unable to consider size-dependent static and dynamic behaviours in microscaled structures [6,7].Lam et al. [8] observed experimentally that the normalized bending stiffness increases by about 2.4 times when the thickness was reduced from 115 to 20 m; Stolken and Evans [9] examined that the plastic work hardening shows a great increase as the microbeam thickness decreases from 50 to 12.5 m.These results demonstrate that the size dependence is significant to certain materials.
During past years, some nonclassical elasticity theories such as the strain gradient, nonlocal, and couple stress theories have been introduced and employed to investigate the microbeams.Nonclassical continuum theories can be classified to couple stress and strain gradient approaches.The couple stress theory includes higher-order stresses, known as the couple stresses.The classical couple stress theory was originated by Mindlin [10] and others including Toupin [11] in 1960s.Some related research works have been performed to model the static and dynamic problems based on the classical couple stress theory [12,13].A new modification to couple stress theory has been proposed by Mindlin [14] in which a new equilibrium equation of high order is considered in addition to the classical equilibrium equations of forces and moments of forces.So far Park and Gao [15] have developed the static bending; Kong et al. [16] have developed free vibration problems of a Euler-Bernoulli beam.Ma et al. [17] have developed the static bending and free vibration problems of a Timoshenko beam.dos Santos and Reddy [18] have analyzed classical and nonclassical frequency ratios for a Timoshenko microbeam.Demir et al. [19] have examined nonclassical frequencies of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique.Akgoz and Civalek [20] have analyzed microsized beams for various boundary conditions based on the strain gradient elasticity theory.In addition, small scale effect of the torsion response of bars and buckling of axially loaded microscaled beams has been presented by Kahrobaiyan et al. [21] and Akgoz and Civalek [22], respectively.
The objective of the present paper is to establish dynamic finite element model for microbeams of constant width and linearly varying depth by using both the basic equations of the strain gradient elasticity theory and weak form formulation of the dynamical equation.The material and microbeam models are assumed to obey the strain gradient theory, as developed by Papargyri-Beskou et al. [23].In the first part of study, the basic equations of gradient elasticity theory are reviewed in Section 2. Then gradient elastic essential boundary conditions for static and dynamic analysis of microbeam obtained with the aid of weak form formulation of equation of motion.By using essential boundary conditions, interpolating functions are calculated for finite element procedure in Section 3. Section 4 is devoted to the construction of stiffness and mass matrix of the microbeam of constant width and linearly varying depth.Four representative numerical examples are solved and solutions are compared with their classical counterparts in Section 5.The paper gives a conclusion in Section 6.

Governing Equation for
Bending.In the present study the following gradient elasticity theory is employed which is suitable for investigating the natural frequencies of microbeams [23]: in which   and   represent the stress and strain tensors,   represents the elastic moduli, and  denotes the simply gradient elastic modulus, representing the internal or characteristic length of the material microstructure.The kinetic and the potential energies of the microbeam shown in Figure 1 are then in the following form: where  is the mass density,  is the modulus of elasticity, () is the moment of inertia about the axis , and () is the cross sectional area.Note that (), () quantities are variable.The variation of the external excitation force is obtained as In order to obtain the partial differential equation governing the motion of the system, above equations substituted into the Hamilton's principle resulting in with the following classical fixed boundary conditions for microbeam: It can be seen from ( 4) that the equation of motion of the microbeams is related to two parts: one associated with () and () as in classical microbeam model and the other associated with the material parameters () 2 .

