Wave propagation in embedded inhomogeneous ...

Waves in Random and Complex Media

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Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects Farzad Ebrahimi , Mohammad Reza Barati & Ali Dabbagh To cite this article: Farzad Ebrahimi , Mohammad Reza Barati & Ali Dabbagh (2017): Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects, Waves in Random and Complex Media, DOI: 10.1080/17455030.2017.1337281 To link to this article: http://dx.doi.org/10.1080/17455030.2017.1337281

Published online: 15 Jun 2017.

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Date: 16 June 2017, At: 01:59

Waves in Random and Complex Media, 2017 https://doi.org/10.1080/17455030.2017.1337281

Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects Farzad Ebrahimi  , Mohammad Reza Barati and Ali Dabbagh Faculty of Engineering, Department of Mechanical Engineering, Imam Khomeini International University, Qazvin, Iran

ABSTRACT

In this article, an analytical approach is developed to study the effects of thermal loading on the wave propagation characteristics of an embedded functionally graded (FG) nanoplate based on refined four-variable plate theory. The heat conduction equation is solved to derive the nonlinear temperature distribution across the thickness. Temperature-dependent material properties of nanoplate are graded using Mori–Tanaka model. The nonlocal elasticity theory of Eringen is introduced to consider small-scale effects. The governing equations are derived by the means of Hamilton’s principle. Obtained frequencies are validated with those of previously published works. Effects of different parameters such as temperature distribution, foundation parameters, nonlocal parameter, and gradient index on the wave propagation response of size-dependent FG nanoplates have been investigated.

ARTICLE HISTORY

Received 15 June 2016 Accepted 25 April 2017

1. Introduction Functionally graded materials (FGMs) are a new class of composite materials which their properties differ from one surface to another and it results in eliminating the stress concentration available in conventional composites. FGMs have many enhanced material properties [1] such as improved thermal resistance, better stress spreading, and low intensity factor which results in a large use of FGMs in several engineering fields including aerospace, civil, nuclear, mechanical, and electronic. In view of these advantages, a number of investigations, dealing with static, buckling, dynamic characteristics of functionally graded (FG) structures, had been published in the scientific literature [2–10]. Nanoscale structures are of an extreme importance in the field of nanomechanics, so it is crucial to account for small-scale effects in their mechanical analysis. The lack of a scale parameter in the classical continuum theory makes it impossible to describe the size effects. Hence, size-dependent continuum theories such as nonlocal elasticity theory of Eringen [11,12] and nonlocal strain gradient theory [13] are developed to consider the small-scale effects. The nonlocal elasticity theory and the nonlocal strain gradient theory are used to analyze the mechanical response of nanostructures in many recent studies [14–23].

CONTACT  Farzad Ebrahimi 

[email protected]

© 2017 Informa UK Limited, trading as Taylor & Francis Group

2 

 F. EBRAHIMI ET AL.

Investigation of size effects on wave propagation behavior of nanoplates resting on elastic medium is performed by Wang et al. [24]. Wang et al. [25] have also investigated the size effects on axial wave propagation of nanoplates employing nonlocal elasticity. Narendar and Gopalakrishnan [26] studied the thermal effects on ultrasonic wave propagation characteristics of nanoplates. Small size effect on the wave propagation of a piezoelectric nanoplate is studied by Zhang et al. [27] via nonlocal elasticity theory. Zhang et al. [28] investigated wave propagation behavior of nanoplates incorporating surface stress effects. Also, Zang et al. [29] showed the size effects on the axial wave propagation of a piezoelectric nanoplate considering surface effects. Recently, a remarkable attention in research community is paid to static and dynamic analysis of nonlocal FG size-dependent structures. The free flexural vibration characteristics of a FG nanoplate are explored by Natarajan et al. [30] via finite element method. Thermal buckling and free vibration characteristics of FG nanobeams subjected to temperature distributions have been investigated by Ebrahimi and Salari [31]. Thermomechanical vibration response of FG nanobeams under linear and nonlinear thermal loadings has been investigated by Ebrahimi et al. [32]. The buckling and vibrational behavior of a FG nanoplate subjected to a thermal loading has been studied in the prebuckling domain by Ansari et al. [33]. Three-dimensional bending and vibration characteristics of FG nanoplates have been studied by Ansari et al. [34]. Employing a new nonlocal refined theory, the vibrational behavior of FG nanoplates is investigated by Belkorissat et al [35]. Nami et al. [36] showed the effects of thermal loading on the buckling responses of FG nanoplates according to the third order plate theory. Free vibration behaviors of FG nanoplates considering small-scale effects are investigated by Daneshmehr et al. [37]. Ansari et al. [38] explored vibration analysis of FG nanoplates via nonlocal three-dimensional theory of elasticity. Moreover, there are lots of theories which are employed in analysis of the static, vibration, buckling, and wave propagation behavior of plates. The simplest theory is called classical plate theory (CPT) in which the shear deformation effects are ignored. Since the application of FGMs increases, more exact plate theories are needed to estimate the response of FG plates, therefore, higher-order shear deformation theories (HSDTs) are suggested. Shear deformation effects are taken into account in HSDTs, so these theories do not need a shear correction factor. Recently, four-­ variable refined plate theories are developed and used to investigate the responses of FG plates. A refined higher-order plate theory is improved by Thai and Choi [39] to investigate the free vibration analysis of embedded FG plates. Thai et al. [40] used a parabolic refined higher-order plate theory for vibration analysis of FG plates. Also, Tounsi et al. [41] presented an exact shear deformation theory for buckling and free vibration of graded sandwich plates. However, in a series of papers thickness stretching effect is considered in the framework of five-variable shear deformation theories. In these theories a new variable is added to capture thickness stretching influences. Thermal buckling behavior of an FG nanoplate resting on an elastic foundation in different types of thermal environments is investigated with a new higher-order refined theory by Barati et al. [42]. Besides above endeavors, Ebrahimi and Barati tried to show influences of various loadings including hygro-thermo-electro-magnetic effects on vibration and buckling responses of FG nanostructures employing higher-order theories [43–67]. Literature survey shows that vibration and buckling analysis of FG nanobeams and FG nanoplates are plenty; however, there are few articles that investigate the wave propagation of such structures. For instance, Zhang et al. [68] developed a nonlocal surface elasticity to

WAVES IN RANDOM AND COMPLEX MEDIA 

 3

Table 1. Temperature-dependent coefficients for Si3 N4 and SUS 304. Material Si3 N4

SUS 304

Properties E (Pa) 𝛼 (K−1 ) 𝜌 (kg/m3 ) 𝜅 (W/mK) ν

P0 348.43e+9 5.8723e−6 2370 13.723 0.24

P−1 0 0 0 0 0

P1 −3.070e−4 9.095e−4 0 −1.032e−3 0

P2 2.160e−7 0 0 5.466e−7 0

P3 −8.946e−11 0 0 −7.876e−11 0

E (Pa) 𝛼 (K−1 ) 𝜌 (kg/m3 ) 𝜅 (W/mK) ν

201.04e+9 12.330e−6 8166 15.379 0.3262

0 0 0 0 0

3.079e−4 8.086e−4 0 −1.264e−3 −2.002e−4

−6.534e−7 0 0 2.092e−6 3.797e−7

0 0 0 −7.223e−10 0

account for surface effects while investigating longitudinal wave propagation behaviors of piezoelectric nanoplates. Zang et al. [29] investigated the surface and thermal effects on the flexural wave propagation of piezoelectric FG nanobeams via nonlocal elasticity theory. The flexural wave propagation analysis of small-scaled FG beams is presented by Li et al. [69] applying a nonlocal strain gradient theory. Ebrahimi and Barati [70] presented a nonlocal strain gradient-based theory to show wave dispersion properties of embedded FG nanobeams utilizing classical beam theory. Moreover, effect of thermal loading is covered in another attempt studying wave propagation problem of a size-dependent FG nanobeam incorporating quasi-3D beam theory by Ebrahimi and Barati [71]. Influences of thermal loading and angular velocity on wave dispersion responses of FG nanobeams are ­investigated by Ebrahimi et al. [72–74]. Wave propagation characteristics of smart magneto-­­electroelastic FG nanobeams and -plates are studied by Ebrahimi et al. [75–79]. Besides, Ke et al. [80] investigated wave propagation responses of small MEE beams employing classical and first-order beam theory. Therefore, it is clear that in spite of all appreciable attempts, there is also some shortcomings in the field of studying wave propagation ­problem of small-scale graded structures. In other words, no article is found covering wave dispersion behaviors of FG nanoplates subjected to nonlinear thermal loading in the framework of higher-order plate theories. In this paper, the nonlocal elasticity is employed to examine the wave propagation behavior of size-dependent FG nanoplates resting on Winkler-Pasternak foundation subjected to thermal loading using a higher-order refined plate theory. The material properties of nanoplate vary through the thickness via Mori–Tanaka scheme and are considered to be temperature-dependent. The governing equations of FG nanoplate are derived by using Hamilton’s principle, and an analytical solution is applied to find the wave frequency and phase velocities. Influences of different parameters such as elastic foundation constants, thermal loading, nonlocality and material composition on wave characteristics of FG nanoplate are investigated.

2.  Theory and formulation 2.1.  Mori–Tanaka FGM nanoplate model Mori–Tanaka homogenization model expresses the effective material properties such as effective local bulk modulus Ke and shear modulus μe in the form [42]:

4 

 F. EBRAHIMI ET AL.

Vc Ke − Km = ( ) ( ) Kc − Km 1 + Vm Kc − Km ∕ Km + 4𝜇m ∕3 Vc 𝜇e − 𝜇m = ( ) ( ) 𝜇c − 𝜇m 1 + Vm 𝜇c − 𝜇m ∕[𝜇m + 𝜇m 9Km + 8𝜇m ∕6(Km + 2𝜇m )]

(1)

(2)

In above equations subscripts m and c denote metal and ceramic, respectively, and their volume fractions are related to each other by following equation: (3)

Vc + Vm = 1 The volume fraction of ceramic is supposed to be: ( ) z 1 p Vc = + h 2

(4)

Here p refers to gradient index which determines distribution of material through thickness of the plate and z is the distance from neutral plane of the FG nanoplate. Therefore, the effective Young’s modulus E, poison’s ratio v, and mass density ρ based on Mori–Tanaka model can be expressed by:

E(z) =

9Ke 𝜇e 3Ke + 𝜇e

(5)

v(z) =

3Ke − 2𝜇e 6Ke + 2𝜇e

(6)

𝜌(z) = 𝜌c Vc + 𝜌m Vm

(7)

The thermal expansion coefficient α and thermal conductivity K can be expressed by:

𝛼e − 𝛼m = 𝛼c − 𝛼m Ke − Km = Kc − Km

1 Ke

−

1 Km

1 Kc

−

1 Km

Vc (Kc −Km )

1 + Vm

(8)

(9)

3Km

According to the following relation, temperature-dependent coefficients of material phases can be written as:

( ) P = P0 1 + P−1 T −1 + P1 T + P2 T 2 + P3 T 3

(10)

where P0, P−1, P1, P2, and P3 are temperature-dependent coefficients which are given in Table 1 for Si3N4 and SUS304. The bottom and top surfaces of FG nanoplate are fully metal (SUS304) and fully ceramic (Si3N4), respectively (Figure 1).

WAVES IN RANDOM AND COMPLEX MEDIA 

 5

In this paper, we assumed temperature to vary nonlinearly through the thickness of plate. Temperature distribution can be achieved by solving the steady-state heat conduction equation with respect to boundary conditions on top and bottom surfaces of nanoplate across the thickness:

−

d dz

( K (z, T )

)

dT dz

(11)

=0

Considering the boundary conditions

T

( ) h = Tc 2

(12a)

( ) h = Tm T − 2

(12b)

By solving above equation we will achieve the following relation for temperature:

( ) ∫− h T = Tm + Tc − Tm h 2 ∫ 2h z

−2

1 K (z,T )

dz

1 K (z,T )

dz

(13)

2.2.  Kinematic relations According to refined shear deformation plate theories, the displacement field of nonlocal FGM plate can be assumed as:

U(x, y, z, t) = u0 (x, y, t) − z

𝜕wb 𝜕w − f (z) s 𝜕x 𝜕x

(14a)

V (x, y, z, t) = v0 (x, y, t) − z

𝜕wb 𝜕w − f (z) s 𝜕y 𝜕y

(14b)

W(x, y, z, t) = wb (x, y, t) + ws (x, y, t)

(14c)

Here u0 is longitudinal displacement and wb, ws are, respectively, deflection made by bending and shear deflection. f (z) is shape function that denotes the distribution of shear stress strain through the plate thickness. In present work we assume ( [or ]) ( [ ]) − z∕ cosh 𝜋2 − 1. Nonzero strains can be calculated by following f (z) = 𝜋h sinh 𝜋zh relations:

⎧ 𝜀 ⎫ ⎧ 𝜀0 ⎪ x ⎪ ⎪ x0 ⎨ 𝜀y ⎬ = ⎨ 𝜀 y ⎪ 𝛾 ⎪ ⎪ 𝛾0 ⎩ xy ⎭ ⎩ xy

⎫ ⎧ kb ⎪ ⎪ xb + z ⎬ ⎨ ky ⎪ ⎪ kb ⎭ ⎩ xy

⎫ ⎧ ks ⎪ ⎪ xs + f (z) ⎬ ⎨ ky ⎪ ⎪ ks ⎭ ⎩ xy

⎫ ⎪ ⎬, ⎪ ⎭

�

� 𝛾yz 𝛾xz

� =

𝛾yz0 𝛾xz0

� (15)

6 

 F. EBRAHIMI ET AL.

where

⎧ 𝜀0 ⎫ ⎧ ⎪ x0 ⎪ ⎪ ⎨ 𝜀y ⎬ = ⎨ ⎪ 𝛾0 ⎪ ⎪ ⎩ xy ⎭ ⎩ ⎧ ks ⎪ xs ⎨ ky ⎪ ks ⎩ xy

𝜕u0 𝜕x 𝜕v0 𝜕x 𝜕u0

+

𝜕v0

𝜕y

𝜕x 𝜕2 w

⎫ ⎧ − 𝜕x 2s ⎪ ⎪ 𝜕 2 ws ⎬ = ⎨ − 𝜕y 2 𝜕 2 ws ⎪ ⎪ ⎭ ⎩ −2 𝜕x𝜕y

⎫ ⎪ ⎬, ⎪ ⎭ ⎫ ⎪ ⎬, ⎪ ⎭

⎧ kb ⎪ xb ⎨ ky ⎪ kb ⎩ xy � 𝛾yz0 𝛾xz0

𝜕2 w

⎫ ⎧ − 𝜕x 2b ⎫ ⎪ ⎪ 𝜕 2 wb ⎪ ⎬ = ⎨ − 𝜕y 2 ⎬, 𝜕 2 wb ⎪ ⎪ ⎪ ⎭ ⎩ −2 𝜕x𝜕y ⎭ � � 𝜕w �

(16a)

s

=

𝜕y 𝜕ws 𝜕x

and

g(z) = 1 −

df (z) dz

(16b)

Now based on Hamilton’s principle we try to find the Euler–Lagrange equations of FG nanoplate in thermal environment: t

∫0

𝛿(U − K + V )dt = 0

(17)

where δU is the variation of strain energy; δK is the variation of kinetic energy and δV is the variation of work done by external (applied) forces. The variation of strain energy is: [ ] 𝛿U = 𝜎ij 𝛿𝜀ij dV = 𝜎x 𝛿𝜀x + 𝜎y 𝛿𝜀y + 𝜏xy 𝛿𝛾xy + 𝜏yz 𝛿𝛾yz + 𝜏xz 𝛿𝛾xz dV (18) ∫ Substituting Equations (15) and (16) into Equation (18) yields: L[ 𝜕 2 𝛿wb 𝜕 2 𝛿wb 𝜕𝛿u0 𝜕 2 𝛿ws 𝜕𝛿v0 𝜕 2 𝛿ws b s s − Mxb − M 𝛿U = Nx − M + N − M y y x y ∫0 𝜕x 𝜕x 𝜕x 2 𝜕x 2 𝜕y 2 𝜕y 2 (19) ) ] ( 2 2 𝜕𝛿ws 𝜕𝛿ws 𝜕 𝛿ws 𝜕 𝛿wb 𝜕𝛿u0 𝜕𝛿v0 s b + − 2Mxy + Qyz + Qxz − 2Mxy dx + Nxy 𝜕y 𝜕x 𝜕x𝜕y 𝜕x𝜕y 𝜕y 𝜕x in which the stress resultants N, M, and Q are defined by:

( ) Ni , Mib , Mis =

Qi =

∫

∫

(1, z, f )𝜎i dA,

g𝜎i dA,

i = (x, y, xy)

i = (x, y, xy)

The first variation of work done by applied forces can be stated as: ) ( ) ) ( ( ) ( L( 𝜕 wb + ws 𝜕𝛿 wb + ws 𝜕 wb + ws 𝜕𝛿 wb + ws 0 0 + Ny 𝛿V = Nx ∫0 𝜕x 𝜕x 𝜕y 𝜕y ( ) )) ( ) ( 2 𝜕 𝛿 wb + w s 𝜕 wb + ws 𝜕 wb + ws ( ) 0 − kw 𝛿 w b + w s + kp dx + 2𝛿Nxy 𝜕x 𝜕y 𝜕x 2

(20a)

(20b)

(21)

WAVES IN RANDOM AND COMPLEX MEDIA 

 7

0 In above relations Nx0 = Ny0 = N T and Nxy = 0:

T

( ) E(z) 𝛼(z, T ) T − T0 dz ∫− h 1 − v h 2

N =

(22)

2

The variation of Kinetic energy is expressed as: {

( )( )] [ ̇ u̇ + v𝛿 ̇ v̇ + w𝛿 ̇ w]𝜌(z)dV ̇ I u̇ 𝛿 u̇ + v̇ 0 𝛿 v̇ 0 + ẇ b + ẇ s 𝛿 ẇ b + 𝛿 ẇ s 𝛿K = [u𝛿 = ∫ ∫0 0 0 0 [ ] [ ] 𝜕𝛿 ẇ b 𝜕 ẇ b 𝜕𝛿 ẇ s 𝜕 ẇ s 𝜕𝛿 ẇ s 𝜕 ẇ s 𝜕𝛿 ẇ s 𝜕 ẇ s ̇ ̇ ̇ ̇ ̇ ̇ ̇ ̇ − I1 u 0 + 𝛿 u 0 + v0 + 𝛿 v − J1 u 0 + 𝛿 u + v0 + 𝛿v 𝜕x 𝜕x 𝜕y 𝜕y 0 𝜕x 𝜕x 0 𝜕y 𝜕y 0 [ [ [ ] ] 𝜕𝛿 ẇ b 𝜕𝛿 ẇ b 𝜕𝛿 ẇ b 𝜕𝛿 ẇ b 𝜕 ẇ s 𝜕𝛿 ẇ s 𝜕 ẇ s 𝜕𝛿 ẇ s 𝜕 ẇ b 𝜕𝛿 ẇ s 𝜕 ẇ s 𝜕𝛿 ẇ b + I2 + + + + K2 + J2 𝜕x 𝜕x 𝜕y 𝜕y 𝜕x 𝜕x 𝜕y 𝜕y 𝜕x 𝜕x 𝜕x 𝜕x ]} 𝜕 ẇ b 𝜕𝛿 ẇ s 𝜕 ẇ s 𝜕𝛿 ẇ b (23) dx + + 𝜕y 𝜕y 𝜕y 𝜕y L

In all of above equations the dot-superscript shows the differentiation with respect to time; and the mass inertias are given by the following equation:

( ) I0 , I 1 , J 1 , I 2 , J 2 , K 2 =

h 2

∫− h

( ) 1, z, f (z), z 2 , zf (z), f 2 (z) 𝜌(z)dz

(24)

2

By setting Equations (18), (20), and (22) into Equation (17) and setting the coefficients of δu0, v0, δwb, and δws to zero, the Euler–Lagrange equations are written as:

𝜕 ẅ 𝜕 ẅ 𝜕Nx 𝜕Nxy + = I0 ü 0 − I1 b − J1 s 𝜕x 𝜕y 𝜕x 𝜕x

(25)

𝜕 ẅ b 𝜕 ẅ − J1 s 𝜕y 𝜕y

(26)

𝜕Nxy 𝜕x 𝜕 2 Mxb 2

+2

b 𝜕 2 Mxy

+

𝜕 2 Myb 2

+

𝜕Ny 𝜕y

= I0 v̈ 0 − I1

( ) ( ) ( ) − N T ∇2 wb + ws − kw wb + ws + kp ∇2 wb + ws

𝜕x𝜕y 𝜕y ( ) ( ) 𝜕 ü 0 𝜕 v̈ 0 = I0 ẅ b + ẅ s + I1 − I2 ∇2 ẅ b − J2 ∇2 ẅ s + 𝜕x 𝜕y 𝜕x

s 𝜕 2 Mxy

𝜕 2 Mys

(27)

( ) ( ) ( ) 𝜕Qxz 𝜕Qyz + − N T ∇2 wb + ws − kw wb + ws + kp ∇2 wb + ws 𝜕x𝜕y 𝜕x 𝜕y 𝜕x 𝜕y (28) ( ) ) ( 𝜕 ü 0 𝜕 ü 0 2 2 = I0 ẅ b + ẅ s + J1 − J2 ∇ ẅ b − K2 ∇ ẅ s + 𝜕x 𝜕y

𝜕 2 Mxs 2

+2

+

2

+

2.3.  The nonlocal elasticity theory Based on Eringen’s nonlocal elasticity model [11,12], the stress state at each point inside the body is assumed to be a function of strains of all points in the neighbor points. Therefore,

8 

 F. EBRAHIMI ET AL.

in this paper for the goal of paying attention to the small-scale effects the nonlocal theory of Eringen is applied. According to this theory, the nonlocal stress tensor components σij at each point x inhomogeneous solids can be written as:

𝜎ij (x) =

( ) ( ) ( ) 𝛼 ||x � − x ||, 𝜏 tij x � dΩ x � ∫

(29)

( ) in above equation tij x ′ are available local stress tensor components at point x which are related to the strain tensor components ɛkl as: (

tij = Cijkl 𝜀kl

)

(30)

The nonlocal kernel 𝛼 ||x � − x ||, 𝜏 considers the effect of strain at the point x′ on the stress at point x in elastic bodies and ||x � − x || is Euclidean distance and τ is a constant value as follows:

𝜏=

e0 a l

(31)

e0a considers the effects of small scale on the response of nanostructures. Eringen et al [11,12] numerically found the functional form of the kernel. By choosing a suitable kernel function the equivalent differential form of Eringen’s theory can be developed as:

( ( ) ) 1 − e0 a ∇2 𝜎kl = tkl

(32)

In the above equation ∇2 is Laplacian operator. So, the consistent equations of nonlocal Eringen’s theory for a FG nanoplate can be written in the following form:

⎧ ⎫ ⎛ ⎪ 𝜎x ⎪ ⎜ Q11 ⎪ 𝜎y ⎪ ⎜ Q12 � � ⎪ 2 ⎪ 1 − 𝜇∇ ⎨ 𝜎xy ⎬ = ⎜ 0 ⎪ 𝜎 ⎪ ⎜⎜ 0 yz ⎪ ⎪ ⎪ 𝜎xz ⎪ ⎜⎝ 0 ⎩ ⎭ ( )2 where 𝜇 = e0 a and Qij is as follows:

Q66

0 0

0

0 Q44

0

0 0 Q55

Q11 = Q22 =

Q12 =

0 0 0 0 0 0

Q12 Q22 0

E(z) 1 − v2

vE(z) 1 − v2

Q44 = Q55 = Q66 =

E(z) 2(1 + v)

0

⎞⎧ 𝜀x ⎟⎪ 𝜀y ⎟⎪ ⎟⎪ 𝛾xy ⎟⎨ 𝛾yz ⎟⎪ ⎟⎪ 𝛾 ⎠⎪ ⎩ xz

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(33)

(34a)

(34b)

(34c)

Force–strain and moment–strain of the nonlocal refined FG plates can be calculated by integrating Equation (32) across the cross section area of the plate:

WAVES IN RANDOM AND COMPLEX MEDIA 

⎧ N x � � 2 ⎪ 1 − 𝜇∇ ⎨ Ny ⎪ N ⎩ xy

⎫ ⎛ A ⎪ ⎜ 11 ⎬ = ⎜ A21 ⎪ ⎜ 0 ⎭ ⎝

A12 A22 0

⎛ ⎜ +⎜ ⎜ 0 ⎝

s B11 s B21

⎧ Mb � � ⎪ x 1 − 𝜇∇2 ⎨ Myb ⎪ Mb ⎩ xy

s B12 s B22

⎫ ⎛ B ⎪ ⎜ 11 ⎬ = ⎜ B21 ⎪ ⎜ 0 ⎭ ⎝

B12 B22 0

⎫ ⎛ Bs ⎪ ⎜ 11 s ⎬ = ⎜ B21 ⎪ ⎜ 0 ⎭ ⎝

0 0 B66

s D12 s D22 0

s B12 s B22 0

⎛ Hs ⎜ 11 s + ⎜ H21 ⎜ 0 ⎝

{

Qx Qy

⎫ ⎛ B ⎪ ⎜ 11 ⎬ + ⎜ B21 ⎪ ⎜ 0 ⎭ ⎝

𝜕u0 𝜕x 𝜕v0 𝜕y 𝜕u0

𝜕v0

+

𝜕y

𝜕x 𝜕 2 ws

⎞⎧ ⎟⎪ ⎟⎨ ⎟⎪ ⎠⎩

𝜕x 𝜕v0 𝜕y 𝜕v0 𝜕x 𝜕 2 ws

⎞⎧ − 𝜕x 2 ⎟⎪ 𝜕 2 ws ⎟⎨ − 𝜕y 2 𝜕 2 ws ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

⎞⎧ ⎟⎪ ⎟⎨ ⎟⎪ ⎠⎩

0 0 s H66

+

𝜕y

𝜕x 𝜕v0 𝜕y 𝜕y

+

𝜕v0 𝜕x 𝜕 2 ws

⎞⎧ − 𝜕x 2 ⎟⎪ 𝜕 2 ws ⎟⎨ − 𝜕y 2 𝜕 2 ws ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

}

( =

As44 0

D12 D22 0

𝜕2 w

⎞⎧ − 𝜕x 2b ⎟⎪ 𝜕 2 wb ⎟⎨ − 𝜕y 2 𝜕 2 wb ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

0 0 D66

𝜕2 w

⎞⎧ − 𝜕x 2b ⎟⎪ 𝜕 2 wb ⎟⎨ − 𝜕y 2 𝜕 2 wb ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

⎫ ⎪ ⎬ ⎪ ⎭ s D12 s D22 0

0 0 s D66

𝜕2 w

⎞⎧ − 𝜕x 2b ⎟⎪ 𝜕 2 wb ⎟⎨ − 𝜕y 2 𝜕 2 wb ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

⎫ ⎪ ⎬ ⎪ ⎭

0 As55

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ ⎪ ⎭

(36)

⎫ ⎛ Ds ⎪ ⎜ 11 s ⎬ + ⎜ D21 ⎪ ⎜ 0 ⎭ ⎝

𝜕u0

𝜕u0

0 0 B66

(35)

⎫ ⎛ D ⎪ ⎜ 11 ⎬ + ⎜ D21 ⎪ ⎜ 0 ⎭ ⎝

𝜕u0

𝜕u0

B12 B22 0

⎫ ⎪ ⎬ ⎪ ⎭

⎞⎧ − 𝜕x 2 ⎟⎪ 𝜕 2 ws ⎟⎨ − 𝜕y 2 𝜕 2 ws ⎟⎪ ⎠⎩ −2 𝜕x𝜕y

0 0 s D66

0 0 s B66

s H12 s H22 0

( ) 1 − 𝜇∇2

⎞⎧ ⎟⎪ ⎟⎨ ⎟⎪ ⎠⎩

0 0 s B66

0

⎛ Ds ⎜ 11 s + ⎜ D21 ⎜ 0 ⎝ ⎧ Ms � � ⎪ x 1 − 𝜇∇2 ⎨ Mys ⎪ Ms ⎩ xy

0 0 A66

 9

⎫ ⎪ ⎬ ⎪ ⎭ (37)

){

𝜕ws

}

𝜕x 𝜕ws

(38)

𝜕y

In Equations (35)–(38) the cross-sectional rigidities are given by following relations:

⎧ A ⎪ 11 ⎨ A12 ⎪ A ⎩ 66

B11 B12 B66

D11 D12 D66

s B11 s B12 s B66

s D11 s D12 s D66

s H11 s H12 s H66

⎫ ⎧ 1 ⎫ h 2 � � ⎪ ⎪ ⎪ 2 2 ⎬ = ∫ h Q11 1, z, z , f (z), zf (z), f (z) ⎨ v ⎬ dz −2 ⎪ ⎪ 1−v ⎪ ⎭ ⎩ 2 ⎭ (39a)

( ) ( ) s s s s s s A22 , B22 , D22 , B22 , D22 , H22 = A11 , B11 , D11 , B11 , D11 , H11

As44 = As55 =

h 2

∫− h 2

[ ]2 Q44 g(z) dz

(39b)

(39c)

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 F. EBRAHIMI ET AL.

Figure 1. Geometry of embedded FG nanoplate.

Figure 2. The effect of different temperature changes on the wave frequency of FG nanoplate (p = 1, μ = 1 nm2). Table 2. Comparison of frequency of FG nanoplates for various nonlocal parameter (p = 5). a/h = 10 μ 0 1 2 4

Natarajan et al. [30] 0.0441 0.0403 0.0374 0.033

a/h = 20 present 0.043803 0.040051 0.037123 0.032791

Natarajan et al. [30] 0.0113 0.0103 0.0096 0.0085

present 0.011255 0.010288 0.009534 0.008418

WAVES IN RANDOM AND COMPLEX MEDIA 

 11

By substituting Equations (35)–(38) into Equations (25)–(28), the governing equations of nanoplate can be directly written in terms of displacements (u0, v0, wb, and ws) as:

( ) 𝜕 2 v0 ) 𝜕 3 wb 𝜕 2 u0 𝜕 3 wb ( 𝜕 3 ws s + A + A + A − B − B + 2B − B 66 12 66 11 12 66 11 𝜕x𝜕y 𝜕x 2 𝜕y 2 𝜕x 3 𝜕x𝜕y 2 𝜕x 3 ) ( ( ) ( s ) 𝜕 3 ws 𝜕 ẅ 𝜕 ẅ s + 1 − 𝜇∇2 −I0 ü 0 + I1 b + J1 s = 0 − B12 + 2B66 2 𝜕x 𝜕x 𝜕x𝜕y

(40)

( ) 𝜕 2 u0 ) 𝜕 3 wb 𝜕 2 v0 𝜕 3 wb ( 𝜕 3 ws s + A + A + A − B − B + 2B − B 66 12 66 22 12 66 22 𝜕x𝜕y 𝜕y 2 𝜕x 2 𝜕y 3 𝜕x 2 𝜕y 𝜕y 3 ) ( ( ) ( s ) 𝜕 3 ws 𝜕 ẅ 𝜕 ẅ s + 1 − 𝜇∇2 −I0 v̈ 0 + I1 b + J1 s = 0 − B12 + 2B66 2 𝜕y 𝜕y 𝜕x 𝜕y

(41)

A11

𝜕 2 u0

A22

B11

𝜕 2 v0

( ( ( ) 𝜕 3 u0 ) 𝜕 3 v0 ) 𝜕 4 wb 𝜕3v 𝜕4 w + B12 + 2B66 + B22 30 + B12 + 2B66 − D11 4b − 2 D12 + 2D66 2 2 𝜕x 𝜕x𝜕y 𝜕y 𝜕x 𝜕y 𝜕x 𝜕x 2 𝜕y 2 ( ) ) 𝜕 4 ws ( ) ( s 𝜕 4 ws 𝜕 4 ws ( 𝜕4 w s s s − 2 D − D + 1 − 𝜇∇2 − I0 ẅ b + ẅ s + 2D − D22 4b − D11 12 22 66 4 2 2 4 𝜕y 𝜕x 𝜕x 𝜕y 𝜕y ) ( (42) ) ̈ ̈ ( ) ( ) 𝜕 u0 𝜕 v0 − I1 + I2 ∇2 wb +J2 ∇2 ws + (kp − N T )∇2 wb + ws − kw wb + ws =0 + 𝜕x 𝜕y 𝜕 3 u0 3

Figure 3. Effect of various temperature changes on phase velocity of FG nanoplate (p = 1, μ = 1).

12 

s B11

 F. EBRAHIMI ET AL.

( s ( s ) 𝜕 3 u0 ) 𝜕 3 v0 ) 𝜕 4 wb 𝜕 3 v0 ( s 𝜕 4 wb s s s s s + B + 2B + B + B + 2B − 2 D + 2D − D 12 66 22 12 66 11 12 66 𝜕x 3 𝜕x𝜕y 2 𝜕y 3 𝜕x 2 𝜕y 𝜕x 4 𝜕x 2 𝜕y 2 ( 4 4 4 4 ) ) 𝜕 ws ( ) ( s ( 𝜕 ws 𝜕 ws 𝜕 wb s s s s − H11 − 2 H12 + 2H66 − H22 + As44 ∇2 ws + 1 − 𝜇∇2 − I0 ẅ b + ẅ s − D22 4 4 2 2 4 𝜕y 𝜕x 𝜕x 𝜕y 𝜕y ) ) ( ( ) ) ( 𝜕 ü 0 𝜕 v̈ 0 2 2 T 2 + J2 ∇ ẅ b + K2 ∇ ẅ s + (kp − N )∇ wb + ws − kw wb + ws + = 0 + − J1 𝜕x 𝜕y 𝜕 3 u0

(43)

3.  Solution procedure By assuming the displacement fields of the waves propagating in the x–y plane with the following form

⎧ u (x, y, t) 0 ⎪ ⎪ v0 (x, y, t) ⎨ ⎪ wb (x, y, t) ⎪ w (x, y, t) s ⎩

⎫ ⎧ U exp �i �k x + k y − 𝜔t �� 2 ⎪ ⎪ �� 1 �� ⎪ ⎪ V exp i k1 x + k2 y − 𝜔t �� �� ⎬=⎨ ⎪ ⎪ Wb exp � i� k1 x + k2 y − 𝜔t�� ⎪ ⎪ W exp i k x + k y − 𝜔t s 1 2 ⎭ ⎩

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(44)

where U; V; Wb and Ws are the coefficients of wave amplitude, k1 and k2 are the wave numbers of wave propagation along x and y directions, respectively, and finally ω is frequency.

Figure 4. Effect of Winkler parameter on phase velocity of FG nanoplate (ΔT = 500, p = 1, μ = 1).

WAVES IN RANDOM AND COMPLEX MEDIA 

 13

Substituting Equation (44) into Equations (40)–(43) gives:

( ) [K ] − 𝜔2 [M] {Δ} = {0}

(45)

In above equation the unknown parameters are

{ }T {Δ} = U, V , Wb , Ws ⎡ ⎢ [K ] = ⎢ ⎢ ⎢ ⎣

a11 a21 a31 a41

a12 a22 a32 a42

a13 a23 a33 a43

a14 a24 a34 a44

⎤ ⎥ ⎥, ⎥ ⎥ ⎦

⎡ ⎢ [M] = ⎢ ⎢ ⎢ ⎣

m11 m21 m31 m41

(46a)

m12 m22 m32 m42

m13 m23 m33 m43

m14 m24 m34 m44

⎤ ⎥ ⎥ (46b) ⎥ ⎥ ⎦

where aij, mij are as written in Appendix 1. The dispersion relations of wave propagation in the FG plate can be developed by setting the following determinant to zero:

| | |[K ] − 𝜔2 [M]| = 0 | |

Figure 5. Effect of Pasternak parameter on the phase velocity of FG nanoplate (p = 1, μ = 1).

(47)

14 

 F. EBRAHIMI ET AL.

By setting k1 = k2 = k, the roots of Equation (47) are written as

𝜔1 = M1 (k),

𝜔2 = M2 (k),

𝜔3 = M3 (k),

𝜔4 = M4 (k)

(48)

The frequencies which are given by Equation (48) correspond with the wave modes M0 , M1 , M2 and M3, respectively. The wave modes M0 and M3 stand for flexural wave, and the wave modes M1 and M2 stand for extensional wave. After calculating Wi (k) for each mode (i = 1, 2, 3, 4), the phase velocities can be defined as:

Ci =

Wi (k) , k

(i = 1, 2, 3, 4)

(49)

The escape frequency of FG nanoplate can be derived by setting k → ∞. It is clear that, after escape frequency, the flexural waves will not propagate anymore.

4.  Results and discussion In this section, wave propagation analysis of an FG nanoplate made of Si3N4 and SUS304 under thermal loading is carried. The material properties are assumed to be temperature-­ dependent in order to achieve more accurate responses. The frequencies of FG nanoplate at gradient index p = 5 are compared with those presented by Natarajan et al. [30] and the results are tabulated in Table 2. It is revealed that presented model and solution can a­ ccurately predict the frequencies of FG nanoscale plates.

Figure 6. Effect of gradient index and temperature change on phase velocity of FG nanoplate.

WAVES IN RANDOM AND COMPLEX MEDIA 

 15

In Figure 2, the variations of wave frequency (ƒ = ω/2π) of temperature-dependent FG nanoplate with respect to different temperature distributions are plotted at p = 1, μ = 1. It is obvious that temperature distribution doesn’t affect the wave frequency at small wave numbers for all modes. But, for higher values of wave number the temperature effect is more considerable. With the increase in temperature, wave frequencies reduce especially at higher values of wave number. So, effect of temperature change on wave frequency depends on the value of wave number. As it can be seen, after k = 10 × 109 the frequencies remain constant in all modes and temperature changes. Figure 3 shows the variation of phase velocity of FG nanoplate for different temperature changes at p = 1, μ = 1 nm2. Although the whole shape of figures are similar in all modes, however, in modes M0 and M1 phase velocity first increases gradually to its maximum value and then starts to attenuate. In modes M2 and M3 the maximum value is in the beginning of the figure and phase velocity decreases in the same form as M0 and M1 modes. In addition, it is worth mentioning that in all modes phase velocity decreases with an increase in temperature change. The effect of different temperature changes becomes less important in large wave numbers. Effect of Winkler parameter on the phase velocity of FG nanoplate is plotted in Figure 4 for each mode at ΔT = 500, p = 1, μ = 1. The similarity of all modes is in independency of phase velocity phase velocities from Winkler parameter in high wave numbers. Another same influence can be the rise in the magnitude of phase velocity with increasing Winkler parameter in small wave numbers for each mode. In small wave numbers an increase in

Figure 7. Effect of nonlocal parameter on the phase velocity of FG nanoplate (p = 1).

16 

 F. EBRAHIMI ET AL.

Winkler parameter results in an increase in phase velocity. So, Winkler parameter has no sensible effect on phase velocity at higher wave numbers. Figure 5 indicates the effects of different Pasternak parameter on the phase velocity of FG nanoplate at ΔT = 500, p = 1, μ = 1. It is observed that by increasing Pasternak parameter, the value of phase velocity increases and this effect is more obvious at smaller wave numbers. In other words, usually for wave numbers bigger than 1 × 109 the effect of this parameter is small enough to be neglected. Figure 6 shows the effects of gradient index and different temperature changes on the phase velocity of FG nanoplates when the coefficients of foundation are set to zero. According to this figure, we can understand that in modes M0 and M1 phase velocity first increases to its maximum value and then becomes gradually smaller in all temperature changes and different gradient indices. In modes M2 and M3 the slope is uniform negative. As a predictable trend, the value of phase velocity becomes smaller as temperature change increases in all modes. Also, it is found that increasing gradient index leads to lower phase velocities. This is due to higher portion of metallic phase with increase of gradient index. In Figure 7 the influence of nonlocality on phase velocities has been plotted at p = 1, kw = kp = 0. The phase velocity becomes small by increasing in value of nonlocal parameter in each mode. This variation is more clear in bigger wave numbers than smaller ones. In all modes, when nonlocal parameter is equal to zero; the phase velocity remains constant after the wave number of almost 1 × 109. Moreover, for nonzero nonlocal parameter, by increasing

Figure 8. Effect of different temperature distributions on the escape frequency of FG nanoplate at different gradient indices.

WAVES IN RANDOM AND COMPLEX MEDIA 

 17

wave number the phase velocity approaches to zero and effect of nonlocal parameter become less important again in this case. Figure 8 illustrates the effect of various temperature changes on the escape frequency of FG nanoplates. To this purpose, the wave number is set to infinity β → ∞. It is clear that with the increase in gradient index the escape frequency decreases, especially at lower gradient indices. It can be observed that by increasing temperature change, the amount of escape frequency decreases; this trend is more significant at higher gradient indices.

5. Conclusion Wave propagation of an FG nanoplate resting on an elastic foundation subjected to nonlinear thermal loading is investigated in the framework of a hyperbolic refined plate theory. Material properties of the FG nanobeam are spatially graded according to Mori–Tanaka distribution. By implementing Hamilton’s principle, the governing differential equations are derived. Finally, through some parametric study, the effect of different parameters such as temperature change, gradient index, nonlocality parameter, and length scale parameter on wave characteristics of temperature-dependent FG nanoplate are investigated. It is found that the wave frequency, phase velocity, and escape frequency decrease with the increase of temperature. Also, nonlocal parameter has a stiffness-softening influence and reduces the phase velocities. The gradient index possesses a considerable reducing impact on the phase velocities. Also, both temperature and gradient index have no notable effect on phase velocities at higher wave numbers. An increase in Winkler and Pasternak coefficients will result in an increase in phase velocity at small wave numbers. Effect of elastic foundation is negligible at larger wave numbers.

Disclosure statement No potential conflict of interest was reported by the authors.

ORCID Farzad Ebrahimi 

 http://orcid.org/0000-0001-9091-4647

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Appendix 1 In Equation (42b) aij and mij , (i, j = 1, 2, 3, 4) are as follows:

a44

( ) a11 = − A11 k12 + A66 k22 ) ( a12 = − A12 + A66 k1 k2 [ ) ] ( a13 = ik1 B11 k12 + B12 + 2B66 k22 [ s 2 ( s ) 2] s k2 a14 = ik1 B11 k1 + B12 + 2B66 ) ( a21 = − A12 + A66 k1 k2 ) ( a22 = − A66 k12 + A22 k22 [( ) 2 ] a23 = ik2 B12 + 2B66 k1 + B22 k22 [( s ) ] s s 2 k12 + B22 k2 a24 = ik2 B12 + 2B66 [ ) ] ( (A1) a31 = −ik1 B11 k12 + B12 + 2B66 k22 [( ) 2 ] 2 a32 = −ik2 B12 + 2B66 k1 + B22 k2 ) ) [( )( ) ]( ( )) ( ( a33 = − D11 k14 + 2 D12 + 2D66 k12 k22 + D22 k24 + N T − kp k12 + k22 − kw 1 + 𝜇 k12 + k22 ) ) [( )( ) ]( ( )) ( s 4 ( s s s 4 k12 k22 + D22 k + N T − kp k12 + k22 − kw 1 + 𝜇 k12 + k22 a34 = − D11 k1 + 2 D12 + 2D66 [ s 22 ( s ) ] s k22 a41 = −ik1 B11 k1 + B12 + 2B66 [( s ) 2 ] s s 2 a42 = −ik2 B12 + 2B66 k1 + B22 k2 ) ) [( )( ) ]( ( )) ( s 4 ( s s s 4 k12 k22 + D22 k2 + N T − kp k12 + k22 − kw 1 + 𝜇 k12 + k22 a43 = − D11 k1 + 2 D12 + 2D66 ) ) [( )( ) ]( ( )) ( ) ( s 4 ( s s s 4 k12 k22 + H22 = − H11 k1 + 2 H12 + 2H66 k2 + N T − kp k12 + k22 − kw 1 + 𝜇 k12 + k22 − k12 + k22 As44

and ( ( )) m11 = −I0 1 + 𝜇 k12 + k22 , m12 = m21 = 0 ( ( 2 )) m13 = I1 1 + 𝜇 k1 + k22 ( ( )) m14 = J1 1 + 𝜇 k12 + k22 ( ( 2 )) m22 = −I0 1 + 𝜇 k1 + k22 ( ( 2 )) m23 = iI1 k2 1 + 𝜇 k1 + k22 ( ( 2 )) m24 = iJ1 k2 1 + 𝜇 k1 + k22 ( ( 2 )) m31 = −iI1 k1 1 + 𝜇 k1 + k22 ( ( 2 )) m32 = −iI1 k2 1 + 𝜇 k1 + k22 ( ))( ( )) ( m33 = − I0 + I2 k12 + k22 1 + 𝜇 k12 + k22 ( ))( ( )) ( m34 = − I0 + J2 k12 + k22 1 + 𝜇 k12 + k22 ( ( 2 )) m41 = −J1 1 + 𝜇 k1 + k22 ( ( )) m42 = −iJ1 k2 1 + 𝜇 k12 + k22 ( 2 ))( ( )) ( m43 = − I0 + J2 k1 + k22 1 + 𝜇 k12 + k22 ( ( 2 ))( ( )) m44 = − I0 + K2 k1 + k22 1 + 𝜇 k12 + k22

(A2)