Size-dependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Pre-stress Loading based on DQM

M. Mohammadimehr^*, H. Mohammadi Hooyeh, H. Afshari, M.R. Salarkia

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

1. 1.      Introduction

Nano technology is one of the most powerful technologies that can produce many materials and devices across a range of applications, such as electronics, biomaterials, medicine, and energy production [1-3]. Vibration analysis of composite beams has been a research topic in many engineering fields because vibration plays an important role in the design of turbine blades, helicopter blades, propeller blades, drill bits, and fluted cutters. In practice, these structures are typically modeled as either Euler or Timoshenko beams. The design of micro- and nano-electro-mechanical systems (MEMS/NEMS) requires widespread use of micro-rods and microbeams with different complex behaviors. Currently, micro-composite beams are employed in micro-turbo machines, ultrasonic piezoelectric micro-motor designs, and medical micro devices.

Recently, many researchers have investigated the mechanical behaviors of micro- and nano-scale materials using beam, plate, and shell theories. Ghorbanpour Arana et al. [4] analyzed the pulsating fluid-induced dynamic instability of double-walled carbon nano-tubes (DWCNTs), based on a sinusoidal strain gradient theory using the differential quadrature method (DQM) and the Bolotin method. Their results depicted that the imposed magnetic field was an effective controlling parameter for dynamic instability of visco-DWCNTs. In another work, Ghorbanpour Arani et al. [5] presented the nonlinear vibration of coupled nano- and microstructures conveying fluid flow based on a Timoshenko beam model under a two-dimensional magnetic field. They expressed that the magnetic field played an important role in the stability of the carbon nano-tubes (CNTs) and controls the stability of the nanosystem.

Simsek [6] studied the free vibration analysis of nanobeams with various boundary conditions, based on the nonlocal elasticity theory for large amplitude. In that research, the effect of nonlocal parameters on the nonlinear frequency ratio was examined. Their results showed that the nonlocal effects should be considered in the analysis of the mechanical behavior of nanostructures. Sahmani and Bahrami [7] analyzed the dynamic stability of microbeams subjected to piezoelectric voltage, using the strain gradient theory (SGT). In their results, for a special value of applied piezoelectric voltage, increasing the dimensionless length scale parameter decreased the difference between stability responses predicted by the classical and non-classical beam models. In addition, Mohammadimehr and Golzari [8] investigated the elliptic phenomenon effect of cross-sections on the torsional buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube (SWCNT). With an increase in the matrix thickness, the tangential and longitudinal strains of SWCNT decreased, and the opposite effect occurred for the interface stress and the dimensionless stress of the outer surface.

Alternately, Mohammadimehr and Rahmati [9] considered the small-scale effects on electro-thermo-mechanical vibration analysis of a single-walled boron nitride nanorod under electric excitation. They represented that the natural frequency decreased with an increase in the small-scale effects or aspect ratios. On the other hand, the small-scale effects were significant for lower aspect ratios and higher natural frequencies. Atabakhshian et al. [10] employed vibration of a smart coupled electro-thermal nanobeam system with an internal flow, based on nonlocal elasticity theory, while Ansari et al. [11] derived free vibration analysis from the evaluation of single and double-walled carbon nanotubes based on nonlocal elastic shell models. They concluded that the small-scale effects in the nonlocal model made nanotubes more flexible. Akgoz and Civalek [12] studied higher-order shear deformation in microbeam models, based on the strain gradient elasticity theory. Their results showed that microbeams derived from the non-classical theories, specifically modified strain gradient theory (MSGT), were stiffer than those based on the classical theory (CT).

Asgharifard Sharabiani and Haeri Yazdi [13] illustrated the nonlinear free vibrations for functionally graded (FG) nanobeams, including their surface effects. The results showed that the surface effects at higher volume fraction indices were either less or more dominant, in small and large amplitude ratios, respectively. Ke et al. [14] investigated the nonlinear vibrations of piezoelectric nanobeams based on the nonlocal and the Timoshenko beam theories. Their results demonstrated that a change in the external electric voltage from a positive value to a negative value led to a decrease in the nonlinear frequency ratio. Ansari et al. [15] analyzed the bending, buckling, and free vibration responses of FG Timoshenko microbeams, and they observed that the critical buckling loads and natural frequencies predicted by the beam models, based on MSGT and CT, provided the maximum and minimum values, respectively. Tounsi et al. [16] illustrated size-dependent bending and vibration analysis of FG microbeams, based on MCST and neutral surface positions. They represented that the inclusion of the couple stress effect makes a microbeam stiffer and decreased the vertical displacement and increased the natural frequency.

Alternately, Nazemnezhad et al. [17] employed an analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. They observed that the effect of the surface density on the variation of the natural frequency of the nanobeam versus the thickness ratio decreases consistently with the increase of the mode number. Nejat Pishkenari et al. [18] examined the surface elasticity and size effects on the vibrational behavior of silicon nanoresonators. They developed a continuum model for nanobeam vibrations that was capable of predicting the results of molecular dynamics (MD) simulations with considerably lower computational effort. Yue et al. [19] proposed a microscale Timoshenko beam model for piezoelectricity using flexoelectricity and surface effects. Their results observed that the change of surface properties not only directly affected the static bending but also significantly changed the natural frequency of the beam. Preethi et al. [20] presented surface and nonlocal effects of the nonlinear analysis of Timoshenko beams using Eringen’s nonlocal theory and the Gurtin-Murdoch approach, where the nonlocal parameters and the positive surface parameters’ values decreased the stiffness of the beam and resulted in larger deflections and lower frequencies.

In this research, a Timoshenko microbeam model, based on the modified strain gradient theory (MSGT) and surface stress effects subjected to pre-stress loading, is presented. The MSGT and surface stress effects were considered together in this study because both of them affect the structure at the microscale. Despite the fact that the surface and small scale effects have been investigated individually in some papers, the novelty of this study lies in the evaluation of size-dependent effects, including three material length scale parameters, and the surface residual stress based on strain gradient, and the surface stress elasticity effects on the dimensionless natural frequency of Timoshenko microbeams, subjected to pre-stress loading and considered simultaneously at a microscale. Moreover, the size-dependent effects increased the dimensionless natural frequency due to increasing flexural rigidity, which then enhanced the stability of the microstructures. The governing equations of motion were obtained using Hamilton’s principle and energy method. The equations were solved using the differential quadrature method (DQM).

1. 2.      The Governing Equations of Timoshenko microbeams

A schematic view of a straight Timoshenko microbeam model based on surface layers, an elastic medium, and pre-stress load is shown in Figure 1. The displacement fields for this model can be stated as [21]

where and are axial and transverse displacements for the neutral axis, respectively, and is the rotational transverse normal angle about the x-axis.

The components of normal ( ) and shear ( ) strains, using Eqs. (1), (2), and (3), are considered as follows:

Figure 1. A schematic view of a Timoshenko microbeam model with a surface layer, elastic medium, and pre-stress load.

The strain energy for the linear isotropic elastic material, based on MSGT, is considered as follows [21,22]:

where and are the Cauchy stress tensor and the strain tensor, respectively. The expressions  and denote the dilatation gradient tensor, the deviatoric stretch gradient tensor, and the symmetric rotation gradient tensor, respectively, which are defined as the following forms [23,24]

where and are the dilatation strain and the displacement vector, respectively, according to Akgöz and Civalek [25]. The Knocker symbol is , and the permutation symbol is :

The constitutive equations for linear, elastic, and isotropic materials are given by the following forms [25]

where is the deviatoric strain, which can be written as follows [25]:

where , , and denote three additional independent material length scale parameters associated with the dilatation gradient tensor, deviatoric stretch gradient tensor, and symmetric rotation gradient tensor, respectively. In addition, the parameters λ and μ are the Lame coefficients which are given as [26,27]

where E and denote Young’s modulus and Poisson’s ratio, respectively.

