Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.


Introduction
As advanced composite materials, functionally graded (FG) materials possess distinctive features of gradually spatial changes in the material properties, which enable FG materials to avoid the stress concentration of conventional laminated composite materials. Besides, FG materials can satisfy the multi-functional requirements by a wide selection of material constituents. model, the effects of the dimensionless material length scale parameter, the length-to-thickness ratio, the gradient index, and the boundary conditions on the vibration characteristics of FG micro-beams are studied. By comparing the results obtained from different micro-beam theories, the effects of transverse shear and normal deformations on the vibration behavior are further investigated.
2 Theoretical formulation 2.1 Model description Figure 1 shows a typical two-phase FG micro-beam, in which the material properties vary continuously and smoothly along the z-direction. According to the Mori-Tanaka scheme, the effective material properties are given as follows [20][21][22]24] : where the bulk modulus and the shear modulus are denoted by K and G, respectively, and the subscripts 1 and 2 represent the two different constituents used in FG materials, respectively. The volume fraction V 1 is where p represents the gradient index governing the spatial variation of material properties. Then, one can obtain the effective Young's modulus E ef and Poisson's ratio µ ef as follows [20] : The effective mass density can be defined as follows:

Kinematic and constitutive relations
A general displacement field, which can capture the transverse shear and normal deformation effects and has the capability of application to various beam theories, is defined as follows: where u 0 and w 0 denote the axial and transverse displacement components on the central line, respectively. u 1 is the rotation of the cross section with the shear effect, and w 1 is introduced to consider the transverse normal deformation effect. The shape functions f (z), g(z), and φ(z) govern the distributions of the strains and the stresses in the z-direction. By choosing appropriate shape functions, the displacement fields corresponding to various beam theories are easily obtained. For example, one can choose for the classical beam theory (CBT); for the first-order beam theory (FBT); and for Reddy's beam theory (RBT). In order to consider the transverse normal deformation effect, a quasi-3D beam theory is developed by choosing [28] f Using the above displacement field, non-zero linear strains can be obtained as follows: in which the notation of prime is used to denote the derivative of shape functions. In the MSGT, some additional higher-order deformation gradients, including the dilatation gradient tensor γ i , the deviatoric stretch gradient tensor η (1) ijk , and the symmetric rotation gradient tensor χ s ij , are introduced to take the size effect into account, which are defined as follows [17] : in which a subscript preceded by a comma represents the differentiation with respect to the subscript. δ ij and e ipq denote the Kronecker delta and the permutation symbol, respectively. Then, substituting Eq. (7) into Eqs. (8)-(10) yields Although FG materials are globally heterogeneous, the locally effective properties at a given point can be assumed to be isotropic. Then, the classical and higher-order stresses are obtained by the following constitutive relations: where the Lamé parameter λ ef is given by The symbols of l 0 , l 1 , and l 2 denote the material length scale parameters associated with those additional higher-order deformation gradients. When l 0 = l 1 = 0 µm, the MSGT can degenerate into the MCST.

Variational formulation
The energy functional for the FG micro-beam is defined as follows: where the strain energy and the kinetic energy are represented by U and T , respectively. U bc is the additional potential energy on the boundaries. The kinetic energy T is achieved by where ρ ef f gdz, The strain energy U is calculated by the following integral equation [17] : With the help of the force and moment resultants achieved by integrating the stresses in the z-direction (see Appendix A), the strain energy U is further written as follows: The essential boundary conditions are enforced via the penalty function method. Introducing the penalty factors k yields the geometrical boundary conditions in the form of additional strain energy as follows [35][36] : It is noted that the penalty factors k i (i = 1, 2, 3, 4, and 7) are associated with the classical boundary conditions obtained from the conventional continuum mechanics theory, and the other penalty factors are used to handle the higher-order boundary conditions resulting from the introduction of some additional higher-order deformation gradients and the corresponding high-order stress terms. The explicit relation between the penalty factors and the boundary conditions is achieved via the Hamilton principle as follows: It can be found that general boundary conditions are easily simulated by choosing appropriate values of penalty factors. In addition, the governing equations of motion are also achieved in this way, which are listed in Eqs. (A10)-(A13) of Appendix A.

Solution procedure
Using Chebyshev orthogonal polynomials as the basis function and implementing a linear coordinate transformation with ξ = 2x/L − 1, the displacement variables given in Eq. (6) are expressed as follows: where A, B, C, and D denote the generalized coordinate variable vectors made up of A m , B m , C m , and D m , respectively. T m (ξ) is the one-dimensional mth-order Chebyshev orthogonal polynomial, which is achieved by By inserting Eqs. (16)- (19) and (29) into Eq. (15) and applying the variational operation, one can obtain the following equation in the form of matrices: The sub-matrices M u0u0 and K u0u0 are taken as examples to further clarify the calculation of the total mass and stiffness matrices. They are calculated by where With the aid of assumption of a harmonic motion, one can obtain the standard characteristic equations from Eq. (31). It is noted that the characteristic equations corresponding to the CBT, the FBT, and the RBT can be achieved by moving the lines and columns of the total mass and stiffness matrices corresponding to the absent displacement variables.
The material length scale parameters are taken as l 0 = l 1 = l 2 = l = 15 µm for the MSGT and l 0 = l 1 = 0 µm, l 2 = l = 15 µm for the MCST. The boundary conditions, including the clamped, simply supported, and free boundary conditions and their arbitrary combinations, are considered [24,31] . The clamped boundary conditions are defined as follows: The simply supported boundary conditions are defined as follows: Note that the free boundary conditions represent no constraints at edges. For the sake of convenience, the dimensionless frequency parameter is defined as Ω = ωL I 10 /A 110 , where A 110 and I 10 represent the values of A 11 and I 1 of a homogeneous metal (Al) beam.

Convergence studies
The accuracy and efficiency of this formulation depend on both the numbers of terms of the Chebyshev orthogonal polynomials taken for the displacement variables and the values of the penalty factors for modeling the boundary conditions. Therefore, those two key parameters need to be studied before going into the analysis. In fact, the larger the number of the orthogonal polynomial terms used in Eq. (29), the more accurate the obtained results. However, for computational efficiency, the appropriate truncated number M needs to be determined to achieve the results with acceptable accuracy. Table 1 presents the first eight dimensionless frequencies of completely free FG micro-beams having h/l = 2, 8, L/h = 5, and p = 1. An excellent convergence behavior can be found from those results. It can be seen that adopting M = 12 can obtain quite stable and acceptable results.
Theoretically, the penalty factors are taken as zero for free constraints whereas infinity for clamped constraints. However, the actual calculation cannot deal with infinite values and taking a very large value for the penalty factor may lead to round-off errors or ill-conditioning. Consequently, the determination of the penalty factors is further carried out. For the interest of convenience, the penalty factors are normalized by , i = 5, 6, 9, , i = 7, 8.  Figure 2 plots the effects of normalized penalty factors on dimensionless frequencies of FG micro-beams. It should be pointed out that appropriate values for the normalized penalty factors of Γ i (i = 1, 2, 3, 5, 6, 7) are in accord with those in Ref. [35], where the MSGT-based FBT was used. For brevity, the corresponding figures related to the above penalty factors are not shown in Fig. 2. In this analysis, the aforementioned six kinds of penalty factors are set to Γ 1 = Γ 3 = Γ 5 = Γ 7 = 10, Γ 2 = Γ 6 = 0 for clamped boundary conditions. For penalty factors of Γ i (i = 4, 8, 9), the 1st and 5th dimensionless frequencies are obtained by changing only one kind of penalty factors from small to large and assigning the others with zeros. It can be observed from Fig. 2 that, as the normalized penalty factor continually increases, all obtained results slightly increase, and finally keep a constant level. Accordingly, the appropriate values of the normalized penalty factors are defined as Γ 4 = Γ 8 = 10, Γ 9 = 0 for clamped boundary conditions.

Verification studies
To validate this formulation, some comparative examples are given. Due to lack of available quasi-3D results for FG micro-beams based on the MSGT, the numerical comparison for FG micro-beams within the frame of the MCST is firstly conducted. The dimensionless fundamental frequencies of simply supported FG micro-beams having L/h = 10 are listed in Table 2. This problem has been analyzed by Trinh et al. [30] via the Navier solution based on various beam theories including the CBT, the FBT, the RBT, and the quasi-3D method. Those calculated results have excellent consistency with the referred date, which not only verifies the accuracy of the present formulation, but also shows that it has the ability to accommodate various beam theories.  Table 3 presents the first ten dimensionless quasi-3D frequencies of simply supported FG micro-beams made of Al 2 O 3 and Al. The used parameters in this analysis are p = 1, L/h = 10, h/l = 1, 4, 20, 100. The quasi-3D solutions provided by Yu et al. [32] by using the isogeometric analysis (IGA) are also listed in this table. Excellent agreement even for higher-order frequencies is observed from the results.
To further validate this formulation, Table 4 presents the dimensionless fundamental frequencies of FG micro-beams subjected to different boundary conditions based on the MSGT. The obtained results are compared with those given by Ansari et al. [21] by using the MSGTbased Timoshenko beam model and the generalized differential quadrature (GDQ) method. It is noted that in Ref. [21], the immovable simply supported boundary conditions were used. In order to better verify the present method, the same boundary conditions are considered. As expected, good agreement is found when adopting the same beam theory for all boundary conditions considered.

Parametric studies
After the success of numerical comparison, the present formulation is used to study the effects of the key parameters on the vibration behaviors of FG micro-beams. In Figs. 3-5, SS presents simply supported-simply supported, CC presents clamped-clamped, CS presents clamped-simply supported, and CF presents clamped-free. Figure 3 depicts the effects of the dimensionless material length scale parameter h/l on the vibration frequencies of FG micro-beams having L/h = 10 and p = 1. For all cases, the dimensionless frequency parameters of FG micro-beams decrease as h/l increases. This lies on the fact that the size effects become weak as h/l increases, leading to the reduction in the flexural rigidity. By comparing the frequencies obtained from the FBT and the RBT, it can be clearly seen that their differences are notable when h/l is small and are diminishing as h/l increases. It may be deduced that the strong size effects lead to failure of the FBT for moderately thick micro-beams with L/h = 10, particularly for clamped-clamped and clamped-simply supported boundary conditions. By making a comparison of the RBT and quasi-3D results, it also can be observed that the quasi-3D results are always smaller than the RBT ones. This is attributable to the normal deformation effect. Another interesting point is that there exists an intersection of two curve lines related to the FBT and quasi-3D results and the value of h/l corresponding to the intersection point is strongly affected by the boundary conditions.  Figure 4 plots the effects of the gradient index p on the dimensionless fundamental frequency parameters for FG micro-beams having h/l = 2 and L/h = 10. As expected, the frequency parameters of FG micro-beams decrease for all cases as p increases. This reason is that the increase in the gradient index p results in the lessening of the volume fraction of SiC, leading to the reduction in the flexural rigidity. It also can be observed that the change of the gradient index p does not affect the order relation of the results from different beam theories when other parameters are fixed. The effects of the length-to-thickness ratio L/h on the vibration frequency parameters of FG micro-beams having h/l = 2 and p = 1 are shown in Fig. 5.
For all cases, the dimensionless frequency parameters of FG micro-beams decrease as the length-to-thickness ratio increases. This is because the increase in L/h leads to the reduction in the flexural rigidity. The differences of the FBT and RBT results decrease as L/h increases, indicating that the FBT is not valid for slightly thick micro-beams, especially for clampedclamped and clamped-simply supported boundary conditions.

Conclusions
This paper develops an MSGT-based quasi-3D micro-beam model for the vibration analysis of FG micro-beams. The effects of transverse shear and normal deformations are considered. The formulation is derived by a Chebyshev-based variational method combined with the penalty function method, in which the displacement field is given in a general form, which can be deduced to the CBT, the FBT, and the RBT. A wide range of examples on the vibration problems of FG micro-beams are presented. The results validate the present formulation. From parametric studies, it is observed that the dimensionless frequencies decrease with the increase in the dimensionless material length scale parameter h/l, the gradient index p, or the lengthto-thickness ratio L/h, and the transverse shear effects play a significant role on vibration behaviors with lower values of h/l and L/h, particularly for clamped-clamped and clampedsimply supported boundary conditions. In addition, comparisons of the quasi-3D and RBT results indicate the transverse normal effects.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link The governing equations of the FG micro-beam are obtained via the Hamilton principle as follows: