Quasi-3D integral model for thermomechanical buckling and vibration of FG porous nanoplates embedded in an elastic medium

International Journal of Mechanical Sciences 157–158 (2019) 320–335

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Quasi-3D integral model for thermomechanical buckling and vibration of FG porous nanoplates embedded in an elastic medium A.F. Radwan∗ Department of Mathematics and Statistics, Higher Institute of Management and Information Technology, Nile for Science and Technology, Kafrelsheikh 33514, Egypt

a r t i c l e

i n f o

Keywords: Thermal buckling Vibration Porosity Elastic medium Nonlocal strain gradient theory Quasi-3D model

a b s t r a c t Mechanical buckling, thermal buckling and free vibration behaviors of functionally graded (FG) porous nanoplates embedded in an elastic medium are investigated via a nonlocal strain gradient theory. Two dimensional (2D) and quasi three dimensional (quasi-3D) sinusoidal shear deformation theories are used in this study. The proposed theories have a displacement field with integral terms which include the effects of both transverse shear and normal deformations. Porosity-dependent material properties of the porous nanoplate are defined via a modified power-law function. Equations of motion are derived based on Hamilton’s principle which includes the effect of the two-parameter elastic foundations. Numerical results are presented to verify the accuracy of the present 2D and quasi-3D shear deformation theories by comparing them with the solutions given in literature with different 2D, quasi-3D and 3D solutions. The effects of many parameters like porosity factor, nonlocal, length scale parameters, plate aspect ratio, side-to-thickness ratio and gradient index on the buckling, thermal buckling and vibration of FG porous nanoplates are all discussed.

1. Introduction Functionally graded material (FGM) is a material with changing composition, microstructure, or porosity across the volume of it [1]. The aim of proceeding a set of specified functions, FGMs are designed with changing properties over the volume of the bulk material [2–6]. In FGMs, the material properties are not uniform toward the perfect material, and the properties count on the spatial position of the material in the bulk structure of the material. FGMs are prepared with varying properties that include changing mechanical, magnetic, thermal, and electrical properties. Also, FGMs are designed as stepwise-graded structures, and some are fabricated to be continuous-graded structures, building on the application areas. There are different types of FGMs, these types are porosity and pore size gradient structured FGMs, microstructural gradient-structured FGMs, and chemical gradient-structured FGMs. In the present study, the author is concerned with the type of porosity (porosity gradient functionally graded material) where the porosity gradient FGM in the material is made to modification with the change in the locative position in the bulk material [7–10]. Porosity gradient materials are generated by the precipitation of dust with a varying mixture of distinct particle sizes and shapes that would help to produce the needed varying porosity with the changing pore sizes and shapes. The porosity gradient in an FGM has also an effect on Young’s modulus and the tensile strength of the material [11–31].

∗

The including of both nonlocal and strain gradient effects are very important in the modeling of small scale structures. Many studies were developed using this theory. Nonlinear thermal loading for wave propagation analysis of an inhomogeneous FG nanoplate was discussed by Ebrahimi et al. [32]. Xu et al. [33] discussed the buckling and bending of nonlocal strain gradient elastic beams. Li and Hu [34] investigated the nonlocal strain gradient theory for buckling analysis of size-dependent nonlinear beams. Ebrahimi and Barati [35] used the nonlocal strain gradient theory to discuss the hygrothermal impact on vibration analysis of viscoelastic FG nanobeams. Plates resting on Winkler-Pasternak’s foundations were discussed through a number of researchers using several techniques. One parameter has been inserted to characterize the attitude of the foundation given in the analysis of Katsikadelis and Armenakas [36], Bezine [37] and El-Zafrany et al. [38] where the deflection in the foundation is proportional to the pressure. The foundation itself is modeled by discrete vertical springs, ignoring the transverse shear strains in the foundation. For two different parameters as in Pasternak’s model, the influence of shear interaction between points in the foundation is taken into account [39–47]. Malekzadeh et al. [48] studied the small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Shahsavari et al. [49] discussed the shear buckling of single-layer graphene sheets in hygrothermal environment resting on an elastic foundation based on different nonlocal strain gradient theories. Panyatong et al. [50] examined the

Corresponding author. E-mail address: [email protected]

https://doi.org/10.1016/j.ijmecsci.2019.04.031 Received 19 December 2018; Received in revised form 27 March 2019; Accepted 16 April 2019 Available online 26 April 2019 0020-7403/© 2019 Published by Elsevier Ltd.

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

nonlocal elasticity and the second-order shear deformation plate theory for the free vibration analysis of FG nanoplates resting on elastic medium. The analysis of buckling load, thermal buckling load and vibration in the nanostructures made of functionally graded materials were undertaken in several studies [51–71]. Bouazza et al. [72] dealt with the nonlocal elasticity theory with four-unknown shear deformation theory for the analysis of thermal buckling of nanoplates embedded in an elastic medium. Sobhy and Zenkour [73] investigated the nonlocal thermal and mechanical buckling of nonlinear orthotropic viscoelastic nanoplates embedded in a visco-Pasternak medium. Bouazza et al. [74] studied the thermal buckling analyses of advanced nanoplates with small-scale effects based on a hyperbolic four-variable refined theory. Ansari et al. [75] examined the buckling and vibration of FG nanoplates based on surface elasticity theory with thermal loading. Farajpour et al. [76] investigated the buckling analysis of orthotropic nanoplates in a thermal environment based on a higher-order nonlocal strain gradient plate theory. Ebrahimi and Barati [77] dealt with the thermal buckling load of embedded magneto-electro-thermo-elastic nonhomogeneous nanoplates based on the nonlocal elasticity theory. Wang et al. [78] studied the thermal buckling analysis of nanoplate considering the small-scale effects. Nami et al. [79] investigated the thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation plate theory. Barati et al. [80] examined the mechanical and thermal buckling analysis of embedded nanosize FG plates based on an inverse cotangential theory with thermal environments. Barati and Shahverdi [81] investigated vibration analysis of FG nanoplates based on a four-variable plate theory with different boundary conditions under non-uniform temperature distributions. He and Ge [82] investigated the cooperative control of a nonuniform gantry crane with constrained tension. Mahmoudpour et al. [83] employed the nonlocal strain gradient theory for nonlinear free and forced vibration of thick FG double layered nanoplates. Ansari et al. [84] investigated the buckling and vibration analysis of FG nanoplates according to surface elasticity theory and subjected to thermal loading. In this study, two models (2D and quasi-3D shear deformation theories with integral form) are used to investigate the mechanical buckling, thermal buckling and vibration analysis of FG porous nanoplates embedded in an elastic medium via the nonlocal strain gradient theory. Porosity-dependent material properties of the porous nanoplate are defined via a modified power-law function. Hamilton’s principle is used to derive the equations of motion which include the influence of the twoparameter elastic foundations. The eigenvalue problem can be solved to obtain the critical buckling load, critical buckling temperature change and natural frequency of FG porous nanoplates embedded in an elastic medium. The numerical results for mechanical buckling, thermal buckling and frequency are obtained and compared well with the results given in the literature. The influences of many parameters like, porosity factor, nonlocal, length scale parameters, aspect ratio, gradient index, and side-to-thickness ratio are all investigated. The present paper is organized as follows: The present paper is organized as follows: Section 1 (introduction) gives a literature survey of the most important previous works that have been conducted. Moreover, the aim of the present study is included in the end of this section. Section 2 (mathematical formulation) presents the deduction of the main and the governing equations for the thermomechanical buckling and vibration of FG porous nanoplates embedded in an elastic medium using 2D and quasi-3D theories. Section 3 (analytical solution) gives the simply-supported boundary conditions and Navier’s solution which used to solve the equilibrium equations of the present study. Section 4 (numerical results) discusses the effects of the nonlocal coefficient, strain gradient parameter and porosity factor on the critical buckling load, thermal buckling load and vibration of FG porous nanoplates. This section contains two subsections. The first one shows the validation of the present quasi-3D model. However, the second presents the parametric results for the FG porous nanoplate. Finally,

Fig. 1. The schematic figure of an FG porous nanoplate embedded in an elastic medium.

general conclusions about the obtained numerical results are introduced in Section 5 (conclusions). 2. Mathematical formulation Fig. 1 shows an FG nanoplate with porosities embedded in an elastic medium of thickness h, width b and length a. Let the graduation of the materials be varied from the upper surface (𝑧 = ℎ2 ) to the lower one

(𝑧 = − ℎ2 ). A non-homogeneity material properties with a porosity volume function 𝛼 (0 ≤ 𝛼 ≪ 1) of the FG porous nanoplate, such as Young’s modulus E(z), thermal expansion coefficient 𝛽(z) and mass density 𝜌(z) vary continuously through the thickness direction using the power law as follows: ( )𝜅 ( ) ( ) 𝑃 (𝑧) = 𝑃𝑚 + 𝑃𝑐 − 𝑃𝑚 𝜅 − 𝛼2 𝑃𝑐 + 𝑃𝑚 , 𝜅 = ℎ𝑧 + 12 , (1) 𝑃 = 𝐸, 𝜌, 𝛽, 0 ≤ 𝜅 ≤ ∞, The 𝜅 is the volume fraction of the Alumina. where 𝜅 is the gradient index, Pm and Pc are the material properties of Aluminium and Alumina respectively, and Poisson’s ratio 𝜈 is constant and equals 0.3. The reaction between the FG porous nanoplate and the foundations is defined as follows: ( 2 ) 𝜕 𝑤𝑏 (𝑥, 𝑦) 𝜕 2 𝑤𝑏 (𝑥, 𝑦) 𝑅0 = 𝐾0 𝑤𝑏 (𝑥, 𝑦) − 𝐾1 + , (2) 𝜕𝑥2 𝜕𝑦2 where R0 , wb , K0 and K1 are the force per unit area, bending component of deflection, Winkler foundation stiffness and the shear layer stiffness, respectively. 2.1. Constitutive equations The four-variable (2D) and five-variable (quasi-3D) shear deformation theories introduce the components of the displacement to represent the effect of the FG porous nanoplate thickness in terms of sinusoidal function. The present 2D and quasi-3D sinusoidal theory are presented as follows [85]: ⎧ 𝜕𝑤𝑏 ⎫ ⎧𝑢1 ⎫ ⎧ 𝑢0 ⎫ ⎪ 𝜕𝑥 ⎪ ⎧𝛾0 Ψ(𝑧) ∫ 𝑤𝑠 d𝑥⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 𝜕𝑤𝑏 ⎪ ⎪ ⎪ 𝑢 𝑣 = − 𝑧 2 0 ⎨ ⎬ ⎨ ⎬ ⎨ 𝜕𝑦 ⎬ + ⎨𝛾1 Ψ(𝑧) ∫ 𝑤𝑠 d𝑦⎬, ⎪𝑢 ⎪ ⎪𝑤 ⎪ ⎪ ⎪ ⎪ ⎪ 𝑠 Φ(𝑧)𝑤𝑏 ⎩ 3⎭ ⎩ 𝑏⎭ ⎭ ⎪ 0 ⎪ ⎩ ⎩ ⎭