Weak Form Formulation of the Equation of Motion and Discretization
In the gradient elastic finite element method, a variational description of the problem is needed first as in common finite element procedure [24].The starting point for deriving the weak form is to multiply the partial differential equation of gradient elastic beam with a test function and integrate over the domain: where () is the test function and integrating by parts Presented equation ( 7) is a weak derivation of the equation of motion and the boundary conditions for the microbeams based on strain gradient elasticity theory and the Euler Bernoulli beam theory.From (7), gradient elastic essential boundary conditions for the microbeam clamped at two ends (see Figure 1) are 3.1.Interpolating Functions for Microbeams.In this section, the gradient elastic interpolating functions are calculated.The weak formulation of the equation of motion in (7) requires that the interpolation functions of one element should be continuous with nonzero derivatives up to three orders; then the essential boundary conditions in ( 8) are satisfied.The nodal degrees of freedom are indicated by the following expressions: Figure 2 shows two-node strain gradient microbeam element.
It should be noted that V  3 and V  6 are strain gradient parameters, which is not shown in Figure 2. Since there is a total of six conditions in an element (three per node), a fifth degree polynomial can be selected for V(): By using ( 9) and ( 10), nodal variables can be expressed in terms of   unknowns in the following compact form: Inverting this matrix equation to express   in terms of , and V  6 and substituting the result into (10), we obtain and the interpolation functions Ω   in ( 12) can be expressed in terms of the local coordinate : where  +1 =   + ℎ  and ℎ  is the length of element.

Developing the Strain Gradient Microbeam Finite Element
Model.In this section the stiffness and mass matrix of a strain gradient microbeam element developed and during an example.The total length of the microbeam is 2, as seen in Figure 3.It has nonuniform cross section ().The modulus of elasticity  is taken as a constant for each element.The microbeam is fixed at two points such that the problem is statically indeterminate.In order to compute the free vibration frequencies, the external excitation force is taken as zero in (6).To illustrate the use of finite elements for gradient elastic microbeams discretized the beam longitudinally with two beam elements, as seen in Figure 4. Considering a section at a distance  from fixed end, as shown in the Figure 1.The variation of the  1 () and  1 () over the first element is as follows: Similarly, the moment of inertia and area of the second element with respect to  axis are
It is noted that during the assembly of the matrices, the components of the natural boundary condition vector will be cancelled.For a nontrivial solution, determinant of the assembled coefficient matrix must be zero as follows: It should be noted that the dimension of the assembled matrix in ( 22) is 3 × 3 while, in classical elasticity, the dimension of the assembled matrix is 2 × 2. The advantage of this determinant is capability of considering any possible combination of slopes.The problem is an eigenvalue problem and determinant of coefficient matrix yields a characteristic equation and the roots of the characteristic equation give the natural frequencies for the microbeam with varying cross section.

Results for the Natural Frequencies
In this section, four numerical examples are presented and discussed to verify the accuracy of the present finite element model based on gradient elasticity.The microbeam considered here is steel ( = 210 GPa,  = 8166 kg/m 3 ).It should be noted that the material scale parameter is taken as ( = 17.6 m) for homogeneous epoxy beam [8].

Effect of the Material Length Scale Parameter and Slope of
Second Element.In order to see the effect of ( 2 ) on the first natural frequency, the gradient elasticity and classical elasticity frequency ratios are presented for various slope values.The dimensionless material length scale parameter is considered as (/ = 0.1, . . ., 0.8).For illustration purpose, the following parameters are used in calculating the numerical results: ( = 200,  = 1 mm, and  =  = 10).Figure 5 shows the variation of slope ( 2 ) schematically; Figure 6 contains plots of the dimensionless depth (/) and material parameter (/) versus first frequency ratios (classical and gradient theories).
As expected, the strain gradient effects are significant when (/) ratio is high.In Figure 7 for various values of (/) are assessed.In other words, last point thickness of the microbeam () is varied when the other thicknesses are keep constant as ( =  = 10).It is clearly seen that the last point thickness decreases as the dimensionless material scale parameter (/) increases.When the dimensionless material parameter (/) and the slope of second element ( 2 ) increase, the first frequency ratios decrease and the difference between the two models is significant, as seen from Figure 7.It is shown that the difference among the predicted frequency values is diminishing when the thickness of the microbeam becomes larger, thereby indicating that the small size effect is only significant when the thickness of the microbeam is comparable to the material length scale parameter.From the results, the reason of differences between the frequencies predicted by two theories can be simply explained by the material length scale parameter in the constitutive equation.