Using Eqs. (3), (4), and (5), the following equations are given by

Using Eqs. (3), (4), and (5), the nonzero components of the deviatoric stretch gradient tensor can be derived as follows:

Substituting Eqs. (19)–(21) into Eq. (11) yields the following form:

Using Eqs. (15) and (21), the higher order stresses for MSGT are obtained as the following forms:

Using Eqs. (14), (16), (19), (20) and (22), we have  

The non-zero stresses are obtained as follows:

In Eq. (25), denotes the shear correction factor which depends on the shape of microbeam cross-section.

Because of a high surface-to-volume ratio, the surface stress effect plays an important role with micro- and nanoscale materials. For this purpose, the constitutive equation of the Gurtin–Murdoch continuum mechanics approach is considered as follows [28]:

where is the residual surface stress under unstrained condition, and and are the surface Lame constants. The components of normal and shear surface stress can be written as follows:

The classical beam theory does not satisfy our model. For solving this problem, it is assumed that the stress component varies linearly through the beam thickness and satisfies the balance conditions on the surfaces. Therefore, can be written as [28]

Using Eq. (27),  can be written as

The normal and shear components, considering bulk and surface effects and using Eq. (29), can be written as

is the total potential energy that includes the strain energy, kinetic energy, and work done by the external loads, which can written as [4]

where

and where , , and are the strain and kinematic energies for bulk and surface effects, respectively. Moreover, and are the work done by the external forces, including the pre-stress load and the elastic foundation, respectively.

Using Hamilton’s principle and a variational method for Timoshenko microbeam model, based on strain gradient theory and the surface stress effects embedded in an elastic medium subjected to pre-stress loading , yields the following equation [4]:

The surface strain energy is obtained as

where

Using the presented equations, the strain energy for bulk and surface effects are explained in Appendix A with details. Using Eqs. (14)–(16), the kinetic energy of the Timoshenko microbeam model for bulk and surface effects can be written as

The work done by external forces, including pre-stress load and elastic foundation, can be written as

where is the pre-stress load, and and are Winkler’s spring and Pasternak’s shear modulli of elastic foundation, respectively.

By substituting Eqs. (A-1), (A-2), (35) and (36) into Eq. (33), one can obtain the governing equations of motion and boundary conditions as follows:

For boundary conditions

where

The dimensionless geometric, mechanical, and surface residual stress, surface mass density, Young’s modulus of surface layer, and three material length scale parameters can be defined as follows [28]:

To use the differential quadrature (DQ) method, first we should convert Eqs. (37) and (38) into dimensionless equations. Thus, substituting Eq. (40) into Eqs. (37) and (38) yields the Eqs. (41a) and (41b).

The dimensionless simply supported (SS) boundary conditions for the microbeam model are considered as follows:

3. Using the DQ method to solve the Timoshenko microbeam model

The (DQ) method was used to solve Eq. (41) and the associated boundary conditions in Eq. (42) to determine the free vibration frequencies of the beam. The basic concept of the DQM was defined as the derivative of a function at a given point that can be approximated as a linear sum of a weighted function at all sample points [29,30]. Using this approximation, the differential equations are then reduced to a set of algebraic equations. This approach is convenient for solving problems governed by fourth- or higher-order differential equations.

 According to this method, the m^th order derivative of the function f(x) with respect to x at a grid point x_i, is approximated by a linear sum of all the functional values in the whole domain as follows [22]:

where x_i is the location of ith sample point in the domain; N is the number of sampling points;  is the functional value at point x_i,  is the weighting coefficient of the m^th order differentiation attached to these functional values. To avoid ill-conditioning, the Lagrange interpolation basis functions are used as the following form [22]:

To determine the unequally-spaced positions of the grid points, the Chebyshev–Gauss–Lobatto polynomials were employed as follows [22]:

The first order weighting matrix can be obtained completely from Eq. (44). Higher-order coefficient matrices can be obtained from the first-order weighting matrix as follows [22]:

Then, substituting Eqs. (44) and (46) into Eqs. (41a) and (42) obtained the following equations of motion using the DQ method

For (SS) boundary conditions, we have:

The general solutions of motion equations are considered as

where is the dimensionless natural frequency. is the fundamental natural frequency, and denotes the density of microbeam.

The stiffness and mass matrices for the Timoshenko microbeam, using strain gradient theory and surface stress effects under pre-stress loading, can be written as

where are the stiffness and mass matrices and the subscripts b and d stand for the boundary and domain points, respectively. By solving Eq. (50), the dimensionless natural frequencies for and their associated vibration mode shapes can be extracted.

4. Numerical Results and Discussion

The mechanical and geometric properties of a Timoshenko microbeam is considered as [28,31]

The material length scale parameter is crucial for the successful application of the MSGT. Dehrouyeh-Semnani and Nikkhah-Bahrami [32] presented an in-depth discussion on how to determine this parameter numerically, and they compared the numerical results obtained to those obtained by experimental testing [32]. They showed the bending rigidity of an epoxy micro-cantilever versus thickness for a modified couple stress model (MCST), using , and the experimental data reported by Lam et al. and found that the results based on the constitutive beam model validated the experimental data, while the Euler-Bernoulli beam model overestimated the bending rigidity of the micro-cantilever. In addition, they depicted that the material length scale parameter of epoxy-based materials on the Euler-Bernoulli beam model equals to . According to the results of Dehrouyeh-Semnani and Nikkhah-Bahrami, the Euler-Bernoulli beam model validated the experimental data very well, but the constitutive beam model underestimated the bending rigidity of the epoxy micro-cantilever. Therefore, in this work, we used the material length scale parameter equal to .

Table 1 gives the dimensionless natural frequencies for the Timoshenko microbeam under various boundary conditions. An excellent agreement was found between the present results and the analytical solutions.

The results, obtained by the present work, are compared with the reported results by Ansari et al. [33] in Figure 2, where they demonstrate good agreement each other. In addition, the trend of the results was the same. On the other hand, increasing the aspect ratio (L/h) reduced the dimensionless natural frequency. Moreover, the stiffness of the Timoshenko microbeam decreased with increasing the aspect ratio.

Figure 2. The dimensionless natural frequency versus aspect ratio.

Table 2 shows the first three dimensionless natural frequencies of the Timoshenko microbeam model for the different values of aspect ratio            (
 
), and surface residual stress (
 
). As shown in Table 2, by increasing the aspect ratio, the value of the first three dimensionless natural frequencies decreases, and the opposite occurs for the surface residual stress. 

The latter subject has been illustrated for dimensionless fundamental natural frequencies in Figure 3. Considering the surface residual stress, the Timoshenko beam at a microscale becomes stiffer, but the effect of this parameter on the dimensionless natural frequency is not noticeable. Therefore, surface residual stress can be ignored in the results. 

Figure 3. The dimensionless fundamental natural frequency versus aspect ratio for different values of

Tables 3 and 4 depict the first three dimensionless natural frequencies of the Timoshenko microbeam model for the values of the aspect ratio       (
 
), surface mass density (
 
), and Young’s modulus of surface layer (
 
), respectively. By increasing of the
 
, the value of the first three dimensionless natural frequencies increases and vice versa for surface mass density. A change in
 
and 
 
led to increase stiffness and mass of the micro structure, respectively. Moreover, the results, shown in Figures 4 and 5, are similar to those shown in Tables 3 and 4. Furthermore, Figures 4 and 5 demonstrate that the effect of 
 
 on the dimensionless natural frequency is more than   
 
. However, the effect of 
 
 on the dimensionless natural frequency is not noticeable, and it can be ignored in the results.

Figure 4. The dimensionless fundamental natural frequency versus aspect ratio for different values of .

