Vibration analysis of porous functionally graded nanoplates

International Journal of Engineering Science 120 (2017) 82–99

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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Vibration analysis of porous functionally graded nanoplates Hossein Shahverdi∗, Mohammad Reza Barati Aerospace Engineering Department & Center of Excellence in Computational Aerospace, AmirKabir University of Technology, Tehran 15875-4413, Iran

a r t i c l e

i n f o

Article history: Received 19 March 2017 Revised 30 May 2017 Accepted 3 June 2017

Keywords: Nanoporous nanoplate Hygro-thermal environment Nonlocal strain gradient theory Four-variable plate theory

a b s t r a c t In this paper, a general nonlocal strain-gradient (NSG) elasticity model is developed for vibration analysis of porous nano-scale plates on an elastic substrate. The present model incorporates two scale coefficients to examine the vibration characteristics much accurately. The application of present nanoplate model as nano-mechanical mass sensors is also investigated. Porosity-dependent material properties of the nanoplate are defined via a modified power-law function and Mori–Tanaka model. Based on Hamilton’s principle, the governing equations of the nanoplate on the elastic substrate under hygro-thermal loading are obtained. These equations are solved for hinged nanoplates via Galerkin’s method. It is demonstrated that nano-pores, temperature change, humidity change, nonlocal-strain gradient parameters, gradient index and attached nanoparticle have a remarkable influence on vibration frequencies of nanoscale plates. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Fast growing developments in materials engineering led to microscopically inhomogeneous spatial composite materials named Functionally graded materials (FGMs) which have extensive applications for various systems and devices, such as aerospace, aircraft, automobile and defense structures and most recently the electronic devices. According to the fact that FG materials have been placed in the category of composite materials, the volume fractions of two or more material constituents such as a pair of ceramic–metal are supposed to change continuously throughout the gradient directions (Akgöz and Civalek, 2014; Kiani, 2016; Lee & Kim, 2013; Zidi, Tounsi, Houari, & Bég, 2014). The FGM materials are made to take advantage of desirable features of its constituent phases, for example, in a thermal protection system, the ceramic constituents are capable of withstanding extreme temperature environments due to their better thermal resistance characteristics, while the metal constituents provide stronger mechanical performance and diminish the possibility of catastrophic fracture (Barati & Shahverdi, 2016a, Khalfi, Houari, & Tounsi, 2014; Matsunaga, 2009; Sobhy, 2016). Hence, possessing novel mechanical properties, FGMs have gained its applicability in several engineering fields, such as biomedical engineering, nuclear engineering and mechanical engineering. Furthermore, noticeable development in the application of structural elements such as FG beams and FG plates with micro or nano length scale in micro/nano electro-mechanical systems (MEMS/ NEMS), due to their outstanding chemical,

∗

Corresponding author. E-mail addresses: [email protected] (H. Shahverdi), [email protected], [email protected] (M.R. Barati).

http://dx.doi.org/10.1016/j.ijengsci.2017.06.008 0020-7225/© 2017 Elsevier Ltd. All rights reserved.

H. Shahverdi, M.R. Barati / International Journal of Engineering Science 120 (2017) 82–99

83

mechanical, and electrical properties, led to a provocation in modeling of micro/nano scale structures (Apppendix A, 2012; Lee et al., 2006; Lü, Chen, & Lim, 2009; Mao et al., 2016; Salehipour, Shahidi, & Nahvi, 2015; Sedighi, Daneshmand, & Abadyan, 2015; Zalesak et al., 2016). The problem in using the classical theory is that the classical continuum mechanics theory does not take into account the size effects in micro/nano scale structures. The classical continuum mechanics over predicts the responses of micro/nano structures. Eringen’s nonlocal elasticity theory (Eringen, 1983; Eringen & Edelen, 1972) is the most commonly used continuum mechanics theory that includes small scale effects with good accuracy to model micro/nano scale devices and systems (Arani & Jalaei, 2016; Barati & Shahverdi, 2016b, Ebrahimi & Barati, 2016a, b, 2017a; Khajeansari, Baradaran, & Yvonnet, 2012; Lei, Adhikari, & Friswell, 2013; Naderi & Saidi, 2014; Nejad & Hadi, 2016a, b; Nejad, ´ c, ´ Karlicˇ ic, ´ Pavlovic, ´ Janevski, & Ciri ´ 2016; Rahmani & Pedram, 2014; Roque, Ferreira, & Reddy, Hadi, & Rastgoo, 2016; Pavlovic, 2011; Shafiei, Kazemi, Safi, & Ghadiri, 2016; Thai, 2012; Thai & Vo, 2012). The nonlocal elasticity theory assumes that the stress state at a reference point is a function of the strain at all neighbor points of the body. Hence, this theory could take into consideration the effects of small scales. Finite element vibration analysis of FG nanosize plates based on classical plate theory (CPT) is conducted by Natarajan, Chakraborty, Thangavel, Bordas, and Rabczuk (2012). Based on the third order plate theory, Daneshmehr and Rajabpoor (2014) examined buckling behavior of nonlocal graded nanoplates under different boundary conditions. Analysis of resonance frequencies of FG micro and nanoplates according to nonlocal elasticity and strain gradient theory is performed by Nami and Janghorban (2014). They used nonlocal and strain-gradient theories separately, and concluded that these theories have different mechanisms in the analysis of nanoplates. Application of three-dimensional nonlocal elasticity theory in static and vibration analysis of FG nanoplate is investigated by Ansari, Shojaei, Shahabodini, and Bazdid-Vahdati (2015) based on classical plate model. Based on the generalized differential quadrature method (GDQM), Daneshmehr, Rajabpoor, and Hadi (2015) analyzed the vibrational behavior of higher-order FG nanoplates using nonlocal stress field theory. Application of four-variable plate theory in vibration analysis of FG nanoplates is examined by Belkorissat, Houari, Tounsi, Bedia, and Mahmoud (2015). They stated that presented plate model have fewer field variables compared with first-order and third-order plate theories. Based on four-variable plate theory, shear deformation effect is captured, while the governing equations are very similar to the classical plate theory. Barati, Zenkour, and Shahverdi (2016) proposed a refined four-variable plate model for thermal buckling analysis of FG nanoplates. Wave propagation, buckling and vibration analyses of smart FG nanoplates under various physical fields are carried out by Ebrahimi and Barati (2016c, d) and Ebrahimi, Dabbagh, and Barati (2016) using different plate theories. A comprehensive investigation of bending, buckling and vibrational behaviors of FG nanoplates on an elastic medium is conducted by Sobhy (2015a). Also, Khorshidi and Fallah (2016) performed buckling analysis of FG nanoplates via a general nonlocal exponential shear deformation plate model. Sobhy and Radwan (2017) presented a new quasi 3D nonlocal plate theory for vibration and buckling of FGM Nanoplates. It is noticeable that all of the aforementioned studies on FG nanoplates have been reported a stiffness-softening mechanism due to the nonlocality. Although nonlocal elasticity theory (NET) of Eringen is a suitable theory for modeling of a nanostructure, it has some shortcomings due to neglecting stiffness-hardening mechanism reported in experimental works and strain gradient elasticity (Lam, Yang, Chong, Wang, & Tong, 2003). By using nonlocal strain gradient theory (NSGT), Lim, Zhang, and Reddy (2015) matched the dispersion curves of nanobeams with those of experimental data. They concluded that NSGT is more accurate for modeling and analysis of nanostructures by considering both stiffness reduction and enhancement effects. Application of NSGT in wave dispersion analysis of FG nanobeams is examined by Li, Hu, and Ling (2015). Also, some investigations are performed using NSGT on vibration and buckling of nanorods, nanotubes and nanobeams (Ebrahimi & Barati, 2016e, 2017b; Li & Hu, 2017; Li, Li, & Hu, 2016; S¸ ims¸ ek, 2016). Also, Farajpour, Yazdi, Rastgoo, and Mohammadi (2016) presented buckling analysis of nanoplates via a nonlocal strain gradient plate model employing exact and differential quadrature methods. In another work, Farajpour, Rastgoo, Farajpour, and Mohammadi (2016) presented nonlocal strain gradient modeling of nano-mechanical vibrating piezoelectric mass sensors. Also, Ebrahimi, Barati, and Dabbagh (2016) applied NSGT for wave propagation analysis of FG nanoplates under thermal loading. Therefore, it is of great importance to analyze the vibration behavior of FG nanoplates via NSGT for the first time. Nanoplates are usually subjected to hygro-thermal environments during their construction or operational life (Alzahrani, Zenkour, & Sobhy, 2013; Sobhy, 2015b). Although the significance, there is no study on hygro-thermal effects on vibration behavior of FG nanoplates supported by an elastic medium. Generally speaking, it is crucial to consider simultaneously moisture and temperature changes for more accurate analysis and design of nanostructures. This paper makes the first attempt to model a compositionally graded nanoporous nanoplate according to the nonlocal strain gradient theory (NSGT). The proposed modeling of nanoplates incorporates a nonlocal stress field parameter as well as a length scale parameter related to strain gradient. Thus, stiffness enhancement or reduction observed in nanostructures are considered. Porosity-dependent material properties of nanoplate are described via a new power-law function. Non-classical boundary conditions related to NSG theory as well as governing equations are obtained using Hamilton’s principle. By solving the governing equations using Galerkin’s method, natural frequencies of the nanoplate are obtained. The results show that vibrational behavior of the nanoplate is significantly influenced by the nonlocality, strain gradient parameter, hygro-thermal loading, material composition, elastic medium and geometrical parameters. Obtained frequencies can be used as benchmark results in the analysis of nanoplates modeled by nonlocal and microstructure-dependent strain-gradient theories.

