Vibration of functionally graded shear and normal deformable porous microplates via finite element method

Journal Pre-proofs Vibration of functionally graded shear and normal deformable porous microplates via finite element method Armagan Karamanli, Metin Aydogdu PII: DOI: Reference:

S0263-8223(19)33517-2 https://doi.org/10.1016/j.compstruct.2020.111934 COST 111934

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Composite Structures

Received Date: Revised Date: Accepted Date:

17 September 2019 11 December 2019 11 January 2020

Please cite this article as: Karamanli, A., Aydogdu, M., Vibration of functionally graded shear and normal deformable porous microplates via finite element method, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111934

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Vibration of functionally graded shear and normal deformable porous microplates via finite element method

Armagan Karamanli1,* 1Faculty

of Engineering and Natural Sciences, Mechatronics Engineering, Bahcesehir University, 34353, Istanbul, Turkey. Metin Aydogdu2

2Department

of Mechanical Engineering, Trakya University, 22030, Edirne, Turkey.

*Corresponding Author: [email protected]

Abstract The size dependent natural frequencies of functionally graded (FG) shear and normal deformable porous square microplates are investigated within this paper for arbitrary boundary conditions. By utilizing the modified couple stress theory, the finite element model is developed based on a shear and normal deformation plate theory and the variational formulation. The material length scale parameter (MLSP) is taken as variable. The effects of the aspect ratio, gradient index, boundary condition, thickness to MLSP ratio, porosity volume fraction and variable MLSP on the dimensionless natural frequencies are investigated for the FG shear and normal deformable porous square microplates. It is found that the difference between the numerical computations employing the constant and variable material length scale parameters is significant. In addition, it is found that with an increment in the aspect ratio, the effect of the MLSP on the natural frequencies increases, especially for the thick microplates. It can be concluded that for the thick microplates (𝑙𝑒𝑛𝑔𝑡ℎ/𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 ≤ 10), the effect of the variable MLSP on the natural frequencies with respect to the changing of the thickness to MLSP is more emphasized than the effect obtained by the constant MLSP for all studied boundary conditions.

Keywords: Free vibration; Functionally graded porous microplate; modified couple stress theory; finite element method; variable material length scale parameter

1. INTRODUCTION Engineering applications developed for severe operating conditions generally require an advanced class of composite materials which can cope with the extremely high temperatures, eliminate the delamination, provide lightweight without any deficiencies on the stiffness and strength and avoid stress concentration. FGMs can be classified as an advanced class of composite materials which can fulfill those harsh requirements. FGMs obtained by two different material phases such as metal and ceramic allow having elastic properties varying smoothly in the required direction by employing a material homogenization rule. Because of these fascinating features, the researchers have been developing advanced theories and solutions techniques to investigate the mechanical behaviors of the FG beams [1-6], plates [716] and shells [17-25]. With the new advances in technology, extreme demands based on the usage of micro/nano electromechanical systems (MEMS/NEMS) such as thin films, actuators, probes, sensors, etc. have been raised from the various industries [26-29]. However, the experiments indicate that the mechanical behavior of the micro/nano elements made of FGMs cannot be investigated by the classical continuum theories (CCTs). More reliable prediction can be obtained by employing the higher-order continuum theories (HOCTs) in which the additional material parameters are necessary [30-32]. In [33], the studies based on the HOCTs were initiated by introducing additional degrees of freedom (DOF) together with ones from CCTs to state the independent microrotation of material particles. Based on the various assumptions on constitutive laws incorporating with the DOF, a good number of theories have been proposed. These HOCTs can be classified in three major categories as the micro-continua, nonlocal elasticity and the strain gradient family [34]. In [35-38], the micro-continua were proposed as 3M theories which are the micropolar, microstretch and micromorphic. In micropolar theory, there are three additional DOF, however 2

in microstretch and micromorphic theories, there are four and nine DOFs, respectively [39]. The nonlocal elasticity was introduced in [40] and further developments were presented in [41– 43]. Since the stress at a point is determined by using the constitutive law accompanying with strains occur around the effective area of the point in the nonlocal elasticity, the small size effects are captured by introducing a nonlocal parameter to the constitutive equations. The strain gradient theories were firstly initiated in [44] by considering the first gradient of strains. Then, with the inclusion of the first and second gradients of strains, a different strain gradient theory was proposed in [45]. Different from the CCTs, in the strain gradient theories additional material parameters i.e. material length scale parameter (MLSP) are required. The classical couple stress theories requiring two additional MLSPs were proposed in [46-49]. Due to the difficulty to determine the MLSPs which are necessary for the HOCTs mentioned above, in [50], the modified couple stress theory (MCST) which makes the couple stress tensor symmetric and consequently requires only one MLSP was proposed. The MCST has been employed for the mechanical analysis of microstructures based on various size dependent shear deformation theories because of its simplicity in the implementation with the requiring less CPU time than the other HOCTs. There are also many interesting studies in the open literature related to development of the microstructure dependent models to understand the mechanical behavior of the small size structures [94-106]. The free vibration behavior of FG microplates is investigated by employing the classical plate theory (CPT) in [51]. In [52], the classical plate theory (CPT) is used to analyze the buckling and post-buckling behaviors of the FG microplates for various boundary conditions (BCs). The analytical solutions are presented for linear and nonlinear mechanical responses of FG microplates using the size dependent Kirchhoff and Mindlin plate theories [53]; in [54] the elastic medium is included to extend the studies. The MCST is utilized to investigate the nonlinear vibration behavior of the FG microplates employing the Mindlin plate theory and 3

differential quadrature method in [55]. Based on the Reddy plate theory (RPT), a size dependent FG microplate model is analyzed by using Navier method in [56]. By employing the Chebyshev-Ritz method, natural frequency analysis of FG quadrilateral microplates is investigated employing thermal effects in [57]. The finite element method (FEM) is utilized for the solutions of nonlinear bending and post-buckling of FG circular piezoelectric microplate problems based on the MCST [58]. The flexural, natural frequency and stability behaviors of FG sandwich microplates are studied by using state-space Levy method in [59]. The manufacturing processes of the MEMS may not be perfect due to poor quality control implementations, technical issues, inexperienced manpower, lack of maintenance, material procurement problems etc. Consequently, the discontinuities such as porosity, microcrack etc. can be found in the FG microplates. The mechanical behavior of FG microplates could be degraded after having a certain level of porosity volume fraction. On the other hand, the MEMS with lightweight based on the optimized porosity distribution may bring some advantages to the structural design in terms of low cost, versatile integration and reasonable strength to weight ratio in the engineering applications. The influence of the porosity distributions for the nonlinear vibration of FG thin plates are investigated in [60]. An analytical solution is presented for the free vibration analysis of FG porous plates based on a first order shear deformation theory [61]. Based on the isogeometric analysis (IGA), the natural frequencies of the imperfect FG microplates are presented by using the MCST [62]. The IGA analysis is also applied in [63] for the thermal stability investigation of the FG imperfect microplates employing the MCST. By employing the modified strain gradient theory and IGA, the flexural and elastic stability behaviors of the FG porous microplates subjecting the thermal and mechanical loads are investigated in [64]. In a very recent study, the bending, natural frequency and elastic stability behaviors of the FG porous microplates are studied to present the effects of the porosities based on the MCST [65]. 4

It should be noted that the MLSP is assigned as constant in the analysis of the microplates mentioned above. However, the MLSP can be considered as a material property which is defined within the context of the MCST. There are very limited studies in the open literature employing the MLSP as a function in the analysis of small-scale structures. In [66], the flexural and free vibration behaviors of FG nanobeams is studied using a strain gradient theory. In addition, the bending and natural frequency analysis of FG microbeams are investigated in [67] by utilizing non-constant MLSP. Using the neutral surface concept, the flexural and dynamic behaviors of FG microbeams are presented in [68] by employing variable MLSP. The natural frequency, elastic and dynamic buckling behaviors of the bi-directional FG microbeams is investigated based on the variable MLSP concept in [69]. Moreover, the structural dynamics and stability analysis of bi-directional FG microbeams having variable porosity distribution and MLSP is presented in [70]. The flexural and natural frequency analysis of circular microplates with variable MLSP is given in [71-72] based on the MCST. The normal and shear deformation theories (Quasi-3D) have been also employed for the structural analysis of FG plates to improve the accuracy. It is noteworthy that in the shear deformation theories mentioned in the studies previous paragraphs, the thickness stretching effect is ignored. However, for the thick plates, the normal deformation is important and should be considered. In [73], analytical solutions obtained from bending and free vibration analysis are presented for simply supported FG shear and normal deformable microplates with constant MLSP. By employing the MCST and a Quasi-3D plate theory, the mechanical behaviors of FG sandwich simply supported microplates under thermal and mechanical loads are investigated in [74]. The NURBS based IGA is applied to obtain the solutions for the mechanical problems of the FG shear and normal deformable microplates for various BCs in [75]. A review is given for the flexural, natural frequency and elastic stability of the FG small-scale structures in [76].

5

Based on the literature review given above, it is clear that the porosity distribution through body of the microplate is important and affects the mechanical behavior of the small-scale FG structures. Moreover, the MLSP should be considered as a variable material property to predict the structural responses of the FG microplates more accurately. In addition, thickness stretching is very important to figure out the structural analysis of thick FG microplates. And finally, the open literature needs more benchmark results on the natural frequency analysis of the FG microplates, especially for different BCs. This work aims to investigate the natural frequency analysis of FG porous microplates with variable MLSP for various BCs including the normal deformation effect by employing the MCST. The FEM is utilized to obtain the numerical computations. The effects of the BCs, aspect ratio, porosity volume fraction (PVF), variable MLSP, thickness stretching, BCs, gradient index on the natural frequencies are examined. 2. FORMULATION 2.1 Functionally graded microplate with porosities A functionally graded porous microplate which is made of ceramic and metal with the dimensions axbxc and coordinate as illustrated in Fig. 1. The modulus of elasticity 'E', mass density 'ρ', Poisson’s ratio 'ν' and the MLSP ‘ℓ’ of the porous microplate change through the thickness (z-axis) direction based on the mixture rule.

Figure 1 Around Here

In the open literature, it is provided that elastic properties of porous solid structures depend on the form, distribution, orientation and shape of pores [77-79]. To introduce the material

6

properties of the homogeneous solids with porosities, the researchers have been devoted a large number of analytical models [80]. Since the conventional volume fraction is used for the presentation of the porosity, the effective material properties are given in the following form [107-108]: 𝑃(𝑧) = 𝑃1 (𝑉1 (𝑧) −

𝛼(𝑧) 𝛼(𝑧) ) + 𝑃2 (𝑉2 (𝑧) − ) 2 2

(1)

where 𝑃1 and 𝑃2 are the material properties, 𝛼 is the PVF, 𝑉1 and 𝑉2 are volume fractions of two constitutes, and the volume fractions are related: 𝑉1 (𝑧) + 𝑉2 (𝑧) = 1

(2𝑎)

1 𝑧 𝑝𝑧 𝑉1 (𝑧) = ( + ) 2 ℎ

(2𝑏) 𝑝𝑧

1 |𝑧| 𝛼(𝑧) = 𝛼0 ( − ) 2 ℎ

(2𝑐)

where 𝛼0 is a coefficient given in the PVF function to control the level of the porosity within ℎ/2

the structure, total PVF can be obtained by using 𝑃𝑉𝐹 = ∫−ℎ/2 𝛼(𝑧)𝑑𝑧, and 𝑝𝑧 is the gradient index. It is clear from a comparison between proposed material distributions given in Eq.1 and [81] that low PVFs given in Eq. 1 can be employed to have more accurate calculations. In the present study total PVF is assumed within the interval of 0 - 0.1. Regarding to the PVF coefficients 0.01, 0.05 and 0.1, the expected differences between the Young’s modulus obtained by using the Eq. 1 and the equation provided by [81] for a isotropic plate can be obtained as 1.34%, 6.62% and 13.11%, respectively. It is noteworthy that with an increment in the gradient index leads to have a decrement in the total PVF. With the decreasing of the total PVF, the difference between the calculations for the Young’s modulus based on the Eq. 1 and the equation presented in [81] decreases, and eventually, more accurate computations can be performed. 7

The effective material properties are to be found by utilizing the Eq.1 and Eq.2. For instance, effective Young’s modulus can be written in the form of: 𝑝𝑧

1 𝑧 𝑝𝑧 𝛼0 1 |𝑧| 𝐸(𝑧) = (𝐸1 − 𝐸2 ) ( + ) + 𝐸2 − ( − ) 2 ℎ 2 2 ℎ

(𝐸1 + 𝐸2 )

(4)

where indices 1 and 2 describe the ceramic and metal constitutes, respectively. Other material properties like 𝜈, 𝜌 and ℓ can be written in the similar form. 2.2 Constitutive Relations Employing the transverse shear-normal deformable plate theory (TSNDPT), the displacement field is obtained as follows [70, 82-88]: 𝜕𝑤𝑏 (𝑥, 𝑦, 𝑡) 4𝑧 3 𝜕𝑤𝑠 (𝑥, 𝑦, 𝑡) 𝑈(𝑥, 𝑦, 𝑧, 𝑡) = 𝑢(𝑥, 𝑦, 𝑡) − 𝑧 − 2 = 𝑢 − 𝑓1 𝑤𝑏,𝑥 − 𝑓2 𝑤𝑠,𝑥 𝜕𝑥 3ℎ 𝜕𝑥

(5𝑎)

𝜕𝑤𝑏 (𝑥, 𝑦, 𝑡) 4𝑧 3 𝜕𝑤𝑠 (𝑥, 𝑦, 𝑡) − 2 = 𝑣 − 𝑓1 𝑤𝑏,𝑦 − 𝑓2 𝑤𝑠,𝑦 𝜕𝑦 3ℎ 𝜕𝑦

(5𝑏)

𝑉(𝑥, 𝑦, 𝑧, 𝑡) = 𝑣(𝑥, 𝑦, 𝑡) − 𝑧

4𝑧 2 𝑊(𝑥, 𝑦, 𝑧, 𝑡) = 𝑤𝑏 (𝑥, 𝑦, 𝑡) + 𝑤𝑠 (𝑥, 𝑦, 𝑡) + (1 − 2 ) 𝑤𝑧 (𝑥, 𝑦, 𝑡) = 𝑤𝑏 + 𝑤𝑠 + 𝑓3 𝑤𝑧 ℎ

(5𝑐)

where 𝑢 and 𝑣 are the in-plane displacements, 𝑤𝑏 is the bending part of the transverse displacement, 𝑤𝑠 is the shear part of the transverse displacement, 𝑤𝑧 is the stretching component of the transverse displacement and t is the time. 𝑓1 , 𝑓2 and 𝑓3 are the shape functions characterizing the contributions of the bending, shear and normal displacement across the thickness of the microplate. It should be noted that, one can obtain the various plate theories using the Eq. 5 as follows, the CPT (𝑓2 = 𝑓3 = 0), the first order shear deformation theory (𝑓2 = 4𝑧 3

0 and 𝑓3 = 1), the third order shear deformation theory (𝑓2 = 3ℎ2 and 𝑓3 = 0). The strains based on the given displacement field in Eq. (5) can be presented as follows:

𝜀𝑥𝑥 =

𝜕𝑈 = 𝑢,𝑥 − 𝑓1 𝑤𝑏,𝑥𝑥 − 𝑓2 𝑤𝑠,𝑥𝑥 𝜕𝑥

(6𝑎) 8

𝜀𝑦𝑦 =

𝜕𝑉 = 𝑣,𝑦 − 𝑓1 𝑤𝑏,𝑦𝑦 − 𝑓2 𝑤𝑠,𝑦𝑦 𝜕𝑦

(6𝑏)

𝜀𝑧𝑧 =

𝜕𝑊 = 𝑓3 ′ 𝑤𝑧 𝜕𝑧

(6𝑐)

𝛾𝑥𝑧 =

𝜕𝑈 𝜕𝑊 + = (1 − 𝑓1 ′ ) 𝑤𝑏,𝑥 + (1 − 𝑓2 ′ )𝑤𝑠,𝑥 + 𝑓3 𝑤𝑧,𝑥 𝜕𝑧 𝜕𝑥

(6𝑑)

𝛾𝑦𝑧 =

𝜕𝑉 𝜕𝑊 + = (1 − 𝑓1 ′ ) 𝑤𝑏,𝑦 + (1 − 𝑓2 ′ )𝑤𝑠,𝑦 + 𝑓3 𝑤𝑧,𝑦 𝜕𝑧 𝜕𝑦

(6𝑒)

𝛾𝑥𝑦 =

𝜕𝑈 𝜕𝑉 + = 𝑢,𝑦 + 𝑣,𝑥 − 2𝑓1 𝑤𝑏,𝑥𝑦 − 2𝑓2 𝑤𝑠,𝑥𝑦 𝜕𝑦 𝜕𝑥

(6𝑓)

Based on the MCST, the rotation vector 𝜃 and the curvature vector 𝜒 can be given in the following form [50]: 𝜃𝑥 𝜃 𝜃 = [ 𝑦] 𝜃𝑧

(7𝑎)

𝜒𝑥𝑥 𝜒𝑦𝑦 𝜒𝑧𝑧 𝜒 = 𝜒𝑧𝑦 𝜒𝑥𝑧 𝜒 [ 𝑥𝑦 ]

(7𝑏)

where 𝜃𝑥 =

1 𝜕𝑊 𝜕𝑉 1 ( − ) = [(1 + 𝑓1 ′ ) 𝑤𝑏,𝑦 + (1 + 𝑓2 ′ )𝑤𝑠,𝑦 + 𝑓3 𝑤𝑧,𝑦 ] 2 𝜕𝑦 𝜕𝑧 2

1 𝜕𝑈 𝜕𝑊 1 𝜃𝑦 = ( − ) = [−(1 + 𝑓1 ′ )𝑤𝑏,𝑥 − (1 + 𝑓2 ′ )𝑤𝑠,𝑥 − 𝑓3 𝑤𝑧,𝑥 ] 2 𝜕𝑧 𝜕𝑥 2 1 𝜕𝑉 𝜕𝑈 1 𝜃𝑧 = ( − ) = (𝑣,𝑥 − 𝑢,𝑦 ) 2 𝜕𝑥 𝜕𝑦 2 𝜒𝑥𝑥 =

𝜕𝜃𝑥 1 = [(1 + 𝑓1 ′ ) 𝑤𝑏,𝑥𝑦 + (1 + 𝑓2 ′ )𝑤𝑠,𝑥𝑦 + 𝑓3 𝑤𝑧,𝑥𝑦 ] 𝜕𝑥 2

(8𝑎)

(8𝑏)

(8𝑐)

(8𝑑) 9

𝜒𝑦𝑦 =

𝜕𝜃𝑦 1 = [−(1 + 𝑓1 ′ )𝑤𝑏,𝑥𝑦 − (1 + 𝑓2 ′ )𝑤𝑠,𝑥𝑦 − 𝑓3 𝑤𝑧,𝑥𝑦 ] 𝜕𝑦 2

(8𝑒)

𝜒𝑧𝑧 =

𝜕𝜃𝑧 =0 𝜕𝑧

(8𝑓)

1 𝜕𝜃𝑥 𝜕𝜃𝑦 𝜒𝑥𝑦 = ( + ) 2 𝜕𝑦 𝜕𝑥 1 𝜒𝑥𝑦 = [(1 + 𝑓1 ′ )(𝑤𝑏,𝑦𝑦 − 𝑤𝑏,𝑥𝑥 ) + (1 + 𝑓2 ′ )(𝑤𝑠,𝑦𝑦 − 𝑤𝑠,𝑥𝑥 ) + 𝑓3 (𝑤𝑧,𝑦𝑦 − 𝑤𝑧,𝑥𝑥 )] 4

(8𝑔)

1 𝜕𝜃𝑦 𝜕𝜃𝑧 1 𝜒𝑦𝑧 = ( + ) = (−𝑓1 ′′ 𝑤𝑏,𝑥 − 𝑓2 ′′ 𝑤𝑠,𝑥 − 𝑓3 ′ 𝑤𝑧,𝑥 + 𝑣,𝑥𝑦 − 𝑢,𝑦𝑦 ) 2 𝜕𝑧 𝜕𝑦 4

(8ℎ)

1 𝜕𝜃𝑥 𝜕𝜃𝑧 1 𝜒𝑥𝑧 = ( + ) = (𝑓1 ′′ 𝑤𝑏,𝑦 + 𝑓2 ′′ 𝑤𝑠,𝑦 + 𝑓3 ′ 𝑤𝑧,𝑦 + 𝑣,𝑥𝑥 − 𝑢,𝑥𝑦 ) 2 𝜕𝑧 𝜕𝑥 4

(8𝑖)

The stress-strain relationship based on the MCST for FG porous microplate is given by [50]: 𝐶11 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜎𝑦𝑧 = 𝜎𝑥𝑧 𝜎 { 𝑥𝑦 } [𝑠𝑦𝑚

𝐶12 𝐶22

𝐶13 𝐶23 𝐶33

0 0 0 𝐶44

0 0 0 0 𝐶55

0 𝜀𝑥𝑥 0 𝜀𝑦𝑦 0 𝜀𝑧𝑧 𝛾 𝑦𝑧 0 𝛾 𝑥𝑧 0 𝛾 𝐶66 ] { 𝑥𝑦 }

𝑚𝑥𝑥 𝜒𝑥𝑥 𝑚𝑦𝑦 𝜒𝑦𝑦 𝐸(𝑧)ℓ(𝑧)2 𝜒𝑧𝑧 𝑚𝑧𝑧 = 𝑚𝑦𝑧 1 + 𝜈(𝑧) 𝜒𝑧𝑦 𝑚𝑥𝑧 𝜒𝑥𝑧 {𝜒𝑥𝑦 } {𝑚𝑥𝑦 }

(9𝑎)

(9𝑏)

where 𝜎𝑖𝑗 are the components of the stress tensor, 𝑚𝑖𝑗 are the components of the symmetric couple stress tensor. And, the elastic constants 𝐶𝑖𝑗 can be presented as follows:

𝐶11 (𝑧) = 𝐶22 (𝑧) = 𝐶33 (𝑧) =

𝐸(1 − 𝑣) (1 − 2𝑣)(1 + 𝑣)

(10𝑎)

𝐶12 (𝑧) = 𝐶13 (𝑧) = 𝐶23 (𝑧) =

𝐸𝑣 (1 − 2𝑣)(1 + 𝑣)

(10𝑏)

10

𝐶44 (𝑧) = 𝐶55 (𝑧) = 𝐶66 (𝑧) =

𝐸 2(1 + 𝑣)

(10𝑐)

2.3 Energy Function By using kinematics and constitutive, the energy function (Π) of the FG porous square microplate is obtained based on the MCST in this section. The strain energy (𝒰) of a functionally graded porous square microplate can be described in the form of [50]:

𝒰=

1 ∫ (𝜎𝑥𝑥 𝜀𝑥𝑥 + 𝜎𝑦𝑦 𝜀𝑦𝑦 + 𝜎𝑧𝑧 𝜀𝑧𝑧 + 𝜎𝑥𝑧 𝛾𝑥𝑧 + 𝜎𝑥𝑦 𝛾𝑥𝑦 + 𝜎𝑧𝑦 𝛾𝑧𝑦 + 𝑚𝑥𝑥 𝜒𝑥𝑥 + 𝑚𝑦𝑦 𝜒𝑦𝑦 2 𝑉

+ 2𝑚𝑥𝑧 𝜒𝑥𝑧 + 2𝑚𝑥𝑦 𝜒𝑥𝑦 + 2𝑚𝑧𝑦 𝜒𝑧𝑦 )𝑑𝑉

(11)

Using the Eq. 9, Eq. 11 can be rewritten as follows:

𝒰=

1 2 2 2 2 2 2 ∫ [𝐶11 {𝜀𝑥𝑥 + 𝜀𝑦𝑦 + 𝜀𝑧𝑧 } + 2𝐶12 {𝜀𝑥𝑥 𝜀𝑦𝑦 + 𝜀𝑦𝑦 𝜀𝑧𝑧 + 𝜀𝑥𝑥 𝜀𝑧𝑧 } + 𝐶44 {𝛾𝑥𝑦 + 𝛾𝑥𝑧 + 𝛾𝑦𝑧 } 2 𝑉

2 2 + 𝑄𝜒 {𝜒𝑥𝑥 + 𝜒𝑦𝑦 + 2𝜒𝑥𝑧 𝜒𝑥𝑧 + 2𝜒𝑥𝑦 𝜒𝑥𝑦 + 2𝜒𝑧𝑦 𝜒𝑧𝑦 }] 𝑑𝑉

(12)

where 𝐸ℓ2 𝑄𝜒 (𝑧) = 1+𝜈

(13)

The kinetic energy (K) based on the studied FG porous square microplate is presented using the TSNDPT in the following form:

𝐾=

1 2 2 2 ∫ 𝜌(𝑈̇ + 𝑉̇ + 𝑊̇ )𝑑𝑉 2

(14)

𝑉

Using the Eq. (5), the kinetic energy (K) can be rewritten as follows:

11

𝐾=

1 ∫ 𝜌[{(𝑢̇ )2 + (𝑓1 )2 (𝑤̇𝑏,𝑥 )2 + (𝑓2 )2 (𝑤̇𝑠,𝑥 )2 − 2𝑓1 𝑢̇ 𝑤̇𝑏,𝑥 − 2𝑓2 𝑢̇ 𝑤̇𝑠,𝑥 + 2𝑓1 𝑓2 𝑤̇𝑏,𝑥 𝑤̇𝑠,𝑥 } 2 𝑉

+ {(𝑣̇ )2 + (𝑓1 )2 (𝑤̇𝑏,𝑦 )2 + (𝑓2 )2 (𝑤̇𝑠,𝑦 )2 − 2𝑓1 𝑣̇ 𝑤̇𝑏,𝑦 − 2𝑓2 𝑣̇ 𝑤̇𝑠,𝑦 + 2𝑓1 𝑓2 𝑤̇𝑏,𝑦 𝑤̇𝑠,𝑦 } + {(𝑤̇𝑏 )2 + (𝑤̇𝑠 )2 + (𝑓3 )2 (𝑤̇𝑧 )2 + 2𝑤̇𝑏 𝑤̇𝑠 + 2𝑓3 𝑤̇𝑏 𝑤̇𝑧 + 2𝑓3 𝑤̇𝑠 𝑤̇𝑧 }]𝑑𝑉

(15)

and, the energy equation (Π) can be given as follows: Π=𝒰−𝐾

(16)

2.4 FEM Formulation Based on the variational formulation given in Eq. (14) and Eq. (15), the FEM formulation of the problem is defined. A 𝐶 1 four node rectangular conforming element (CE) is developed. The displacement functions 𝑢(𝑥, 𝑦, 𝑡), 𝑣(𝑥, 𝑦, 𝑡), 𝑤𝑏 (𝑥, 𝑦, 𝑡), 𝑤𝑠 (𝑥, 𝑦, 𝑡) and 𝑤𝑧 (𝑥, 𝑦, 𝑡) can be described in the following form: 16

𝑢(𝑥, 𝑦, 𝑡) = ∑ 𝑢𝑗 𝜑𝑗 (𝑥, 𝑦)𝑒 𝑖𝜔𝑡 ,

(17𝑎)

𝑗=1 16

𝑣(𝑥, 𝑦, 𝑡) = ∑ 𝑣𝑗 𝜑𝑗 (𝑥, 𝑦)𝑒 𝑖𝜔𝑡 ,

(17𝑏)

𝑗=1 16

𝑤𝑏 (𝑥, 𝑦, 𝑡) = ∑ 𝑤𝑏 𝑗 𝜑𝑗 (𝑥, 𝑦)𝑒 𝑖𝜔𝑡 ,

(17𝑐)

𝑗=1 16

𝑤𝑠 (𝑥, 𝑦, 𝑡) = ∑ 𝑤𝑠 𝑗 𝜑𝑗 (𝑥, 𝑦)𝑒 𝑖𝜔𝑡 ,

(17𝑑)

𝑗=1 16

𝑤𝑧 (𝑥, 𝑦, 𝑡) = ∑ 𝑤𝑧 𝑗 𝜑𝑗 (𝑥, 𝑦)𝑒 𝑖𝜔𝑡 ,

(17𝑒)

𝑗=1

12

where 𝜔 is the natural frequency and 𝑖 2 = −1. The unknowns per node can be defined in the following form: 𝑢𝑗 = [𝑢, 𝑢,𝑥 , 𝑢,𝑦 , 𝑢,𝑥𝑦 ]

(18𝑎)

𝑣𝑗 = [𝑣, 𝑣,𝑥 , 𝑣,𝑦 , 𝑣,𝑥𝑦 ]

(18𝑏)

𝑤𝑏 𝑗 = [𝑤𝑏 , 𝑤𝑏,𝑥 , 𝑤𝑏,𝑦 , 𝑤𝑏,𝑥𝑦 ]

(18𝑐)

𝑤𝑠 𝑗 = [𝑤𝑠 , 𝑤𝑠,𝑥 , 𝑤𝑠,𝑦 , 𝑤𝑠,𝑥𝑦 ]

(18𝑑)

𝑤𝑧 𝑗 = [𝑤𝑧 , 𝑤𝑧,𝑥 , 𝑤𝑧,𝑦 , 𝑤𝑧,𝑥𝑦 ]

(18𝑒)

For a rectangular element as illustrated in Fig. 2, the interpolation functions (𝜑𝑗 ) for the jth node can be (𝜉, 𝜂) given in terms of the natural coordinates [89] as follows: 𝜑𝑗 =

1 2 2 (𝜉 + 𝜉𝑗 ) (𝜉𝜉𝑗 − 2)(𝜂 + 𝜂𝑗 ) (𝜂𝜂𝑗 − 2), 16

𝜑𝑗 =

1 2 2 𝜉𝑗 (𝜉 + 𝜉𝑗 ) (1 − 𝜉𝜉𝑗 )(𝜂 + 𝜂𝑗 ) (𝜂𝜂𝑗 − 2), 16

𝑗 = 2, 6, 10, 14

(19𝑏)

𝜑𝑗 =

1 2 2 𝜂𝑗 (𝜉 + 𝜉𝑗 ) (𝜉𝜉𝑗 − 2)(𝜂 + 𝜂𝑗 ) (1 − 𝜂𝜂𝑗 ), 16

𝑗 = 3, 7, 11, 15

(19𝑐)

𝜑𝑗 =

1 2 2 𝜉𝑗 𝜂𝑗 (𝜉 + 𝜉𝑗 ) (1 − 𝜉𝜉𝑗 )(𝜂 + 𝜂𝑗 ) (1 − 𝜂𝜂𝑗 ), 16

𝑗 = 1, 5, 9, 13

𝑗 = 4, 8, 12, 16

(19𝑎)

(19𝑑)

Figure 2 Around Here

The FEM model for the natural frequencies of FG porous microplate is obtained by using the Eq. (16) and (17) in the following form:

13

[𝐾11 ]

(

𝑇

[𝐾12 ]

[𝐾12 ] [𝐾22 ]

[𝐾13 ]𝑇

[𝐾23 ]𝑇

[𝐾13 ] [𝐾23 ] [𝐾33 ]

[𝐾14 ]𝑇

[𝐾24 ]𝑇

[𝐾34 ]𝑇

[𝐾14 ] [𝐾24 ] [𝐾34 ] [𝐾44 ]

[[𝐾15 ]𝑇

[𝐾25 ]𝑇

[𝐾35 ]𝑇

[𝐾45 ]𝑇

[𝑀11 ]

[𝐾15 ] [𝐾25 ] [𝐾35 ] [𝐾45 ] [𝐾55 ]]

[𝑀12 ]𝑇

[𝑀12 ] [𝑀22 ]

− 𝜔2 [𝑀13 ]𝑇

[𝑀23 ]𝑇

[𝑀13 ] [𝑀23 ] [𝑀33 ]

[𝑀14 ]𝑇

[𝑀24 ]𝑇

[𝑀34 ]𝑇

[𝑀14 ] [𝑀24 ] [𝑀34 ] [𝑀44 ]

[[𝑀15 ]𝑇

[𝑀25 ]𝑇

[𝑀35 ]𝑇

[𝑀45 ]𝑇

{𝑢𝑗 } [𝑀15 ] { 0} {𝑣𝑗 } [𝑀25 ] { 0} {𝑤 } 𝑏 [𝑀35 ] 𝑗 = {0} (20) { 0} [𝑀45 ] {𝑤𝑠 } 𝑗 {{0}} [𝑀55 ]] {𝑤 } { 𝑧 𝑗 })

where [𝐾𝑘𝑙 ] and [𝑀𝑘𝑙 ] are the stiffness and mass matrices, respectively. As the natural frequency ω is obtained, related vibration shapes of the FG porous microplate are plotted by using the Eq.20. The components of the stiffness and mass matrices are presented in the form of: 1 𝐾11 (𝑖, 𝑗) = ∫ [𝐴11 𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝐴44 𝜑𝑖,𝑦 𝜑𝑗,𝑦 + 𝐴𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 )] 𝑑𝑥𝑑𝑦 8

(21𝑎)

1 𝐾12 (𝑖, 𝑗) = ∫ [𝐴12 𝜑𝑖,𝑥 𝜑𝑗,𝑦 + 𝐴44 𝜑𝑖,𝑦 𝜑𝑗,𝑥 − 𝐴𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑦 + 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑥 )] 𝑑𝑥𝑑𝑦 8

(21𝑏)

𝐴𝑒

𝐴𝑒

𝐾13 (𝑖, 𝑗) = ∫ [−𝐵11 𝜑𝑖,𝑥 𝜑𝑗,𝑥𝑥 − 𝐵12 𝜑𝑖,𝑥 𝜑𝑗,𝑦𝑦 − 2𝐸44 𝜑𝑖,𝑦 𝜑𝑗,𝑥𝑦 𝐴𝑒

1 + 𝑇𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 8

(21𝑐)

𝐾14 (𝑖, 𝑗) = ∫ [−𝐶𝐶11 𝜑𝑖,𝑥 𝜑𝑗,𝑥𝑥 − 𝐶𝐶12 𝜑𝑖,𝑥 𝜑𝑗,𝑦𝑦 − 2𝐹44 𝜑𝑖,𝑦 𝜑𝑗,𝑥𝑦 𝐴𝑒

1 + 𝑊𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 8

(21𝑑)

14

1 𝐾15 (𝑖, 𝑗) = ∫ [𝐺12 𝜑𝑖,𝑥 𝜑𝑗 + 𝑁𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 8

(21𝑒)

1 𝐾22 (𝑖, 𝑗) = ∫ [𝐴11 𝜑𝑖,𝑦 𝜑𝑗,𝑦 + 𝐴44 𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝐴𝜒 (𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 )] 𝑑𝑥𝑑𝑦 8

(21𝑓)

𝐴𝑒

𝐴𝑒

𝐾23 (𝑖, 𝑗) = ∫ [−𝐵11 𝜑𝑖,𝑦 𝜑𝑗,𝑦𝑦 − 𝐵12 𝜑𝑖,𝑦 𝜑𝑗,𝑥𝑥 − 2𝐸44 𝜑𝑖,𝑥 𝜑𝑗,𝑥𝑦 𝐴𝑒

1 + 𝑇𝜒 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥 )] 𝑑𝑥𝑑𝑦 8

(21𝑔)

𝐾24 (𝑖, 𝑗) = ∫ [−𝐶𝐶11 𝜑𝑖,𝑦 𝜑𝑗,𝑦𝑦 − 𝐶𝐶12 𝜑𝑖,𝑦 𝜑𝑗,𝑥𝑥 − 2𝐹44 𝜑𝑖,𝑥 𝜑𝑗,𝑥𝑦 𝐴𝑒

1 + 𝑊𝜒 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥 )] 𝑑𝑥𝑑𝑦 8

(21ℎ)

1 𝐾25 (𝑖, 𝑗) = ∫ [𝐺12 𝜑𝑖,𝑦 𝜑𝑗 + 𝑁𝜒 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦 − 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥 )] 𝑑𝑥𝑑𝑦 8

(21𝑖)

𝐴𝑒

𝐾33 (𝑖, 𝑗) = ∫ [𝐷11 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 ) + 𝐷12 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 𝐴𝑒

1 + 𝐵44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 4𝐻44 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 + 𝐵𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 1 + 𝐵𝜒 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝐻𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑗) 8 𝐾34 (𝑖, 𝑗) = ∫ [𝐸11 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 ) + 𝐸12 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 𝐴𝑒

1 + 𝐶𝐶44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 4𝐺44 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 + 𝐶𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 1 + 𝐶𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝑃𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑘) 8

15

1 𝐾35 (𝑖, 𝑗) = ∫ [−𝐻12 (𝜑𝑖,𝑥𝑥 𝜑𝑗 + 𝜑𝑖,𝑦𝑦 𝜑𝑗 ) + 𝑁44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 𝐹𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 𝐴𝑒

1 + 𝐹𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝑅𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑙) 8 𝐾44 (𝑖, 𝑗) = ∫ [𝐹11 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 ) + 𝐹12 (𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 𝐴𝑒

1 + 𝐷44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 4𝐿44 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 + 𝐷𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 1 + 𝐷𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝐿𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑚) 8 1 𝐾45 (𝑖, 𝑗) = ∫ [−𝐿12 (𝜑𝑖,𝑥𝑥 𝜑𝑗 + 𝜑𝑖,𝑦𝑦 𝜑𝑗 ) + 𝑃44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 𝐺𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 𝐴𝑒

1 + 𝐺𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝑆𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑛) 8 1 𝐾55 (𝑖, 𝑗) = ∫ [𝐺11 𝜑𝑖 𝜑𝑗 + 𝑀44 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 ) + 𝐸𝜒 𝜑𝑖,𝑥𝑦 𝜑𝑗,𝑥𝑦 2 𝐴𝑒

1 + 𝐸𝜒 (𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑦𝑦 + 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑥𝑥 − 𝜑𝑖,𝑥𝑥 𝜑𝑗,𝑦𝑦 − 𝜑𝑖,𝑦𝑦 𝜑𝑗,𝑥𝑥 ) 8 1 + 𝑀𝜒 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )] 𝑑𝑥𝑑𝑦 (21𝑜) 8 𝑀11 (𝑖, 𝑗) = ∫ 𝐼0 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦

(21𝑝)

𝐴𝑒

𝑀13 (𝑖, 𝑗) = − ∫ 𝐼1 𝜑𝑖 𝜑𝑗,𝑥 𝑑𝑥𝑑𝑦

(21𝑞)

𝐴𝑒

𝑀14 (𝑖, 𝑗) = − ∫ 𝐽1 𝜑𝑖 𝜑𝑗,𝑥 𝑑𝑥𝑑𝑦

(21𝑟)

𝐴𝑒

16

𝑀22 (𝑖, 𝑗) = ∫ 𝐼0 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦

(21𝑠)

𝐴𝑒

𝑀23 (𝑖, 𝑗) = − ∫ 𝐼1 𝜑𝑖 𝜑𝑗,𝑦 𝑑𝑥𝑑𝑦

(21𝑡)

𝐴𝑒

𝑀24 (𝑖, 𝑗) = − ∫ 𝐽1 𝜑𝑖 𝜑𝑗,𝑦 𝑑𝑥𝑑𝑦

(21𝑢)

𝐴𝑒

𝑀33 (𝑖, 𝑗) = ∫ 𝐼0 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦 + ∫ 𝐼2 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )𝑑𝑥𝑑𝑦 𝐴𝑒

(21𝑣)

𝐴𝑒

𝑀34 (𝑖, 𝑗) = ∫ 𝐼0 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦 + ∫ 𝐽3 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )𝑑𝑥𝑑𝑦 𝐴𝑒

(21𝑤)

𝐴𝑒

𝑀35 (𝑖, 𝑗) = ∫ 𝐽2 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦

(21𝑥)

𝐴𝑒

𝑀44 (𝑖, 𝑗) = ∫ 𝐼0 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦 + ∫ 𝐾1 (𝜑𝑖,𝑥 𝜑𝑗,𝑥 + 𝜑𝑖,𝑦 𝜑𝑗,𝑦 )𝑑𝑥𝑑𝑦 𝐴𝑒

(21𝑦)

𝐴𝑒

𝑀45 (𝑖, 𝑗) = ∫ 𝐽2 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦

(21𝑧)

𝐴𝑒

𝑀55 (𝑖, 𝑗) = ∫ 𝐾2 𝜑𝑖 𝜑𝑗 𝑑𝑥𝑑𝑦

(21𝑎𝑎)

𝐴𝑒

𝑀12 (𝑖, 𝑗) = 0, 𝑀15 (𝑖, 𝑗) = 0 𝑎𝑛𝑑 𝑀25 (𝑖, 𝑗) = 0

(21𝑎𝑏)

where +

(𝐴11 , 𝐵11 , 𝐶𝐶11 , 𝐷11 , 𝐸11 , 𝐹11 , 𝐺11 ) = ∫

ℎ 2

ℎ − 2

2

𝐶11 (1, 𝑓1 , 𝑓2 , 𝑓1 2 , 𝑓1 𝑓2 , 𝑓2 2 , 𝑓3 ′ ) 𝑑𝑧

(22𝑎)

(𝐴12 , 𝐵12 , 𝐶𝐶12 , 𝐷12 , 𝐸12 , 𝐹12 , 𝐺12 , 𝐻12 , 𝐿12 ) +

=∫

ℎ 2

ℎ − 2

𝐶12 (1, 𝑓1 , 𝑓2 , 𝑓1 2 , 𝑓1 𝑓2 , 𝑓2 2 , 𝑓3 ′ , 𝑓1 𝑓3 ′ , 𝑓2 𝑓3 ′ )𝑑𝑧

(22𝑏)

17

(𝐴44 , 𝐵44 , 𝐶𝐶44 , 𝐷44 , 𝐸44 , 𝐹44 , 𝐺44 , 𝐻44 , 𝐿44 , 𝑀44 , 𝑁44 , 𝑃44 ) +

=∫

ℎ 2

ℎ − 2

𝐶44 [1, (1 − 𝑓1 ′ )2 , (1 − 𝑓1 ′ )(1 − 𝑓2 ′ ), (1 − 𝑓2 ′ )2 , 𝑓1 , 𝑓2 ,

𝑓1 𝑓2 , 𝑓1 2 , 𝑓2 2 , 𝑓3 2 , (1 − 𝑓1 ′ )𝑓3 , (1 − 𝑓2 ′ )𝑓3 ]𝑑𝑧

(22𝑐)

(𝐴𝜒 , 𝐵𝜒 , 𝐶𝜒 , 𝐷𝜒 , 𝐸𝜒 , 𝐹𝜒 , 𝐺𝜒 , 𝐻𝜒 , 𝐿𝜒 , 𝑀𝜒 , 𝑁𝜒 , 𝑃𝜒 , 𝑅𝜒 , 𝑆𝜒 , 𝑇𝜒 , 𝑊𝜒 ) +ℎ/2

=∫

𝑄𝜒 [1, (1 + 𝑓1 ′ )2 , (1 + 𝑓1 ′ )(1 + 𝑓2 ′ ), (1 + 𝑓2 ′ )2 , 𝑓3 2 , (1 + 𝑓1 ′ )𝑓3 ,

−ℎ/2 2

2

2

(1 + 𝑓2 ′ )𝑓3 , 𝑓1 ′′ , 𝑓2 ′′ , 𝑓3 ′ , 𝑓3 ′ , 𝑓1 ′′ 𝑓2 ′′ , 𝑓1 ′′ 𝑓3 ′ , 𝑓2 ′′ 𝑓3 ′ , 𝑓1 ′′ , 𝑓2 ′′ ]𝑑𝑧 +

(𝐼0 , 𝐼1 , 𝐼2 , 𝐽1 , 𝐽2 , 𝐽3 , 𝐾1 , 𝐾2 ) = ∫

ℎ 2

ℎ − 2

𝜌 (1, 𝑓1 , 𝑓1 2 , 𝑓2 , 𝑓3 , 𝑓1 𝑓2 , 𝑓2 2 , 𝑓3 2 )𝑑𝑧

(22𝑑)

(22𝑒)

The BCs for the studied problems are described below: Simply Support (S): 𝑣 = 𝑤𝑏 = 𝑤𝑏,𝑦 = 𝑤𝑠 = 𝑤𝑠,𝑦 = 𝑤𝑧 = 𝑤𝑧,𝑦 = 0 𝑎𝑡 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 𝑎 𝑢 = 𝑤𝑏 = 𝑤𝑏,𝑥 = 𝑤𝑠 = 𝑤𝑠,𝑥 = 𝑤𝑧 = 𝑤𝑧,𝑥 = 0 𝑎𝑡 𝑦 = 0 𝑎𝑛𝑑 𝑦 = 𝑎 Clamped (C): 𝑢 = 𝑣 = 𝑤𝑏 = 𝑤𝑏,𝑥 = 𝑤𝑏,𝑦 = 𝑤𝑠 = 𝑤𝑠,𝑥 = 𝑤𝑠,𝑦 = 𝑤𝑧 = 𝑤𝑧,𝑥 = 𝑤𝑧,𝑦 = 0 𝑎𝑡 𝑥 = 0, 𝑎 𝑎𝑛𝑑 𝑦 = 0, 𝑎

3. NUMERICAL RESULTS Section 3 is dedicated to investigate the natural frequencies of FG porous square microplates with respect to variation of BC, gradient index, thickness to MLSP, porosity volume fraction and aspect ratio. Moreover, the mode shapes are also illustrated. Unless otherwise stated, the followings are used in the studied examples [67,86]:

18

SiC (Ceramic): 𝐸1 = 𝐸𝑐 = 427 𝐺𝑃𝑎, 𝜈1 = 𝜈𝑐 = 0.17, 𝜌𝑐 = 3100 𝑘𝑔/𝑚3 and ℓ1 = ℓ𝑐 = 22.5𝜇𝑚, Al (Metal): 𝐸2 = 𝐸𝑚 = 70 𝐺𝑃𝑎 , 𝜈2 = 𝜈𝑚 = 0.3 a, 𝜌𝑚 = 2702 𝑘𝑔/𝑚3 and ℓ2 = ℓ𝑚 = 15𝜇𝑚. The total thickness of the microplate is 15μm. In the representation of the numerical results by using ℓ1 = ℓ2 which means that the MLSPs of the two constitutes are made equal, MLSP is set to 15𝜇𝑚. The DFF is presented within the study in the form of:

𝜆=

𝜔𝑎2 𝜌𝑐 √ ℎ 𝐸𝑐

(23)

3.1 Convergence and Verification Studies The convergence and verification analysis are performed by employing the BCs which are used in the extensive analysis for the developed FEM code. These BCs are SSSS, SCSC, CCCC and CFFF. The computed results along with the results from open literature including the Quasi-3D theories are given in Tables 1-5. In Table 1, the DFFs of SSSS 𝐴𝑙/𝐴𝑙2 𝑂3 FG square plates for various aspect ratios and gradient indexes are calculated based on the various uniform mesh sizes (4x4, 6x6, 8x8 and 10x10), gauss rules (2x2, 3x3 and 4x4) and compared with the computed results from previous studies. Excellent agreement is established while the uniform mesh size is set to 6x6. It is concluded that 8x8 mesh size can produce acceptable results for the natural frequency analysis of FG porous square microplates. Moreover, the DFFs could not be calculated by employing 2x2 Gauss rule. The numerical results obtained by 3x3 and 4x4 Gauss rules are very close the results

19

of previous studies. For the sake of accuracy, 4x4 Gauss quadrature rule is used for the evaluation of the [𝐾𝑘𝑙 ] and [𝑀𝑘𝑙 ].

Table 1 Around Here

The DFFs are investigated for the SCSC 𝐴𝑙/𝐴𝑙2 𝑂3 FG square plates for various aspect ratios and gradient indexes based on the results obtained by employing different shear deformable plate theories and computed results are presented in Table 2. Again, computed results show are good agreement along with the previous calculated results. However, the mesh size should be set to 10x10 for the problems with SCSC BC to obtained more reliable results.

Table 2 Around Here

The natural frequencies of the CFFF 𝐴𝑙/𝐴𝑙2 𝑂3 FG square plates for various aspect ratios and gradient indexes are also studied and calculated DFFs are compared with the results given in [90]. It is worth noting that by employing the 8x8 uniform mesh size, good agreement can be captured as shown in Table 3. It is important to note that in the open literature, the number of studies employing the CFFF BC is very limited for FG plates.

Table 3 Around Here

20

In Table 4, the FEM code is verified by analyzing the free vibration problem of SSSS 𝐴𝑙/𝐴𝑙2 𝑂3 FG microplates for various thickness to MLSP ratios, aspect ratios and gradient indexes. In this analysis, the non-conforming element (NCE) is employed only for the in-plane displacement functions as well. The transverse displacement functions are presented with the conforming elements. It should be noted that for the presentation of the shape functions based on the inplane displacements, the NCE can be enough and the most suitable choice. Moreover, employing NCE for the in-plane displacements decreases the CPU time, if it is compared with CPU time obtained by employing the CE. In addition, more accurate results may be obtained by using NCE for the in-plane displacement functions. The computed results are also compared with numerical calculations based on the RPT and Quasi-3D theories and IGA. It is observed that with the 8x8 uniform mesh size, the computed results show excellent agreement with the ones given in [75]. Moreover, the difference between the results obtained by employing NCE and CE for the in-plane displacement functions is negligible for the studied problem.

Table 4 Around Here

And finally, the free vibration of CCCC 𝐴𝑙/𝐴𝑙2 𝑂3 FG square microplates for various aspect ratios and mode numbers (first six) is analyzed by employing various uniform mesh sizes (6x6, 8x8, 9x9, 10x10 and 11x11) and obtained results are presented in Table 5. The comparisons are carried out for the numerical results which are obtained by employing different Quasi-3D plate theories and Mori-Tanaka homogenization scheme. The computed results in terms of the DFFs based on 10x10 uniform mesh size are in excellent agreement with the solutions based on the shear and normal deformable plate theories. Moreover, for the higher order frequencies, 10x10 uniform mesh size still produces agreed results with the previous studies. The uniform mesh 21

size is set to 8x8 for SSSS and CFFF BCs. However, to get more accurate results, it is set to 10x10 for SCSC and CCCC BCs for extensive analysis.

Table 5 Around Here

3.2 Natural frequencies of FG porous square microplates with variable MLSP The free vibration behavior of FG porous square microplates is analyzed based on the TSNDPT and MCST by employing FEM formulation. In this section, the effects of the gradient index, aspect ratio, porosity volume fraction, BC, variable MLSP and thickness to MLSP are investigated for analysis natural frequencies. The computed DFFs are presented through the Tables 6-9. In Table 6, the investigations are done for the SSSS FG porous microplates and the computed results show that the DFF increases, with the increasing of the length/thickness ratio, as expected. Besides, it is clear that the DFFs decreases for all problems with the increasing of the gradient index. It should be noted that the small size effect becomes more pronounced with the increasing of the aspect ratio. And the most important output of this example can be the computed results with variable MLSP. It is found that the numerical results obtained by employing ℓ1 ≠ ℓ2 (variable MLSP) are always higher than the ones obtained by using ℓ1 = ℓ2 (constant MLSP). Since the MLSP of the ceramic phase (ℓ1 = 22.5𝜇𝑚) is greater than the MLSP of the metal phase (ℓ2 = 15𝜇𝑚), the stiffness of the microplate with variable MLSP always becomes higher than the one of microplate with constant MLSP (ℓ1 = ℓ2 = 15𝜇𝑚). By employing variable MLSP approach in the structure, the higher values of effective MLSP which makes the structure stiffer than the case with constant MLSP, can be always obtained. While

22

the porosity volume fraction increases for the macro FG plates, the DFFs increase. However, with the increasing of the small size effect, this affect is causing a decrement in the DFFs.

Table 6 Around Here

The effects of the thickness to MLSP, variable MLSP, length/thickness ratio, porosity volume fraction and gradient index for the DFFs of the FG imperfect square microplates with SCSC, CCCC and CFFF BCs presented in Tables 7-9 can be seen explicitly the same with the obtained behavior of SSSS BC. Moreover, the effect of the gradient index tends to be more pronounced with the inclusion of the variable MLSP than the inclusion of the constant MLSP. In addition, this effect is more visible for the microplates with CCCC BC and the least affected one is the CFFF FG microplate. A general observation from the Tables 6-9 indicates that the involvement of couple stress with variable MLSP increases the stiffness of the FG microplate and yields higher DFFs that the constant MLSP. Again, the most affected microplate in terms of stiffness increasing is the microplate with CCCC BC. As can be observed, the effect of the boundary conditions on the DFF changes depending on the constraints on the edges. The order of the DFF from highest to lowest is CCCC, SCSC, SSSS and CFFF. All the numerical results can be used as benchmark for the studies to de done in future.

Table 7 Around Here

Table 8 Around Here

23

Table 9 Around Here

In Figures 3-6, the effects of the aspect ratio of the FG porous microplates on the DFFs are examined for various thickness to MLSPs and BCs. The computed results are plotted by using not only for constant MLSP but also variable MLSP including the variable MLSPs. As expected, with an increment in the aspect ratio leads to an increment in the DFFs. Nevertheless, the influence of the aspect ratio decreases as the size effect decreases in all studied problems. For the thick microplates (𝑎/ℎ ≤ 10), the difference between the results obtained with respect to various thickness to MLSPs is smaller than the results computed for thin microplates. The influence of the 𝑎/ℎ variation on the DFFs is more pronounced for thick microplates. And, the least affected microplate in terms of the variation of the DFFs based on the aspect ratio changing is found as the microplate with CFFF BC. Besides, as can be detected, the most affected one is the microplate with CCCC BC. The difference between the DFFs obtained based on the various thickness to MLSP is smaller than the those of obtained by using variable MLSP. And finally, for the thick microplates, the influence of the variable MLSP regarding to the changing of the thickness to MLSP on the DFFs is more detectable than the influence of the constant MLSP for all BCs.

Figure 3 Around Here

Figure 4 Around Here

Figure 5 Around Here 24

Figure 6 Around Here The effects of the variation of the thickness to MLSP ratio and gradient index are illustrated in Figs. 7-10 for various BCs. As can be seen, increasing the gradient index leads to a decrement in the stiffness of the microplate, eventually the DFF decrease for all cases. However, the decreasing speed of the DFFs decreases as the gradient index increases. It is found that with the increasing of the small size effect, the influence of the materials gradient index for the decreasing of the DFFs increases. In addition, this effect is more detectable for variable MLSP that the one of constant MLSP.

Figure 7 Around Here

Figure 8 Around Here

Figure 9 Around Here

Figure 10 Around Here

Several transverse vibration modes shapes of the FG porous microplates with variable MLSP are plotted in Figures 7-10 for various BCs. For the sake of better presentation, the modes shapes are plotted with 17x17 uniform mesh size.

25

Figure 11 Around Here

Figure 12 Around Here

Figure 13 Around Here

Figure 14 Around Here

In Figure 15, the 2D contours of the third transverse free vibration mode shapes of FG porous microplates are plotted employing the SSSS, SCSC and CCCC BCs. For better illustration 17x17 uniform mesh size is employed. If the 2D contours of SSSS and SCSC microplates are compared, it is observed that the transverse deflections are mode concentrated on the middle of the SCSC microplate. However, for SSSS BC, these deflections are more evenly distributed along the left and right edges. For all clamped edges, the transverse deflections become concentrated on the diagonal (form the left to the right bottom) of the microplate.

Figure 15 Around Here

4. CONCLUSIONS By employing the MCST, the free vibration behavior of shear and normal deformable FG porous microplates is investigated for arbitrary BCs accompanying the influence of the variable MLSP. Based on a TSNDPT, the total potential energy is formulated and then the FEM model 26

is developed. After introducing the stiffness and mass matrices, the DFFs and modes shapes are obtained to investigate the effects of the aspect ratio, gradient index, normal deformation, BC, porosity volume fraction and variable MLSP. The important findings of this study can be summarized as follows: 

As the aspect ratio increases, the effect of the small size becomes more pronounced for all BCs.



The computed DFFs employing the variable MLSP are greater than the results obtained by constant MLSP.



As the porosity volume fraction increases for the macro FG plates, the DFFs increase. However, with the strong size effect, this affect is resulting a decrement in the DFFs.



The effect of the gradient index tends to be more pronounced with the inclusion of the variable MLSP than the inclusion of the constant MLSP



For the thick microplates (𝑎/ℎ ≤ 10), the influence of the variable MLSP regarding to the variation of the thickness to MLSP is more detectable than the influence of the constant MLSP for all BCs.

Since the difference between the results obtained by constant and variable MLSPs cannot be ignored, the determination of the MLSP is crucial to obtain reliable results. It is clear that regarding to the determination of the MLSP exactly for MCST, there is no open source in the literature. In most of the studies it is assumed as a constant independently for the type of the material.

27

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35

Table 1. Convergence and validation studies on the DFFs (𝜆 =

𝜔𝑎2 ℎ

𝜌

𝑐 √𝐸 ) of SSSS 𝐴𝑙/𝐴𝑙2 𝑂3 𝑐

FG square plates for various gradient indexes and aspect ratios. BC

𝑎/ℎ

Theory

Number of Element 4x4

6x6 Quasi-3D (Present) 8x8 5 10x10

Gauss Quadrature 2x2 3x3 4x4 2x2 3x3 4x4 2x2 3x3 4x4 2x2 3x3 4x4

FSDT - Meshless [90] RPT - Levy [92] RPT- IGA [75] Quasi-3D - Navier [91] Quasi-3D - IGA [75]

SSSS

4x4

6x6 Quasi-3D (Present) 10

8x8

10x10 FSDT - Meshless [90] RPT - Levy [92] Quasi-3D - Navier [91]

2x2 3x3 4x4 2x2 3x3 4x4 2x2 3x3 4x4 2x2 3x3 4x4

pz = 0

pz = 1

pz = 2

pz = 5

pz = 10

0 5.2988 5.2990 0 5.3034 5.3034 0 5.3040 5.3040 0 5.3041 5.3041 5.2791 5.2813 5.2813 5.3025 5.3090 0 5.7741 5.7742 0 5.7766 5.7766 0 5.7769 5.7769 0 5.7769 5.7769 5.7619 5.7694 5.7770

0 4.1426 4.1445 0 4.1463 4.1479 0 4.1468 4.1485 0 4.1470 4.1487 4.0746 4.0781 4.0781 4.1000 4.1524 0 4.4831 4.4850 0 4.4856 4.4874 0 4.4860 4.4878 0 4.4862 4.4879 4.4106 4.4192 4.4260

0 3.7615 3.7638 0 3.7650 3.7672 0 3.7656 3.7677 0 3.7658 3.7679 3.6923 3.6805 0 4.0954 4.0978 0 4.0979 4.1002 0 4.0984 4.1006 0 4.0985 4.1008 4.0059 4.0090 -

0 3.4566 3.4593 0 3.4603 3.4628 0 3.4609 3.4634 0 3.4611 3.4636 3.4461 3.3938 0 3.8383 3.8413 0 3.8408 3.8436 0 3.8413 3.8441 0 3.8414 3.8442 3.7806 3.7682 -

0 3.2885 3.2913 0 3.2921 3.2948 0 3.2927 3.2954 0 3.2928 3.2955 3.3062 3.2514 3.2519 3.2650 3.3126 0 3.6766 3.6797 0 3.6789 3.6819 0 3.6793 3.6823 0 3.6794 3.6824 3.6510 3.6368 3.6420

Table 2. Convergence and validation studies on the DFFs (𝜆 =

𝜔𝑎2 ℎ

𝜌

𝑐 √𝐸 ) of SCSC 𝐴𝑙/𝐴𝑙2 𝑂3 𝑐

FG square plates for various gradient indexes and aspect ratios. BC

𝑎/ℎ

Theory

Quasi-3D (Present) 5

Number of Element 4x4 6x6 8x8 10x10 12x12

FSDT - Meshless [90] RPT - Levy [92] HSDT - Levy [59]

SCSC

Quasi-3D (Present) 10

4x4 6x6 8x8 10x10 12x12

FSDT - Meshless [90] RPT - Levy [92]

20

Quasi-3D (Present)

RPT - Levy [92]

4x4 6x6 8x8 10x10 12x12

pz = 0

pz = 1

pz = 2

pz = 5

pz = 10

7.2459 7.2561 7.2486 7.2419 7.2374 6.7722 7.1104 7.110 8.3571 8.3741 8.3608 8.3465 8.3351 8.0849 8.2258 8.7424 8.7642 8.7497 8.7330 8.7188 8.6155

5.7505 5.7517 5.7398 5.7302 5.7235 5.3069 5.5509 5.552 6.5581 6.5617 6.5435 6.5270 6.5143 6.2321 6.3264 6.8321 6.8381 6.8184 6.7996 6.7846 6.5898

5.2143 5.2092 5.1947 5.1836 5.1760 4.8052 4.9920 4.993 6.0063 6.0020 5.9805 5.9621 5.9483 5.6574 5.7300 6.2814 6.2789 6.2555 6.2348 6.2187 5.9856

4.6896 4.6853 4.6736 4.6647 4.6586 4.4133 4.5128 4.514 5.5743 5.5719 5.5536 5.5379 5.5260 5.2987 5.3422 5.9027 5.9027 5.8825 5.8643 5.8500 5.6491

4.4145 4.4139 4.4056 4.3990 4.3945 4.1864 4.2845 4.286 5.3005 5.3037 5.2903 5.2781 5.2686 5.0889 5.1367 5.6359 5.6424 5.6276 5.6132 5.6016 5.4597

37

Table 3. Convergence and validation studies on the DFFs (𝜆 =

𝜔𝑎2 ℎ

𝜌

𝑐 √𝐸 ) of CFFF 𝐴𝑙/𝐴𝑙2 𝑂3 𝑐

FG square plates for various BCs, gradient indexes and aspect ratios. BC

𝑎/ℎ

5

Theory

Quasi-3D (Present)

Number of Element 4x4 6x6 8x8 10x10

FSDT - Meshless [90] Quasi-3D (Present) CFFF

10

4x4 6x6 8x8 10x10

FSDT - Meshless [90] FSDT - Ritz [93]

100

Quasi-3D (Present) CPT - Ritz [93]

4x4 6x6 8x8 10x10

pz = 0

pz = 1

pz = 2

pz = 5

pz = 10

1.0178 1.0263 1.0279 1.0283 1.0106 1.0412 1.0495 1.0506 1.0503 1.0361 1.0383 1.0495 1.0580 1.0590 1.0586 1.0567

0.7933 0.7982 0.7986 0.7982 0.7738 0.8100 0.8148 0.8148 0.8140 0.7910 0.8160 0.8208 0.8207 0.8199 -

0.7268 0.7302 0.7300 0.7294 0.7028 0.7436 0.7470 0.7464 0.7454 0.7190 0.7496 0.7531 0.7525 0.7514 -

0.6815 0.6849 0.6849 0.6844 0.6631 0.7013 0.7048 0.7044 0.7036 0.6811 0.7084 0.7120 0.7116 0.7107 -

0.6517 0.6558 0.6563 0.6560 0.6403 0.6718 0.6760 0.6761 0.6756 0.6591 0.6790 0.6833 0.6834 0.6828 -

38

Table 4. Verification studies based on the DFFs (𝜆 =

𝜔𝑎2 ℎ

𝜌

𝑐 √𝐸 ) of SSSS 𝐴𝑙/𝐴𝑙2 𝑂3 FG microplates for various thickness to MLSP ratios, gradient 𝑐

indexes and aspect ratios. pz = 0 𝑎/ℎ

pz = 1

RPT IGA [75]

Quasi-3D - IGA [75]

NCE

CE

0

5.2813

5.3090

5.3040

0.2

5.7496

5.7622

0.4

6.9667

1

pz = 10

RPT IGA [75]

Quasi-3D - IGA [75]

RPT IGA [75]

Quasi-3D - IGA [75]

NCE

CE

NCE

CE

5.3041

4.0781

4.1521

4.1486

4.1487

3.2519

3.3126

3.2954

3.2955

5.7690

5.7690

4.4959

4.5542

4.5593

4.5592

3.5312

3.5740

3.5773

3.5773

6.9438

6.9785

6.9785

5.5620

5.5865

5.6123

5.6123

4.2584

4.2627

4.2987

4.2987

9.9791

9.9791

10.0867

10.0887

8.3019

8.3019

8.3891

8.3840

5.9073

5.9073

5.9697

5.9661

0

5.9199

5.9235

5.9219

5.9219

4.5228

4.5919

4.5906

4.5906

3.7623

3.8129

3.8079

3.8079

0.2

6.4009

6.4030

6.4027

6.4027

4.9556

5.0179

5.0175

5.0174

4.0299

4.0761

4.0733

4.0732

0.4

7.6646

7.6633

7.6662

7.6662

6.0714

6.1203

6.1223

6.1222

4.7428

4.7794

4.7804

4.7803

1

13.5330

13.5202

13.5350

13.5350

11.0882

11.1082

11.1186

11.1185

8.1384

8.1510

8.1587

8.1587

0

5.9712

5.9723

5.9713

5.9713

4.5579

4.6263

4.6254

4.6254

3.8058

3.8533

3.8515

3.8515

0.2

6.4534

6.4544

6.4535

6.4535

4.9922

5.0546

5.0538

5.0537

4.0724

4.1168

4.1151

4.1151

0.4

7.7215

7.7222

7.7216

7.7216

6.1124

6.1635

6.1629

6.1628

4.7837

4.8215

4.8202

4.8202

1

13.6178

13.6177

13.6179

13.6179

11.1554

11.1832

11.1832

11.1832

8.1842

8.2060

8.2055

8.2055

ℓ/ℎ

Present Quasi-3D

Present Quasi-3D

Present Quasi-3D

5

20

100

39

Table 5. Verification studies based on the first six dimensionless frequencies (𝜆 =

𝜔𝑎2 ℎ

𝜌

𝑚 √𝐸 ) of CCCC 𝐴𝑙/𝐴𝑙2 𝑂3 FG square microplates (Mori𝑚

Tanaka scheme, ℓ/ℎ = 0.2, pz = 1). 𝑎/ℎ

Theory

Number of Element

Mode 1

2

3

4

5

6

Quasi-3D (Zenkour) – IGA [75]

12.7213

22.6661

22.6661

27.9021

27.9021

31.1450

Quasi-3D – IGA [75]

13.1029

22.9300

22.9300

27.8791

27.8791

31.3005

6x6

12.9628

23.5122

23.5122

28.7549

28.7549

31.7545

8x8

12.9402

23.4936

23.4936

28.5800

28.5800

31.8604

9x9

12.9302

23.4836

23.4836

28.5157

28.5157

31.8821

10x10

12.9149

23.4624

23.4624

28.4473

28.4473

31.8772

11x11

12.9150

23.4673

23.4673

28.4183

28.4183

31.9005

Quasi-3D (Zenkour) – IGA [75]

15.2379

28.9665

28.9665

41.2211

47.8649

47.9221

Quasi-3D – IGA [Thuc]

15.4413

29.3267

29.3267

41.5955

48.2657

48.4504

6x6

15.3002

29.6001

29.6001

41.4473

49.3404

49.8446

8x8

15.2543

29.5428

29.5428

41.5981

49.0850

49.6124

9x9

15.2324

29.5156

29.5156

41.6179

49.0033

49.5411

10x10

15.2050

29.4671

29.4671

41.5995

48.9127

49.4584

11x11

15.1966

29.4574

29.4574

41.6169

48.8865

49.4388

5 Present Quasi-3D

10 Present Quasi-3D

40

Table 6. DFFs of the SSSS FG porous microplates for various gradient indexes, aspect ratios, porosity volume fractions and thickness to MLSPs 𝑎/ℎ

𝑀𝐿𝑆𝑃

ℓ1 = ℓ2

5

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

10

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

20

ℓ1 ≠ ℓ2

𝛼0 = 0 pz

ℓ2 /ℎ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 5.1916 5.7270 7.0917 8.9061 10.4253 10.6117 5.1916 6.3321 8.9061 10.5147 10.8286 11.2091 5.6112 6.1564 7.5592 9.4445 11.5781 13.8449 5.6112 6.7763 9.4445 12.6993 16.1880 19.7842 5.7377 6.2863 7.7009 9.6063 11.7664 14.0642 5.7377 6.9110 9.6063 12.9026 16.4420 20.0964

1 3.8643 4.2891 5.3630 6.7799 8.0582 8.2014 3.8643 4.5830 6.2563 8.0525 8.2455 8.4795 4.1667 4.6026 5.7137 7.1933 8.8580 10.6204 4.1667 4.9024 6.6365 8.7959 11.1347 13.5586 4.2576 4.6972 5.8200 7.3179 9.0056 10.7945 4.2576 4.9989 6.7508 8.9375 11.3094 13.7711

2 3.4291 3.8010 4.7415 5.9832 7.0654 7.1907 3.4291 4.0143 5.3917 7.0416 7.1983 7.3883 3.7243 4.1010 5.0651 6.3540 7.8080 9.3499 3.7243 4.3133 5.7257 7.5089 9.4545 11.4792 3.8146 4.1929 5.1639 6.4658 7.9374 9.5001 3.8146 4.4047 5.8263 7.6275 9.5970 11.6496

𝛼0 = 0.05 pz 5 3.0735 3.3754 4.1365 5.1440 5.8710 5.9749 3.0735 3.5016 4.5161 5.7978 5.9468 6.0876 3.4122 3.7031 4.4585 5.4863 6.6610 7.9174 3.4122 3.8192 4.8279 6.1434 7.6084 9.1515 3.5208 3.8074 4.5591 5.5903 6.7745 8.0445 3.5208 3.9197 4.9231 6.2448 7.7232 9.2840

0 5.2336 5.7296 7.0074 8.7244 10.5392 10.7126 5.2336 6.3112 8.7733 10.6293 10.9250 11.2832 5.6533 6.1579 7.4697 9.2509 11.2808 13.4463 5.6533 6.7531 9.3018 12.4366 15.8102 19.2950 5.7797 6.2873 7.6096 9.4092 11.4633 13.6576 5.7797 6.8869 9.4606 12.6342 16.0556 19.5952

1 3.8655 4.2818 5.3371 6.7331 8.0615 8.2019 3.8655 4.5723 6.2227 8.0567 8.2464 8.4762 4.1689 4.5955 5.6863 7.1427 8.7841 10.5238 4.1689 4.8916 6.6003 8.7332 11.0460 13.4446 4.2601 4.6903 5.7920 7.2660 8.9298 10.6950 4.2601 4.9880 6.7138 8.8731 11.2181 13.6535

2 3.4280 3.7976 4.7329 5.9686 7.0622 7.1867 3.4280 4.0105 5.3823 7.0388 7.1947 7.3837 3.7240 4.0980 5.0563 6.3384 7.7855 9.3205 3.7240 4.3098 5.7158 7.4920 9.4307 11.4485 3.8146 4.1902 5.1550 6.4498 7.9143 9.4699 3.8146 4.4014 5.8162 7.6102 9.5725 11.6181

𝛼0 = 0.1 pz 5 3.0733 3.3752 4.1360 5.1432 5.8704 5.9742 3.0733 3.5014 4.5157 5.7971 5.9462 6.0870 3.4122 3.7029 4.4581 5.4855 6.6598 7.9159 3.4122 3.8191 4.8275 6.1427 7.6074 9.1501 3.5209 3.8073 4.5587 5.5895 6.7733 8.0429 3.5209 3.9196 4.9227 6.2441 7.7222 9.2826

0 5.2806 5.7379 6.9285 8.5465 10.3878 10.8248 5.2806 6.2956 8.6443 10.7548 11.0328 11.3691 5.7008 6.1656 7.3863 9.0620 10.9859 13.0480 5.7008 6.7358 9.1638 12.1764 15.4330 18.8045 5.8272 6.2946 7.5247 9.2169 11.1631 13.2516 5.8272 6.8689 9.3198 12.3686 15.6700 19.0932

1 3.8666 4.2746 5.3116 6.6870 8.0650 8.2026 3.8666 4.5618 6.1897 8.0611 8.2475 8.4733 4.1711 4.5887 5.6593 7.0928 8.7112 10.4283 4.1711 4.8810 6.5648 8.6715 10.9586 13.3322 4.2627 4.6836 5.7646 7.2150 8.8549 10.5967 4.2627 4.9775 6.6775 8.8097 11.1282 13.5376

2 3.4268 3.7942 4.7244 5.9542 7.0589 7.1826 3.4268 4.0067 5.3730 7.0360 7.1911 7.3791 3.7237 4.0951 5.0475 6.3229 7.7632 9.2915 3.7237 4.3064 5.7060 7.4754 9.4072 11.4183 3.8146 4.1874 5.1462 6.4341 7.8914 9.4400 3.8146 4.3981 5.8063 7.5932 9.5484 11.5871

5 3.0732 3.3749 4.1355 5.1425 5.8698 5.9736 3.0732 3.5011 4.5153 5.7964 5.9456 6.0864 3.4122 3.7028 4.4577 5.4847 6.6586 7.9143 3.4122 3.8190 4.8271 6.1420 7.6063 9.1487 3.5210 3.8073 4.5583 5.5887 6.7721 8.0413 3.5210 3.9196 4.9224 6.2434 7.7211 9.2812

41

Table 7. DFFs of the SCSC FG porous microplates for various gradient indexes, aspect ratios, porosity volume fractions and thickness to MLSPs 𝑎/ℎ

𝑀𝐿𝑆𝑃

ℓ1 = ℓ2

5

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

10

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

20

ℓ1 ≠ ℓ2

𝛼0 = 0 pz

ℓ2 /ℎ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 7.0876 7.8985 9.9086 11.7282 12.3982 13.1483 7.0876 8.7955 11.7282 12.7636 13.9718 15.3258 8.0543 8.8590 10.9161 13.6682 16.7746 20.0691 8.0543 9.7696 13.6682 18.4048 23.4564 24.4460 8.3797 9.1834 11.2542 14.0414 17.2000 20.5594 8.3797 10.0981 14.0414 18.8612 24.0353 29.3766

1 5.3463 5.9846 7.5589 9.0807 9.5990 10.1791 5.3463 6.4258 8.8686 9.5758 10.3501 11.2205 6.0487 6.6882 8.3112 10.4666 12.8883 15.4496 6.0487 7.1292 9.6616 12.8068 16.2080 18.4564 6.2843 6.9251 8.5630 10.7502 13.2163 15.8314 6.2843 7.3652 9.9222 13.1169 16.5843 20.1844

2 4.7345 5.3043 6.7020 7.9723 8.4267 8.9353 4.7345 5.6355 7.6779 8.3268 8.9642 9.6807 5.4203 5.9767 7.3921 9.2769 11.3988 13.6461 5.4203 6.2928 8.3670 10.9736 13.8116 16.0805 5.6572 6.2090 7.6265 9.5289 11.6810 13.9678 5.6572 6.5183 8.5942 11.2270 14.1079 17.1121

𝛼0 = 0.05 pz 5 4.1540 4.6460 5.8236 6.6401 7.0190 7.4433 4.1540 4.8590 6.4237 6.8453 7.3297 7.8742 4.9286 5.3732 6.5068 8.0291 9.7575 11.6003 4.9286 5.5545 7.0655 9.0075 11.1577 13.2539 5.2170 5.6404 6.7472 8.2621 10.0007 11.8651 5.2170 5.8068 7.2825 9.2213 11.3892 13.6784

0 7.1468 7.8978 9.7819 11.7998 12.4313 13.1368 7.1468 8.7608 11.8155 12.8030 13.9538 15.2450 8.1128 8.8576 10.7816 13.3828 16.3392 19.4876 8.1128 9.7320 13.4571 18.0203 22.9194 24.5755 8.4371 9.1810 11.1170 13.7499 16.7539 19.9622 8.4371 10.0592 13.8251 18.4660 23.4678 28.6415

1 5.3479 5.9746 7.5240 9.0738 9.5835 10.1538 5.3479 6.4114 8.8239 9.5640 10.3269 11.1843 6.0519 6.6784 8.2724 10.3949 12.7837 15.3128 6.0519 7.1139 9.6104 12.7179 16.0824 18.4429 6.2883 6.9154 8.5228 10.6752 13.1064 15.6871 6.2883 7.3498 9.8688 13.0235 16.4518 20.0135

2 4.7324 5.2994 6.6907 7.9659 8.4178 8.9236 4.7324 5.6303 7.6657 8.3199 8.9544 9.6676 5.4199 5.9727 7.3798 9.2551 11.3673 13.6050 5.4199 6.2881 8.3532 10.9500 13.7781 16.0691 5.6576 6.2054 7.6140 9.5061 11.6477 13.9242 5.6576 6.5139 8.5800 11.2022 14.0726 17.0665

𝛼0 = 0.1 pz 5 4.1536 4.6456 5.8229 6.6394 7.0181 7.4422 4.1536 4.8586 6.4231 6.8445 7.3288 7.8732 4.9285 5.3730 6.5062 8.0280 9.7559 11.5981 4.9285 5.5543 7.0650 9.0066 11.1562 13.2525 5.2171 5.6403 6.7466 8.2610 9.9989 11.8628 5.2171 5.8067 7.2820 9.2203 11.3877 13.6764

0 7.2131 7.9053 9.6628 11.8828 12.4763 13.1380 7.2131 8.7335 11.9139 12.8542 13.9483 15.1768 8.1795 8.8653 10.6563 13.1043 15.9074 18.9063 8.1795 9.7032 13.2527 17.6394 22.3698 24.7279 8.5030 9.1882 10.9895 13.4657 16.3121 19.3660 8.5030 10.0293 13.6161 18.0749 22.9015 27.9053

1 5.3495 5.9648 7.4897 9.0672 9.5684 10.1290 5.3495 6.3973 8.7800 9.5526 10.3042 11.1490 6.0553 6.6688 8.2343 10.3242 12.6804 15.1775 6.0553 7.0992 9.5602 12.6305 15.9586 18.4299 6.2925 6.9061 8.4834 10.6014 12.9979 15.5446 6.2925 7.3350 9.8165 12.9317 16.3213 19.8451

2 4.7302 5.2945 6.6795 7.9595 8.4090 8.9120 4.7302 5.6252 7.6537 8.3130 8.9446 9.6547 5.4194 5.9686 7.3678 9.2336 11.3361 13.5642 5.4194 6.2834 8.3397 10.9267 13.7452 16.0578 5.6579 6.2019 7.6017 9.4836 11.6149 13.8812 5.6579 6.5096 8.5661 11.1778 14.0379 17.0216

5 4.1531 4.6451 5.8223 6.6386 7.0172 7.4412 4.1531 4.8582 6.4226 6.8438 7.3280 7.8722 4.9284 5.3728 6.5057 8.0269 9.7543 11.5960 4.9284 5.5541 7.0645 9.0056 11.1548 13.2511 5.2172 5.6402 6.7461 8.2599 9.9972 11.8605 5.2172 5.8067 7.2815 9.2193 11.3862 13.6744

42

Table 8. DFFs of the CCCC FG porous microplates for various gradient indexes, aspect ratios, porosity volume fractions and thickness to MLSPs 𝑎/ℎ

𝑀𝐿𝑆𝑃

ℓ1 = ℓ2

5

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

10

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

20

ℓ1 ≠ ℓ2

𝛼0 = 0 pz

ℓ2 /ℎ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 8.2879 9.3258 11.8319 15.0543 18.5859 21.5643 8.2879 10.4508 15.0543 20.4023 22.3025 23.5312 9.8034 10.8099 13.3674 16.7749 20.6118 24.6740 9.8034 11.9438 16.7749 22.6226 28.8614 35.2673 10.3587 11.3576 13.9291 17.3875 21.3048 25.4699 10.3587 12.4938 17.3875 23.3645 29.7786 36.3983

1 6.2819 7.0937 9.0515 11.5623 14.3109 17.1493 6.2819 7.6530 10.6782 14.2898 17.3281 18.1081 7.3870 8.1837 10.1966 12.8618 15.8509 19.0085 7.3870 8.7345 11.8751 15.7632 19.9600 24.2978 7.7899 8.5843 10.6145 13.3254 16.3817 19.6224 7.7899 9.1308 12.3018 16.2625 20.5610 25.0233

2 5.5398 6.2738 8.0275 10.2654 12.7117 15.2211 5.5398 6.7032 9.2604 12.3213 15.2474 15.8883 6.6104 7.3076 9.0709 11.4090 14.0347 16.8111 6.6104 7.7074 10.2930 13.5263 17.0377 20.6813 7.0151 7.7001 9.4589 11.8185 14.4870 17.3222 7.0151 8.0856 10.6632 13.9298 17.5030 21.2287

𝛼0 = 0.05 pz 5 4.7827 5.4369 6.9448 8.8316 10.8826 12.8821 4.7827 5.7248 7.7262 10.1011 12.6144 13.2663 5.9624 6.5332 7.9666 9.8706 12.0202 14.3050 5.9624 6.7708 8.6814 11.1071 13.7783 16.5773 6.4529 6.9831 8.3639 10.2481 12.4070 14.7208 6.4529 7.1935 9.0349 11.4454 14.1371 16.9780

0 8.3609 9.3227 11.6745 14.7318 18.1074 21.5704 8.3609 10.4060 14.8177 19.9892 22.3021 23.4727 9.8758 10.8073 13.2001 16.4218 20.0743 23.9575 9.8758 11.8964 16.5136 22.1484 28.1845 34.3956 10.4289 11.3535 13.7578 17.0250 20.7509 24.7290 10.4289 12.4443 17.1184 22.8739 29.0747 35.4872

1 6.2837 7.0818 9.0109 11.4907 14.2096 17.0208 6.2837 7.6361 10.6270 14.2043 17.2808 18.0495 7.3909 8.1716 10.1495 12.7752 15.7248 18.8436 7.3909 8.7160 11.8133 15.6562 19.8089 24.1039 7.7949 8.5724 10.5651 13.2331 16.2464 19.4448 7.7949 9.1119 12.2361 16.1476 20.3979 24.8130

2 5.5367 6.2677 8.0145 10.2440 12.6820 15.1962 5.5367 6.6970 9.2466 12.2989 15.2242 15.8624 6.6095 7.3026 9.0561 11.3829 13.9969 16.7619 6.6095 7.7015 10.2766 13.4981 16.9977 20.6297 7.0155 7.6957 9.4437 11.7906 14.4463 17.2687 7.0155 8.0802 10.6458 13.8994 17.4597 21.1727

𝛼0 = 0.1 pz 5 4.7820 5.4362 6.9440 8.8305 10.8811 12.8799 4.7820 5.7242 7.7256 10.1002 12.6130 13.2641 5.9621 6.5329 7.9659 9.8693 12.0183 14.3024 5.9621 6.7705 8.6808 11.1059 13.7766 16.5750 6.4530 6.9830 8.3632 10.2467 12.4049 14.7180 6.4530 7.1934 9.0343 11.4442 14.1353 16.9756

0 8.4423 9.3292 11.5257 14.4145 17.6296 20.9849 8.4423 10.3700 14.5869 19.5774 22.3264 23.4390 9.9581 10.8160 13.0439 16.0769 19.5411 23.2411 9.9581 11.8598 16.2605 21.6786 27.5082 33.5211 10.5097 11.3613 13.5985 16.6716 20.2024 23.9893 10.5097 12.4063 16.8583 22.3885 28.3724 34.5745

1 6.2854 7.0701 8.9711 11.4199 14.1093 16.8933 6.2854 7.6196 10.5766 14.1200 17.2350 17.9925 7.3949 8.1600 10.1034 12.6899 15.6002 18.6806 7.3949 8.6981 11.7528 15.5509 19.6599 23.9126 7.8001 8.5610 10.5167 13.1423 16.1129 19.2694 7.8001 9.0937 12.1718 16.0347 20.2373 24.6057

2 5.5335 6.2616 8.0015 10.2229 12.6525 15.1645 5.5335 6.6908 9.2331 12.2769 15.2013 15.8367 6.6086 7.2975 9.0415 11.3571 13.9596 16.7131 6.6086 7.6958 10.2604 13.4703 16.9583 20.5789 7.0159 7.6914 9.4286 11.7630 14.4060 17.2159 7.0159 8.0749 10.6288 13.8695 17.4170 21.1175

5 4.7813 5.4356 6.9432 8.8294 10.8795 12.8776 4.7813 5.7237 7.7250 10.0992 12.6117 13.2619 5.9619 6.5326 7.9652 9.8680 12.0163 14.2998 5.9619 6.7703 8.6802 11.1048 13.7748 16.5727 6.4532 6.9829 8.3625 10.2453 12.4028 14.7152 6.4532 7.1933 9.0337 11.4430 14.1334 16.9731

43

Table 9. DFFs of the CFFF FG porous microplates for various gradient indexes, aspect ratios, porosity volume fractions and thickness to MLSPs 𝑎/ℎ

𝑀𝐿𝑆𝑃

ℓ1 = ℓ2

5

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

10

ℓ1 ≠ ℓ2

ℓ1 = ℓ2

20

ℓ1 ≠ ℓ2

𝛼0 = 0 pz

ℓ2 /ℎ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.9955 1.0949 1.3226 1.6122 1.9346 2.2764 0.9955 1.1981 1.6122 2.1036 2.6308 3.1777 1.0161 1.1164 1.3474 1.6399 1.9648 2.3092 1.0161 1.2212 1.6399 2.1351 2.6664 3.2179 1.0216 1.1224 1.3555 1.6497 1.9757 2.3208 1.0216 1.2282 1.6497 2.1464 2.6786 3.2309

1 0.7414 0.8239 1.0062 1.2339 1.4858 1.7519 0.7414 0.8753 1.1489 1.4760 1.8289 2.1963 0.7562 0.8397 1.0250 1.2553 1.5093 1.7776 0.7562 0.8918 1.1689 1.4988 1.8544 2.2250 0.7602 0.8442 1.0312 1.2628 1.5177 1.7866 0.7602 0.8967 1.1758 1.5068 1.8632 2.2344

2 0.6640 0.7366 0.8961 1.0951 1.3152 1.5478 0.6640 0.7735 0.9987 1.2694 1.5626 1.8687 0.6788 0.7521 0.9138 1.1148 1.3366 1.5710 0.6788 0.7892 1.0167 1.2894 1.5847 1.8932 0.6829 0.7565 0.9197 1.1218 1.3443 1.5793 0.6829 0.7939 1.0231 1.2968 1.5926 1.9016

𝛼0 = 0.05 pz 5 0.6097 0.6668 0.7945 0.9552 1.1341 1.3242 0.6097 0.6876 0.8528 1.0549 1.2763 1.5093 0.6274 0.6839 0.8120 0.9734 1.1532 1.3444 0.6274 0.7041 0.8695 1.0724 1.2950 1.5294 0.6322 0.6889 0.8179 0.9803 1.1607 1.3523 0.6322 0.7091 0.8755 1.0793 1.3024 1.5372

0 1.0029 1.0948 1.3084 1.5826 1.8896 2.2159 1.0029 1.1941 1.5904 2.0637 2.5731 3.1023 1.0235 1.1162 1.3330 1.6101 1.9195 2.2484 1.0235 1.2171 1.6179 2.0950 2.6083 3.1420 1.0290 1.1221 1.3409 1.6197 1.9302 2.2599 1.0290 1.2240 1.6276 2.1062 2.6204 3.1549

1 0.7419 0.8228 1.0019 1.2262 1.4746 1.7371 0.7419 0.8735 1.1434 1.4665 1.8153 2.1787 0.7568 0.8386 1.0207 1.2475 1.4980 1.7627 0.7568 0.8900 1.1633 1.4891 1.8407 2.2072 0.7608 0.8430 1.0268 1.2549 1.5063 1.7716 0.7608 0.8950 1.1702 1.4971 1.8495 2.2165

2 0.6640 0.7361 0.8947 1.0927 1.3117 1.5434 0.6640 0.7730 0.9972 1.2669 1.5590 1.8640 0.6789 0.7516 0.9125 1.1124 1.3331 1.5665 0.6789 0.7887 1.0152 1.2868 1.5810 1.8885 0.6830 0.7561 0.9184 1.1194 1.3408 1.5747 0.6830 0.7934 1.0215 1.2942 1.5889 1.8968

𝛼0 = 0.1 pz 5 0.6097 0.6668 0.7944 0.9551 1.1339 1.3239 0.6097 0.6875 0.8527 1.0548 1.2761 1.5091 0.6274 0.6839 0.8119 0.9733 1.1530 1.3441 0.6274 0.7041 0.8694 1.0723 1.2948 1.5292 0.6322 0.6889 0.8179 0.9802 1.1605 1.3520 0.6322 0.7091 0.8755 1.0792 1.3023 1.5370

0 1.0113 1.0958 1.2953 1.5540 1.8451 2.1558 1.0113 1.1912 1.5695 2.0245 2.5157 3.0270 1.0320 1.1172 1.3197 1.5812 1.8748 2.1879 1.0320 1.2141 1.5969 2.0556 2.5507 3.0663 1.0374 1.1231 1.3275 1.5906 1.8854 2.1992 1.0374 1.2209 1.6063 2.0666 2.5626 3.0791

1 0.7425 0.8216 0.9977 1.2186 1.4635 1.7226 0.7425 0.8718 1.1379 1.4571 1.8019 2.1614 0.7574 0.8374 1.0164 1.2398 1.4868 1.7480 0.7574 0.8883 1.1578 1.4796 1.8272 2.1896 0.7614 0.8419 1.0225 1.2472 1.4950 1.7568 0.7614 0.8932 1.1646 1.4876 1.8359 2.1989

2 0.6641 0.7357 0.8933 1.0904 1.3084 1.5389 0.6641 0.7724 0.9957 1.2643 1.5554 1.8593 0.6790 0.7512 0.9111 1.1100 1.3297 1.5620 0.6790 0.7882 1.0137 1.2843 1.5774 1.8838 0.6831 0.7557 0.9170 1.1170 1.3374 1.5702 0.6831 0.7929 1.0201 1.2916 1.5853 1.8921

5 0.6097 0.6668 0.7943 0.9550 1.1338 1.3237 0.6097 0.6875 0.8527 1.0547 1.2760 1.5089 0.6274 0.6839 0.8119 0.9732 1.1529 1.3439 0.6274 0.7041 0.8694 1.0722 1.2947 1.5290 0.6322 0.6889 0.8178 0.9801 1.1604 1.3518 0.6322 0.7091 0.8754 1.0791 1.3021 1.5368

44

z y

h/2

x

h/2

a

Figure 1. Geometry and co-ordinate of a FG plate

𝜼 (-1,1) 4

(1,1) 3 𝝃

1 (-1,-1)

2 (1,-1)

Figure 2. Four node rectangular element in its natural coordinate.

46

𝑆𝑆𝑆𝑆 − ℓ1 = ℓ2

𝑆𝑆𝑆𝑆 − ℓ1 ≠ ℓ2 Figure 3. Variation of the DFFs of SSSS FG porous microplates with respect to various aspect ratios and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 47

𝑆𝐶𝑆𝐶 − ℓ1 = ℓ2

𝑆𝐶𝑆𝐶 − ℓ1 ≠ ℓ2 Figure 4. Variation of the DFFs of SCSC FG porous microplates with respect to various aspect ratios and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 48

𝐶𝐶𝐶𝐶 − ℓ1 = ℓ2

𝐶𝐶𝐶𝐶 − ℓ1 ≠ ℓ2 Figure 5. Variation of the DFFs of CCCC FG porous microplates with respect to various aspect ratios and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 49

𝐶𝐹𝐹𝐹 − ℓ1 = ℓ2

𝐶𝐹𝐹𝐹 − ℓ1 ≠ ℓ2 Figure 6. Variation of the DFFs of CFFF FG porous microplates with respect to various aspect ratios and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 50

𝑆𝑆𝑆𝑆 − ℓ1 = ℓ2

𝑆𝑆𝑆𝑆 − ℓ1 ≠ ℓ2

Figure 7. Variation of the DFFs of SSSS FG porous microplates with respect to various gradient indexes and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 51

𝑆𝐶𝑆𝐶 − ℓ1 = ℓ2

𝑆𝐶𝑆𝐶 − ℓ1 ≠ ℓ2 Figure 8. Variation of the DFFs of SCSC FG porous microplates with respect to various gradient indexes and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 52

𝐶𝐶𝐶𝐶 − ℓ1 = ℓ2

𝐶𝐶𝐶𝐶 − ℓ1 ≠ ℓ2 Figure 9. Variation of the DFFs of CCCC FG porous microplates with respect to various gradient indexes and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 53

𝐶𝐹𝐹𝐹 − ℓ1 = ℓ2

𝐶𝐹𝐹𝐹 − ℓ1 ≠ ℓ2 Figure 10. Variation of the DFFs of CFFF FG porous microplates with respect to various gradient indexes and thickness to MLSPs (𝑝𝑧 = 1, 𝛼0 = 0.05). 54

𝑀𝑜𝑑𝑒 (1,1)

𝑀𝑜𝑑𝑒 (2,1)

𝑀𝑜𝑑𝑒 (1,2)

𝑀𝑜𝑑𝑒 (2,2)

Figure 11. Transverse vibration mode shapes of SSSS FG porous microplates with variable MLSP (𝑝𝑧 = 1, ℓ1 ≠ ℓ2 , 𝛼0 = 0.05, 𝑎/ℎ = 5, ℓ2 /ℎ = 0.4).

55

𝑀𝑜𝑑𝑒 (1,1)

𝑀𝑜𝑑𝑒 (1,2)

𝑀𝑜𝑑𝑒 (2,1)

𝑀𝑜𝑑𝑒 (2,2)

Figure 12. Transverse vibration mode shapes of SCSC FG porous microplates with variable MLSP (𝑝𝑧 = 1, ℓ1 ≠ ℓ2 , 𝛼0 = 0.05, 𝑎/ℎ = 5, ℓ2 /ℎ = 0.4).

56

𝑀𝑜𝑑𝑒 1

𝑀𝑜𝑑𝑒 3

𝑀𝑜𝑑𝑒 2

𝑀𝑜𝑑𝑒 4

Figure 13. Transverse vibration mode shapes of CCCC FG porous microplates with variable MLSP (𝑝𝑧 = 1, ℓ1 ≠ ℓ2 , 𝛼0 = 0.05, 𝑎/ℎ = 5, ℓ2 /ℎ = 0.4).

57

𝑀𝑜𝑑𝑒 1

𝑀𝑜𝑑𝑒 2

𝑀𝑜𝑑𝑒 4

𝑀𝑜𝑑𝑒 3

𝑀𝑜𝑑𝑒 5 Figure 14. Transverse vibration mode shapes of CFFF FG porous microplates with variable MLSP (𝑝𝑧 = 1, ℓ1 ≠ ℓ2 , 𝛼0 = 0.05, 𝑎/ℎ = 5, ℓ2 /ℎ = 0.4). 58

𝑆𝑆𝑆𝑆

𝑆𝐶𝑆𝐶

𝐶𝐶𝐶𝐶 Figure 15. The third transverse vibration mode shapes of SSSS, SCSC and CCCC FG porous microplates with variable MLSP (𝑝𝑧 = 1, ℓ1 ≠ ℓ2 , 𝛼0 = 0.05, 𝑎/ℎ = 5, ℓ2 /ℎ = 0.4). 59