Effect of the Material Length Scale Parameter and Slope
of First Element.In the second example, first frequency ratios are examined for various values of the slope ( 1 ).The dimensionless material length scale parameters are taken as (/ = 0.01, . . ., 0.08).The following parameters are used in computing the numerical solutions: ( = 300,  = 1 mm, and  =  = 5).Figure 8 shows the variation of slope ( 1 ) schematically.The effects of  1 and dimensionless material length scale parameter on the first frequency ratio are demonstrated in Figures 9 and 10. Figure 9 contains plots of the dimensionless depth (/) and material parameter (/) versus frequency ratios (classical and gradient theories).
It is interesting to note that as the dimensionless material parameter (/) and the slope of first element ( 1 ) increase, the frequency ratios also increases, as seen from Figure 10.

Effect of the Material Length Scale Parameter and Slopes.
This subsection deals with the effect of slopes and slope ratios on the free vibration frequencies of a microbeam.The following parameters are used in this example: ( = 300,  = 1mm), and the difference between the two models is examined.In this example, the microbeam midpoint thickness () is varied when both first and last point depth are keep constant as ( = 25,  = 3).Namely, the slope of first element ( 1 ) increases, when the slope of second element ( 2 ) decreases.Figure 11 shows the variation of  1 and  2 .The effects of midpoint depth, slopes ( 1 ,  2 ), and slope ratios ( 1 / 2 ,  2 / 1 ) on the frequency ratios are demonstrated in Figures 12 and 13.It can be concluded that the slopes ( 1 ,  2 ) play the essential role in increasing or decreasing the first natural frequency for the microbeam.Also, it can be inferred that the frequencies of microbeam can be controlled by choosing proper values of ( 1 ) and ( 2 ), as seen from Figure 13.However, the small size effects are almost diminishing as  the depths of the microbeam is far greater than the material length scale parameter.

Effect of the Material Length Scale Parameter on the Second
Mode.To investigate the effects of the length scale parameter on the first two frequencies of microbeam, frequency ratios are plotted in Figures 14-15 for various values of length scale parameter.According to these figures the first two frequency ratios are less than unity.It can be concluded from the figures that the effect of the length scale parameter on Mode-2 is more than on Mode-1.The difference between gradient and classical elasticity results for vibration frequencies is changing with slopes and material scale parameter.

Conclusion
In this study, an analysis has been performed for the evaluation of the natural frequencies of microbeams of constant width and linearly varying depth.The equation of motion and boundary conditions are obtained by a combination of the weak formulation and the basic equations of gradient elasticity.A finite element formulation is proposed for computation of natural frequencies and its numerical implementation is then verified obtaining a good agreement with classical data available in the scientific literature.Four numerical examples are solved and the small size effects on the natural frequencies are assessed for various slope values.Numerical results are presented in a series of figures to examine the effect of material parameter on the natural frequencies.The results reveal that for the microbeam with varying cross section comparable to its material length scale parameter, the effect of strain gradient is significant.As expected, it is also found that the present model exhibit size dependence when the depth of the microbeam approaches the material scale parameter.In addition the slopes have a great influence on the dynamic behavior of microbeam.It can be concluded that the frequencies of microbeam can be controlled by choosing proper values of depths.The present solution is very useful to analyze microbeams with linear varying cross section.

Figure 1 :
Figure 1: Schematic diagram of Cartesian coordinate system for a beam fixed at two ends.

Figure 3 :
Figure 3: Geometry of a constant width and linearly varying depth microbeam.

2 Figure 5 :Figure 6 :
Figure 5: Schematic diagram for the variation of  2 and .

Figure 11 :
Figure 11: Schematic diagram for the variation of  1 and  2 .