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Fig. 1. Configuration of nanoporous inhomogeneous nanoplate on elastic substrate.

2. Nonlocal strain gradient nanoplate model The proposed nonlocal strain gradient theory (Lim et al., 2015) takes into account both nonlocal stress field and the strain gradient effects by introducing two scale parameters. This theory defines the stress field as:

σi j = σi(j0) − ∇σi(j1)

(1)

in which the stresses σi(j0 ) and σi(j1 ) are corresponding to strain εi j and strain gradient ∇ εi j , respectively as:

σi(j0) =



V

σi(j1) = l 2

 (x )dx Ci jkl α0 (x, x , e0 a )εkl

 V

 (x )dx Ci jkl α1 (x, x , e1 a )∇ εkl

(2a) (2b)

in which Ci jkl are the elastic coefficients and e0 a and e1 a capture the nonlocal effects and l captures the strain gradient effects. When the nonlocal functions α0 (x, x , e0 a ) and α1 (x, x , e1 a ) satisfy the developed conditions by Eringen, the constitutive relation of nonlocal strain gradient theory has the following form:

[1 − (e1 a )2 ∇ 2 ][1 − (e0 a )2 ∇ 2 ]σi j = Ci jkl [1 − (e1 a )2 ∇ 2 ]εkl − Ci jkl l 2 [1 − (e0 a )2 ∇ 2 ]∇ 2 εkl in which

∇2

(3a)

denotes the Laplacian operator. Considering e1 = e0 = e, the general constitutive relation in Eq. (3) becomes:

[1 − (ea )2 ∇ 2 ] σi j = Ci jkl [1 − l 2 ∇ 2 ]εkl

(3b)

To consider hygro-thermal effects Eq. (4) can be written as (Ebrahimi & Barati, 2017b):

[1 − (ea )2 ∇ 2 ] σi j = Ci jkl [1 − l 2 ∇ 2 ](εkl − γi j T − βi jC )

(3c)

where γi j and βi j are thermal and moisture expansion coefficients, respectively. 3. FG plate model based on neutral surface position Consider a rectangular (a × b) porous nanoplate of uniform thickness h made of FGM as shown in Fig. 1. Also, the configuration of attached nanoparticles is shown in Fig. 2. A FG material can be specified by the variation in the volume fractions. Due to this variation, the neutral axis of FG nanoplate may not coincide with its mid-surface which leads to bendingextension coupling. By using neutral axis, this coupling is eliminated. The effective material properties Pf can be stated as:



Pf = P c V c −

ξ



2



+ Pm V m −

ξ 2



(4)

where subscripts m and c denote metal and ceramic, respectively and the volume fraction of the ceramic is associated to that of the metal in the following relation:

Vc + Vm = 1, Vc =

z

h

+

1 2

p

(5)

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85

Fig. 2. Distribution of nanoparticles and their mass.

Based on the modified power-law model, Young’ modulus E, density ρ , thermal expansion coefficient γ and moisture expansion coefficient β are described as:

E ( z ) = ( Ec − Em )

z

h

ρ ( z ) = ( ρ c − ρm ) γ (z ) = (γc − γm )

+

z h

z

β (z ) = (βc − βm )

h

z h

1 2

p

+

1 2

+

1 2

+

1 2

+ Em −

p p p

ξ 2

+ ρm − + γm − + βm −

( Ec + Em )

ξ 2

ξ 2

ξ 2

(6a)

( ρc + ρm )

(6b)

(γc + γm )

(6c)

(βc + βm )

(6d)

in p is inhomogeneity or power-law index. Also, ξ is the porosity volume fraction. Here, Mori–Tanaka homogenization technique is also employed to model the effective material properties of the FG nanoplate. According to Mori–Tanaka homogenization technique the local effective material properties of the FG nanoplate such as effective local bulk modulus Ke and shear modulus μe can be calculated (Barati & Shahverdi, 2016a):

Ke − Km Vc = Kc − Km 1 + Vm (Kc − Km )/(Km + 4μm /3 )

(7a)

μe − μm Vc = μc − μm 1 + Vm (μc − μm )/[(μm + μm (9Km + 8μm )/(6(Km + 2μm ))]

(7b)

Therefore from Eq. (4), the effective Young’s modulus (E) based on Mori–Tanaka scheme can be expressed by:

E (z ) =

9Ke μe 3Ke + μe

(7c)

Also, the mass density of FG nanoplate can be expressed in the form of (6b) according to the rule of mixture. The displacement field according to the four-variable plate model considering the exact position of the neutral surface can be expressed by:

u1 (x, y, z, t ) = u(x, y, t ) − (z − z∗ )

∂ wb ∂ ws − [ f (z ) − z∗∗ ] ∂x ∂x

(8a)

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u2 (x, y, z, t ) = v(x, y, t ) − (z − z∗ )

∂ wb ∂ ws − [ f (z ) − z∗∗ ] ∂y ∂y

u3 (x, y, z, t ) = w(x, y, t ) = wb (x, y, t ) + ws (x, y, t ) where

 h/2 −h/2

z∗ =  h/2

E (z ) zdz

−h/2

E (z ) dz

 h/2 , z∗∗ =

(z ) f (z )dz

−h/2 E  h/2 −h/2

E (z ) dz

(8b) (8c)

(8d)

Also, u and v are in-plane displacements and wb and ws denote the bending and shear transverse displacement, respectively. The shape function of transverse shear deformation is considered as:

f (z ) = −

z 5z 3 + 4 3 h2

(9)

According to the present plate theory with four unknown, the nonzero strains are obtained as:

∂u ∂ 2 wb ∂ 2 ws − (z − z∗ ) − [ f (z ) − z∗∗ ] 2 ∂x ∂x ∂ x2 2 ∂v ∂ wb ∂ 2 ws εy = − (z − z∗ ) − [ f (z ) − z∗∗ ] 2 ∂y ∂y ∂ y2 2 ∂ u ∂v ∂ wb ∂ 2 ws γxy = + − 2 (z − z∗ ) − 2[ f (z ) − z∗∗ ] ∂y ∂x ∂ x∂ y ∂ x∂ y ∂ ws ∂ ws γyz = g(z ) , γ = g( z ) ∂ y xz ∂x εx =

(10)

Also, the extended Hamilton’s principle expresses that:



t

0

δ (U − T + V ) dt = 0

(11)

here, U is strain energy, T is kinetic energy and V is work done by external forces. The first variation of the strain energy can be calculated as:

δU =



V

(σxx δ εxx + σxx(1) δ∇ εxx + σyy δ εyy + σyy(1) δ∇ εyy + σxy δ γxy + σxy(1) δ∇ γxy

(1 ) +σyz δ γyz + σyz δ ∇ γyz + σxz δ γxz + σxz(1) δ∇ γxz ) dV

(12)

in which σ are the components of the stress tensor and ε are the components of the strain tensor. Substituting Eqs. (8) and (10) into Eq. (12) yields:

 2 2 ∂δ u ∂ w ∂δ w ∂δv ∂ w ∂δ w b ∂ δ wb s ∂ δ ws δU = Nxx + − Mxx − Mxx + Nyy + ∂x ∂x ∂x ∂y ∂y ∂y ∂ x2 ∂ x2 0 0   2 2 2 ∂δ u ∂δv ∂ w ∂δ w ∂ w ∂δ w b ∂ δ wb s ∂ δ ws b ∂ δ wb −Myy − Myy + Nxy + + + − 2Mxy ∂y ∂x ∂x ∂y ∂y ∂x ∂ x∂ y ∂ y2 ∂ y2 2 ∂δ ws ∂δ ws s ∂ δ ws −2Mxy + Qyz + Qxz dydx ∂ x∂ y ∂y ∂x 

in which:

Nxx =

a



h/2

−h/2

 b Myy =

h/2

−h/2

 s Mxx =

h/2

−h/2

 b Mxx =

h/2

−h/2

 Nyy =

h/2

−h/2

 Nxy =



h/2

−h/2

b





(0 ) (1 ) (σxx0 − ∇σxx(1) )dz = Nxx − ∇ Nxx (0 ) (1 ) (σxy0 − ∇σxy(1) )dz = Nxy − ∇ Nxy (1 ) (0 ) (1 ) 0 (σyy − ∇σyy )dz = Nyy − ∇ Nyy (1 ) b( 0 ) b( 1 ) 0 z(σxx − ∇σxx )dz = Mxx − ∇ Mxx (1 ) s (0 ) s (1 ) 0 f (σxx − ∇σxx )dz = Mxx − ∇ Mxx (1 ) b( 0 ) b( 1 ) 0 z(σyy − ∇σyy )dz = Myy − ∇ Myy

(13)

H. Shahverdi, M.R. Barati / International Journal of Engineering Science 120 (2017) 82–99

 s Myy =

h/2

−h/2

 b Mxy =

h/2

−h/2

 s Mxy =

h/2

−h/2

 Qxz =

h/2

−h/2

 Qyz =

h/2

−h/2

where

Ni(j0) = Mibj(0) = Misj(0) = (0 ) Qxz = (0 ) Qyz =



(1 ) s (0 ) s (1 ) 0 f (σyy − ∇σyy )dz = Myy − ∇ Myy (1 ) b( 0 ) b( 1 ) 0 z(σxy − ∇σxy )dz = Mxy − ∇ Mxy (1 ) s (0 ) s (1 ) 0 f (σxy − ∇σxy )dz = Mxy − ∇ Mxy (0 ) (1 ) 0 g(σxz − ∇σxz(1) )dz = Qxz − ∇ Qxz (1 ) 0 g(σyz − ∇σyz )dz = Qyz(0) − ∇ Qyz(1)

h/2

−h/2



h/2

−h/2



h/2

−h/2



h/2

−h/2



87

h/2

−h/2

(σi(j0) )dz, Ni(j1) =



h/2 −h/2

z(σibj (0) )dz, Mibj(1) = f (σisj(0) )dz, Misj(1) = i (0 ) g(σxz )dz, Qxz(1) = i (0 ) g(σyz )dz, Qyz(1) =







(σi(j1) )dz h/2

−h/2 h/2 −h/2



(14a)

h/2 −h/2 h/2 −h/2

z(σibj (1) )dz f (σisj(1) )dz

i (1 ) g(σxz )d z i (1 ) g(σyz )d z

(14b)

in which (ij = xx, xy, yy). The first variation of the work done by applied forces can be written as:

δV =







∂ (wb + ws ) ∂δ (wb + ws ) ∂ (wb + ws ) ∂δ (wb + ws ) + Ny0 ∂x ∂x ∂y ∂y 0 ∂ ( wb + ws ) ∂ ( wb + ws ) +2δ Nxy − (kw − q particle )(wb + ws )δ (wb + ws ) ∂x ∂y   ∂ (wb + ws ) ∂δ (wb + ws ) ∂ (wb + ws ) ∂δ (wb + ws ) +k p + dydx ∂x ∂x ∂y ∂y a

0

b

Nx0

0

(15a)

0 are in-plane applied loads; k and k are Winkler and Pasternak constants. where Nx0 , Ny0 , Nxy w p Also, qparticle is the transverse force due to attached nanoparticles (such as a buckyball and molecular or bacterium) with mass mc which is attached at the location x = x0 , y = y0 :

q particle = −

N

m j δ (x − x j , y − y j )

j=1

∂ 2w ∂t2

(15b)

in which mj is the mass of jth attached particle and N is the number of concentrated masses. Also, δ (x − x j ) is Dirac delta function defined by:



δ (x − x j ) =

∞

x = x j

0

x = xj

(15c)

The first variation of the kinetic energy can be written in the following form:

δK =

  ∂ u ∂δ u ∂v ∂δv ∂ (wb + ws ) ∂δ (wb + ws ) ∂ u ∂δ wb ∂ wb ∂δ u ∂v ∂δ wb + + − I1 + + ∂t ∂t ∂t ∂t ∂t ∂t ∂ t ∂ x∂ t ∂ x∂ t ∂ t ∂ t ∂ y∂ t 0 0      ∂ wb ∂δv ∂ u ∂δ ws ∂ ws ∂δ u ∂v ∂δ ws ∂ ws ∂δv ∂ wb ∂δ wb ∂ wb ∂δ wb + − I3 + + + + I2 + ∂ y∂ t ∂ t ∂ t ∂ x∂ t ∂ x∂ t ∂ t ∂ t ∂ y∂ t ∂ y∂ t ∂ t ∂ x∂ t ∂ x∂ t ∂ y ∂ t ∂ y ∂ t     ∂ ws ∂δ ws ∂ ws ∂δ ws ∂ wb ∂δ ws ∂ ws ∂δ wb ∂ wb ∂δ ws ∂ ws ∂δ wb +I5 + + I4 + + + dydx ∂ x∂ t ∂ x∂ t ∂ y∂ t ∂ y∂ t ∂ x∂ t ∂ x∂ t ∂ x∂ t ∂ x∂ t ∂ y ∂ t ∂ y ∂ t ∂ y∂ t ∂ y ∂ t



a



b

  I0

in which

(I0 , I1 , I2 , I3 , I4 , I5 ) =



h/2 −h/2

(1, z − z∗ , (z − z∗ )2 , f − z∗∗ , (z − z∗ )( f − z∗∗ ), ( f − z∗∗ )2 )ρ (z )dz

(16)

(17)

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H. Shahverdi, M.R. Barati / International Journal of Engineering Science 120 (2017) 82–99

By inserting Eqs. (13)–(16) into Eq. (11) and setting the coefficients of δ u, δ v, δ wb and δ ws to zero, the following Euler– Lagrange equations can be obtained.

∂ Nx ∂ Nxy ∂ 2u ∂ 3 wb ∂ 3 ws + = I0 2 − I1 − I 3 ∂x ∂y ∂t ∂ x∂ t 2 ∂ x∂ t 2

(18)

∂ Nxy ∂ Ny ∂ 2v ∂ 3 wb ∂ 3 ws + = I0 2 − I1 −I ∂x ∂y ∂t ∂ y∂ t 2 3 ∂ y∂ t 2

(19)

N b ∂ 2 Mxy ∂ 2 Myb

∂ 2 Mxb ∂ 2w +2 + − m j δ ( x − x j , y − y j ) 2 − ( N T + N H )∇ 2 ( wb + ws ) − kw ( wb + ws ) 2 2 ∂ x∂ y ∂x ∂y ∂t j=1    2   2  ∂ 2 ( wb + ws ) ∂ 3v ∂ 3u 2 ∂ wb 2 ∂ ws +k p ∇ 2 (wb + ws ) = I0 + I + − I ∇ − I ∇ 1 2 4 ∂t2 ∂ x∂ t 2 ∂ y∂ t 2 ∂t2 ∂t2

(20)

N s ∂ 2 Mxy ∂ 2 Mys ∂ Qxz ∂ Qyz

∂ 2 Mxs ∂ 2w + 2 + + + − m j δ ( x − x j , y − y j ) 2 − ( N T + N H )∇ 2 ( wb + ws ) ∂ x∂ y ∂x ∂y ∂ x2 ∂ y2 ∂t j=1    2   2  ∂ 2 ( wb + ws ) ∂ 3v ∂ 3u 2 2 ∂ wb 2 ∂ ws −kw (wb + ws ) + k p ∇ (wb + ws ) = I0 + I3 + − I4 ∇ − I5 ∇ (21) ∂t2 ∂ x∂ t 2 ∂ y∂ t 2 ∂t2 ∂t2

0 = 0 and hygro-thermal resultant can be expressed by: where Nx0 = Ny0 = N T + N H , Nxy

 NT =

−h/2

 NH =

h/2

h/2

−h/2

E (z ) 1−v

γ (z ) (T − T0 ) dz

E (z ) 1−v

β (z ) (C − C0 ) dz

(22)

in which C = C + C0 and T = T + T0 are uniform moisture and temperature changes, and T0 are reference moisture and temperature. The classical and non-classical boundary conditions can be obtained in the derivation process when using the integrations by parts. The non-classical boundary conditions are:

∂ 2 wb ∂ x2 ∂ 2 wb Specify ∂ y2 ∂ 2 ws Specify ∂ x2 ∂ 2 ws Specify ∂ y2 Specify

b( 1 ) or Mxx =0 b( 1 ) or Myy =0 s (1 ) or Mxx =0 s (1 ) or Myy =0

(23)

Based on the NSGT, the constitutive relations of presented higher order FG nanoplate can be stated as:

⎧ ⎫ ⎛1 σx ⎪ ⎪ ⎪ ⎨ σy ⎪ ⎬ ⎜v E (z ) 2 ⎜ (1 − μ∇ 2 ) σxy = ( 1 − λ∇ ) ⎝0 ⎪ ⎪ 1 − v2 ⎪ 0 ⎩ σyz ⎪ ⎭ σxz 0

v

0 0 ( 1 − v )/2 0 0

1 0 0 0

0 0 0 ( 1 − v )/2 0

⎞⎧

⎫

0 εx − γ T − β C ⎪ ⎪ ⎪ ⎪ 0 ⎟⎨ εy − γ T − β C ⎬ ⎟ 0 ⎠⎪ γxy ⎪ ⎪ ⎪ 0 ⎩ γyz ⎭ ( 1 − v )/2 γxz

(24)

Integrating Eq. (24) over the plate’s cross-section area, one can obtain the force-strain and the moment-strain of the nonlocal refined FG plates can be obtained as follows:

⎧ ⎫ ⎨ Nx ⎬

⎛

1

(1 − μ∇ 2 ) Ny = A(1 − λ∇ 2 )⎝v ⎩ ⎭ Nxy

0

v 1 0

⎞⎧ ∂ u ⎪ ⎨ ∂x ⎠ ∂v 0 ⎪ ∂y (1 − v )/2 ⎩ ∂∂ uy + 0

⎫ ⎪ ⎬ ⎪

∂v ⎭ ∂x

(25)

H. Shahverdi, M.R. Barati / International Journal of Engineering Science 120 (2017) 82–99

⎧ b Mx ⎪ ⎪ ⎨

⎫ ⎪ ⎪ ⎬

⎛

1

⎜ (1 − μ∇ 2 ) Myb = D(1 − λ∇ 2 )⎜v ⎝ ⎪ ⎪ ⎪ ⎩ b⎪ ⎭

0

Mxy

⎞⎧ ∂ 2 wb ⎫ ⎛ ⎪ − ∂ x2 ⎪ 0 1 ⎪ ⎪ ⎪ ⎪ ⎟⎨ ∂ 2 w ⎬ ⎜ 2 ⎜ ⎟ − 2b 0 ⎠⎪ ∂ y ⎪ + E (1 − λ∇ )⎝v ⎪ ⎪ ⎪ ⎪ (1 − v )/2 ⎩ −2 ∂ 2 wb ⎭ 0

v 1 0

∂ x∂ y

89

⎞⎧ ∂ 2 ws − ⎪ ⎪ ∂ x2 ⎟⎨ ∂ 2 w ⎟ − 2s ⎠⎪ ∂ y ⎪ ⎩ 2

v

0

1

0

0

( 1 − v )/2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪

⎭ s −2 ∂∂ xw ∂y (26)

⎧ s⎫ Mx ⎪ ⎪ ⎪ ⎪ ⎨ ⎬

⎛

1

⎜ (1 − μ∇ 2 ) Mys = E (1 − λ∇ 2 )⎜v ⎝ ⎪ ⎪ ⎪ ⎩ s⎪ ⎭ Mxy

0

⎞⎧ ∂ 2 wb ⎫ ⎛ ⎪ − ∂ x2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎟⎨ ∂ 2 w ⎬ ⎜ 2 ⎜ ⎟ − 2b 0 ⎠⎪ ∂ y ⎪ + F (1 − λ∇ )⎝v ⎪ ⎪ ⎪ ⎪ (1 − v )/2 ⎩ −2 ∂ 2 wb ⎭ 0

v

0

1 0

∂ x∂ y

⎞⎧ ∂ 2 ws − ⎪ ⎪ ∂ x2 ⎟⎨ ∂ 2 w ⎟ − 2s ⎠⎪ ∂ y ⎪ ⎩ 2

v

0

1

0

0

( 1 − v )/2

⎫ ⎪ ⎪ ⎬

s −2 ∂∂ xw ∂y

⎪ ⎪ ⎭

(27)



(1 − μ∇

2

Q ) x Qy





= A44 (1 − λ∇

2

1 ) 0

 ∂ ws 

0 1

∂x

∂ ws ∂y

(28)

in which:

 A=  F =

h/2 −h/2 h/2 −h/2

E (z ) dz, D = 1 − v2



h/2 −h/2

E (z ) ( f − z ) dz, 1 − v2 ∗∗ 2

 h/2 2 E (z ) (z − z∗ ) E (z )(z − z∗ )( f − z∗∗ ) dz, E = dz 2 1−v 1 − v2 −h/2  h/2 E (z ) A44 = g2 dz −h/2 2 (1 + v )

(29)

The governing equations in terms of the displacements for a NSGT refined four-variable FG nanoplate can be derived by substituting Eqs. (25)–(28), into Eqs. (18)–(21) as follows:



A(1 − λ∇ 2 )

   ∂ 2u ∂ 3 wb ∂ 3 ws ∂ 2u 1 − v ∂ 2u 1 + v ∂ 2v 2 + + + ( 1 − μ∇ ) −I + I + I =0 0 1 3 2 ∂ y2 2 ∂ x∂ y ∂ x2 ∂t2 ∂ x∂ t 2 ∂ x∂ t 2

   ∂ 2v ∂ 3 wb ∂ 3 ws ∂ 2v 1 − v ∂ 2v 1 + v ∂ 2u 2 A(1 − λ∇ ) + + + (1 − μ∇ ) −I0 2 + I1 +I =0 2 ∂ x2 2 ∂ x∂ y ∂ y2 ∂t ∂ y∂ t 2 3 ∂ y∂ t 2

(30)



2

(31)



  4  ∂ 4 wb ∂ 4 wb ∂ 4 ws ∂ 4 ws ∂ 4 wb ∂ ws 2 + 2 + − E ( 1 − λ∇ ) + 2 + ∂ x4 ∂ x2 ∂ y2 ∂ y4 ∂ x4 ∂ x2 ∂ y2 ∂ y4    ∂ 2 ( wb + ws ) ∂ 3u ∂ 3v ∂ 2 wb +(1 − μ∇ 2 ) −I0 − I1 ( + ) + I2 ∇ 2 2 2 2 ∂t ∂ x∂ t ∂ y∂ t ∂t2   2  N

∂ 2w 2 ∂ ws T H 2 2 +I4 ∇ − m j δ ( x − x j , y − y j ) 2 − ( N + N )∇ ( wb + ws ) − kw ( wb + ws ) + k p ∇ ( wb + ws ) = 0 ∂t2 ∂t j=1

−D(1 − λ∇ 2 )

(32)



  4  ∂ 4 wb ∂ 4 wb ∂ 4 ws ∂ 4 ws ∂ 4 wb ∂ ws 2 −E (1 − λ∇ ) +2 2 2 + − F (1 − λ∇ ) +2 2 2 + ∂ x4 ∂x ∂y ∂ y4 ∂ x4 ∂x ∂y ∂ y4  2     ∂ 2 ( wb + ws ) ∂ 3v ∂ ws ∂ 2 ws ∂ 3u 2 +A44 (1 − λ∇ 2 ) + + ( 1 − μ∇ ) −I − I + 0 3 ∂ x2 ∂ y2 ∂t2 ∂ x∂ t 2 ∂ y∂ t 2  2   2  N

∂ 2w ∂ wb ∂ ws +I4 ∇ 2 + I5 ∇ 2 − m j δ ( x − x j , y − y j ) 2 − ( N T + N H )∇ 2 ( wb + ws ) 2 2 ∂t ∂t ∂t j=1  − kw ( wb + ws ) + k p ∇ 2 ( wb + ws ) = 0 2

(33)

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4. Solution procedure In this section, Galerkin’s method is implemented to solve the governing equations of nonlocal strain gradient based FG nanoplates. Thus, the displacement field can be calculated as:

u=

v=

∞

∞

m=1 n=1 ∞

∞

Umn

∂ Xm (x ) Y ( y )e iωn t ∂x n

(34)

∂ Yn (y ) iωn t e ∂y

(35)

Vmn Xm (x )

m=1 n=1

wb =

∞

∞

Wbmn Xm (x )Yn (y )e iωn t

(36)

Wsmn Xm (x ) Yn (y )e iωn t

(37)

m=1 n=1

ws =

∞

∞

m=1 n=1

where (Umn ,Vmn ,Wbmn ,Wsmn ) are the unknown coefficients and the functions Xm and Yn satisfy the boundary conditions. The classical and non-classical boundary condition based on the present plate model are:

wb = ws = 0,

∂ 2 wb ∂ 2 ws ∂ 2 wb ∂ 2 ws = = = =0 ∂ x2 ∂ x2 ∂ y2 ∂ y2 ∂ 4 wb ∂ 4 ws ∂ 4 wb ∂ 4 ws = = = =0 ∂ x4 ∂ x4 ∂ y4 ∂ y4

(38)

By substituting Eqs. (34)–(37) into Eqs. (30)–(33), and using the Galerkin’s method, one obtains

⎧⎛ ⎪ ⎨ k1,1 ⎜k2,1 ⎝ ⎪ ⎩ k3,1 k4,1

in which

k2,1

k1,3 k2,3 k3,3 k4,3

⎞

⎛

k1,4 m1,1 k2,4 ⎟ 2 ⎜m2,1 +ω ¯ n⎝ k3,4 ⎠ m3,1 k4,4 m4,1

m1,2 m2,2 m3,2 m4,2

m1,3 m2,3 m3,3 m4,3

⎞⎫⎧

⎫

m1,4 ⎪⎪ Umn ⎪ ⎬⎨ ⎬ m2,4 ⎟ Vmn =0 ⎠ m3,4 ⎪⎪Wbmn ⎪ ⎭⎩ ⎭ m4,4 Wsmn

(39)

     b  a  5  b a 3 ∂ 3 Xm ∂ Xm ∂ Xm ∂ Xm ∂ Xm ∂ 2Yn ∂ Xm Y Y d xd y − λ Y Y d xd y + Y d xd y ∂ x3 n ∂ x n ∂ x5 n ∂ x n ∂ x3 ∂ y2 ∂ x n 0 0 0 0 0 0   b  a      b a 1−v ∂ Xm ∂ 2Yn ∂ Xm ∂ 3 Xm ∂ 2Yn ∂ Xm +A Yn dxdy − λ Y dxdy 2 2 ∂x ∂y ∂x ∂ x3 ∂ y2 ∂ x n 0 0 0 0    b a ∂ Xm ∂ 4Yn ∂ Xm + Y dxdy (40) ∂ x ∂ y4 ∂ x n 0 0  b  a  2    b  a  4 1+v ∂ Xm ∂ Yn ∂ Yn ∂ Xm ∂ Yn ∂ Yn =A X dxdy − λ X dxdy 2 ∂ x2 ∂ y m ∂ y ∂ x4 ∂ y m ∂ y 0 0 0 0     b a ∂ 2 Xm ∂ 3Yn ∂ Yn + X dxdy (41) ∂ x2 ∂ y3 m ∂ y 0 0  b  a  3  b  a    1+v ∂ Xm ∂ 2Yn ∂ Xm ∂ Xm ∂ 2Yn ∂ Xm =A Y dxdy − λ Y dxdy 2 ∂ x ∂ y2 ∂ x n ∂ x3 ∂ y2 ∂ x n 0 0 0 0     b a ∂ Xm ∂ 4Yn ∂ Xm + Y dxdy (42) ∂ x ∂ y4 ∂ x n 0 0

k1,1 = A

k1,2

k1,2 k2,2 k3,2 k4,2

 b  a 

 b  a  2      b a ∂ 3Yn ∂ Yn ∂ 5Yn ∂ Yn ∂ Xm ∂ 3Yn ∂ Yn k2,2 = A Xm X dxdy − λ X dxdy + Xm X dxdy ∂ y3 m ∂ y ∂ x2 ∂ y3 m ∂ y ∂ y5 m ∂ y 0 0  b a  2  0 0  b a  4 0 0 1−v ∂ Xm ∂ Yn ∂ Yn ∂ Xm ∂ Yn ∂ Yn +A X dxdy − λ X dxdy 2 ∂y m ∂y 2 ∂ x ∂ x4 ∂ y m ∂ y 0 0 0 0    b a 2 ∂ Xm ∂ 3Yn ∂ Yn + X dxdy ∂ x2 ∂ y3 m ∂ y 0 0  b  a 

(43)

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91

 b a 

    b a 2  b a ∂ 4Yn ∂ 4 Xm ∂ Xm ∂ 2Yn k2,3 = k3,2 = −E Y X Y dxdy + 2 X Y dxdy + Xm X Y dxdy ∂ x4 n m n ∂ x2 ∂ y2 m n ∂ y4 m n 0 0 0 0   b  a 0 60      b a 4   b a ∂ Xm ∂ Xm ∂ 2Yn ∂ 2 Xm ∂ 4Yn −λ Y X Y dxdy + 3 X Y dxdy + 3 X Y dxdy ∂ x6 n m n ∂ x4 ∂ y2 m n ∂ x2 ∂ y4 m n 0 0 0 0 0 0    b a ∂ 6Yn + Xm X Y dxdy ∂ y6 m n 0 0 (44)

    b a 2  b a ∂ 4Yn ∂ 4 Xm ∂ Xm ∂ 2Yn Y X Y d xd y + 2 X Y d xd y + X X Y dxdy m ∂ x4 n m n ∂ x2 ∂ y2 m n ∂ y4 m n 0 0 0 0       0 b 0 a  6  b a  b a ∂ Xm ∂ 4 Xm ∂ 2Yn ∂ 2 Xm ∂ 4Yn −λ Y X Y dxdy + 3 X Y dxdy + 3 X Y dxdy ∂ x6 n m n ∂ x4 ∂ y2 m n ∂ x2 ∂ y4 m n 0 0 0 0 0 0          b a     b a b a ∂ 6Yn ∂ 2 Xm + Xm X Y dxdy − Kw XmYn XmYn dxdy − μ Y X Y dxdy ∂ y6 m n ∂ x2 n m n 0 0 0 0 0 0          b a    b a b a ∂ 2Yn ∂ 2Yn ∂ 2 Xm T H + Xm X Y d xd y − N + N − K Y X Y d xd y + X X Y dxdy m n p n m n m m n ∂ y2 ∂ x2 ∂ y2 0 0 0 0 0        0 b  a  4  b a  b a ∂ 4Yn ∂ Xm ∂ 2 Xm ∂ 2Yn −μ Yn XmYn dxdy + 2 XmYn dxdy + Xm X Y dxdy 4 2 2 ∂x ∂x ∂y ∂ y4 m n 0 0 0 0 0 0

k3,3 = −D

 b a 

(45)

 b a 

    b a 2  b a ∂ 4Yn ∂ 4 Xm ∂ Xm ∂ 2Yn k4,4 = −F Y X Y dxdy + 2 X Y dxdy + Xm X Y dxdy ∂ x4 n m n ∂ x2 ∂ y2 m n ∂ y4 m n 0 0 0 0      0 b 0 a  6  b a 4   b a ∂ Xm ∂ Xm ∂ 2Yn ∂ 2 Xm ∂ 4Yn −λ Y X Y dxdy + 3 X Y dxdy + 3 X Y dxdy ∂ x6 n m n ∂ x4 ∂ y2 m n ∂ x2 ∂ y4 m n 0 0 0 0 0 0         b a     b a b a ∂ 6Yn ∂ 2Yn ∂ 2 Xm + Xm X Y d xd y + A Y X Y d xd y + X X Y dxdy m n n m n m m n 44 ∂ y6 ∂ x2 ∂ y2 0 0 0 0 0 0          b a  b a  b a ∂ 4Yn ∂ 4 Xm ∂ 2 Xm ∂ 2Yn −λ Y X Y d xd y + 2 X Y d xd y + X X Y d xd y m ∂ x4 n m n ∂ x2 ∂ y2 m n ∂ y4 m n 0 0  0 b 0 a       0 b 0 a  2  b a 2 ∂ Yn ∂ Xm −Kw XmYn XmYn dxdy − μ Y X Y dxdy + Xm X Y dxdy ∂ x2 n m n ∂ y2 m n 0 0 0 0 0     b  a  2 0    b a  b a 4 ∂ 2Yn ∂ Xm ∂ Xm − NT + NH − Kp Y X Y d xd y + X X Y d xd y − μ Y X Y dxdy n m n m m n n m n ∂ x2 ∂ y2 ∂ x4 0 0 0 0 0 0      b a 2   b a ∂ 4Yn ∂ Xm ∂ 2Yn +2 XmYn dxdy + Xm X Y dxdy 2 2 ∂x ∂y ∂ y4 m n 0 0 0 0 (46)

m1,1 = +I0

 b  a  0

0

∂ Xm ∂ Xm Y Y )dxdy − μ ∂x n ∂x n

 b  a  0

0

    b a ∂ 3 Xm ∂ Xm ∂ Xm ∂ 2Yn ∂ Xm Y Y d xd y + Y d xd y n n n ∂x ∂ x ∂ y2 ∂ x ∂ x3 0 0 (47)

m1,2 = +I0

 b  a       b a 2 ∂ Yn ∂ Yn ∂ 3Yn ∂ Yn ∂ Xm ∂ Yn ∂ Yn Xm X dxdy − μ Xm X dxdy + X dxdy ∂y m ∂y ∂ y3 m ∂ y ∂ x2 ∂ y m ∂ y 0 0 0 0

 b  a  0

0

(48)

m3,1 = −I1

 b  a  0

0

 b  a  3      b a ∂ Xm ∂ Xm ∂ Xm ∂ Xm ∂ Xm ∂ 2Yn ∂ Xm Yn Yn dxdy − μ Y Y d xd y + Y d xd y ∂x ∂x ∂ x ∂ y2 ∂ x n ∂ x3 n ∂ x n 0 0 0 0 (49)

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m4,1 = −I3

 b  a  0

     b  a  3  b a ∂ Xm ∂ Xm ∂ Xm ∂ Xm ∂ Xm ∂ 2Yn ∂ Xm Yn Yn dxdy − μ Y Y d xd y + Y d xd y n n n ∂x ∂x ∂x ∂ x ∂ y2 ∂ x ∂ x3 0 0 0 0

0

(50)

m3,2 = −I1

 b  a  Xm 0

0

 b  a       b a 2 ∂ Yn ∂ Yn ∂ 3Yn ∂ Yn ∂ Xm ∂ Yn ∂ Yn Xm dxdy − μ Xm X d xd y + X d xd y ∂y ∂y ∂ y3 m ∂ y ∂ x2 ∂ y m ∂ y 0 0 0 0 (51)

m4,2 = −I3

 b  a  Xm 0

0

 b  a       b a 2 ∂ Yn ∂ Yn ∂ 3Yn ∂ Yn ∂ Xm ∂ Yn ∂ Yn Xm dxdy − μ Xm X d xd y + X d xd y ∂y ∂y ∂ y3 m ∂ y ∂ x2 ∂ y m ∂ y 0 0 0 0 (52)

 b  a 



 b  a 



 b a





∂ 2Yn ∂ 2 Xm Yn XmYn dxdy + Xm X Y dxdy 2 ∂x ∂ y2 m n 0 0 0 0    0b  0a  2       b a  b a ∂ 2Yn ∂ Xm ∂ 4 Xm −I2 Y X Y dxdy + Xm X Y dxdy − μ Y X Y dxdy ∂ x2 n m n ∂ y2 m n ∂ x4 n m n 0 0 0 0 0 0      b a 2   b a ∂ 4Yn ∂ Xm ∂ 2Yn +2 XmYn dxdy + Xm X Y dxdy 2 2 ∂x ∂y ∂ y4 m n 0 0  b  a  2     b  a0  0   4mc x0 y0 ∂ Xm 2 2 + sin π sin π XmYn XmYn dxdy − μ Y X Y dxdy ab a b ∂ x2 n m n 0 0 0 0     b a ∂ 2Yn + Xm X Y dxdy ∂ y2 m n 0 0

m3,3 = +I0

XmYn XmYn dxdy − μ



 b  a 

    b a ∂ 2Yn ∂ 2 Xm Y X Y d xd y + X X Y d xd y m ∂ x2 n m n ∂ y2 m n 0 0   b  0a 0 4   b  a 0 2 0     b a 2 ∂ Yn ∂ Xm ∂ Xm −I4 Y X Y dxdy + Xm X Y dxdy − μ Y X Y dxdy ∂ x2 n m n ∂ y2 m n ∂ x4 n m n 0 0 0 0 0 0      b a 2   b a ∂ 4Yn ∂ Xm ∂ 2Yn +2 XmYn dxdy + Xm X Y dxdy 2 2 ∂x ∂y ∂ y4 m n 0 0  b  a  2     b  a0  0   4mc x0 y0 ∂ Xm 2 2 + sin π sin π XmYn XmYn dxdy − μ Y X Y dxdy ab a b ∂ x2 n m n 0 0 0 0     b a ∂ 2Yn + Xm X Y dxdy ∂ y2 m n 0 0

m3,4 = m4,3 = +I0

 b  a 

(53)

XmYn XmYn dxdy − μ

(54)

 b  a  2     b a ∂ 2Yn ∂ Xm (XmYn XmYn )dxdy − μ Y X Y d xd y + X X Y d xd y m ∂ x2 n m n ∂ y2 m n 0 0 0  0 b  a 4   0b  0a 2     b a 2 ∂ Xm ∂ Yn ∂ Xm −I5 ( 2 Yn XmYn dxdy + Xm X Y dxdy − μ ( 4 Yn XmYn dxdy ∂x ∂ y2 m n ∂x 0 0 0 0 0 0     b a 2   b a ∂ 4Yn ∂ Xm ∂ 2Yn +2 X Y dxdy + Xm X Y dxdy )) ∂ x2 ∂ y2 m n ∂ y4 m n 0 0  b  a  2     b  a0  0   4mc x0 y0 ∂ Xm 2 2 + sin π sin π XmYn XmYn dxdy − μ Y X Y dxdy n m n ab a b ∂ x2 0 0 0 0    b a ∂ 2Yn + Xm X Y dxdy ∂ y2 m n 0 0

m4,4 = +I0

 b 

a

(55)

Also, non-dimensional parameters are defined as:



ωˆ = ωa

ρc Ec

, Kw =

k p a2 kw a4 Ec h3 , Kp = , Dc = Dc Dc 12(1 − v2c )

(56)

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93

Table 1.  Comparison of non-dimensional fundamental natural frequency ω ˆ = ωh ρc /Gc of Mori–Tanaka based FG nanoplates with simply-supported boundary conditions (p = 5). a /h

μ

a /b = 1

a/b = 2

Natarajan et al. (2012)

Present

Natarajan et al. (2012)

Present

10

0 1 2 4

0.0441 0.0403 0.0374 0.0330

0.0438 0.0400 0.0371 0.0328

0.1055 0.0863 0.0748 0.0612

0.1043 0.0854 0.0741 0.0606

20

0 1 2 4

0.0113 0.0103 0.0096 0.0085

0.0112 0.0102 0.0095 0.0084

0.0279 0.0229 0.0198 0.0162

0.0277 0.0227 0.0197 0.0161

Fig. 3. Comparison between the natural frequencies of power-law and Mori–Tanaka porous FGM nanoplates for different nonlocal parameters (a/h = 10, λ = 0.1, Kw = 0, Kp = 0, C = 0%, ξ = 0.2).

Finally, setting the coefficient matrix to zero gives the natural frequencies. The function Xm for simply-supported boundary conditions is defined by:

Xm (x ) = sin (λm x )

λm =

mπ a

(57)

The function Yn can be obtained by replacing x, m and a, respectively by y, n and b. It is proved by many researchers that the function presented in Eq. (57) can satisfy the boundary conditions of nanostructures modeled via nonlocal strain gradient theory. To avoid the presentation of duplicated results, interested readers are referred to related papers (Farajpour, Yazdi et al., 2016; Li et al., 2016). It should be also noted that with the consideration of the exact position of the neutral surface, the in-plane displacements could vanish from the governing equations. Because bending-extension coupling is eliminated with the consideration of the exact position of the neutral surface. However, in this section, the general solution of the governing of a FG nanoplate was presented. But, it is possible to discard u and v from the governing equations. 5. Numerical results and discussions In this paper, free vibration characteristics of porous functionally graded (FG) nanoplates embedded on an elastic medium are investigated based on nonlocal strain gradient theory and a higher order plate theory by presenting Galerkin’s solution for the first time. The material properties of FG nanoplate are assumed to vary gradually along the thickness and are estimated through the power-law and Mori–Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with the plate. Natural frequencies of the present nanoscale plate

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Fig. 4. Variation of dimensionless frequency of perfect nanoplate versus temperature rise for different nonlocal and strain gradient parameters (a/h = 15, Kw = 0, Kp = 0, C = 0%).

model are compared with those of classical plate theory obtained by Natarajan et al. (2012) using finite element method (FEM) and the results are presented in Table 1. The length of nanoplate is assumed as a = 50 nm. Also, material properties of nanoplate (alumina and aluminum) are considered as: • Ec = 380 GPa, ρc = 3800 kg/m3 , vc = 0.3, αc = 7 × 10−6 1/◦ C, βc = 0.001 (wt.% H2 o )−1 • Em = 70 GPa, ρm = 2707 kg/m3 , vm = 0.3, αm = 23 × 10−6 1/◦ C, βm = 0.44 (wt.% H2 o )−1 Fig. 3 shows the comparison between the natural frequencies of power-law (P-FGM) and Mori–Tanaka (MT-FGM) porous nanoplates for different nonlocal parameters when a/h = 10, λ = 0.1, Kw = 0, Kp = 0, C = 0% and ξ = 0.2. It can be seen that MT-FGM model gives lower vibration frequencies than P-FGM model at a fixed nonlocal parameter, porosity volume fraction and material gradient index. This is due to the fact that FG plates become more flexible according to Mori–Tanaka homogenization scheme than with respect to power-law model at a constant material gradient index. In other words, Mori–Tanaka model produces lower values for Young’s modulus than the power-law model. Fig. 4 presents the effects of nonlocal (μ) and stain gradient (λ) parameters as well as temperature change on natural frequency of FG nanoplates at a/h = 15, Kw = 0 and Kp = 0. The case μ = λ = 0 is corresponding to the macro scale plates

H. Shahverdi, M.R. Barati / International Journal of Engineering Science 120 (2017) 82–99

95

Fig. 5. Variation of dimensionless frequency of nanoplate versus temperature rise for different porosities (p = 1, μ = 0.2, λ = 0.1, Kw = 0, Kp = 0).

without size-dependent behavior. The case λ = 0 incorporates only nonlocal effects without strain gradient mechanism. One can see that temperature rise results in decrease of plate stiffness and vibration frequency. At a certain temperature, the natural frequency of nanoplate becomes zero. At this critical temperature, the nanoplate is buckled and doesn’t oscillate. It is found that natural frequencies and critical buckling temperatures of FG nanoplates are significantly influenced by the value of nonlocal and strain gradient parameters. In fact, nonlocal parameter introduces a stiffness-softening mechanism, while strain gradient parameter provides a stiffness-hardening mechanism. In other words, increasing nonlocal parameter leads to smaller frequencies and critical temperatures. In contrast, increasing strain gradient parameter yields larger frequencies and critical temperatures. When λ < μ, obtained frequency is smaller than that of nonlocal elasticity theory. However, at λ > μ obtained frequencies becomes larger than nonlocal elasticity theory. In respite of the significance, such conclusions are not reported in previous investigations on the vibration of nanoplates. It is suggested that both nonlocal and strain gradient effects should be considered for more accurate analysis of nanoplates. Also, all these observations are affected by the gradation of material properties or inhomogeneity index (p). In fact, an increase of inhomogeneity index (p) is proportional to a higher metal constituent which leads to smaller frequencies and critical buckling temperatures. In Fig. 5, the influence of porosity volume fraction on dimensionless frequencies of a FG nanoplate versus temperature change is plotted when p = 1, μ = 0.2, λ = 0.1, Kw = 0 and Kp = 0. When there is no hygro-thermal loading, porosities inside

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Fig. 6. Variation of dimensionless frequency of porous nanoplates versus moisture percentage for various power-law indices (a/h = 10, T = 100, μ = 0.2, λ = 0.1).

the material lead to smaller frequencies by reducing the stiffness of nanoplate. However, when the nanoplate is subjected to thermal or hygro-thermal loading, the frequencies becomes larger with the effect of porosities. This is due to the reduction in the thermal or hygro-thermal resultants in the presence of porosities. Therefore, a porous FG nanoplate has larger critical buckling temperatures than a perfect one. These observations are consistent with the previous studies on FG macro-scale structures. Another investigation on the effects of moisture percentage rise and inhomogeneity index on the variation of vibration frequencies of porous FG nanoplates is performed in Fig. 6 for different elastic foundation parameters. As mentioned, moisture percentage rise leads to lower bending rigidity of nanoplates and smaller frequencies. Another interesting observation is that there is a large gap as increasing of the inhomogeneity index. In fact, with the increase of inhomogeneity index (p), the influence of hygro-thermal loading increases. This is due to the excellent characteristics of ceramic to block the moisture. Therefore, increasing the metal portion with increasing of inhomogeneity index reveals that the humidity effect is increased in the structures. Consequently, FG materials are distinct from the conventional composite materials in the hygro-thermal mechanism, and the structure is more affected by the moisture at larger power-law indices. However, the elastic medium

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97

Fig. 7. Frequency shift of the nano-mechanical mass sensor versus nanoparticle mass for different porosity volume fractions (μ = 0.2, p = 1, a = 50 nm, a/h = 10, Kw = 10, K p = 0.5 ).

Fig. 8. Frequency shift of nanoporous nano-mechanical mass sensor versus nanoparticle mass for various numbers of nanoparticles (μ = 0.2, λ = 0.1, ζ = 0.1, p = 1, Kw = 25, Kp = 5, a/h = 10 ).

has an increasing effect on natural frequencies of FG nanoplates. In fact, increasing in Winkler and Pasternak constants yields enhancement of bending rigidity of FG nanoplate. Porosity effect on the frequency shift of inhomogeneous nano-mechanical mass sensors with respect to the attached mass is demonstrated in Fig. 7 at p = 1, μ = 0.2, λ = 0.1, Kw = 10 and Kp = 0.5. A nanoparticle is attached to the center of the nanoplate. The frequency shift is defined as f = ω − ω0 , in which ω and ω0 are the frequency of nanoplate with and without attached mass, respectively. Increasing in nanoparticle mass leads to increment in the frequency shift. This phenomenon agrees with the previously published papers on graphene-based mass sensors. It is also well-known that the

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variation of frequency is apparent when the attached mass is larger than 10−21 g. In fact, the mass sensitivity of the nanomechanical sensor can reach at least 10−21 g, which has the same order as reported in previously published papers on graphene-based mass sensors. It is also seen that porosities inside the material lead to higher frequency shifts by reducing the stiffness of nano-mechanical mass sensors. Therefore, a porous nano-mechanical mass sensor has larger frequency shifts than a perfect one. Hence, control of porosities inside the material is crucial for enhancing the mass detection of nanomechanical mass sensors. The frequency shift of nanoporous nano-mechanical mass sensor versus nanoparticle mass for various numbers of nanoparticles is presented in Fig. 8 at μ = 0.2, λ = 0.1, ζ = 0.1, p = 1, Kw = 25 and Kp = 5. It is considered that the total mass of nanoparticles is 10−20 g. One can see that the frequency shift is significantly affected by the distribution of nanoparticles. When one attached mass is placed at the center of nanoplate, the nano-mechanical mass sensor has maximum frequency shift. Another observation is that two attached masses at the position (ξ ,ɳ) = (0.25,0.5), (0.75,0.5) provide smaller frequency shift than both one attached mass and four attached masses. Because the total mass of nanoparticles is identical, then their location and effective mass have a dominant effect on the frequency shift. 6. Conclusions This paper was concerned with the modeling and vibration analysis of nanoplates as nano-mechanical mass sensors according to the nonlocal strain gradient theory. The proposed generalized theory introduced two scale parameters for the prediction of vibration frequencies of nanoplates much accurately. The formulation of nanoplate was based on a higher order shear deformation theory with four field variables. 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