nanoplates resting on elastic foundation using modified couple stress theory

Accepted Manuscript

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory Hamzeh Salehipour , Hassan Nahvi , AliReza Shahidi , Hamid Reza Mirdamadi PII: DOI: Reference:

S0307-904X(17)30156-7 10.1016/j.apm.2017.03.007 APM 11648

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

26 January 2016 22 February 2017 5 March 2017

Please cite this article as: Hamzeh Salehipour , Hassan Nahvi , AliReza Shahidi , Hamid Reza Mirdamadi , 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.007

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ACCEPTED MANUSCRIPT Highlight 

Development of a model based on the 3-D elasticity and modified couple stress theories for functionally graded micro/nano plates.

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Exponential law is used for variation of material properties through the plate

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thickness. Exact closed-form solutions are presented for the bending of functionally graded micro/nano plates.

The effects of the length scale parameter and gradient index on the bending of

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

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functionally graded micro/nano plates are investigated.

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3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

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Hamzeh Salehipour1, Hassan Nahvi2, AliReza Shahidi2, Hamid Reza Mirdamadi2 1

Department of Mechanical Engineering, Ilam University, Ilam 69315-516, Iran

2

Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran

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Abstract

This paper addresses a 3D elasticity analytical solution for static deformation of a simplysupported rectangular micro/nanoplate made of both homogeneous and functionally graded

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(FG) material within the framework of modified couple stress theory. The plate is assumed to be resting on a Winkler-Pasternak elastic foundation, and its modulus of elasticity is assumed

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to vary exponentially along thickness. By expanding displacement components in double Fourier series along in-plane coordinates and imposing relevant boundary conditions, the

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boundary value problem (BVP) of plate system, including its governing partial differential

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equations (PDEs) of equilibrium are reduced to BVP consisting only ordinary ones (ODEs). Parametric studies are conducted among displacement and stress components developed in

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the plate and FG material gradient index, length scale parameter, and foundation stiffnesses. From the numerical results, it is concluded that the out-of-plane shear stresses are not necessarily zero at the top and bottom surfaces of plate. The results of this investigation may

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Corresponding author E-mail addresses: [email protected] (H. Salehipour), [email protected] (H. Nahvi), [email protected] (A. R. Shahidi), [email protected] (H. R. Mirdamadi)

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ACCEPTED MANUSCRIPT serve as a benchmark to verify further bending analyses of either homogeneous or FG micro/nanoplates on elastic foundation.

Keywords: analytical solution, 3D elasticity, functionally graded micro/nanoplate, static

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deformation, Winkler-Pasternak foundation, modified couple stress

1. Introduction

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FG materials refer to a nonhomogeneous class of materials, in which material properties are varied gradually and continuously from point to point. The concept of FG materials was first considered in Japan in 1984 during a space plane project to develop a new generation of

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composite materials that can resist high thermal environments without stress concentration problem commonly found in laminated composite structures [1]. In recent years, the usage of

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FG materials is extended to micro/nanoscale structures and systems, such as micro/nano

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electromechanical systems (MEMS/NEMS) [2, 3], shape memory alloy thin films [4], and atomic force microscopes [5]. In the quest for a proper design of micro/nano structures,

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understanding their mechanical behaviors is one of the first important issues. The experimental observations reveal that at micro-nanometer scale, the mechanical

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behaviors of structures are size-dependent. Since the classical continuum theory neglects the size-dependency, several non-classical continuum theories have been developed. In this regard, classical couple stress [6-9], strain gradient [10], nonlocal elasticity [11], surface elasticity [12], and modified couple stress theories [13] may be mentioned. Recently, a modified version of nonlocal elasticity theory was developed by Salehipour et al. [14] for FG micro/nanoscale materials. These size-dependent theories have been used frequently by 3

ACCEPTED MANUSCRIPT researchers to study different mechanical aspects of micro/nanostructures, mainly beams and plates, made of FG materials. For example, Reddy and Berry [15] proposed nonlinear sizedependent models of classical laminate plate (CLPT) and first-order shear deformation plate (FSDT) theories for axisymmetric bending of circular plates. Reddy and Kim [16] utilized modified couple stress theory and nonlinear strains of the von Karman to develop a size-

