Research Article | Open Access
Vibration Analysis of a Postbuckled Microscale FG Beam Based on Modified Couple Stress Theory
On the basis of modified couple stress theory, the postbuckling behavior of the Euler-Bernoulli microscale FG beams is investigated by means of an exact solution method. The modified couple stress theory as a nonclassical continuum theory is capable of interpreting the size dependencies which become more significant at micro/nanoscales. The Von-Karman type nonlinear strain-displacement relationships are employed. The thermal effects are also incorporated into formulation. The governing equation of motion and the corresponding boundary conditions are derived using Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A closed-form solution is obtained for the postbuckling deformation which is beyond the critical buckling load. To study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first three buckled configurations. The natural frequencies of the lowest vibration modes around each of the first three buckled configurations are obtained. The influences of power-law exponent, boundary condition, length scale parameter, and thermal environment changes on the static deflection and free vibration frequencies are studied. A comparison is also made between the present results and those obtained via the classical beam theories.
Since the first introduction by Japanese scientists in 1984, functionally graded materials (FGMs) have been under worldwide development during the past years. FGMs have received considerable attention for a large variety of engineering applications [1–4]. They are now being developed for general use as structural components in extremely high temperature environments, such as rocket engine components, space plan bodies, engine components, turbine blades, and other engineering and technological applications. They are a mixture of two constituent materials whose volume fractions vary smoothly and continuously from one surface to another. This leads to a continuous variation in mechanical properties. The best known FGM is compositionally graded from ceramic to metal, to incorporate the heat, wear, and oxidation resistance of ceramics with the toughness, strength, machinability, and bending capability of metals. This continuously compositional variation of the constituents provides FGMs with a solution to the several problems which laminated composites encounter with such as high magnitude shear stresses, debonding, and delamination under large interlaminar and thermal loads. With the ever increasing usage of these materials, it is of high interest to understand the buckling and vibrational behavior of FGMs, which have a vast range of applications in engineering and technology.
In almost two decades, there has been much effort in terms of publications dealing with buckling and vibrational analysis of beams based upon different beam theories [5–14]. Nayfeh and Emam  presented a closed-form solution for the postbuckling analysis of isotropic beams based on the Euler-Bernoulli beam theory (EBT). They also studied the free vibration behavior of the buckled isotropic beams in the postbuckling domain which means investigating the vibration characteristics taking place in the vicinity of a buckled configuration. Afterwards, they extended their work and found an exact solution for the postbuckling behavior of symmetrically laminated composite beams . They investigated the critical buckling load and free vibration in the postbuckling region. To study the effect of shear deformation and rotary inertia, Emam  extended the previous model proposed in  to examine these significant effects on the static postbuckling of symmetrically laminated composite beams within the framework of the Timoshenko beam theory (TBT). However, he did no attempt on the nonlinear vibration analysis in the postbuckling domain with considering the shear deformation effect.
Several researches can be found in the literature concerning with buckling and vibration of FG beams. Based upon TBT, Xiang and Yang  studied the free and forced vibration of FG beams under heat conduction. They implemented the differential quadrature method (DQM) for solving the governing equations of motion. Sina et al.  presented an analytical solution based on the first order shear deformation theory (FSDT) for analyzing the free vibrational behavior of FG beams. TBT was employed to study postbuckling and nonlinear vibration of edge cracked FG beams [20, 21]. Fallah and Aghdam  studied the large amplitude free vibration and postbuckling of FG beams rested on nonlinear elastic foundation and subjected to axial load by means of an analytical method based on the variational approach. Their analysis is based upon the EBT assumptions together with the Von-Karman strain-displacement relations. They also studied thermomechanical buckling and nonlinear free vibration of FG beams . Very recently, Rahimi et al.  studied the postbuckling and free vibrations of FG beams by means of an exact solution. The Von-Karman type nonlinear strain-displacement relationships have been employed. Based on TBT, they examined the effects of the transverse shear deformation and rotary inertia on both buckling load and natural frequencies of vibration.
While almost all of the above investigations have been carried out on the basis of conventional continuum mechanics theory which are not capable of capturing size effect in micro/nanoscales, developing size-dependent elasticity theories has been a major issue. Several microtorsion and micro-bending experiments have reported the size-dependent deformation behavior of micro- and nanostructures. So far, several size-dependent continuum theories including the couple stress elasticity [25, 26], nonlocal elasticity [27–31], strain gradient elasticity [32–34], and the surface elasticity [35, 36] theories have been proposed.
In recent years, FGMs have shown to have advantages applications while using in microstructures such as microelectromechanical systems. Due to this fact, there is a need for further study on their mechanical behavior. To this aim, based on the modified couple stress theory (MCST) , Ke et al.  investigated the nonlinear free vibration of the size-dependent FG Timoshenko microbeams. They deduced that, when the thickness of the FG microbeam is comparable to the material length scale parameter, both linear and nonlinear frequencies increase prominently. Asghari et al.  studied the static bending and free vibration of the FG Timoshenko beams on the basis of MCST. Ansari et al.  analyzed the nonlinear free vibration of the FGM Timoshenko microbeams based on the strain gradient theory (SGT). The higher-order governing differential equations were obtained on the basis of Hamilton's principle. These equations along with the associated boundary conditions are discretized then using the generalized DQM. They compared the obtained results with those reported via MCST and arrived at excellent agreement. They observed that the difference between the SGT and MCST is more significant for lower values of dimensionless length scale parameter. Nateghi and Salamat-talab  investigated the thermal effects on the buckling and free vibration of FG microbeams based on MCST and classical and first order shear deformation beam theories. Using generalized differential quadrature (GDQ) method, they obtained the buckling load and natural frequency of FG microbeams with different boundary conditions. Ansari et al.  in one of their newest works studied the thermal postbuckling of the Timoshenko FG microbeams based on the MSGT and by implementing GDQ method. They also investigated the influence of geometrical imperfection on the buckling deformation of microbeams in prebuckling and postbuckling domains.
All of the aforementioned studies were based upon a single mode approximation and neglected the modal interactions. Considering the higher mode approximation, Ghayesh and his coassociates numerically investigated the nonlinear dynamics of both straight and initially curved microbeams based on the MCST and under the assumptions of EBT [43–45] and TBT . They derived the nonlinear partial differential equations of motion using Hamilton’s principle. The Galerkin scheme is then applied to these nonlinear partial differential equations, resulting in a set of nonlinear ordinary differential equations with coupled terms. Finally, the discretized equations of motion are solved via the pseudoarclength continuation technique to obtain the frequency-response and force-response curves. They also studied the nonlinear size-dependent behavior of an electrically actuated MEMS resonator based on the MCST and the same numerical approach as in their previous works . In one of their newest works, they investigated the nonlinear resonant behavior of a microbeam over its buckled configuration . They assumed the system to be subjected to an axial load along with a distributed transverse harmonic load. Based on the MCST within the framework of EBT, the nonlinear equation of motion was obtained by employing Hamilton’s principle. After discretizing into a set of ordinary differential equations using the Galerkin approach, the pseudoarclength continuation technique is implemented for solving the nonlinear equations. First, the postbuckling configuration is obtained and then the nonlinear resonant response of the system over the buckled state is examined.
To the best of the authors’ knowledge, no work has been performed on finding exact solution for the thermal postbuckling and vibration analyses of FG microbeams based on MCST and different boundary conditions. In this regard, the present research aims to present a closed-form solution for the postbuckling configuration of FG microbeams based on MCST. The thermal effects are also incorporated into formulation. FG microbeam is modeled based on EBT assumptions. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. The Von-Karman type nonlinear strain-displacement relationships are employed. Free vibration of the beam in the vicinity of the buckled configuration is also investigated. The influences of power-law exponent, some commonly used boundary conditions, thermal changes, and length scale parameter on the static deflection and free vibration frequencies are studied.
2. Modified Couple Stress Theory (MCST)
Based on the MCST , the strain energy density of a linear elastic material for infinitesimal deformations can be written as in which , , , and stand for the components of the Cauchy stress tensor , infinitesimal strain tensor , deviatoric part of the couple stress tensor , and the symmetric part of the curvature tensor , respectively. For isotropic materials, these components can be given as in which denotes the length scale parameter. In the above relations, and define the components of infinitesimal displacement and rotation vectors, respectively, and
In addition, in the above formulas, and denote Lame’s constants and can be evaluated through the following relations:
3. Material Properties of the FG Beam
A FG microbeam made of ceramic and metal with length and thickness is considered here as depicted in Figure 1 in a Cartesian coordinate system. It is supposed that the materials at bottom surface and top surface of the beam are metal rich and ceramic rich, respectively. Effective material characteristics of the FGM beam such as Young’s modulus , Poisson’s ratio , thermal expansion coefficient , and thermal conductivity can be obtained as follows:Here, the subscripts and refer to metal and ceramic phases, respectively. is the volume fraction function which can be defined as below for the power-law distribution: where represents the power-law index.
4. Problem Formulation
According to the EBT, the plane sections remain plane and normal to the axis of the beam. Therefore, the displacement components at any arbitrary point along the thickness can be given as  in which and define the axial and transverse displacements of the midplane in the and directions, respectively. According to the above displacement field, the only nonzero component of the Von-Karman strain tensor is the axial strain which can be written as below at a distance from the midplane: in which prime indicates the differentiation with respect to . The symmetric part of the curvature tensor and also deviatoric part of the couple stress tensor are given by  Using (2) and (11), the Cauchy axial stress can be written as In the next step, the governing equations and boundary conditions are obtained using Hamilton’s principle as where , , and denote the kinetic energy, strain energy, and the external work done by the applied forces, respectively, and can be obtained as below: where dot denotes the differentiation with respect to time. In (16), is the beam strain energy due to the bending and change of the stretch with respect to the initial configuration and is the strain energy due to thermal stress. Therefore, the total normal stress in the FG microbeam is the summation of and the thermal stress with the following definition: In this investigation, initial uniform temperature () is a stress-free state.
Taking the first variation of the total strain energy leads to
Also, the first variations of the kinetic energy and external work functions can be written as In (20), and denote the axial and transverse applied loads per unit length, respectively. Also, , , and stand for the resultant normal force, end shearing force, and end bending moment, respectively.
Now, using Hamilton’s principle given by (14) and using the fundamental lemma of calculus, one arrives at the following governing differential equations of motion for a FG microbeam based on MCST:
Moreover, all possible boundary conditions at both ends of the beam can be obtained asIn the above equations, the extensional, coupling, and bending stiffness coefficients are defined asIn (21) and (22), stands for the total mass of the microbeam defined as and denotes the normal resultant force due to the thermal loading which can be obtained as Suppose the axial body force to be zero; the axial inertia is negligible and is independent of . Integrating both sides of (21) from to and using the condition at both ends of the beam give the following relation: with
Therefore, (22) can be recast to where
Using (28) together with introducing the nondimensional variables Equation (27) can be rewritten in the following form: where is the radius of gyration and for a rectangular cross section its value would be . The other parameters are given as It should be said that, for isotropic beams or symmetrically laminated ones, the coupling stiffness vanishes. The nondimensional geometric boundary conditions can be expressed as follows.
For clamped-clamped boundary condition for clamped-simply supported and for simply supported-simply supported
5. Buckling Problem
The governing equation of the linear buckling problem for a FG beam can be achieved from (18) by dropping the nonlinear and inertia terms. The result is where The closed-form solution for (38) gets the following form: After imposing the associated boundary conditions, the closed-form solutions for FG microbeams with clamped-clamped, clamped-simply supported, and simply supported-simply supported boundary conditions are obtained. Imposing (35) on (40) yields the characteristic equation and the closed-form solution for the buckling configuration of a clamped-clamped FG beam as
In a similar manner as above, by imposing (36) on (40), we arrive at the solution for clamped-simply supported Also, imposing (37) gives the solution for simply supported-simply supported boundary condition as in which is a constant. The critical buckling loads which are the roots of the characteristic equation for the three above-mentioned boundary conditions are evaluated and listed in Table 1.
6. Free Vibration Analysis in the Thermal Postbuckling Domain
To investigate the vibrational behavior in the postbuckling domain, we consider the vibrations taking place around a buckled configuration which are obtained from the previous section. To this aim assume the solutions for transverse deflection to be in the following form: where is a small dynamic disturbance in the vicinity of buckled configuration . Substituting (44) into (33) results in Discarding the nonlinear terms from the right-hand side of (45), we reach  The corresponding boundary conditions are as those given in (35)–(37). The dynamic disturbance can be considered as
For clamped-clamped end supports for clamped-simply supported and for simply supported-simply supported The general solution of (48) reads as
After substituting the above equation into (48), the homogeneous part of the solution, , and the particular one, , should satisfy the following equations, respectively: The characteristic equation of (53) is which can be readily solved to give the following roots: where The homogeneous part of the solution can be written as The coefficient of in (54) is a constant for given and . Therefore, the particular solution can be assumed as which upon substituting into (54) yields a relation for and () as Finally, the general solution of (48) can be written as
In order to evaluate (), one should impose the corresponding set of boundary conditions on the general solution of (61). Since an important part of our research initiative is concerned with free vibration analysis of microscale FG beams in the postbuckling domain and under different boundary conditions, the detailed analysis for any kind of boundary condition is given in separate subsections.
6.1. Clamped-Clamped End Condition
Applying the boundary conditions associated with the clamped-clamped support (49), one arrives at
Equations (62) and (60) represent an eigenvalue problem for the vibration of the clamped-clamped buckled beam. Setting the coefficient determinant equal to zero yields the characteristic equation whose roots are the natural frequencies for clamped-clamped microscale FG beams.
6.2. Simply Supported-Simply Supported End Condition
Applying the boundary conditions associated with simply supported-simply supported beam (see (51)), we arrive at Equations (63)–(66) along with (60) result in an eigenvalue problem governing the natural frequencies of vibration for the simply supported-simply supported buckled beam. Considering the first mode of vibration, where , and using (60) result in the following equation: in which
Therefore, we reach which yields the natural frequency of the first vibration mode upon using the values of and obtained in the linear buckling analysis. Considering and using (61), the corresponding mode shape reads as For other modes where , one should evaluate the roots of the characteristic equation. Equations (63)–(66) constitute a system of four equations. In order to find a nontrivial solution, the determinant of the coefficient matrix must be set to zero. Therefore, which yields There are three possibilities for the above relation which are listed as
Using (57), (74), and (75) and solving for , we arrive at where and are two integers denoting the vibration and buckling modes, respectively. The case yields the positive amount of and, consequently, is held, while, for , is negative and is satisfied. Solving the eigenvalue problem results in As a result (61) gives Substituting (76) into (79) results in which states that there is no physical mode shape corresponding to the negative eigenvalue.
6.3. Clamped-Simply Supported End Condition
Applying the boundary conditions of a clamped-simply supported beam (see (50) to (61)), one arrives at Equations (82) and (60) represent an eigenvalue problem for the vibration of the clamped-simply supported buckled beam.
7. Thermal Loading
Consider a thin FG beam for which the temperature values on ceramic-rich and metal-rich surfaces are specified by and , respectively. Assuming a linear temperature distribution, the temperature distribution for the given boundary conditions is obtained by solving the heat conduction equation along the beam thickness: Solving the above equation under the associated boundary conditions gives This temperature distribution is the one which must be substituted in the thermal stress relation (see (17)).
8. Results and Discussion
In this section, the results obtained from thermal postbuckling analysis of FG microbeams under different boundary conditions are presented in terms of nondimensional deflection and natural frequencies. A FG microbeam composed of metal and ceramic is considered here with material properties of constituents listed in Table 2. The top and bottom surfaces are assumed to be ceramic and metal rich, respectively. Letting , the formulation based on MCST is summarized to the classical Euler-Bernoulli FG beam (available in the literature, e.g., in ) and the present approach exactly results in the achievements of .
The variation of nondimensional thermal postbuckling deflection against the temperature parameter () for the first three buckled configurations of a microscale FG beam is given in Figures 2, 3, and 4. The results obtained are plotted for different values of power-law index and various types of end conditions. A linear temperature distribution is considered. and refer to isotropic beams made of pure ceramic and metal, respectively. As would be observed, with an increase in the power-law index, the critical temperature parameter decreases and the deflection increases. Also, the critical temperature parameter values associated with the clamped-clamped support are the most amongst the three aforementioned end conditions which clarify the high stability of a FG beam with this type of boundary condition. In order to examine the length scale effects, the variation of the nondimensional thermal deflection versus the temperature parameter for the first buckling mode of a FG beam with different values of and under three types of boundary conditions is illustrated in Figure 5. Figures 6 and 7 are the similar figures plotted for the second and third buckling modes, respectively. As can be seen, the nondimensional deflection decreases as increases. Another point deduced here is that, for higher values of , the difference between the results estimated by the MCST and the classical beam theory becomes greater and this difference is more prominent in the case of clamped-clamped boundary condition. As illustrated by Figures 5–7, the critical temperature parameter increases as the ratio increases and the curves shift to the right and the beam becomes stiffer. Further, the plots clarify the inability of the classical beam theory in predicting the critical temperature parameter for the microbeams.
The variation of the nondimensional natural frequencies associated with the first three vibration modes around the first three buckled configurations versus the temperature parameter for a clamped-clamped FG beam with different values of the power-law index is graphically illustrated in Figure 8. The similar plots for clamped-simply supported and simply supported-simply supported FG beams are exhibited by Figures 9 and 10, respectively. As shown by these figures, in the postbuckling domain, the natural frequency increases as the temperature parameter increases and this increase is more apparent for the first vibration mode rather than the two other. Also, with an increase in the power index in all three buckling modes, the natural frequency of the first vibration mode increases as the temperature parameter increases which indicates the higher resistance of the ceramic beam compared with the metal kind. However, with an increase in the power index there cannot be observed significant effect on the natural frequencies of higher vibration modes. Also, the influence of power index becomes more notable in the higher buckling modes. As can be seen in Figure 9, for the second vibration mode, an increment in the power index has no significant effect on the natural frequency for lower values of temperature parameters. However, for higher values of temperature parameters, this effect becomes more visible. In view of (77) it can be concluded that, for simply supported-simply supported boundary condition, only the natural frequency associated with the first vibration mode is a function of temperature parameter and this can be easily seen in Figure 10.
The variations of the nondimensional natural frequency of vibration with temperature parameter for different values of nondimensional length scale parameter and various end conditions are depicted in Figures 11, 12, and 13. As can be seen, for all first three buckling modes the natural frequency of the first vibration mode decreases as the temperature parameter and nondimensional length scale parameter increase. However, for the second and third vibration modes the trend becomes reverse and the natural frequencies increase as the temperature parameter and ratio increase. This fact once again states the validation of the present nonclassical theory over the classical one. In all of the graphical illustrations, the temperature parameter increase has no significant effect on the natural frequencies of higher vibration modes. Also, by reducing the amount of nondimensional length scale parameter, the MCST approaches the classical beam theory.
An exact solution is obtained for the thermal postbuckling behavior of the microscale functionally graded Euler-Bernoulli beams on the basis of modified couple stress theory. First, the buckling problem was solved to find the buckling configuration of the beams in terms of the critical temperature for a beam with different boundary conditions. Results achieved show that, with an increase in the power-law index, the critical temperature parameter decreases and the deflection increases. Also, the critical temperature parameter values associated with the clamped-clamped support are the most amongst the three aforementioned end conditions which clarify the high stability of a beam with this type of boundary condition. It has been shown that the nondimensional deflection decreases as the nondimensional length scale parameter, , increases. Also, the critical temperature parameter increases as increases and the beam becomes stiffer. Then the linear vibration problem is investigated around the first three buckled configurations and the lowest natural frequencies are obtained for FG microbeams with different boundary conditions. The natural frequency increases as the temperature parameter increases and this increase is more observed in the first vibration mode rather than the other two. Also, with an increase in the power index in all three buckling modes, the natural frequency of the first vibration mode increases, but there cannot be observed any significant effect on the natural frequencies of higher vibration modes.
It has been found that for all first three buckling modes the natural frequency of the first vibration mode decreases as the temperature parameter as well as nondimensional length scale parameter increases. However, for the second and third vibration modes the trend becomes reverse and the frequency values increase. For great values of nondimensional length scale parameter, there can be seen an apparent difference between the frequency values reported by the present nonclassical theory and those achieved via the classical one. Furthermore, this difference between the two mentioned theories is more prominent in the case of clamped-clamped boundary condition. This fact illustrates the insufficient precision of classical theory in predicting the mechanical behavior of microscale beams and shows the necessity of using nonclassical theories.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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