Document Type : Research Paper
Authors
Department of Mechanics, Thai Nguyen University of Technology, Thainguyen, Vietnam
Abstract
Keywords
Main Subjects
Buckling and Nonlinear Vibration of Functionally Graded Porous Microbeam Resting on Elastic Foundation
Department of Mechanics, Thai Nguyen University of Technology, Thainguyen, Vietnam
KEYWORDS 

ABSTRACT 
Microbeams; Nonlocal strain gradient theory; Functionally graded porous; Buckling; Nonlinear vibration. 
The buckling and nonlinear free vibration problems of functionally graded porous (FGP) microbeam resting on an elastic foundation are presented through the nonlocal strain gradient theory (NSGT) and the EulerBernoulli beam theory (EBT) with the vonKármán’s geometrical nonlinearity. The microbeam is made up of metal and ceramic in which the material properties are assumed to be varied continuously in the thickness direction through a simple exponential law. Two porosity distribution models, including even and uneven distributions, are considered. The governing equation of motion is derived by employing Hamilton’s principle. The analytical expressions of the critical buckling force and nonlinear frequency of the FGP microbeam with simply supported (SS) boundary conditions (BCs) are obtained by utilizing the Galerkin technique and the equivalent linearization method (ELM). The reliability of the obtained results has been checked. Effects of the powerlaw index, the porosity distribution factor, the lengththickness ratio, the material length scale parameter (MLSP), the nonlocal parameter (NP), and the coefficients of the elastic foundation on the buckling and nonlinear free vibration responses of the FGP microbeam are investigated and discussed in this work. 
By combining materials into a uniform volume, functionally graded (FG) materials showed prominent advantages compared with the traditional composite materials. Nowadays, FG materials are widely used in many fields, especially in aerospace engineering, nuclear engineering, biomedical engineering, and optical engineering [13]. Fallah and Aghdam [4] investigated the postbuckling and nonlinear free vibration behaviors of FG beams on nonlinear elastic foundations. Using various higherorder shear deformation beam theories, Thai and Vo [5] presented the analysis of the linear free vibration and bending responses of FG beams. The multiscale method was applied by Yan et al. [6] to study the stability of an axially moving FG beam with timedependent velocity. In manufacturing FG materials, it is difficult to avoid the appearance of microvoids or pores inside the FG materials. The existence of pores reduces the weight but increases the ability to absorb the energy of FG materials [7, 8]. Therefore, the influence of porosity on the mechanical behavior of FG structures needs to be studied in detail. Wattanasakulpong and Chaikittiratana [9] first studied the influence of porosity on the vibration behavior of beams. In this work, the authors introduced two models of porosity distribution in FG materials, including even and uneven distributions. They found that the frequencies of the beam were reduced by increasing the porosity volume fraction. Akbaş [10] presented the geometrically nonlinear analysis of FGP Timoshenko beams using the finite element method in conjunction with the NewtonRaphson method. The nonlinear vibration behavior of a SS axially FG EulerBernoulli beam subjected to a moving harmonic force was examined by Alimoradzadeh et al. [11] using the Galerkin technique and the variational method. Besides analyzing the mechanical behavior of FG beams, the static and dynamic analysis of FG plates has also been reported by several authors [1218].
Because of the applications of micro/nano sized structures in micro/nanoelectromechanical systems, many scientists are interested in analyzing the mechanical behaviors of these structures. The classical elasticity theory (CET) without the length scale parameters (LSPs) is not suitable for modeling micro/nano sized structures. Several higherorder theories of elasticity containing the LSPs have been introduced to model the micro/nano sized structures, such as the nonlocal elasticity theory (NET) [19, 20] the strain gradient theory (SGT) [2125]. To date, some versions of the SGT have been proposed, for example, the modified strain gradient theory (MSGT) [26] and the modified couple stress theory (MCST) [27]. Sizedependent behaviors of micro/nano sized structures can be observed in two different directions using the NET and the SGT. A stiffness softening effect can be observed by using the NET [19, 20], while a stiffness hardening effect can be observed by utilizing the SGT [2125] and its versions [26, 27]. Some works presented the static and dynamic analysis of microand nanobeams reported based on these above higherorder elasticity theories [2840]. In addition, Hou et al. [41] investigated the buckling and bending behaviors of FG microcylindrical imperfect beams using the MCST. Based on the MCST and EBT, the nonlinear free vibration response of the FG nonuniform microtube was studied by Huang et al. [42] utilizing the homotopy perturbation and the differential quadrature methods. The semianalytical solutions for the nonlinear and linear forced vibration problems of FG nonuniform cylindrical microbeams based on the EBT and MCST were carried out by Xu et al. [43] by applying the differential quadrature method.
Recently, both the NET and the SGT were combined in a generalized higherorder elasticity theory, the NSGT; this elasticity theory was proposed by Lim et al. [44]. The NSGT considered that the stress is a sum of the nongradient nonlocal stress and higherorder gradient stress. In the framework of the NSGT, depending on the relationship between the NP and MLSP, the micro/nanosized structures arise the stiffness softening effect or stiffness hardening effect [4549]. Many works related to the bending, stability, and vibration analysis of the FG micro/nanostructures were reported by using the NSGT. The nonlinear vibration behavior of FG EulerBernoulli nanobeams was investigated by Şimşek [50], utilizing the novel Hamiltonian approach. The bending, buckling, and vibration behaviors of viscoelastic FG curved EulerBernoulli nanobeam resting on an elastic foundation were investigated by Allam and Radwan [51]. Nonlinear vibration behavior of an electrostatic FG EulerBernoulli nanoresonator taking into account the effect of surface stress was investigated by Esfahani et al. [52]. The nonlinear vibration behavior of FG nanobeams was studied by Hieu et al. [53] utilizing the NSGT considering thickness effect. The nonlinear vibration and stability characteristics of FGP EulerBernoulli microbeams under electrostatic force were examined by Dang and Do [54]. Dang et al. [55] investigated the stability and nonlinear vibration behaviors of FG nanotubes conveying fluid. The effect of magnetic field on the nonlinear vibration of electrostatically actuated FG microbeam was reported by Hieu et al. [56]. Tang and Qing [57] studied the buckling and free vibration behaviors of the FG Timoshenko beam using the Laplace transform method. The forced vibration behavior of laminated FG graphene plateletreinforced composite microbeams under external harmonic forces was investigated by Wu et al. [58] using the refined hyperbolic shear deformation beam theory. Esen et al. [59] examined the dynamical behavior of FG nanobeam reinforced by carbon nanotubes subjected to a moving point load. Moreover, recently, the effects of magnetic and thermal fields on FG Timoshenko nanobeam buckling and free vibration behaviors were examined by Esen et al. [60].
The influence of porosity on the mechanical behavior of FG microbeams is an important topic that needs to be studied. The novelty of this work is to present the analytical analysis for the buckling and nonlinear free vibration problems of the FGP microbeam resting on the elastic foundation based on the NSGT and the EBT for the first time. The microbeam is composed of a mixture of metal and ceramic, in which the material properties are assumed to change continuously in the direction of thickness according to the simple exponential law. Two porosity distribution models, including even and uneven distributions, are examined to consider the porosity effect. Hamilton’s principle is applied to establish the governing equation of motion. The analytical expressions of the critical buckling force and nonlinear frequency of the FGP microbeam with SS BCs are carried out. Numerical illustrations are performed to check the accuracy and evaluate the impact of some important parameters on the nonlinear free vibration and stability behaviors of the microbeam.
A model of an FGP microbeam resting on an elastic foundation is considered in Fig. 1. The FGP microbeam has the length L, width b, and height h. A Winkler–Pasternak type elastic foundation with two layers is considered, including the Winkler layer with foundation coefficient kL and the Pasternak layer with foundation coefficient kS. The microbeam is composed of metal and ceramic with porosity distribution. The material properties of the microbeam are assumed to vary in the thickness direction according to the simple exponential law. In this work, two kinds of porosity distributions, including even and uneven distribution, are considered (see Fig. 2). The material properties, Young’s modulus , and the mass density , can be estimated by [9]:

(1) 

(2) 
for the even porosity distribution (or FGMI), and [9]:

(3) 

(4) 
For the uneven porosity distribution (FGMII). In the above equations, subscripts “c” and “m” represent ceramic and metal phases, respectively; k ( ) is the powerlaw index which governs the change of the volume fraction of ceramic and metal phases; is the thickness coordinate from the geometry middle surface of the FGP microbeam, and ( ) refers to the porosity distribution factor. When (without porosity), the FGP microbeam becomes the perfect FG microbeam. It is noted that the perfect FG microbeam becomes a fully ceramic microbeam if k is equal to zero and nearly a metal microbeam for a very large value of k.
Fig. 1. Modeling of a FGP microbeam resting on an elastic foundation
Fig. 2. Porosity distribution models: (a) even porosity distribution (FGMI), (b) uneven porosity
distribution (FGMII)
As can see from Fig. 2, the FGMI model has porosities uniformly distributed over the microbeam’s crosssection. At the same time, the FGMII model has porosities spreading mostly around the middle area of the microbeam’s crosssection, and the amount of porosity decreases linearly to zero at the upper and lower surfaces of the microbeam’s crosssection.
The total stress of the FGP microbeam based on the NSGT is defined as [44]:

(5) 
where and are the nonlocal stress and the higherorder nonlocal stress, respectively, these stresses are functions of the classical strain ( ) and the strain gradient ( ) [44]:

(6) 

(7) 
in which, is the MLSP which describes the effect of strain gradient stress field; and are the two NPs describing the effect of the nonlocal stress field; and are the two nonlocal kernel functions defined by Eringen [19, 20].
For one dimensional problem, the general nonlocal strain gradient constitutive equation takes a form [44]:

(8) 
where is the Laplace operator. When considering , Eq. (8) is reduced to:

(9) 
The nonlocal constitutive equations for the NET [19, 20] and the SGT [25] can be recovered from Eq. (9) by letting and , respectively.
According to the EBT and the vonKármán’s geometrical nonlinearity, the nonzero strain of the FGP microbeam is [46, 48, 50]:

(10) 
where u and w represent the axial and transverse displacements of any point on the geometry middle surface of the FGP microbeam, respectively. The virtual strain energy of the FGP microbeam based on the NSGT is given as [46, 48, 50]:

(11) 
Combination of Eqs. (10) and (11), leads to:

(12) 
where denotes the axial force resultant, denotes the bending moment resultant, indicates the nonclassical axial force, and is the nonclassical moment; these quantities are defined as:

(13) 
herein, h. The axial force resultant and the bending moment resultant are expressed as, respectively:

(14) 

(15) 
where
, 
(16) 
The virtual kinetic energy of the FGP microbeam is given as [48]:

(17) 
where:
, 
(18) 
The virtual external work caused by the transverse distributed force and the elastic foundation reaction can be calculated by:

(19) 
To get the motion equation, Hamilton’s principle is employed [4650]:

(20) 
Substituting Eqs. (12), (17), and (19) into Eq. (20), the following equations can be achieved:

(21) 

(22) 
and BCs at x = 0 and x = L:

(23) 

(24) 

(25) 

(26) 

(27) 
From Eqs. (14), (15), (21), and (22), the expressions of the axial force and bending moment resultants can be obtained as:

(28) 

(29) 
Therefore, the equations of motion for the FGP microbeam can be obtained by putting Eqs. (28) and (29) into Eqs. (14) and (15) as:

(30) 

(31) 
The following result can be obtained if the axial inertia terms in Eq. (30) are neglected:

(32) 
where C is the integral constant which can be determined by the following BCs:

(33) 
in which, P_{0} is the initial compressive axial force. Employing the BCs (33) and integrating both sides of Eq. (32) from 0 to L, it can be obtained:

(34) 
From Eqs. (28), (32), and (34), the expression of the axial force resultant can be derived as:

(35) 
Now, substituting Eq. (35) into Eq. (31), the motion equation of the FGP microbeam in terms of the transverse displacement (w) based on the NSGT can be derived as:

(36) 
For the SS FGP microbeam, the classical and nonclassical BCs at and can be expressed as [48]:

(37) 
which was investigated by Şimşek [50].
Without considering the elastic foundation ( ) and the influence of the axial compressive force ( ), and the second mass moment of inertia are ignored ( ), Eq. (36) is reduced to:

(38) 
For the homogeneous microbeam, i.e. , , and , Eq. (36) becomes:

(39) 
which was studied by Dang [46].
For the convenience of calculations, the dimensionless variables are introduced as follows:

(40) 
Using Eq. (40), the motion equation of the FGP microbeam can be rewritten in the following dimensionless form:

(41) 
Where

(42) 
Thus, the classical and nonclassical BCs at and becomes:

(43) 
For buckling analysis, ignoring the terms related to time and the transverse distributed force in Eq. (41), the equation for buckling analysis can be achieved as:

(44) 
With SS BCs, the solution of Eq. (44) can be assumed as:

(45) 
where n denotes the number of halfwaves, substituting the solution (45) into Eq. (44), leads to:

(46) 
in which

(47) 
By letting in Eq. (46), the dimensionless critical buckling force can be achieved as:

(48) 
Considering Eqs. (40) and (42), the physical form of the critical buckling force can be achieved as:

(49) 
For the homogeneous microbeam and without the elastic foundation, the critical buckling force becomes:

(50) 
which is the same as a result achieved by Li and Hu [47]. For the classical homogeneous microbeam, the critical buckling force (50) reduces to:

(51) 
It can be observed that the dimensionless MLSP (β) increases the critical buckling force, while the dimensionless NP (α) decreases the critical buckling force. The elastic foundation coefficients lead to an increase in the critical buckling force.
For free nonlinear vibration analysis, letting and , Eq. (41) becomes:

(52) 
which is the nonlinear partial differential equation (PDE); the exact solution of this equation is very difficult to find. Therefore, the approximate solution is an effective choice. First, the Galerkin technique will be applied to convert the nonlinear PDE (52) into the nonlinear ordinary differential equation (ODE). To apply the Galerkin technique, the solution of the nonlinear PDE (52) is assumed to have the form:

(53) 
where, the timedependent function needs to be found, the shape function is chosen so that the solution (53) satisfies the BCs (43). For the SS FGP microbeam, the shape function can be chosen as follows [46, 50]:

(54) 
Now, applying the Galerkin technique to the nonlinear PDE (52), the following nonlinear ODE can be obtained:

(55) 
where:

(56) 

(57) 
Eq. (55) is the nonlinear ODE which is assumed to satisfy the following initial conditions:

(58) 
where is the dimensionless vibrational amplitude of the FGP microbeam.
It can see that Eq. (55) is the cubicDuffing nonlinear oscillator which can be solved by many different analytical methods [61]. In this section, the ELM and a weighted averaging value [6264] will be used to find the approximate analytical solution of Eq. (55). According to the ELM, the replacement of nonlinear equation (55) by a linear equation is performed, in which the weighted averaging value is employed to estimate the coefficients of the linear equation. Accordingly, the amplitudefrequency relationship of the FGP microbeam can be achieved as:

(59) 
Using the shape function in Eq. (54), the integrals in Eqs. (56) and (57) can be calculated as:

(60) 
Substituting the results in Eq. (60) into Eqs. (56) and (58); and then replacing the obtained results into Eq. (59), the expression of nonlinear frequency of the FGP microbeam can be achieved as follows:

(61) 
The linear frequency of the FGP microbeam can be obtained from the nonlinear frequency (61) by letting as:

(62) 
The elastic foundation coefficients ( and ) lead to an increase in the FGP microbeam frequencies. The FGP microbeam frequencies increase by increasing the dimensionless MLSP ( ) or decreasing the value of the dimensionless NP ( ).
To verify the accuracy of present results and predict the buckling and free vibration behaviors of the FGP microbeam, a microbeam composed of Aluminum (Almetal) and Alumina (Alumina Al_{2}O_{3}ceramic) is considered. Table 1 shows the material properties of Al and Al_{2}O_{3}.
For the classical perfect FG beam, the present results are compared with those obtained by Thai and Vo [5]. The comparison is shown in Table 2. A very good agreement between the obtained linear frequency and the one achieved by Thai and Vo [5] can be observed in Table 2.
Table 1 The material properties of Al and Al_{2}O_{3}
Materials 
E (GPa) 
ρ (kg/m^{3}) 
Al 


Al_{2}O_{3} 


Table 2 The dimensionless linear frequencies
of the FG beam
k 
L/h=5 

L/h=20 

Ref. [5] 
Present 

Ref. [5] 
Present 

0 
5.3953 
5.3953 

5.4777 
5.4777 
0.5 
4.5931 
4.5932 

4.6641 
4.6641 
1 
4.1484 
4.1485 

4.2163 
4.2163 
2 
3.7793 
3.7796 

3.8472 
3.8472 
5 
3.5949 
3.5952 

3.6628 
3.6628 
10 
3.4921 
3.4923 

3.5547 
3.5547 
To study the buckling response of the FGP microbeam, the critical buckling force ratio ( ) and the scale ratio ( ) are defined as follows:

(63) 
where is the dimensionless classical critical buckling force of the full metal microbeam.
The effects of the dimensionless MLSP and the dimensionless NP on the critical buckling force ratio of the FGP microbeam are presented in Fig. 3. It can be seen that the classical results ( and ) are equal to the NSG results if . If (i.e., ), the critical buckling force ratio increases as the dimensionless MLSP increases. The critical buckling force ratio decreases as the dimensionless MLSP increases if (i.e., ). The obtained result is suited to the result obtained by Dang [46], Li, and Hu [47]. Employing the NSGT, the FGP microbeam exert the softening and hardening effects corresponding to and , respectively.
The effect of the powerlaw index k on the buckling behavior of the FGP microbeam is presented in Fig. 4. It can be observed that k has the effect of decreasing the value of . The FGP microbeam will be more flexible as k increases; this leads to a decrease . As can see from this figure, the critical buckling forces of the FGMII microbeam are always larger than those of the FGMI microbeam. This result can be explained as follows: with porosities distributed mainly in the middle surface of the microbeam’s crosssection and the amount of porosity seeming to decrease linearly to zero at the upper and lower surfaces of the microbeam’s crosssection, the FGMII microbeam have a greater stiffness than the FGMI microbeam.


Fig. 3. The variation of R_{cr} to for some values of β 

Fig. 4. Variation of P_{cr} to k with and 

Fig. 5. Variation of P_{cr} to δ 
Fig. 5 reveals the influence of the porosity distribution factor δ on the variation of the dimensionless critical buckling force . The critical buckling force decreases linearly when the porosity distribution factor increases. The fact the stiffness of FGP microbeams reduces by increasing the porosity distribution factor, and thus, the critical buckling force decreases. The critical buckling forces of the FGMI microbeam are always smaller than those of the FGMII microbeam (it is the same as Fig. 4).


Fig. 6. Variation of P_{cr} to K_{L} (a) and K_{S} (b) 
The effects of the elastic foundation coefficients on the variation of the critical buckling force of the FGP microbeam can be seen in Fig. 6. The coefficients of the elastic foundation lead to an increase in the value . This result is completely consistent because the elastic foundation enhances the stiffness of the FGP microbeam.
To observe the effects of the dimensionless MLSP and NP on the nonlinear free vibration behavior of the FGP microbeam, Fig. 7 shows the variation of the nonlinear frequency to the scale ratio for some values of the dimensionless MLSP . The nonlinear frequencies of the FGP classical microbeams (i. e., ) are equal to those of the FGP NSG microbeams if (namely, ). It can also be observed that the scale ratio leads to a decrease of the value of . It is a fact that decreases when the dimensionless NP increases due to the softening effect observed in the NET. If (namely, ), the dimensionless MLSP has the effect of increasing the value of . This means that the MLSP makes the hardening effect as in the SGT. If (namely, ), the MLSP has the effect of decreasing the value of ; again, the softening effect can be observed.
The effect of the powerlaw index k on the variation of can be observed in Fig. 8. It can be seen that k has an interesting impact on the vibration behavior of the FGP microbeam. This figure indicates that increases strongly with small values of k ( ) and decreases slowly with larger values of k.


Fig. 7. Variation of to for some values β 

Fig. 8. Variation of to k 

Fig. 9. Variation of the nonlinear frequency to the porosity distribution factor δ 
Fig. 9 shows the effect of δ on the variation of of the FGP microbeam. It can be concluded that decreases when δ increases. When the value of δ increases from 0.04 to 0.08, the nonlinear frequency reduces about 1.52% for FGMII microbeams and 3.28% for FGMI microbeams. The nonlinear frequencies of the FGMII microbeams are always larger than those of the FGMI microbeams. This result can be explained as the stiffness of the FGMII microbeam is larger than that of the FGMI microbeam. This result completely agrees with the results obtained by Wattanasakulpong and Chaikittiratana [9], Dang and Do [54].

Fig. 10. Variation of the nonlinear frequency 

Fig. 11. Variation of frequencies ω to Q_{0} 

Fig. 12. Variation of frequencies ω to Q_{0} 
The effect of the lengththickness ratio L/h on the variation of is shown in Fig. 10. From this figure, it can be concluded that increase when the lengththickness ratio increases. This result is consistent with those obtained by Thai and Vo [5], Dang [46].
Figures 11 and 12 show the variations of the frequencies (ω) of the FGP microbeam to the dimensionless Winkler parameter K_{L} and the dimensionless Pasternak parameter K_{S}, respectively. The elastic foundation makes the FGP microbeam stiffer, so the frequencies of the FGP microbeam increase as the coefficients of the elastic foundation increase. Also, these figures increase when the dimensionless initial amplitude increases. When considering the effect of geometrical nonlinearity, the microbeam’s frequency increases rapidly as the initial amplitude increases.
The EBT is developed based on the NSGT to investigate the buckling and nonlinear free vibration problems of the imperfect FG microbeam with porosities resting on an elastic foundation. The analytical expressions of the critical buckling force and the nonlinear frequency of the SS FGP microbeam are obtained using the Galerkin technique and the ELM. Numerical illustrations are performed, through which the accuracy of the results is verified, and the impact of some important parameters on the microbeam’s behavior is evaluated. Some points can be concluded as follows:
Acknowledgments
This research is supported by the Thai Nguyen University of Technology (TNUT).
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