Figure 5. The dimensionless fundamental natural frequency versus aspect ratio for different values of
 
 

Figures 6a and 6b show the influence of pre-stress load on the dimensionless first and third natural frequencies versus aspect ratio, respectively. These results demonstrated that the effect of pre-stress load on the greater mode is negligible for higher aspect ratios, and this effect was similar to the lower aspect ratios for all modes. Clearly, the stiffness of microbeam increased at lower aspect ratios. In this figure, the effect of the positive pre-stress load on the natural frequency was higher than that of the negative pre-stress load. Consequently, positive and negative pre-stress loads led to increase and decrease stiffness of the Timoshenko microbeam, respectively. These results are the same for dimensionless natural frequencies.

(a)

(b)

Figure 6. The influence of pre-stress load on the dimensionless first (a) and third (b) natural frequencies versus aspect ratios .

To consider the size-dependent effects (l denotes the material length scale parameter), the parameter at a microscale is taken into account, and it is non-zero for MSGT (
 
) or MCST (
 
), while at a macro scale, it is zero for CT                           (
 
). 

Figures 7a and 7b are plotted to illustrate the influence of various material length scale theories including modified strain gradient (MSGT)                      (
 
), modified couple stress (MCST)               (
 
), and classical theories (CT)                    (
 
) on the dimensionless first and third natural frequencies versus 
 
, respectively. 

(a)

(b)

Figure 7. The influence of various material length scale theories on the dimensionless (a) first and (b) third natural frequencies versus h/l.

The effect of the material length scale parameters on the dimensionless natural frequencies for MSGT was higher than that of the other states, such as MCST and CT. This indicates that considering three material length scale parameters
 
 led to increase stiffness of the Timoshenko microbeam model, and therefore the dimensionless natural frequencies for MSGT enhanced.

Figures 8 and 9 present the influence of transverse and shear constants of the elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios. The dimensionless natural frequency increased with an increase in the transverse and shear constants of the elastic foundation, while the elastic foundation increased stiffness of the microstructure. 

Figure 8. The influence of the transverse constant of the elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios , .

Figure 9. The influence of shear constant of elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios , .

Moreover, increasing the transverse and shear constants of the elastic foundation were directly related to the stiffness of the Timoshenko microbeam and the dimensionless natural frequency.

5. Conclusions

Size-dependent effects on the free vibration analysis of the Timoshenko microbeam model, based on MSGT and surface stress effects subjected to pre-stress loading embedded in an elastic medium, were investigated. The Gurtin–Murdoch continuum mechanical approach was considered, and the set of governing equations were derived using a variational method and solved using DQM. Effects of pre-stress load, surface residual stress, surface mass density, Young’s modulus of surface layer, material length scale parameters, and elastic foundation coefficients were studied. 

The results of this article can be listed as follows:

·         By increasing the aspect ratio, the values of natural frequencies decreased while the opposite occurred for surface residual stress. In addition, when increasing the value of
 
, the value of the natural frequencies increased, while the surface mass density decreased. Variations in 
 
 and 
 
 led to increase stiffness and mass matrices for the micro structures, respectively. The numerical results showed that the effect of surface residual stress was more than the surface mass density or Young’s modulus of the surface layer.

·         The effect of pre-stress loading in higher modes was negligible for higher aspect ratios, and this effect was similar to lower aspect ratios across all modes.

·         The effect of the three material length scale parameters on the natural frequencies for MSGT was higher than that of the other theories. Application of each of the three material length scale parameters
 
, increased the natural frequencies for MSGT, which was due to the increasing stiffness of the Timoshenko microbeam model.

• Natural frequencies increased with an increase in the transverse and shear constants of the elastic foundation. Consequently, applying the elastic foundation values led to increase stiffness of the Timoshenko microbeam model.
• Comparison between the material length scale parameters and the surface effect confirmed that natural frequencies are more affected by the material length scale parameters than surface effects.

Acknowledgments

The authors would like to thank the reviewers for their reports that improved the clarity of this article. Moreover, the authors are grateful to the Iranian Nanotechnology Development Committee for their financial support. We are also grateful to the University of Kashan for supporting this work through Grant no. 463855/1.

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Appendix

The strain energies for bulk and surface effects are written as follows: