Buckling and Nonlinear Vibration of Functionally Graded Porous Micro-beam Resting on Elastic Foundation

1. H. Dang ^* , T. H. Nguyen

Department of Mechanics, Thai Nguyen University of Technology, Thainguyen, Vietnam

1.     Introduction

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 [1-3]. Fallah and Aghdam [4] investigated the post-buckling and nonlinear free vibration behaviors of FG beams on nonlinear elastic foundations. Using various higher-order shear deformation beam theories, Thai and Vo [5] presented the analysis of the linear free vibration and bending responses of FG beams.   The multi-scale method was applied by Yan et al. [6] to study the stability of an axially moving FG beam with time-dependent velocity. In manufacturing FG materials, it is difficult to avoid the appearance of micro-voids 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 Newton-Raphson method. The nonlinear vibration behavior of a S-S axially FG Euler-Bernoulli 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 [12-18].

Because of the applications of micro-/nano- sized structures in micro-/nano-electromechanical 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 higher-order 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) [21-25]. 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]. Size-dependent 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 [21-25] and its versions [26, 27]. Some works presented the static and dynamic analysis of micro-and nano-beams reported based on these above higher-order elasticity theories [28-40]. In addition, Hou et al. [41] investigated the buckling and bending behaviors of  FG micro-cylindrical imperfect beams using the MCST. Based on the MCST and EBT, the nonlinear free vibration response of the FG non-uniform micro-tube was studied by Huang et al. [42] utilizing the homotopy perturbation and the differential quadrature methods. The semi-analytical solutions for the nonlinear and linear forced vibration problems of FG non-uniform cylindrical micro-beams 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 higher-order 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 non-gradient nonlocal stress and higher-order gradient stress. In the framework of the NSGT, depending on the relationship between the NP and MLSP, the micro-/nano-sized structures arise the stiffness softening effect or stiffness hardening effect [45-49]. Many works related to the bending, stability, and vibration analysis of the FG micro-/nano-structures were reported by using the NSGT. The nonlinear vibration behavior of FG Euler-Bernoulli nano-beams was investigated by Şimşek [50], utilizing the novel Hamiltonian approach. The bending, buckling, and vibration behaviors of viscoelastic FG curved Euler-Bernoulli nano-beam resting on an elastic foundation were investigated by Allam and Radwan [51]. Nonlinear vibration behavior of an electrostatic FG Euler-Bernoulli nano-resonator taking into account the effect of surface stress was investigated by Esfahani et al. [52]. The nonlinear vibration behavior of FG nano-beams was studied by Hieu et al. [53] utilizing the NSGT considering thickness effect. The nonlinear vibration and stability characteristics of FGP Euler-Bernoulli micro-beams under electrostatic force were examined by Dang and Do [54]. Dang et al. [55] investigated the stability and nonlinear vibration behaviors of  FG nano-tubes conveying fluid. The effect of magnetic field on the nonlinear vibration of electrostatically actuated FG micro-beam 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 platelet-reinforced composite micro-beams 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 nano-beam reinforced by carbon nanotubes subjected to a moving point load. Moreover, recently, the effects of magnetic and thermal fields on FG Timoshenko nano-beam buckling and free vibration behaviors were examined by Esen et al. [60].

The influence of porosity on the mechanical behavior of FG micro-beams 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 micro-beam resting on the elastic foundation based on the NSGT and the EBT for the first time. The micro-beam 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 micro-beam with S-S 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 micro-beam.

2. Model and Formulations
   • Modeling of the FGP Micro-beam

A model of an FGP micro-beam resting on an elastic foundation is considered in Fig. 1. The FGP micro-beam 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 micro-beam is composed of metal and ceramic with porosity distribution. The material properties of the micro-beam 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]:

for the even porosity distribution (or FGM-I), and [9]:

For the uneven porosity distribution (FGM-II). In the above equations, subscripts “c” and “m” represent ceramic and metal phases, respectively; k ( ) is the power-law 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 micro-beam, and  ( ) refers to the porosity distribution factor. When  (without porosity), the FGP micro-beam becomes the perfect FG micro-beam. It is noted that the perfect FG micro-beam becomes a fully ceramic micro-beam if k is equal to zero and nearly a metal micro-beam for a very large value of k.

Fig. 1. Modeling of a FGP micro-beam resting on an elastic foundation

Fig. 2. Porosity distribution models: (a) even porosity distribution (FGM-I), (b) uneven porosity
distribution (FGM-II)

As can see from Fig. 2, the FGM-I model has porosities uniformly distributed over the micro-beam’s cross-section. At the same time, the FGM-II model has porosities spreading mostly around the middle area of the micro-beam’s cross-section, and the amount of porosity decreases linearly to zero at the upper and lower surfaces of the micro-beam’s cross-section.

• The NSGT

The total stress of the FGP micro-beam based on the NSGT is defined as [44]:

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

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]:

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

The nonlocal constitutive equations for the NET [19, 20] and the SGT [25] can be recovered from Eq. (9) by letting   and , respectively.

• The Governing Equation for the FGP Micro-beam

According to the EBT and the von-Kármán’s geometrical nonlinearity, the nonzero strain of the FGP micro-beam is [46, 48, 50]:

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

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

where  denotes the axial force resultant,  denotes the bending moment resultant,  indicates the non-classical axial force, and  is the non-classical moment; these quantities are defined as:

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

where

The virtual kinetic energy of the FGP micro-beam is given as [48]:

where:

The virtual external work caused by the transverse distributed force  and the elastic foundation reaction  can be calculated by:

To get the motion equation, Hamilton’s principle is employed [46-50]:

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

and BCs at x = 0 and x = L:

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

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

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

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

in which, P₀ 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:

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

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

For the S-S FGP micro-beam, the classical and nonclassical BCs at  and  can be expressed as [48]:

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:

For the homogeneous micro-beam, i.e. ,  ,  and , Eq. (36) becomes:

which was studied by Dang [46].

For the convenience of calculations, the dimensionless variables are introduced as follows:

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

Where

Thus, the classical and nonclassical BCs at   and   becomes:

3. Solution Procedure
   • Buckling Analysis

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:

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

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

in which

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

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

For the homogeneous micro-beam and without the elastic foundation, the critical buckling force becomes:

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

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.

• Nonlinear Vibration Analysis

For free nonlinear vibration analysis, letting  and , Eq. (41) becomes:

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:

where, the time-dependent function  needs to be found, the shape function  is chosen so that the solution (53) satisfies the  BCs (43). For the S-S FGP micro-beam, the shape function can be chosen as follows [46, 50]:

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

where:

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

where  is the dimensionless vibrational amplitude of the FGP micro-beam.

It can see that Eq. (55) is the cubic-Duffing nonlinear oscillator which can be solved by many different analytical methods [61]. In this section, the ELM and a weighted averaging value [62-64] 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 amplitude-frequency relationship of the FGP micro-beam can be achieved as:

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

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 micro-beam can be achieved as follows:

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

The elastic foundation coefficients ( and ) lead to an increase in the FGP micro-beam frequencies. The FGP micro-beam frequencies increase by increasing the dimensionless MLSP ( ) or decreasing the value of the dimensionless NP ( ).

4. Numerical Results

To verify the accuracy of present results and predict the buckling and free vibration behaviors of the FGP micro-beam, a micro-beam composed of Aluminum (Al-metal) and Alumina (Alumina Al₂O₃-ceramic) is considered. Table 1 shows the material properties of Al and Al₂O₃.

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 Al₂O₃

Table 2 The dimensionless linear frequencies
of the FG beam

• Buckling

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

where  is the dimensionless classical critical buckling force of the full metal micro-beam.

The effects of the dimensionless MLSP  and the dimensionless NP  on the critical buckling force ratio  of the FGP micro-beam 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 micro-beam exert the softening and hardening effects corresponding to  and , respectively.

The effect of the power-law index k on the buckling behavior of the FGP micro-beam is presented in Fig. 4. It can be observed that k has the effect of decreasing the value of . The FGP micro-beam will be more flexible as k increases; this leads to a decrease . As can see from this figure, the critical buckling forces of the FGM-II micro-beam are always larger than those of the FGM-I micro-beam. This result can be explained as follows: with porosities distributed mainly in the middle surface of the micro-beam’s cross-section and the amount of porosity seeming to decrease linearly to zero at the upper and lower surfaces of the micro-beam’s cross-section, the FGM-II micro-beam have a greater stiffness than the FGM-I micro-beam.

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 micro-beams reduces by increasing the porosity distribution factor, and thus, the critical buckling force decreases. The critical buckling forces of the FGM-I micro-beam are always smaller than those of the FGM-II micro-beam (it is the same as Fig. 4).

The effects of the elastic foundation coefficients on the variation of the critical buckling force of the FGP micro-beam 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 micro-beam.

• Nonlinear Vibration

To observe the effects of the dimensionless MLSP and NP on the nonlinear free vibration behavior of the FGP micro-beam, 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 micro-beams (i. e., ) are equal to those of the FGP NSG micro-beams 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 power-law 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 micro-beam. This figure indicates that  increases strongly with small values of k ( ) and decreases slowly with larger values of k.

Fig. 9 shows the effect of δ on the variation of  of the FGP micro-beam. 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 FGM-II microbeams and 3.28% for FGM-I microbeams. The nonlinear frequencies of the FGM-II micro-beams are always larger than those of the FGM-I micro-beams. This result can be explained as the stiffness of the FGM-II microbeam is larger than that of the FGM-I microbeam. This result completely agrees with the results obtained by Wattanasakulpong and  Chaikittiratana [9], Dang and Do [54].

The effect of the length-thickness ratio L/h on the variation of  is shown in Fig. 10. From this figure, it can be concluded that increase when the length-thickness 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 micro-beam to the dimensionless Winkler parameter K_L and the dimensionless Pasternak parameter K_S, respectively. The elastic foundation makes the FGP micro-beam stiffer, so the frequencies of the FGP micro-beam 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 micro-beam’s frequency increases rapidly as the initial amplitude increases.

5. Conclusion

The EBT is developed based on the NSGT to investigate the buckling and nonlinear free vibration problems of the imperfect FG micro-beam with porosities resting on an elastic foundation. The analytical expressions of the critical buckling force and the nonlinear frequency of the S-S FGP micro-beam 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 micro-beam’s behavior is evaluated. Some points can be concluded as follows:

• An increase of the power-law index k leads to a reduction of the critical buckling force, while k has an interesting effect on the variation of the nonlinear frequency of the micro-beam.
• When , the MLSP leads to an increase of both the critical buckling force and the nonlinear frequency. Moreover, on the other hand when , the MLSP reduces the critical buckling force and the nonlinear frequency.
• The length-thickness ratio leads to an increase in the value of the nonlinear frequency.
• The elastic foundation makes the FGP micro-beam stiffer, and consequently, both the critical buckling force and the nonlinear frequency increase as increasing the coefficients of the elastic foundation.

Acknowledgments

This research is supported by the Thai Nguyen University of Technology (TNUT).

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