(3)

where ui , (i=1,2,3) are functions of (x, y, z, t), while u0 , v0 , wb , ws , 𝑤𝑠𝑏 are all functions of (x, y, t) only and 𝛾 0 , 𝛾 1 are constants depending on the geometry. And ( ) ( ) Ψ(𝑧) = 𝜋ℎ sin 𝜋𝑧 Φ(𝑧) = cos 𝜋𝑧 (4) , . ℎ ℎ The above Eq. (3) gives the stretching effect and namely a quasi-3D sinusoidal theory, while if the stretching effect is neglected (𝑤𝑠𝑏 = 0) the 321

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

2D sinusoidal theory is developed. According to Eq. (3) the six-strain expressions can be derived as follows:

𝛿 =

0 1 2 ⎧ 𝜀1 ⎫ ⎧ 𝜀1 ⎫ ⎧ 𝜀1 ⎫ ⎧ 𝜀1 ⎫ ⎪ ⎪ ⎪ 0⎪ ⎪ 1⎪ ⎪ 2⎪ ⎨ 𝜀2 ⎬ = ⎨ 𝜀2 ⎬ − 𝑧⎨ 𝜀2 ⎬ + Ψ(𝑧)⎨ 𝜀2 ⎬, ⎪𝛾 ⎪ ⎪𝛾 0 ⎪ ⎪𝛾 1 ⎪ ⎪𝛾 2 ⎪ ⎩ 12 ⎭ ⎩ 12 ⎭ ⎩ 12 ⎭ ⎩ 12 ⎭

𝛾𝑖3 = Φ(𝑧)𝛾𝑖03 ,

𝜀3 = Φ‵ (𝑧)𝜀03 ,

follows: (𝜎𝑖𝑗 𝛿𝜖𝑖𝑗 )d𝑣, [ ] 𝜕 2 𝑤𝑏 𝜕 2 𝑤𝑏 𝜕 2 𝑤𝑏 𝛿 = −𝐹11 − 2𝐹12 − 𝐹22 + 𝑅0 𝛿𝑢3 dΩ, ∫Ω 𝜕 𝑥𝜕 𝑦 𝜕𝑥2 𝜕𝑦2 [ ]

(5)

(𝑖 = 1, 2),

𝛿 =

where 𝜀01 = 𝜀11 =

𝜕𝑢0 , 𝜕𝑥 𝜕 2 𝑤𝑏 , 𝜕𝑥2

𝜀21 = 𝛾0 𝑤𝑠 ,

𝜀02 = 𝜀12 =

𝜕𝑣0 , 𝜕𝑦

𝜕2 𝑤

𝑏 𝜕𝑦2

,

𝜀22 = 𝛾1 𝑤𝑠 ,

0 = 𝛾 ∫ 𝑤 d𝑥 + 𝛾13 0 𝑠

𝜕𝑤𝑠𝑏 𝜕𝑥

0 = 𝛾12

𝜕𝑣0 𝜕𝑥

+

𝜕2 𝑤

𝜕𝑢0 , 𝜕𝑦

1 =2 𝑏 𝛾12 , 𝜕 𝑥𝜕 𝑦

2 = 𝛾 𝜕 ∫ 𝑤 d𝑥 + 𝛾 𝜕 ∫ 𝑤 d𝑦, 𝛾12 0 𝜕𝑦 𝑠 1 𝜕𝑥 𝑠 𝜕𝑤𝑠𝑏

0 = 𝛾 ∫ 𝑤 d𝑦 + 𝛾23 1 𝑠

𝜕𝑦

(6)

1 𝜕𝑤𝑠 𝑤 d𝑦 = − . ∫ 𝑠 𝛾1 𝜕𝑦

(7)

2 𝜕 1 𝜕 𝑤𝑠 𝑤 d𝑦 = − , 𝜕𝑥 ∫ 𝑠 𝛾1 𝜕 𝑦𝜕 𝑥

𝛿𝑤𝑏 ∶ 𝛿𝑤𝑠 ∶

(8)

𝛿𝑤𝑠𝑏 ∶

where 𝛾0 = 𝜃𝑚2 and 𝛾1 = 𝜙2𝑛 and will be defined after Eq. (23). According to the nonlocal strain gradient theory which was presented by Aifantis [87] and Lim et al. [88], the constitutive relations of an FG porous nanoplate can be expressed as follows: ⎧𝜎 𝑥 ⎫ ⎡𝑐11 )⎪ ⎪ ( )⎢ 1 − 𝜇∇2 ⎨𝜎𝑦 ⎬ = 1 − 𝜁 ∇2 ⎢𝑐12 ⎪𝜎 ⎪ ⎢𝑐 ⎣ 13 ⎩ 𝑧⎭ ⎧𝜎𝑦𝑧 ⎫ ⎡𝑐44 ( )⎪ ⎪ ( )⎢ 1 − 𝜇∇2 ⎨𝜎𝑥𝑧 ⎬ = 1 − 𝜁 ∇2 ⎢ 0 ⎪𝜎 ⎪ ⎢0 ⎣ ⎩ 𝑥𝑦 ⎭ (

𝑐12 𝑐22 𝑐23

𝑐13 ⎤⎧𝜀1 ⎫ ⎥⎪ ⎪ 𝑐23 ⎥⎨𝜀2 ⎬, ⎪ 𝑐33 ⎥⎦⎪ ⎩𝜀 3 ⎭

0 𝑐55 0

0 ⎤⎧𝛾23 ⎫ ⎥⎪ ⎪ 0 ⎥⎨𝛾13 ⎬, ⎪ 𝑐66 ⎥⎦⎪ ⎩𝛾12 ⎭

𝛿𝑣0 ∶ 𝛿𝑤𝑏 ∶ 𝛿𝑤𝑠 ∶

𝑐11 = 𝑐22

𝑐12

𝑐44 = 𝑐55 = 𝑐66

+

𝜕 2 𝑀𝑦

=

𝜕𝑁 𝜕𝑁𝑥 𝜕 𝑤̈ 𝜕 𝑤̈ + 𝜕𝑦𝑥𝑦 = 𝐼0 𝑢̈ 0 − 𝐼1 𝜕𝑥𝑏 − 𝐼2 𝜕𝑥𝑠 , 𝜕𝑥 𝜕𝑁𝑥𝑦 𝜕𝑁 𝜕 𝑤̈ 𝜕 𝑤̈ + 𝜕𝑦𝑦 = 𝐼0 𝑣̈ 0 − 𝐼1 𝜕𝑦𝑏 − 𝐼2 𝜕𝑦𝑠 , 𝜕𝑥 2𝑀 2𝑀 2 2 𝜕 𝜕 𝜕 𝑀𝑥 𝜕 𝑤 𝜕2 𝑤 𝜕2 𝑤 + 2 𝜕 𝑥𝜕𝑥𝑦 + 𝜕𝑦2 𝑦 + 𝐹11 𝜕𝑥2𝑏 + 2𝐹12 𝜕 𝑥𝜕 𝑦𝑏 + 𝐹22 𝜕𝑦2𝑏 𝜕𝑥2 ( 𝑦 ) 𝜕 𝑢̈ 𝜕 𝑣̈ 𝐼0 𝑤̈ 𝑏 + 𝐼1 𝜕𝑥0 + 𝜕𝑦0 − 𝐼2 ∇2 𝑤̈ 𝑏 − 𝐼4 ∇2 𝑤̈ 𝑠 , ( ) 𝜕2 𝑆 𝜕𝑄 𝜕𝑄 𝜕 𝑢̈ 𝜕 𝑣̈ −𝛾0 𝑆𝑥 − 𝛾1 𝑆𝑦 + 2 𝜕 𝑥𝜕𝑥𝑦 − 𝜕𝑥𝑥𝑧 − 𝜕𝑦𝑦𝑧 = 𝐼2 𝜕𝑥0 + 𝜕𝑦0 𝑦 −𝐼4 ∇2 𝑤̈ 𝑏 − 𝐼5 ∇2 𝑤̈ 𝑠 ,

− 𝑅𝑓 =

where N, M, S and Q are defined by: [

] 𝑁𝑙 , 𝑀𝑙 , 𝑆𝑙 = ∫

𝑄𝑖 = ∫

ℎ 2 − ℎ2 ℎ 2 − ℎ2

𝑁𝑧 = ∫

(10)

𝐸(𝑧) = . 2(1 + 𝜈)

(𝜕 𝑥𝜕 𝑦 𝜕 𝑢̈ 𝐼1 𝜕𝑥0

(15)

While, for the 2D shear deformation theory, the constitutive constants cij may be displayed as follows: 𝐸(𝑧)𝜈 = , 1 − 𝜈2

𝜕 2 𝑀𝑥𝑦

𝜕 2 𝑤𝑏 𝜕 2 𝑤𝑏 𝜕 2 𝑤𝑏 2 + 𝐹11 𝜕𝑥2 + 2𝐹12 𝜕 𝑥𝜕 𝑦 + 𝐹22 𝜕𝑦2 − 𝑅𝑓 𝜕𝑦) 𝜕 𝑣̈ 𝐼0 𝑤̈ 𝑏 + + 𝜕𝑦0 − 𝐼2 ∇2 𝑤̈ 𝑏 + 𝐼3 𝑤̈ 𝑠𝑏 − 𝐼4 ∇2 𝑤̈ 𝑠 , ( ) 𝜕2 𝑆 𝜕𝑄 𝜕𝑄 𝜕 𝑢̈ 𝜕 𝑣̈ −𝛾0 𝑆𝑥 − 𝛾1 𝑆𝑦 + 2 𝜕 𝑥𝜕𝑥𝑦 − 𝜕𝑥𝑥𝑧 − 𝜕𝑦𝑦𝑧 = 𝐼2 𝜕𝑥0 + 𝜕𝑦0 𝑦 −𝐼4 ∇2 𝑤̈ 𝑏 − 𝐼5 ∇2 𝑤̈ 𝑠 , 𝜕 2 𝑤𝑠 𝜕 2 𝑤𝑠 𝜕 2 𝑤𝑠 𝜕𝑄 𝜕𝑄 𝑁𝑧 − 𝜕𝑥𝑥𝑧 − 𝜕𝑦𝑦𝑧 + 𝐹11 𝜕𝑥2𝑏 + 2𝐹12 𝜕 𝑥𝜕 𝑦𝑏 + 𝐹22 𝜕𝑦2𝑏 = −𝐼3 𝑤̈ 𝑏 −𝐼6 𝑤̈ 𝑠𝑏 ,

+2

𝛿𝑢0 ∶

(9)

𝑐11 (𝑧) = 𝑐22 (𝑧) = 𝑐33 (𝑧) =

𝐸(𝑧) = , 1 − 𝜈2

𝜕 2 𝑀𝑥 𝜕𝑥2

(14)

while, by setting 𝑤𝑠𝑏 = 0 (𝑖.𝑒., 𝜀3 = 0, 𝜎𝑧 = 0) in the displacement field Eq. (3) then the equations of motion associated with the 2D shear deformation theory are indicated as follows:

( )2 where 𝜇 = 𝑒0 𝑙 is the nonlocal coefficient revealing the small scale effect and 𝜁 denotes the strain gradient length scale parameter in nanostructures. The constitutive constants cij according to the quasi-3D model are depicted as follows: 1−𝜈 𝐸(𝑧), (1 − 2𝜈)(1 + 𝜈) 𝜈 𝑐12 (𝑧) = 𝑐23 (𝑧) = 𝑐13 (𝑧) = 𝐸(𝑧), (1 − 2𝜈)(1 + 𝜈) 1 𝑐𝑖𝑖 (𝑧) = 𝐺(𝑧) = 𝐸(𝑧), (𝑖 = 4, 5, 6). 2(1 + 𝜈)

𝜕 𝑁𝑥 𝜕 𝑁𝑥𝑦 𝜕 𝑤̈ 𝑏 𝜕 𝑤̈ 𝑠 + = 𝐼0 𝑢̈ 0 − 𝐼1 − 𝐼2 , 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕 𝑁𝑥𝑦 𝜕 𝑁𝑦 𝜕 𝑤̈ 𝑏 𝜕 𝑤̈ 𝑠 + = 𝐼0 𝑣̈ 0 − 𝐼1 − 𝐼2 , 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑦

𝛿𝑣0 ∶

By using Eq. (7), then we get: 2 𝜕 1 𝜕 𝑤𝑠 𝑤 d𝑥 = − , 𝜕𝑦 ∫ 𝑠 𝛾0 𝜕 𝑥𝜕 𝑦

𝜌(𝑧) 𝑢̈ 1 𝛿 𝑢̈ 1 + 𝑢̈ 2 𝛿 𝑢̈ 2 + 𝑢̈ 3 𝛿 𝑢̈ 3 d𝑣,

𝛿𝑢0 ∶

By using Navier’s solution, the integrals in Eq. (6) are solved and can be given as follows [86]: 1 𝜕𝑤𝑠 𝑤 d𝑥 = − , ∫ 𝑠 𝛾0 𝜕𝑥

∫𝑣

(13)

by substituting Eq. (13) in the Hamilton’s principle Eq. (12), then the equations of motion associated with the displacement field Eq. (3) and the constitutive Eq. (9) can be given for a quasi-3D model as follows:

𝜀03 = 𝑤𝑠𝑏 .

,

∫𝑣

ℎ 2 − ℎ2

[

Φ(𝑧)𝜎𝑖 d𝑧,

] 1, 𝑧, Ψ 𝜎𝑙 d𝑧,

(𝑙 = 𝑥, 𝑦, 𝑥𝑦),

(𝑖 = 𝑥𝑧, 𝑦𝑧),

(16)

Φ‵ 𝜎𝑧 d𝑧,

and the inertias Ii are presented as follows: [

(11)

] 𝐼0 , 𝐼1 , 𝐼2 , 𝐼3 , 𝐼4 , 𝐼5 , 𝐼6 = ∫

ℎ 2 − ℎ2

[ ] 𝜌(𝑧) 1, 𝑧, 𝑧2 , Φ, 𝑧Ψ, Ψ2 , Φ d𝑧.

(17)

The in-plane edge loads Fij are written as follows: 2.2. Equations of motion

𝑇 , 𝐹 = 𝑝 + 𝑁 𝑇 , 𝐹 = 0, 𝐹11 = 𝑝11 + 𝑁11 22 22 12 22

Hamilton’s principle [89–91] which is the dynamic version of the principle of virtual works is given as: 𝑡

∫0

𝛿( +  + )d𝑡 = 0,

(18)

in which (p11 , p22 ) are the normal in-plane forces due to the mechanical 𝑇 , 𝑁 𝑇 ) are the normal in-plane forces due to the thermal load and (𝑁11 22 load which are distributed along the edges parallel to (y; x), and are given as follows:

(12)

where  is the strain energy,  is the work done by external forces and  is the kinetic energy and the variation of them can be given as

𝑝11 = 𝑝22 = −𝑝, 322

ℎ 2 − ℎ2

𝑇 = 𝑁𝑇 = ∫ 𝑁11 22

𝐸(𝑧) 𝛽(𝑧)Δ𝑇 d𝑧. 1−𝜈

(19)

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

Integrating Eq. (16) over the FG porous nanoplate thickness by using Eq. (5) and Eq. (9) then the stress resultants are given as follows: ⎧ 𝑁𝑥 ⎫ ⎡ 11 ⎪ ⎪ ⎢ ⎪ 𝑁𝑦 ⎪ ⎢ ⎪ ⎪ ⎢ ⎪𝑀𝑥 ⎪ ⎢ ( ) ( ) ⎪ ⎪ ⎢ 2 2 𝑀 1 − 𝜇∇ ⎨ 𝑦 ⎬ = 1 − 𝜁 ∇ ⎢ ⎪𝑆 ⎪ ⎢ ⎪ 𝑥⎪ ⎢ ⎪ 𝑆𝑦 ⎪ ⎢ ⎪ ⎪ ⎢ ⎪ 𝑁𝑧 ⎪ ⎢symm. ⎣ ⎩ ⎭

12

13

14

15

16

22

23

24

25

26

33

34

35

36

44

45

46

55

56 66

⎧ 𝑁𝑥𝑦 ⎫ ⎡ 11 ( )⎪ )⎢ ⎪ ( 2 2 1 − 𝜇∇ ⎨𝑀𝑥𝑦 ⎬ = 1 − 𝜁 ∇ ⎢ ⎪𝑆 ⎪ ⎢symm. ⎣ ⎩ 𝑥𝑦 ⎭

12 22

} { [ ( ) 𝑄𝑦𝑧 ( ) 44 = 1 − 𝜁 ∇2 1 − 𝜇∇2 𝑄𝑥𝑧 0

0

(20)

0 13 ⎤⎧𝛾12 ⎫ ⎥⎪ 1 ⎪ 23 ⎥⎨𝛾12 ⎬, 2 ⎪ 33 ⎥⎦⎪ ⎩𝛾12 ⎭

]{

55

17 ⎤⎧𝜀01 ⎫ ⎥⎪ ⎪ 27 ⎥⎪𝜀02 ⎪ ⎥⎪ ⎪ 37 ⎥⎪𝜀11 ⎪ ⎥⎪ ⎪ 47 ⎥⎨𝜀12 ⎬, 57 ⎥⎪𝜀21 ⎪ ⎥⎪ ⎪ 67 ⎥⎪𝜀21 ⎪ ⎥⎪ ⎪ 77 ⎥⎪𝜀03 ⎪ ⎦⎩ ⎭

0 𝛾23 1 𝛾13

} ,

where ℎ

𝑖𝑗 = ∫ 2ℎ 𝑐𝑖𝑗 d𝑧, −2 [ ] ℎ 15 , 16 = ∫ 2ℎ −2 [ ] ℎ 25 , 26 = ∫ 2ℎ −2 [ ] ℎ 35 , 36 = ∫ 2ℎ −2 [ ] ℎ 45 , 46 = ∫ 2ℎ −2 [ ] ℎ 55 , 56 = ∫ 2ℎ ℎ

−2

[ ] [ ] ℎ (𝑖, 𝑗 = 1, 2), 13 , 14 , 24 = ∫ 2ℎ 𝑧 𝑐11 , 𝑐12 , 𝑐22 d𝑧, −2 [ ] ℎ Ψ(𝑧) 𝑐11 , 𝑐12 d𝑧, 𝑖7 = ∫ 2ℎ 𝑐𝑖3 Φ‵ (𝑧)d𝑧, (𝑖 = 1, 2), −2 [ ] [ ] [ ] ℎ Ψ(𝑧) 𝑐12 , 𝑐22 d𝑧, 33 , 34 = ∫ 2ℎ 𝑧2 𝑐11 , 𝑐12 d𝑧, −2 [ ] ℎ 𝑧Ψ(𝑧) 𝑐11 , 𝑐12 d𝑧, 44 = ∫ 2ℎ 𝑧2 𝑐22 d𝑧, −2 [ ] [ ] [ ] ℎ 𝑧Ψ(𝑧) 𝑐12 , 𝑐22 d𝑧, 37 , 47 = ∫ 2ℎ 𝑧Φ‵ (𝑧) 𝑐13 , 𝑐23 d𝑧, −2 [ ] [ ] [ ] ℎ 57 , 67 = ∫ 2ℎ Ψ(𝑧)Φ‵ (𝑧) 𝑐13 , 𝑐23 d𝑧, Ψ2 (𝑧) 𝑐11 , 𝑐12 d𝑧,

(21)

−2

ℎ

66 = ∫ 2ℎ 𝑧2 𝑐22 Ψ2 (𝑧)d𝑧, 77 = ∫ 2ℎ 𝑧2 𝑐33 (Φ‵ (𝑧))2 d𝑧, −2 [ ] [ ] −2 [ ] [ ] ℎ ℎ 2 11 , 12 , 13 = ∫ ℎ 𝑐66 1, 𝑧, Ψ(𝑧) d𝑧, 22 , 23 = ∫ 2ℎ 𝑧𝑐66 𝑧, Ψ(𝑧) d𝑧, −2 −2 [ ] [ ] ℎ ℎ 33 = ∫ 2ℎ 𝑐66 Ψ2 (𝑧)d𝑧, 44 , 55 = ∫ 2ℎ Φ2 (𝑧) 𝑐44 , 𝑐55 d𝑧. −2

−2

3. Analytical solution The present boundary conditions for this study associated with the equilibrium equations of simply-supported FG porous nanoplates embedded in an elastic medium are presented as follows: 𝑁𝑥 = 𝑀𝑥 = 𝑆𝑥 = 𝑣0 = 𝑤𝑏 = 𝑤𝑠 = 𝑤𝑠𝑏 = 0 𝑁𝑦 = 𝑀𝑦 = 𝑆𝑦 = 𝑢0 = 𝑤𝑏 = 𝑤𝑠 = 𝑤𝑠𝑏 = 0

at 𝑥 = 0, 𝑎, at 𝑦 = 0, 𝑏.

(22)

The following solution form for u0 , v0 , wb , ws and 𝑤𝑠𝑏 which satisfies the boundary conditions using the Navier’s solution, is assumed as follows: ⎧ ⎫ ⎧ 𝑚𝑛 ⎫ ⎪ 𝑢0 ⎪ ⎪ 𝑈0 cos(𝜃𝑚 𝑥) sin(𝜙𝑛 𝑦) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝑣0 ⎪ ⎪ 𝑉0𝑚𝑛 sin(𝜃𝑚 𝑥) cos(𝜙𝑛 𝑦) ⎪ ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ ⎪ ∑ ∑ ⎪ 𝑚𝑛 ⎪ 𝑖𝜔𝑡 ⎨𝑤𝑏 ⎬ = ⎨𝑊𝑏 sin(𝜃𝑚 𝑥) sin(𝜙𝑛 𝑦)⎬𝑒 , ⎪ ⎪ 𝑚=1 𝑛=1 ⎪ ⎪ ⎪ 𝑤𝑠 ⎪ ⎪𝑊𝑠𝑚𝑛 sin(𝜃𝑚 𝑥) sin(𝜙𝑛 𝑦)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 𝑠⎪ ⎪ 𝑚𝑛 ⎪ ⎪𝑤𝑏 ⎪ ⎪𝑊𝑏𝑠 sin(𝜃𝑚 𝑥) sin(𝜙𝑛 𝑦)⎪ ⎩ ⎭ ⎩ ⎭

(23)

where 𝜃𝑚 = 𝑚𝜋 , 𝜙𝑛 = 𝑛𝜋 and 𝑈0𝑚𝑛 , 𝑉0𝑚𝑛 , 𝑊𝑏𝑚𝑛 , 𝑊𝑠𝑚𝑛 and 𝑊𝑏𝑠𝑚𝑛 are unknown parameters and 𝜔 is the eigenfrequency. Incorporating Eq. (20) into 𝑎 𝑏 Eq. (14) with the help of Eqs. (6) and (23) yields the eigenvalue equation that will give the mechanical buckling, thermal buckling and frequency of 323

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

the FG porous nanoplate as follows: ⎛⎡11 ⎜⎢ ⎜⎢21 ⎜⎢ ⎜⎢31 ⎜⎢ ⎜⎢ 41 ⎜⎢51 ⎝⎣

12

13

14

22

23

24

32

33

34

42

43

44

52

53

54

15 ⎤ ⎡11 ⎥ ⎢ 25 ⎥ ⎢ 0 ⎥ ⎢ 35 ⎥ − 𝜔2 ⎢13 ⎢ 45 ⎥ ⎥ ⎢ 14 ⎥ ⎢ 0 55 ⎦ ⎣

0

13

14

22

23

24

23

33

34

24

34

44

0

35

0

0 ⎤⎞⎧ 𝑈0𝑚𝑛 ⎫ ⎧0⎫ ⎥⎟⎪ ⎪ ⎪ ⎪ 0 ⎥⎟⎪ 𝑉0𝑚𝑛 ⎪ ⎪0⎪ ⎥⎟⎪ ⎪ ⎪ ⎪ 35 ⎥⎟⎨𝑊𝑏𝑚𝑛 ⎬ = ⎨0⎬, 0 ⎥⎟⎪𝑊𝑠𝑚𝑛 ⎪ ⎪0⎪ ⎥⎟⎪ ⎪ ⎪ ⎪ 𝑚𝑛 ⎪ ⎪ ⎪ 55 ⎥⎦⎟⎠⎪ 𝑊 ⎩ 𝑏𝑠 ⎭ ⎩0⎭

(24)

where ( )( ) 11 = 𝜃𝑚2 + 𝜙2𝑛 11 𝜃𝑚2 + 11 𝜙2𝑛 𝜁 + 11 𝜃𝑚2 + 11 𝜙2𝑛 ( )( ) ( ) 12 = 𝜃𝑚 𝜙𝑛 𝜃𝑚2 + 𝜙2𝑛 12 + 11 𝜁 + 𝜃𝑚 𝜙𝑛 12 + 11 , ( 2 )( ) ( ) 13 = −𝜃𝑚 𝜃𝑚 + 𝜙2𝑛 13 𝜃𝑚2 + 14 𝜙2𝑛 + 212 𝜙2𝑛 𝜁 − 𝜃𝑚 13 𝜃𝑚2 + 14 𝜙2𝑛 + 212 𝜙2𝑛 , 14 = −𝜃𝑚 (𝜙2𝑛 + 𝜃𝑚2 )(15 𝜃𝑚2 + 16 𝜙2𝑛 + 213 𝜙2𝑛 )𝜁 − 𝜃𝑚 (15 𝜃𝑚2 + 16 𝜙2𝑛 + 213 𝜙2𝑛 ), 15 = −17 𝜃𝑚 (𝜙2𝑛 + 𝜃𝑚2 )𝜁 − 17 𝜃𝑚 , 22 = (𝜙2𝑛 + 𝜃𝑚2 )(22 𝜙2𝑛 + 11 𝜃𝑚2 )𝜁 + 22 𝜙2𝑛 + 11 𝜃𝑚2 , 23 = −𝜙𝑛 (𝜙2𝑛 + 𝜃𝑚2 )(23 𝜃𝑚2 + 24 𝜙2𝑛 + 212 𝜃𝑚2 )𝜁 − 𝜙𝑛 (23 𝜃𝑚2 + 24 𝜙2𝑛 + 212 𝜃𝑚2 ), 24 = −𝜙𝑛 (𝜙2𝑛 + 𝜃𝑚2 )(25 𝜃𝑚2 + 26 𝜙2𝑛 + 213 𝜃𝑚2 )𝜁 − 𝜙𝑛 (25 𝜃𝑚2 + 26 𝜙2𝑛 + 213 𝜃𝑚2 ), (25)

25 = −27 𝜙𝑛 (𝜙2𝑛 + 𝜃𝑚2 )𝜁 − 27 𝜙𝑛 ,

33 = −(−1 + (−𝜙2𝑛 − 𝜃𝑚2 )𝜇 2 )𝐾0 − (−(𝜙2𝑛 + 𝜃𝑚2 )2 𝜇 2 − 𝜙2𝑛 − 𝜃𝑚2 )𝐾1 + (𝜙2𝑛 + 𝜃𝑚2 )2 (−𝑝 +   )𝜇 2 +(𝜙2𝑛 + 𝜃𝑚2 )(33 𝜃𝑚4 + 234 𝜙2𝑛 𝜃𝑚2 + 44 𝜙4𝑛 + 422 𝜙2𝑛 𝜃𝑚2 )𝜁 + 33 𝜃𝑚4 + 234 𝜃𝑚2 𝜙2𝑛 + 44 𝜙4𝑛 +422 𝜃𝑚2 𝜙2𝑛 + 𝜙2𝑛   + 𝜃𝑚2   − 𝜙2𝑛 𝑝 − 𝜃𝑚2 𝑝,

34 = (𝜙2𝑛 + 𝜃𝑚2 )(35 𝜃𝑚4 + 36 𝜙2𝑛 𝜃𝑚2 + 45 𝜙2𝑛 𝜃𝑚2 + 46 𝜙4𝑛 + 423 𝜙2𝑛 𝜃𝑚2 )𝜁 + 35 𝜃𝑚4 +36 𝜙2𝑛 𝜃𝑚2 + 45 𝜃𝑚2 𝜙2𝑛 + 46 𝜙4𝑛 + 423 𝜃𝑚2 𝜙2𝑛 , 35 = (𝜙2𝑛 + 𝜃𝑚2 )(37 𝜃𝑚2 + 47 𝜙2𝑛 )𝜁 + 37 𝜃𝑚2 + 47 𝜙2𝑛 , 44 = (𝜙2𝑛 + 𝜃𝑚2 )(55 𝜃𝑚4 + 256 𝜙2𝑛 𝜃𝑚2 + 66 𝜙4𝑛 + 433 𝜙2𝑛 𝜃𝑚2 + 44 𝜙2𝑛 + 55 𝜃𝑚2 )𝜁 + 433 𝜃𝑚2 𝜙2𝑛 +256 𝜙2𝑛 𝜃𝑚2 + 66 𝜙4𝑛 + 55 𝜃𝑚2 + 44 𝜙2𝑛 + 55 𝜃𝑚4 , 45 = −(𝜙2𝑛 + 𝜃𝑚2 )(44 𝜙2𝑛 + 55 𝜃𝑚2 − 57 𝜃𝑚2 − 67 𝜙2𝑛 )𝜁 − 44 𝜙2𝑛 − 55 𝜃𝑚2 + 57 𝜃𝑚2 + 67 𝜙2𝑛 , ( ) 55 = (𝜙2𝑛 + 𝜃𝑚2 )(44 𝜙2𝑛 + 55 𝜃𝑚2 + 77 )𝜁 + 44 𝜙2𝑛 + Φ(𝑧)(𝜙2𝑛 + 𝜃𝑚2 )(𝑝 −   ) 1 + 𝜇 2 (𝜙2𝑛 + 𝜃𝑚2 ) +55 𝜃𝑚2 + 77 .

11 = 𝐼0 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 + 𝐼0 , 14 = 23 = 33 = 34 = 44 =

13 = −𝐼1 𝜃𝑚 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 − 𝜃𝑚 𝐼1 , 2 2 2 −𝐼2 𝜃𝑚 (𝜙𝑛 + 𝜃𝑚 )𝜇 − 𝜃𝑚 𝐼2 , 22 = 𝐼0 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 + 𝐼0 , 2 2 2 −𝐼1 𝜙𝑛 (𝜙𝑛 + 𝜃𝑚 )𝜇 + 𝜙𝑛 𝐼1 , 24 = −𝐼2 𝜙𝑛 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 − 𝜙𝑛 𝐼2 , (𝜙2𝑛 + 𝜃𝑚2 )(𝐼2 𝜙2𝑛 + 𝐼2 𝜃𝑚2 + 𝐼0 )𝜇 2 + 𝐼2 𝜙2𝑛 + 𝐼2 𝜃𝑚2 + 𝐼0 , 𝐼4 (𝜙2𝑛 + 𝜃𝑚2 )2 𝜇 2 + 𝐼4 𝜙2𝑛 + 𝐼4 𝜃𝑚2 , 35 = 𝐼3 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 + 𝐼3 , 𝐼5 (𝜙2𝑛 + 𝜃𝑚2 )2 𝜇 2 + 𝐼5 𝜙2𝑛 + 𝐼5 𝜃𝑚2 , 55 = 𝐼6 (𝜙2𝑛 + 𝜃𝑚2 )𝜇 2 + 𝐼6 .

(26)

For the mechanical buckling and thermal buckling loads (𝜔 = 0), while for vibration analysis (𝑝 = Δ𝑇 = 0).

4.1. Validation of the present quasi-3D model In this section, the results of mechanical buckling, thermal buckling and vibration analysis of simply-supported homogeneous and FG nanoplates embedded in an elastic medium are compared with those available in the literature to discuss the accuracy of the proposed model. In the case of a quasi-3D solution, the buckling load and thermal buckling load are evaluated at the plate center 𝑧 = 0. Table 1 shows the comparison of non-dimensional thermal buckling load ΓT of a homogeneous nanoplate embedded in an elastic medium (ℎ = 0.34) nm. Good agreements can be clearly observed with the solutions of Zenkour and Sobhy [92]. Increasing the ratio a/h leads to decrease in the thermal buckling load ΓT . Also, the results for ΓT increase with the presence of Winkler or Pasternak elastic medium but it decreases with the increase of the aspect ratio a/b. It is found that the presence of the nonlocal parameter (𝜇 = 1) has a significant effect on the results, where it leads to a considerable reduction in the thermal buckling load. Comparison of non-dimensional vibration 𝜔0 of FG plates is shown in Table 2. In this table, the frequency for different values of gradient index (𝜅), aspect ratio (b/a) and side-to-thickness ratio (a/h) is calcu-

4. Numerical results The effects of the nonlocal coefficient, strain gradient parameter and porosity factor on the critical buckling load, thermal buckling load and fundamental frequency of FG porous nanoplates embedded in an elastic medium are investigated in this section. The following dimensionless quantities are used in the present results as follows: Γ= 𝜔0

𝑎2 𝑝, 𝐷 √

Γ𝑇 = 103 𝛼𝑐 Δ𝑇 ,

= 𝜔ℎ 𝜌𝑚 ∕𝐸𝑚 ,

𝜅𝑊 =

𝑎4 𝐾 , 𝐷 0

𝜅𝑃 =

√ 𝜔∗ = 𝜔ℎ 𝜌𝑐 ∕𝐸𝑐 ,

𝑎2 𝐾 , 𝐷 1

𝐷=

ℎ3

𝐸𝑐 . 12(1−𝜈 2 )

𝜔̄ = 𝜔 𝜋𝑏 2 2

√

𝜌𝑐 ℎ ∕ 𝐷 ,

(27)

The material properties of Alumina are: Ec = 380 GPa, 𝛽𝑐 = 23 × 10−6 K−1 and 𝜌𝑐 = 3800 kg/m3 , whereas the characteristics of Aluminium are given as: Em = 70 GPa, 𝛽 m = 7 ×10−6 K−1 , 𝜌𝑚 = 2707 kg/m3 and 𝑎 = 10 nm. 324

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

Table 1 Comparison of non-dimensional thermal buckling load ΓT of a homogeneous nanoplate embedded in an elastic medium (ℎ = 0.34) nm. Local (𝜇 = 0) 𝜅w

𝜅p

𝑏 𝑎

0

0

1

2

3

100

0

1

2

3

100

10

1

2

3

Theory

𝜀z

𝑎 ℎ

Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present Ref. [92] Present Present

=0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0

0.5050 0.5050 0.5069 0.3159 0.3159 0.3171 0.2809 0.2808 0.2819 0.6349 0.6349 0.6369 0.5237 0.5237 0.5249 0.5147 0.5147 0.5157 0.8913 0.8913 0.8933 0.7802 0.7801 0.7813 0.7711 0.7711 0.7721

= 50

Nonlocal (𝜇 = 1)

25

10

𝑎 ℎ

2.0065 2.0065 2.0184 1.2583 1.2582 1.2645 1.1192 1.1191 1.1245 2.5261 2.5261 2.5380 2.0896 2.0896 2.0958 2.0544 2.0544 2.0597 3.5517 3.5517 3.5637 3.1153 3.1152 3.1215 3.0801 3.0801 3.0854

11.9793 11.9793 12.2115 7.6396 7.6396 7.7431 6.8165 6.8165 6.9013 15.2268 15.2267 15.4593 12.8355 12.8355 12.9393 12.6619 12.6619 12.7469 21.6370 21.6370 21.8702 19.2458 19.2458 19.3498 19.0722 19.0722 19.1574

0.4728 0.4727 0.4746 0.3030 0.3030 0.3041 0.2706 0.27057 0.2716 0.6027 0.6026 0.6045 0.5108 0.5108 0.5119 0.5044 0.5044 0.5054 0.8591 0.8590 0.8609 0.7672 0.7672 0.7683 0.7609 0.7609 0.7618

= 50

25

10

1.5759 1.5759 1.5853 1.0747 1.0747 1.0801 0.9717 0.9717 0.9763 2.0955 2.0955 2.1049 1.9061 1.9061 1.9114 1.9069 1.9069 1.9116 3.1212 3.1211 3.1305 2.9317 2.9317 2.9370 2.9326 2.9326 2.9372

4.4244 4.4244 4.5101 3.6956 3.6956 3.7457 3.4981 3.4981 3.5416 7.6719 7.6719 7.7579 8.8915 8.8915 8.9419 9.3435 9.3435 9.3873 14.0821 14.0821 14.1689 15.3018 15.3018 15.3524 15.7538 15.7538 15.7977

Table 2 Comparison of non-dimensional vibration 𝜔0 of FG plates. 𝑏∕𝑎 = 1

𝑏∕𝑎 = 2

a/h

Theory

𝜀z

𝜅=0

1

2

5

𝜅=0

1

2

5

10

Ref. [93] Ref. [94] Present Present Ref. [93] Ref. [94] Present Present Ref. [93] Ref. [94] Present Present

≠0 =0 =0 ≠0 ≠0 =0 =0 ≠0 ≠0 =0 =0 ≠0

0.1135 0.1137 0.1135 0.1138 0.4169 0.4183 0.4155 0.4182 1.8470 1.8543 1.8304 1.8600

0.0870 0.0883 0.0869 0.0884 0.3222 0.3271 0.3207 0.3269 1.4687 1.4803 1.4474 1.4838

0.0789 0.0806 0.0788 0.0808 0.2905 0.2965 0.2893 0.2969 1.3095 1.3224 1.2906 1.3274

0.0741 0.0756 0.0740 0.0756 0.2676 0.2726 0.2666 0.2725 1.1450 1.1565 1.1312 1.1578

0.0719 0.0719 0.0718 0.0719 0.2713 0.2721 0.2707 0.2720 0.9570 1.3075 1.2926 1.3098

0.0550 0.0558 0.0549 0.0558 0.2088 0.2121 0.2082 0.2120 0.7937 1.0371 1.0146 1.0384

0.0499 0.0510 0.0499 0.0512 0.1888 0.1928 0.1883 0.1931 0.7149 0.9297 0.9072 0.9326

0.0471 0.0480 0.0470 0.0479 0.1754 0.1789 0.1750 0.1788 0.6168 0.8248 0.8063 0.8251

5

2

lated using the present quasi-3D and 2D solutions and compared with the solution given by Jin et al. [93] and Mantari [94]. It is found that good agreement results occurred. Table 3 shows the comparison of non-dimensional vibration 𝜔∗ of simply-supported FG plates for the first three modes (m, n). The results for different values of the gradient index (𝜅) is realized by the present quasi-3D and 2D solutions and compared with the 2D solution which was presented by Hosseini-Hashemi et al. [95] and quasi-3D solution which was presented by Abulnour et al. [96]. It is noted that the present models give results identical to the published ones. Investigations of the vibration, buckling and thermal buckling loads solutions for FG nanoplates without elastic medium or embedded in an elastic medium are shown in Tables 4, 5 and 6, respectively. In this com( ) 𝜅 𝑃 parison the power low 𝑃 (𝑧) = 𝑃𝑚 𝑃 𝑐 is used. Tables 4, 5 and 6 show

4.2. Parametric results for an FG porous nanoplate After confirming the validity of the current theories and formulations, some new results are obtained to show the effects of the porosity factor, nonlocal, length scale, foundation, side-to-thickness ratio a/h, nanoplate aspect ratio a/b, gradient index 𝜅 parameters on the mechanical buckling, thermal buckling loads and vibration analysis of FG porous nanoplates embedded in an elastic medium. Tables 7, 8, and 9 present the influences of the porosity factor 𝛼, length scale 𝜁 and nonlocal 𝜇 parameters on the critical buckling load Γ, thermal buckling load ΓT and the vibration 10𝜔0 of a square FG porous nanoplate embedded in Pasternak medium using the present 2D and quasi-3D solutions, respectively. Table 7 shows the effects of porosity factor, length scale and nonlocal parameters on the buckling load Γ of a square FG porous nanoplate embedded in Pasternak medium. It is noted that the buckling load Γ decreases with the increase of the porosity factor and the nonlocal parameter 𝜇 but it increases by the increase of the length scale parameter 𝜁 . Table 8 illustrates the effects of porosity factor, length scale and nonlocal parameters on the thermal buckling load ΓT

𝑚

the comparison of non-dimensional vibration 𝜔̄ , buckling Γ and thermal buckling ΓT of square FG nanoplates without an elastic medium, embedded in an Winkler foundation and embedded in Winkler-Pasternak foundations, respectively. For the present quasi-3D solution, the results are identical to the 2D solution given by Sobhy [97].

325

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International Journal of Mechanical Sciences 157–158 (2019) 320–335

Table 3 Comparison of non-dimensional vibration 𝜔∗ of FG plates for the first three modes (m, n). 𝑎∕ℎ = 5

𝑎∕ℎ = 10

𝑎∕ℎ = 20

𝜅

Theory

𝜀z

(1,1)

(1,2)

(2,2)

(1,1)

(1,2)

(2,2)

(1,1)

0

Ref. [95] Ref. [96] Present Present Ref. [95] Ref. [96] Present Present Ref. [95] Ref. [96] Present Present Ref. [95] Ref. [96] Present Present Ref. [95] Ref. [96] Present Present

=0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0 =0 ≠0

0.2113 0.2126 0.2113 0.2126 0.1807 0.1829 0.1807 0.1828 0.1631 0.1663 0.1631 0.1662 0.1378 0.1411 0.1378 0.1410 0.1301 0.1320 0.1299 0.1319

0.4623 0.4674 0.4624 0.4674 0.3989 0.4052 0.3989 0.4051 0.3607 0.3687 0.3607 0.3686 0.2980 0.3052 0.2975 0.3050 0.2771 0.2817 0.2768 0.2814

0.6688 0.6783 0.6693 0.6783 0.5803 0.5911 0.5805 0.5909 0.5254 0.5381 0.5255 0.5379 0.4284 0.4389 0.4277 0.4386 0.3948 0.4018 0.3945 0.4015

0.0577 0.0579 0.0577 0.0579 0.0490 0.0495 0.0490 0.0494 0.0442 0.0450 0.0442 0.0449 0.0381 0.0390 0.0380 0.0389 0.0364 0.0369 0.0363 0.0368

0.1377 0.1383 0.1377 0.1383 0.1174 0.1186 0.1174 0.1186 0.1059 0.1078 0.1059 0.1078 0.0903 0.0924 0.0902 0.0923 0.0856 0.0868 0.0855 0.0868

0.2113 0.2126 0.2113 0.2126 0.1807 0.1829 0.1807 0.0499 0.1631 0.1663 0.1631 0.1662 0.1378 0.1411 0.1377 0.1410 0.1301 0.1320 0.1299 0.1319

0.0148 0.0148 0.0148 0.0148 0.0125 0.0126 0.0125 0.1828 0.0113 0.0115 0.0113 0.0115 0.0098 0.0100 0.0098 0.0100 0.0094 0.0095 0.0094 0.0095

0.5

1

4

10

Table 4 Comparison of non-dimensional vibration 𝜔̄ , buckling load Γ and thermal buckling load ΓT of a square FG nanoplates (𝜅𝑤 = 𝜅𝑝 = 0, 𝑎 = 10ℎ). 𝜇=0

𝜇=2

𝜅

Theory

𝜀z

𝜔̄

Γ

ΓT

𝜔̄

Γ

ΓT

0

Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present

=0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0

1.9318 1.9318 1.9377 1.4969 1.4969 1.5182 1.2572 1.2572 1.2844 1.2087 1.2087 1.2258 1.1609 1.1609 1.1707

18.6876 18.6876 19.0499 10.0638 10.0638 10.4699 6.2593 6.2593 6.5976 5.5200 5.5200 5.7397 4.9677 4.9677 5.1128

11.9793 11.9793 12.2115 7.5745 7.5745 7.8801 5.6795 5.6795 5.9864 5.3680 5.3680 5.5816 5.0102 5.0102 5.1565

1.4441 1.4441 1.4485 1.1189 1.1189 1.1349 0.9397 0.9398 0.9601 0.9035 0.9035 0.9163 0.8678 0.8678 0.8752

10.4425 10.4425 10.6449 5.6235 5.6235 5.8505 3.4976 3.4976 3.6867 3.0845 3.0845 3.2073 2.7759 2.77597 2.8570

6.6939 6.6939 6.8237 4.2326 4.2326 4.4033 3.1737 3.1737 3.3452 2.9996 2.9996 3.1190 2.7997 2.7997 2.8814

0.5

2.5

5.5

10.5

Table 5 Comparison of non-dimensional vibration 𝜔̄ , buckling load Γ and thermal buckling load ΓT of a square FG nanoplates embedded (𝜅𝑤 = 100, 𝑎 = 10ℎ). 𝜇=0

𝜇=2

𝜅

Theory

𝜀z

𝜔̄

Γ

ΓT

𝜔̄

Γ

ΓT

0

Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present

=0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0

2.1780 2.1780 2.1802 1.8354 1.8354 1.8493 1.6910 1.6910 1.7077 1.6738 1.6738 1.6818 1.6499 1.6499 1.6517

23.7537 23.7537 24.1165 15.1298 15.1298 15.5364 11.3254 11.3254 11.6640 10.5860 10.5861 10.8062 10.0338 10.0338 10.1793

15.2267 15.2267 15.4593 11.3874 11.3874 11.6933 10.2763 10.2763 10.5835 10.2945 10.2945 10.5086 10.1196 10.1196 10.2664

1.7598 1.7598 1.7597 1.5427 1.5427 1.5501 1.4704 1.4704 1.4792 1.4686 1.4686 1.4715 1.4585 1.4586 1.4571

15.5086 15.5086 15.7115 10.6896 10.6896 10.9170 8.5637 8.5637 8.7531 8.1506 8.1506 8.2738 7.8420 7.8420 7.9235

9.9414 9.9414 10.0715 8.0455 8.0455 8.2166 7.7705 7.7705 7.9423 7.9261 7.9261 8.0459 7.9091 7.9091 7.9913

0.5

2.5

5.5

10.5

326

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International Journal of Mechanical Sciences 157–158 (2019) 320–335

Table 6 Comparison of non-dimensional vibration 𝜔̄ , buckling load Γ and thermal buckling load ΓT of a square FG nanoplates resting on Winkler-Pasternak foundations (𝜅𝑤 = 100, 𝜅𝑝 = 50, 𝑎 = 10ℎ). 𝜇=0

𝜇=2

𝜅

Theory

𝜀z

𝜔̄

Γ

ΓT

𝜔̄

Γ

ΓT

0

Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present Ref. [97] Present Present

=0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0 =0 =0 ≠0

3.8377 3.8377 3.8209 3.8077 3.8077 3.7963 3.9339 3.9339 3.9233 4.0031 4.0031 3.9855 4.0349 4.0349 4.0114

73.7537 73.7537 74.1215 65.1298 65.1298 65.5404 61.3254 61.3254 61.6675 60.5860 60.5861 60.8102 60.0338 60.0338 60.1839

47.2780 47.2780 47.5137 49.0196 49.0196 49.3285 55.6447 55.6447 55.9551 58.9175 58.9175 59.1355 60.5472 60.5472 60.6986

3.6167 3.6167 3.5968 3.6753 3.6753 3.6584 3.8432 3.8433 3.8269 3.9206 3.9206 3.8982 3.8553 3.8554 3.8554

65.5086 65.5086 65.7165 60.6896 60.6896 60.9210 58.5637 58.5637 58.7565 58.1506 58.1506 58.2778 57.8420 57.8420 57.9281

41.9927 41.9927 42.1259 45.6777 45.6777 45.8517 53.1389 53.1389 53.3138 56.5491 56.5491 56.6728 58.3367 58.3367 58.4235

0.5

2.5

5.5

10.5

Table 7 Non-dimensional critical buckling load Γ of a square FG porous nanoplate embedded in a Pasternak medium under different porosity factor, length scale and nonlocal parameters (𝜅 = 𝑎∕ℎ = 10, 𝜅𝑤 = 100, 𝜅𝑝 = 10). Present 2D

Present quasi-3D

𝜁

𝜇

𝛼=0

0.1

0.2

0.3

𝛼=0

0.1

0.2

0.3

0

0 1 2 0 1 2 0 1 2 0 1 2

20.5493 19.6454 18.1301 21.6316 20.5493 18.7349 22.7139 21.4532 19.3397 23.7963 22.3571 19.9445

19.2056 18.5232 17.3792 20.0227 19.2056 17.8358 20.8398 19.8880 18.2924 21.6569 20.5704 18.7490

17.6704 17.2411 16.5214 18.1845 17.6704 16.8086 18.6986 18.0998 17.0959 19.2127 18.5291 17.3832

15.5266 15.4507 15.3234 15.6175 15.5266 15.3742 15.7084 15.6025 15.4249 15.7993 15.6784 15.4758

20.7599 19.8215 18.2483 21.8837 20.7599 18.8763 23.0074 21.6984 19.5042 24.1311 22.6369 20.1321

19.4125 18.6961 17.4953 20.2702 19.4125 17.9746 21.1279 20.1288 18.4539 21.9857 20.8452 18.9332

17.8714 17.4091 16.6340 18.4251 17.8714 16.9434 18.9787 18.3338 17.2527 19.5323 18.7961 17.5621

15.6113 15.5214 15.3707 15.7189 15.6113 15.4309 15.8265 15.7012 15.4910 15.9341 15.7910 15.5511

1

2

3

Table 8 Non-dimensional thermal buckling load ΓT of a square FG porous nanoplate embedded in a Pasternak medium under different porosity factor, length scale and nonlocal parameters (𝜅 = 𝑎∕ℎ = 10, 𝜅𝑤 = 100, 𝜅𝑝 = 10). Present 2D

Present quasi-3D

𝜁

𝜇

𝛼=0

0.1

0.2

0.3

𝛼=0

0.1

0.2

0.3

0

0 2 3 0 2 3 0 2 3 0 2 3

18.2479 16.0996 15.1325 19.2090 16.6367 15.4786 20.1702 17.1738 15.8248 21.1313 17.7108 16.1709

24.7739 22.4179 21.3573 25.8279 23.0069 21.7369 26.8819 23.5959 22.11654 27.9359 24.1849 22.4962

38.0875 35.6108 34.4957 39.1955 36.2299 34.8948 40.3036 36.8491 35.2939 41.4119 37.4683 35.6929

80.7683 79.7114 79.2356 81.2412 79.9756 79.4059 81.7140 80.2399 79.5762 82.1869 80.5041 79.7465

18.4350 16.2046 15.2005 19.4328 16.7622 15.5599 20.4307 17.3198 15.9193 21.4285 17.8774 16.2787

25.0407 22.5676 21.4523 26.1471 23.1859 21.8527 27.2535 23.8041 22.2512 28.3599 24.4224 22.6497

38.5207 35.8535 34.6527 39.7140 36.5203 35.0825 40.9073 37.1871 35.5123 42.1006 37.8539 35.9421

81.2089 79.9577 79.3944 81.7687 80.2705 79.5960 82.3285 80.5833 79.7976 82.8883 80.8961 79.9992

1

2

3

of a square FG porous nanoplate embedded in Pasternak medium. One can note that the thermal buckling load ΓT increases with the increase of the porosity factor and length scale parameters but it decreases by the increase of the nonlocal parameter 𝜇. Table 9 illustrates the effects of porosity factor, length scale and nonlocal parameters on the vibration 10𝜔0 of a square FG porous nanoplate embedded in Pasternak medium. One can note that the vibration 10𝜔0 increases with the increase of the porosity factor and length scale parameters but it decreases by the increase of the nonlocal parameter 𝜇. Next, parametric studies are performed and typical results are exhibited in Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 for buckling load (see,

Figs. 2 and 3), thermal buckling load (see, Figs. 4, 5, 6, 7) and vibration analysis (see, Figs. 8, 9, 10, 11, 12). Fig. 2 exhibits the variation of the buckling load against the side-to-thickness ratio a/h of a FG porous square nanoplate resting on elastic foundations for various values of the nonlocal and length scale parameters (𝛼 = 0.1, 𝜅 = 2, 𝜅𝑤 = 100, 𝜅𝑝 = 10). It can be noted that the buckling load increases by increasing the ratio a/h and decreasing the nonlocal parameter 𝜇, whereas it has an opposite behavior for the length scale parameter 𝜁 . Fig. 3 shows the variation of the buckling load against the aspect ratio a/b of an FG porous nanoplate resting on elastic foundations for various values of the nonlocal and length scale parameters (𝛼 = 0.1, 𝜅 = 327

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

Table 9 Non-dimensional vibration 10𝜔0 of a square FG porous nanoplate embedded in a Pasternak medium under different porosity factor, length scale and nonlocal parameters (𝜅 = 𝑎∕ℎ = 10, 𝜅𝑤 = 100, 𝜅𝑝 = 10). Present 2D

Present quasi-3D

𝜁

𝜇

𝛼=0

0.1

0.2

0.25

𝛼=0

0.1

0.2

0.25

0

0 1 2 0 1 2 0 1 2 0 1 2

1.3831 1.3524 1.2991 1.4191 1.3831 1.3206 1.4542 1.4133 1.3418 1.4885 1.4427 1.3626

1.4211 1.3956 1.3517 1.4511 1.4211 1.3694 1.4804 1.4462 1.3868 1.5092 1.4708 1.4041

1.4594 1.4414 1.4106 1.4806 1.4594 1.4230 1.5015 1.4772 1.4353 1.5221 1.4946 1.4474

1.4731 1.4606 1.4389 1.4879 1.4731 1.4477 1.5024 1.4855 1.4563 1.5167 1.4976 1.4648

1.3823 1.3506 1.2957 1.4193 1.3823 1.3179 1.4553 1.4133 1.3397 1.4905 1.4435 1.3612

1.4214 1.3949 1.3489 1.4526 1.4214 1.3675 1.4831 1.4475 1.3857 1.5129 1.4731 1.4037

1.4623 1.4431 1.4100 1.4849 1.4623 1.4234 1.5072 1.4812 1.4365 1.5291 1.4999 1.4495

1.4779 1.4642 1.4406 1.4941 1.4779 1.4501 1.5099 1.4914 1.4595 1.5257 1.5048 1.4688

1

2

3

Fig. 2. Effect of (a) the nonlocal and (b) length scale parameters on the buckling load through the side-to-thickness ratio a/h of an FG porous square nanoplate embedded in an elastic medium.

Fig. 3. Effect of (a) the nonlocal and (b) length scale parameters on the buckling load through the aspect ratio a/b of an FG porous square nanoplate embedded in an elastic medium.

328

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International Journal of Mechanical Sciences 157–158 (2019) 320–335

Fig. 4. Effect of (a) the gradient index and (b) porosity factor on the thermal buckling load through the nonlocal parameter 𝜇 of an FG porous square nanoplate embedded in an elastic medium.

Fig. 5. Effect of (a) the gradient index and (b) porosity factor on the thermal buckling load through the side-to-thickness ratio a/h of an FG porous square nanoplate embedded in an elastic medium.

2, 𝜅𝑤 = 100, 𝜅𝑝 = 10, 𝑎 = 10ℎ). It is seen that the buckling load decreases up to the ratio 𝑎∕𝑏 = 1 then it increases by increasing the ratio a/b and decreasing the nonlocal parameter 𝜇, whereas it has an opposite behavior for the length scale parameter 𝜁 . Fig. 4 illustrates the effect of the gradient index and porosity factor on the thermal buckling load through the nonlocal parameter 𝜇 of an FG porous square nanoplate embedded in an elastic medium (𝜁 = 2, 𝜅𝑤 = 100, 𝜅𝑝 = 10, 𝑎 = 10ℎ). It can be observed that the thermal buckling load decreases by increasing the nonlocal parameter 𝜇 and decreasing the gradient index 𝜅, whereas it has the same behavior for the porosity factor 𝛼. Fig. 5 shows the effect of the gradient index and porosity factor on the thermal buckling load against the side-to-thickness ratio a/h

of an FG porous square nanoplate embedded in an elastic medium (𝜇 = 𝜁 = 2, 𝜅𝑤 = 100, 𝜅𝑝 = 10). It is observable that the thermal buckling load decreases by increasing the ratio a/h. It is also noted that the increment in the gradient index 𝜅 increases the strength of the plate that leads to an increment in thermal buckling temperature (see, Fig. 5a). Moreover, the thermal buckling increase with the increase of the porosity factor 𝛼 as shown in Fig 5b. Fig. 6 gives the variation of the thermal buckling load against the aspect ratio a/b of an FG porous nanoplate embedded in an elastic medium for various values of the gradient index and porosity factor (𝜇 = 𝜁 = 2, 𝜅𝑤 = 100, 𝜅𝑝 = 10, 𝑎 = 10ℎ). It is clear that the thermal buckling load decreases up to the ratio 𝑎∕𝑏 = 1 then it increases by increasing the ratio a/b as shown in Fig 6a and Fig 6b.

329

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International Journal of Mechanical Sciences 157–158 (2019) 320–335

Fig. 6. Effect of (a) the gradient index and (b) porosity factor on the thermal buckling load through the aspect ratio a/b of an FG porous square nanoplate embedded in an elastic medium.

Fig. 7. Effect of (a) the length scale and (b) foundation parameters on the thermal buckling load through the nonlocal parameter 𝜇 of an FG perfect square nanoplate embedded in an elastic medium.

Fig. 7 illustrates the effect of the length scale and foundation parameters on the thermal buckling load through the nonlocal parameter 𝜇 of an FG perfect (𝛼 = 0) square nanoplate embedded in an elastic medium (𝜅 = 2, 𝑎 = 10ℎ). It is noticed that the thermal buckling load decreases by increasing the nonlocal parameter 𝜇 but it increases with the increase of the length scale parameter 𝜁 as shown in Fig 7a. It is also observed that the Pasternak parameter 𝜅 P has a greater effect on the thermal buckling than the Winkler parameter 𝜅 W as shown in Fig 7b. In addition, the thermal buckling increases with the increase of both Pasternak and Winkler parameters. Fig. 8 shows the effect of the nonlocal parameter on the vibration through the aspect ratio a/b of an FG porous nanoplate

without elastic foundation and embedded in a Winkler foundation (𝛼 = 0.1, 𝜅 = 2, 𝜁 = 1, 𝑎 = 10ℎ). It should be noted that the vibration increases by increasing the ratio a/b and decreasing the nonlocal parameter 𝜇. Fig. 9 illustrates the effect of the length scale and foundation parameters on the vibration against the aspect ratio a/b of an FG porous nanoplate embedded in an elastic medium (𝛼0.1, 𝜅 = 2, 𝜇 = 1, 𝑎 = 10ℎ). It is found that the vibration increases by increasing the ratio a/b and increasing the length scale parameter 𝜁 . In addition, it has the same behavior for the foundation coefficients 𝜅 w and 𝜅 p . Fig. 10 shows the effect of the gradient index and length scale parameters on the vibration versus the nonlocal parameter 𝜇 of an FG 330

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Fig. 8. Effect of the nonlocal parameter on the vibration through the aspect ratio a/b of an FG porous nanoplate (a) without and (b) resting on Winkler foundation.

Fig. 9. Effect of (a) the length scale and (b) foundation parameters on the vibration through the aspect ratio a/b of an FG porous nanoplate embedded in an elastic medium.

porous square nanoplate embedded in an elastic medium (𝛼 = 0.1, 𝑎 = 10ℎ). It can be observed that the vibration decreases by increasing the gradient index 𝜅 and decreasing the nonlocal parameter 𝜇 as shown in Fig. 10a. It is also remarked that the vibration increase with the increase of the length scale parameter 𝜁 as shown in Fig. 10b. Fig. 11 illustrates the effect of the length scale and nonlocal parameters on the vibration versus the side-to-thickness ratio a/h of an FG porous square nanoplate embedded in an elastic medium (𝛼 = 0.1, 𝜅 = 2, 𝜅𝑤 = 100, 𝜅𝑝 = 10). It is observed that the vibration increases by increasing the side-to-thickness ratio a/h and increasing the length scale

of 𝜁 as shown in Fig. 11a. In addition, it has the opposite behavior for the nonlocal parameter 𝜇 as shown in Fig. 11b. Finally, Fig. 12 demonstrates the effect of the gradient index and foundation parameters on the vibration versus the side-to-thickness ratio a/h of an FG porous nanoplate embedded in an elastic medium (𝜁 = 𝜇 = 1, 𝛼 = 0.1). It is noticed that the vibration decreases by increasing the gradient index 𝜅 and increasing the side-to-thickness ratio a/h as shown in Fig 12a. It is also observed that the vibration has the opposite behavior for the parameters of the elastic foundation 𝜅 w and 𝜅 p as shown in Fig 12b.

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Fig. 10. Effect of (a) the gradient index and (b) length scale parameters on the vibration through the nonlocal parameter 𝜇 of an FG porous square nanoplate embedded in an elastic medium.

Fig. 11. Effect of (a) the length scale and (b) nonlocal parameters on the vibration through the side-to-thickness ratio a/h of an FG porous square nanoplate embedded in an elastic medium.

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Fig. 12. Effect of (a) the gradient index and (b) foundation parameters on the vibration through the side-to-thickness ratio a/h of an FG porous nanoplate embedded in an elastic medium.

5. Conclusions

[5] Belabed Z, Houari MSA, Tounsi A, Mahmoud SR, Bég OA. An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Compos Part B 2014;60:274–83. [6] Mantari JL, Soares CG. A novel higher-order shear deformation theory with stretching effect for functionally graded plates. Compos Part B 2013;45: 268–281. [7] Yahia SA, Atmane H, Houari MSA, Tounsi A. Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct Eng Mech 2015;53:1143–65. [8] Akbaş ŞD. Thermal effects on the vibrations of functionally deep beams with porosity. J Appl Mech 2017;9:1750076. [9] Akbaş ŞD. Post-buckling responses of functionally graded beams with porosities. Steel Compos Struct 2017;24:579–89. [10] Akbaş ŞD. Vibration and static analysis of functionally graded porous plates. J Appl Comput Mech 2017;3:199–207. [11] Rezaei AS, Saidi AR. Exact solution for free vibration of thick rectangular plates made of porous materials. Compos Struct 2015;134:1051–60. [12] Rad AB, Shariyat M. Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations. Compos Struct 2015;125:558–74. [13] Rezaei AS, Saidi AR. Application of Carrera unified formulation to study the effect of porosity on natural frequencies of thick porous-cellular plates. Compos B 2016;91:361–70. [14] Shafiei N, Mousavi A, Ghadiri M. On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. Int J Eng Sci 2016;106:42–56. [15] Chen D, Kitipornchai S, Yang J. Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Struct 2016;107:39–48. [16] Shafiei N, Mirjavadi SS, Mohasel Afshari B, Rabby S, Kazemi M. Vibration of two-dimensional imperfect functionally graded (2d-FG) porous nano-/micro-beams. Comput Methods Appl Mech Eng 2017;322:615–32. [17] Wu D, Liu A, Huang Y, Huang Y, Pi Y, Gao W. Dynamic analysis of functionally graded porous structures through finite element analysis. Eng Struct 2018;165:287–301. [18] Arshid E, Khorshidvand AR. Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method. Thin-Walled Struct 2018;125:220–33. [19] Chen D, Kitipornchai S, Yang J. Dynamic response and energy absorption of functionally graded porous structures. Mater Des 2018;140:473–87. [20] Barati MR. A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates. Eur J Mech A Solids 2018;67:215– 230. [21] Khorshidvand AR, Joubaneh EF, Jabbari M, Eslami MR. Buckling analysis of a porous circular plate with piezoelectric sensorctuator layers under uniform radial compression. Acta Mech 2014;225:179–93. [22] Mojahedin A, Joubaneh EF, Jabbari M. Thermal and mechanical stability of a circular porous plate with piezoelectric actuators. Acta Mech 2014;225:3437– 3452. [23] Joubaneh EF, Mojahedin A, Khorshidvand AR, Jabbari M. Thermal buckling analysis of porous circular plate with piezoelectric sensor-actuator layers under uniform thermal load. J Sandwich Struct Mater 2015;17:3–25. [24] Barati MR, Sadr MH, Zenkour AM. Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation. Int J Mech Sci 2016;117:309–20.

Mechanical buckling, thermal buckling loads and free vibration analysis of FG porous nanoplate embedded in an elastic medium are all investigated using the nonlocal strain gradient theory. Two integral models as 2D and quasi-3D are developed by making a further simplifying assumption to the existing higher shear deformation theories. A law distribution is developed to show the mechanical properties of the FG porous nanoplate that vary smoothly and continuously from one surface to another. The equations of motion for the present problem are derived based on Hamilton’s principle for the 2D and quasi-3D models. The results obtained by the 2D and quasi-3D solutions are compared with those available in the literature to check the accuracy of these integral models. Moreover, detailed parametric studies are carried out to investigate the influences of the porosity factor, normal strain, geometrical parameters, nonlocal parameter, length scale parameter, gradient index and the foundation parameters on the buckling load, thermal buckling load and free vibration of FG porous nanoplates. On the basis of the results of these studies, it is concluded that the gradient index has increasing influences on the thermal buckling load, while it has a decreasing impact on the buckling load and free vibration of FG porous nanoplates. The large values of the foundation parameters increase the buckling, thermal buckling loads and free vibration of the FG porous nanoplate. With the increase of the nonlocal parameter, a decrement occurs for the stiffness of the FG porous nanoplates whereas it has an opposite behavior for the length scale parameter. Finally, with the increase of porosity factor, a decrement occurs for the buckling load while an increase occurs for the thermal buckling load and free vibrations behaviors. The present study can be extended in the future to investigate the bending response of multi-layered composites and soft core sandwich plates with general boundary conditions subjected to hygrothermal loading.

References [1] Niino M, Hirai T, Watanabe R. The functionally gradient materials. J Jpn Soc Compos Mater 1987;13:257–64. [2] Matsunaga H. Free vibration and stability of functionally graded plates according to a 2d higher order deformation theory. Compos Struct 2008;82:499–512. [3] Zhao X, Lee YY, Liew KM. Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib 2009;319:918–39. [4] Reddy JN, Chin CD. Thermo-mechanical analysis of functionally graded cylinders and plates. J Therm Stress 1998;21:593–626. 333

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

[25] Mojahedin A, Jabbari M, Khorshidvand AR, Eslami MR. Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory. Thin-Walled Struct 2016;99:83–90. [26] Feyzi MR, Khorshidvand AR. Axisymmetric post-buckling behavior of saturated porous circular plates. Thin-Walled Struct 2017;112:149–58. [27] Rezaei AS, Saidi AR. Buckling response of moderately thick fluid-infiltrated porous annular sector plates. Acta Mech 2017;228:3929–45. [28] Cong PH, Chien TM, Khoa ND, Duc ND. Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy’s HSDT. Aerosp Sci Technol 2018;77:419–28. [29] Chen D, Yang J, Kitipornchai S. Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams. Compos Sci Technol 2017;142:235–45. [30] Kitipornchai S, Chen D, Yang J. Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater Des 2017;116:656–65. [31] Yang J, Chen D, Kitipornchai S. Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method. Compos Struct 2018;193:281–94. [32] Ebrahimi F, Barati MR, Dabbagh A. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 2016;107:169–82. [33] Xu XJ, Wang XC, Zheng ML, Ma Z. Bending and buckling of nonlocal strain gradient elastic beams. Compos Struct 2017;160:366–77. [34] Li L, Hu Y. Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 2015;97:84–94. [35] Ebrahimi F, Barati MR. Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Compos Struct 2017;159:433–44. [36] Katsikadelis JT, Armenakas AE. Plates on elastic foundation by BIE method. J Eng Mech 1984;110:1086–105. [37] Bezine G. A new boundary element method for bending of plates on elastic foundations. Int J Solids Struct 1988;24:557–65. [38] El-Zafrany A, Fadhil S, Al-Hosani K. A new fundamental solution for boundary element analysis of this plates on Winkler foundation. Int J Numer Meth Eng 1995;38:887–903. [39] Pasternak PL. New calculation method for flexible substructures on a two-parameter elastic foundation. In: Gos. Izdat. Literatury po Stroitelstvu i Arkhitekture, Moscow; 1954. p. 1–56. [40] Jaiswal OR, Iyengar RN. Dynamic response of a beam on elastic foundation of finite depth under a moving force. Acta Mech 1993;96:67–83. [41] Chudinovich I, Constanda C. Integral representations of the solutions for a bending plate on an elastic foundation. Acta Mech 2000;139:33–42. [42] Dumir PC. Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations. Acta Mech 2003;71:233–44. [43] Chien RD, Chen CS. Nonlinear vibration of laminated plates on an elastic foundation. Thin-Walled Struct 2006;44:852–60. [44] Tsiatas GC. Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech 2010;209:141–52. [45] Zenkour AM, Allam MNM, Shaker MO, Radwan AF. On the simple and mixed first-order theories for plates resting on elastic foundations. Acta Mech 2011;220:33–46. [46] Shen HS, Zhu ZH. Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations. Eur J Mech A/Solids 2012;35:10–21. [47] Zenkour AM, Radwan AF. On the simple and mixed first-order theories for functionally graded plates resting on elastic foundations. Meccanica 2013;48:1501–16. [48] Malekzadeh P, Setoodeh AR, Alibeygi Beni A. Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Compos Struct 2011;93:2083–9. [49] Shahsavari D, Karami B, Mansouri S. Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. Eur J Mech A-Solids 2018;67:200–14. [50] Panyatong M, Chinnaboon B, Chucheepsakul S. Free vibration analysis of FG nanoplates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity. Compos Struct 2016;153:428–41. [51] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J 2002;40:162–9. [52] Matsunaga H. Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory. Compos Struct 2005;68:439–54. [53] Carrera E. Transverse normal strain effects on thermal stress analysis of homogeneous and layered plates. AIAA J 2005;43:2232–42. [54] Shen HS. Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. Int J Mech Sci 2007;49:466–78. [55] Matsunaga H. Thermal buckling of functionally graded plates according to a 2d higher-order deformation theory. Compos Struct 2009;90:76–86. [56] Zenkour AM, Sobhy M. Thermal buckling of various types of FGM sandwich plates. Compos Struct 2010;93:93–102. [57] Carrera E. An assessment of mixed and classical theories for the thermal stress analysis of orthotropic multilayered plates. J Therm Stress 2010;23:797–831. [58] Bodaghi M, Saidi AR. Thermoelastic buckling behavior of thick functionally graded rectangular plates. Arch Appl Mech 2011;81:1555–72. [59] Kiani Y, Eslami MR. Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation. Arch Appl Mech 2012;82:891–905.

[60] Fazzolari FA, Carrera E. Thermo-mechanical buckling analysis of anisotropic multilayered composite and sandwich plates by using refined variable-kinematics theories. J Therm Stress 2012;36:321–50. [61] Chamis CC. Simplified composite micromechanics equations for hygral, thermal and mechanical properties. NASA TM 83320; 1983. [62] Hopkins DA, Chamis CC. A unique set of micromechanics equations for high temperature metal matrix composites. NASA TM 87154; 1985. [63] Birman V. Stability of functionally graded hybrid composite plates. Compos Eng 1995;5:913–21. [64] Ma LS, Wang TJ. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. Int J Solids Struct 2004;41:85–101. [65] Zenkour AM. A comprehensive analysis of functionally graded sandwich plates: Part 2buckling and free vibration. Int J Solids Struct 2005;42:5224–42. [66] Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211–18. [67] Najafizadeh MM, Heydari HR. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory. Eur J Mech/Solids 2004;23:1085–100. [68] Morimoto T, Tanigawa Y, Kawamura R. Thermal buckling of functionally graded rectangular plates subjected to partial heating. Int J Mech Sci 2006;48:926– 937. [69] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based on higher order theory. J Therm Stresses 2011;25:603–25. [70] Zenkour AM, Radwan AF. Analysis of multilayered composite plates resting on elastic foundations in thermal environment using a hyperbolic model. J Barzilian Society Mech Sci Eng 2017;39:2801–16. [71] Sayyad A.S., Ghugal Y.M.. Effects of non-linear hygro-thermo-mechanical loading on bending of FGM rectangular plates resting on two parameter elastic foundation using four unknown plate theory. J Therm Stress10.1080/01495739.2018.1469962 [72] Bouazza M, Becheri T, Boucheta A, Benseddiq N. Thermal buckling analysis of nanoplates based on nonlocal elasticity theory with four-unknown shear deformation theory resting on winklerP. Int J Comput Methods Eng Sci Mech 2016;17:362–73. [73] Sobhy M, Zenkour AM. Nonlocal thermal and mechanical buckling of nonlinear orthotropic viscoelastic nanoplates embedded in a visco-pasternak medium. Int J Appl Mech 2018;10:17588251. [74] Bouazza M, Zenkour AM, Benseddiq N. Closed-form solutions for thermal buckling analyses of advanced nanoplates according to a hyperbolic four-variable refined theory with small-scale effects. Acta Mech 2018;229:2251–65. [75] Ansari R, Ashrafi MA, Pourashraf T, Sahmani S. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronautica 2015;109:42–51. [76] Farajpour A, Haeri Yazdi MR, Rastgoo A, Mohammadi M. A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech 2016;227:1849–67. [77] Ebrahimi F, Barati MR. Nonlocal thermal buckling analysis of embedded magneto-electro-thermo-elastic nonhomogeneous nanoplates. Iranian J Sci Techno Trans Mech Eng 2016;40:243–64. [78] Wang YZ, Cui HT, Li FM, Kishimoto K. Thermal buckling of a nanoplate with small-scale effects. Acta Mech 2013;224:1299–307. [79] Nami MR, Janghorban M, Damadam M. Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory. Aerospace Sci Techno 2015;41:7–15. [80] Barati MR, Zenkour AM, Shahverdi H. Thermo-mechanical buckling analysis of embedded nanosize FG plates in thermal environments via an inverse cotangential theory. Compos Struct 2016;141:203–12. [81] Barati MR, Shahverdi H. A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions. Struct Eng Mech 2016;60:707–27. [82] He W, Ge SS. Cooperative control of a nonuniform gantry crane with constrained tension. Automatica 2016;66:146–54. [83] Mahmoudpour E, Hosseini-Hashemi SH, Faghidian SA. A nonlocal strain gradient theory for nonlinear free and forced vibration of embedded thick FG double layered nanoplates. Struct Eng Mech 2018;68:103–19. [84] Ansari R, Ashrafi MA, Pourashraf T, Sahmani S. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronautica 2015;109:42–51. [85] Zaoui FZ, Tounsi A, Ouinas Dj. Free vibration of functionally graded plates resting on elastic foundations based on quasi-3d hybrid-type higher order shear deformation theory. Smart Struct Syst Int J 2017;20(4):509–24. [86] Mantari JL, Granados EV. An original FSDT to study advanced composites on elastic foundation. Thin-Walled Struct 2016;107:80–9. [87] Aifantis EC. On the gradient approach-relation to Eringen’s nonlocal theory. Int J Eng Sci 2011;49:1367–77. [88] Lim CW, Zhang G, Reddy JN. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 2015;78:298–313. [89] Mantari JL, Oktem AS, Guedes Soares C. Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher-order shear deformation theory. Compos Part B 2012;43:3348– 3360. [90] Barati MR, Shahverdi H. Dynamic modeling and vibration analysis of double-layered multi-phase porous nanocrystalline silicon nanoplate systems. Eur J Mech Solid 2017;66:256–68.

334

A.F. Radwan

International Journal of Mechanical Sciences 157–158 (2019) 320–335

[91] Guerroudj HZ, Yeghnem R, Kaci A, Zaoui FZ, Benyoucef S, Tounsi A. Eigenfrequencies of advanced composite plates using an efficient hybrid quasi-3d shear deformation theory. Smart Struct Syst Int J 2018;22(1):121–32. [92] Zenkour AM, Sobhy M. Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler-Pasternak elastic substrate medium. Physica E, 2013;53:251–9. [93] Jin G, Su Z, Shi S, Ye T, Gao S. Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Compos Struct 2014;108:565–77. [94] Mantari JL. A refined theory with stretching effect for the dynamics analysis of advanced composites on elastic foundation. Mech Mater 2015;86:31–43.

[95] Hosseini-Hashemi SH, Rokni Damavandi Taher H, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Appl Math Model 2010;34:1276–91. [96] Abualnour M, Houari MSA, Tounsi A, Adda Bedia EA, Mahmoud SR. A novel quasi-3d trigonometric plate theory for free vibration analysis of advanced composite plates. Compos Struct 2018;184:688–97. [97] Sobhy M. A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos Struct 2015;134:966–80.

335