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dependent third-order plate model. In another work, they used the linear form of this model to study bending, buckling, and free vibration of rectangular plates with simply-supported boundary conditions [17]. Sharafkhani et al. [18] investigated the effects of nonlinear

transverse electrostatic and dynamic loading on responses of circular microplates using

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Galerkin-based step-by-step linearization method. Natarajan et al. [19] studied flexural free vibration of Reissner–Mindlin plates using nonlocal theory and the solution method of isogeometric based finite element. Based on modified couple stress and Mindlin plate theories,

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and employing differential quadrature method, Ke et al. [20] carried out mechanical response of annular microplates. Based on the modified couple stress theory, Thai et al. [21-23]

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proposed new models based on CLPT, FSDT, third-order shear deformation (TSDT), and

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sinusoidal shear deformation plate theories, and they analyzed bending and free vibration of rectangular microplates with simply-supported edges. Sahmani and Ansari [24] analyzed free

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vibration of rectangular microplates using strain gradient and TSDT. Bending, buckling, and vibration of sigmoid FG nanoplates were studied by Jung and Han [25, 26] using nonlocal

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elasticity and FSDT. Hosseini-Hashemi et al. [27] presented an analytical closed-form solution to determine exactly the natural frequencies of circular/annular nonlocal Mindlin plates having arbitrary boundary conditions. Using nonlocal elasticity and a higher-order shear deformation plate theory (HSDT), Daneshmehr et al. [28] carried out size-dependent instability of nanoplates under biaxial in-plane loadings. They used generalized differential quadrature (GDQ) method to solve governing buckling equations. Ansari et al. [29] 4

ACCEPTED MANUSCRIPT employed GDQ to consider nonlinear vibration of Mindlin microplates. Recently, Salehipour et al. [30-31] used the size-dependent theories of both nonlocal elasticity and modified couple stress. They presented 3D closed-form solutions for free vibration of simply-supported micro/nanoplates. In another work, they also reported the effects of elastic foundation on the nonlocal 3D static deformation of FG micro/nanoplates using an analytical exact solution

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[32]. Researches of Jung and Han [33], Akgoz and Civalek [34-37], Wang et al. [38], Zhang et al. [39], He et al. [40], Gurses et al. [41], Nateghi et al. [42] and Bedroud et al. [43] are some of the other noticeable works in this field.

Based on the 3D elasticity theory, there is no research for static bending of FG

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micro/nanoplates. But, several works have been carried out to study static bending of FG macroplates, for example [44-48], that due to similarities to present research are introduced here. Kashtalyan [44] presented an exact closed-form solution for 3D deformation of FG

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rectangular simply-supported plates subjected to sinusoidal transverse loading. In other works, Kashtalyan and Menshykova [45, 46] analytically investigated 3D deformation of FG

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coating/substrate plates and plates with FG interlayer with simply-supported boundary

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conditions. They assumed sinusoidal transverse loading. Using state space method, Huang et al.[47] presented benchmark solutions for bending deformation of rectangular FG simply-

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supported plates under arbitrary transverse loading. Bending analysis of rectangular transversely isotropic FG simply-supported plates were carried out by Woodward and

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Kashtalyan [48].

According to the best authors’ knowledge, there could not be found any 3D elasticity

solution based on modified couple stress theory in the literature for static bending analysis of either homogeneous or FG micro/nanoplates. Based on the 3D elasticity, an analytical exact solution is conducted for two-directional bending of both homogeneous and FG rectangular micro/nanoplates subjected to transverse loading. To investigate small scale effect on bending 5

ACCEPTED MANUSCRIPT behavior, the modified couple stress theory is implemented. The plate is assumed to be simply-supported at its all edges and resting on an isotropic Winkler-Pasternak elastic foundation. The variation of modulus of elasticity along the thickness direction is assumed to be in accordance with an exponential law. To verify the accuracy and stability of presented approach, some comparative results are presented. Furthermore, the effects of length scale

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parameter, material gradient index, and foundation stiffnesses are examined and discussed on the displacement and stress fields.

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2. Theoretical formulation

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2.1. Modified couple stress theory

According to the modified couple stress theory proposed by Yang et al. [13], the strain

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energy in an elastic material depends on both components of strain and curvature tensors, as

1 ( :   m :  )dV 2 V

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(1)

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U 

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follows:

where  is the Cauchy stress tensor,  is the small strain tensor, m is the deviatory part of couple stress tensor, and  is the symmetric curvature tensor. These tensors are expressed as

 C :

(2a)

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ACCEPTED MANUSCRIPT





1 T u   u  2

m

E ( z )l 2  1 



1    2





(2b)

(2c)

 



(2d)

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in which u is the displacement vector, E (z ) and  are the Young's modulus and Poisson's

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ratio, l is the length scale parameter, and  is the infinitesimal rotation vector, defined as

1    u 2

(3)



 

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1       E (z ) C    (1  )(1  2 )  0  0   0

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Moreover, C is the fourth-order elasticity isotropic tensor, expressed in 3D elasticity as

1 



0 0 0 0 1  2 0

0  0  0   0  0   1  2 

(4)

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0 0 0

1  0 0 0

0 0 0 1  2 0 0

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2.2. Governing equilibrium equations A rectangular FG micro/nanoplate is considered resting on a Winkler-Pasternak elastic foundation at the bottom surface. The plate is assumed to have a total length of a, width b, and thickness h along the directions of Cartesian coordinates (x, y, z) as shown in Fig. 1. By neglecting body forces and external couples, 3D equilibrium equations for a FG micro/nanoplate are derived as [31]: 7

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2 2 2 2  xx  xy  xz 1  m yy 1  2 m zz 1  m xy 1  2 m xz 1  m yz 1  m yz          0 (5a) x y z 2 y z 2 y z 2 x z 2 x y 2 z 2 2 y 2

 xy

 yy

 yz

2 2 1  2 m xx 1  2 m zz 1  m xy 1  2 m xz 1  2 m xz 1  m yz          0 (5b) x y z 2 x z 2 x z 2 y z 2 z 2 2 x 2 2 x y

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2 2 2 2  xz  yz  zz 1  2 m xx 1  m yy 1  m xy 1  m xy 1  2 m xz 1  m yz          0 (5c) x y z 2 x y 2 x y 2 y 2 2 x 2 2 y z 2 x z

By expressing  ij and m ij in terms of displacement components, from Eqs. (2-4), the

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following equations are obtained:

  2v E (z )  2 1  2u  E (z )  2w  u       2(1  )  1  2 x 2  2(1  )(1  2 )  x y x z

 E (z )  u w       2(1   )  z x 

(6a)

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 E (z ) 2   4u  4u  2   2u  2u   4u 2 2 2  l  4  4  2  2  2 2 2 2   v   2w     8(1  )  y z x  y z  y z x y x z  2 2 2 3 2 E (z ) 2     u u u v   2  w   l   2 2 2 2 2     w  2  8(1  )  z  x y z  x y z x  z   E (z ) 2   2u  2u   v w    l  2  2      0 8(1  )  y z x  y z  

 E (z )  v w       2(1   )  z y   E (z ) 2   2  4v  4v  4v  2   2v  2v  2 2  l   u    2     2w      4 4 2 2 2  2 2  8(1  )  x y x z x z y  x z  y z  (6b) E (z ) 2   3u    2v  2v  2v    2  2w    l    2 2  2   w  2  8(1  )  x y z z  x 2 y 2 z  y  z   E (z ) 2   2v  2v   u w    l  2  2     0 8(1  )  x z y  x z  

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  2u E (z )  2 1  2v  E (z )  2w  v       2(1  )  1  2 y 2  2(1  )(1  2 )  x y y z

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ACCEPTED MANUSCRIPT E (z )  2 1  2w  w   2(1  )  1  2 z 2

   2u E (z )  2v        2(1   )(1  2 )  x z y z    u v   E (z ) 2   2 E (z ) w (1   )    l   2u      (1  )(1  2 )  z  x  y 8(1   )  x  z   



E (z ) 2   2  u v     2w  2w   l       0   8(1  )  z 2  x y  z  x 2 y 2  

  

(6c)

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2  4w  4w  4w  2   2w  2w   2v     2     y z x 4 y 4 x 2y 2 z 2  x 2 y 2

It is assumed that the material of micro/nanoplate is isotropic and FG; and Young's

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modulus varies exponentially from the bottom to the top surface, as follows:

E  E 0 exp( z )

(7)

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where ϕ is the gradient index of variation and E0 is the Young's modulus at the bottom

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2.3. Solution procedure

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surface. In addition, Poisson’s ratio is considered to be constant.

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The equilibrium equations (6) can be solved for the simply-supported boundary conditions. To this end, the displacement components are expanded in double Fourier series along the in-

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plane directions of x and y:

 u     1 (z ) m cos( m x )sin(  n y )      v     2 (z )  n sin( m x ) cos(  n y )  w  n 1 m 1   (z )sin( x )sin(  y )  m n    3 

(8)

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ACCEPTED MANUSCRIPT in which  m  m  a and n  n b . The displacement components of Eq. (8) satisfy the following simply-supported boundary conditions [31]:

1 m xy 1 m xz  0 2 z 2 y

at

x  0, a

(9a)

u  w  0, m xy  m yz   yy 

1 m xy 1 m yz  0 2 z 2 x

at

y  0, b

(9b)

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v  w  0, m xy  m xz   xx 

By substituting the displacement components from Eq. (8) and Young's modulus from Eq. (7)

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into the equations of equilibrium, i.e., Eqs. (6), the following simplified ODEs are obtained:

A1  1(4) (z )  2 A1  1(3) (z )   A 2   m2 A1  1 (z )   A 3   m2  A1  1 (z )  A 4  1 (z )   n2 A1 2 (z )   n2 A1 2 (z )   n2 A 6 2 (z )  A1 3(3) (z ) 

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2 A1 3 (z )  A 7 3 (z )  A8 3 (z )  0

(10a)

A1  2(4) (z )  2 A1  2(3) (z )   A 2   n2 A1  2 (z )   A 3   n2 A1  2 (z )  A 4  2 (z )   m2 A1 1 (z )   m2  A1 1 (z )   m2 A 6 1 (z )  A1 3(3) (z ) 

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(10b)

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2 A1 3 (z )  A 7 3 (z )  A8 3 (z )  0



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B1 3 (z )   B1 3 (z )  B 2 3 (z )   m2 B 3  1(3) (z )   B 3  1 (z )  B 4  1 (z )  B 5  1 (z )





  n2 B 3  2(3) (z )   B 3  2 (z )  B 4  2 (z )  B 5  2 (z )  0

where

10

 (10c)

ACCEPTED MANUSCRIPT

A1  B 3  

E 0l 2 8(1   )

A2 

E0 E l 2 E l 2 2  0 mn  0 2(1  ) 4(1  ) 8(1  )

A3 

E 0 E l 2 mn  0 2(1   ) 4(1   )

E 0 E l 2 mn  0 2(1  ) 8(1  )

B1 

E 0 (1  ) E l 2  0 mn (1   )(1  2 ) 8(1   )

B2  

2 E 0 mn E 0 l 2 mn  2(1   ) 8(1   )

E 0 E0 E l 2   0 mn (1  )(1  2 ) 2(1  ) 8(1  ) E 0 B5   (1  )(1  2 )

(11a)

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B4  

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A8 

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A 4  E 0 (1  )  m2   E 0 E 0 l 2 mn E 0 l 2 2    n2              (1   )(1  2 )   n2   2(1   ) 8(1   ) 8(1   )   m2  A5  A 6  1 E l 2 2 E 0 E0 E l 2   0 mn    0   A 7  (1  )(1  2 ) 2(1   ) 8(1   )  1  8(1   )

(11b)

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 mn   m2  n2

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and

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Eqs. (10a-c) have the set of following general solution:

 1 (z )  C       2 (z )   D  exp( z )  (z )   F   3   

(12)

11

ACCEPTED MANUSCRIPT Inserting Eq. (12) in Eqs. (10a-b), the following relationships are obtained between the coefficients C, D, and F:

A1 3  2 A1 2  A 7   A8 F A1 4  2 A1 3   mn A1  A 2   2   mn A1  A3    A 4   n2 A 6

(13)

CR IP T

C D 

By substituting Eq. (12) into Eq. (10c) and using Eq. (13), the following characteristic equation is established:

AN US

A1  B 1   mn B 3   6  3 A1  B 1   mn B 3   5   mn A1B 1  A 2 B 1  2 2 A1B 1  A1B 2

 mn A 7 B 3   mn A1B 4  2 mn  2 A1B 3   4   2 mn  A1B 1  A 3B 1   A 2 B 1  2 A1B 2  mn A8 B 3   mn  A 7 B 3  2 mn  A1B 4   mn A1B 5   3   A 4 B 1   n2 A 6 B 1   mn  2 A1B 1

(14)

 A 3B 1   mn A1B 2  A 2 B 2   mn  A8 B 3   mn A 7 B 4  2 mn  A1B 5     A 4 B 1   A 6 B 1 2

2 n

ED

M

 mn  A1B 2  A 3 B 2   mn A8 B 4   mn A 7 B 5    A 4 B 2   n2 A 6 B 2   mn A8 B 5  0

The six roots i (i  1, 2,...,6) can be real or complex-valued. When all the roots are real-

PT

valued, one gets:

CE

 1 (z )   2 (z )   1F1 exp(1z )   2 F2 exp(2 z )   3F3 exp(3z )  4 F4 exp(4 z )   5 F5 exp(5 z )   6 F6 exp(6 z )

AC

 3 (z )  F1 exp(1z )  F2 exp(2 z )  F3 exp(3z )  F4 exp(4 z )  F5 exp(5 z )  F6 exp(6 z )

where

12

(15a)

(15b)

ACCEPTED MANUSCRIPT

i 

A  1

3 i

 2 A1 i2  A 7  i  A8 

A1 i4  2 A1 i3    mn A1  A 2   i2   mn A1  A3   i  A 4   n2 A 6

, (i  1, 2,..., 6)

(16)

The displacement components in Eq. (8) should satisfy the following boundary conditions at

 1 m zz 1 m yy 1 m xy 1 m yz     0  xz   2 y 2 y 2 x 2 z  z 0,h 

z 0, h

 m yz

z 0, h

0

M

m xz

AN US

 1 m zz 1 m xx 1 m xy 1 m xz     0  yz   2 x 2 x 2 y 2 z  z 0,h 

CR IP T

the top and bottom surfaces of plate:

ED

   2w  2w 1 m xz 1 m yz     kw w  k p  2  2  zz  2 y 2 x y  x  

  0      z 0

  1 m xz 1 m yz   Q (x , y )  0  zz  2 y 2 x   z h

(17a)

(17b)

(17c)

(17d)

CE

PT

(17e)

AC

where the constants k w and k p , respectively, are the Winkler and Pasternak modulus of foundation, and Q (x , y ) is the transverse load on the plate top surface. By using Eqs. (2-4), the boundary conditions of Eqs. (17) can be written in terms of displacements, as

13

ACCEPTED MANUSCRIPT   u w  E (z )  z  x   

2 3  3u    3u  3w  1   3w  E ( z )l    v        2   x y z y 2z   y 2z x y 2  2  x y 2 

 3v  3u  3w  1   3u  3w  3v  3u  2  3   3    x y z x z x  2  z x z 2 x y z y 2z

   

(18a)

  v w  E (z )l 2    3v  3u E ( z )          2   x 2z x y z  z y  

CR IP T

E (z )l 2   2u  2w  2v  2u    2 0  2 4 x z x y y  z 0,h  z

   3w  3v     2 2   x y x z

 1   3w   3  2  y

 1   3w  3v  3v  3u          2 z 3 x 2z x y z    2  y z

 3v  3u  3w   y 2 z x y z x 2y



E (z )l 2   2w  2v  2v  2u     0   2 4 x 2 x y  z 0,h  y z z

(18b)

AN US



  2w  2v  2u  2v  E ( z )l   2  0 x y z 2  z 0,h  y z x

(18c)

  2w  2u  2v  2u  E ( z )l 2   2  2 0 x y z  z 0,h  x z y

(18d)

PT

ED

M

2

CE

 E (z )   u v   E (z )l 2   3w w  3v  3u  3v            1      z 8  y 2z x 2y x y 2 y z 2   x y    (1  2 ) 

AC

   2w  2w E (z )l 2   3w  3u  3v  3u      1   k w  k   w    p  2 8  x 2 z x y 2 x 2y x z 2   x y 2  

  0      z 0 (18e)

 E (z )   u v   E (z )l 2   3w w  3v  3u  3v        2  2   1      z 8  y z x y x y 2 y z 2   x y    (1  2 )   E (z )l 2   3w  3u  3v  3u       1   Q ( x , y )   2 2 2 2   8  x z x y x y x z   14

0 z h

(18f)

ACCEPTED MANUSCRIPT

Substituting the displacement components from Eqs. (8) and (15) into Eqs. (18) yields the following simplified algebraic equations:

E (z )  4 11  4  l 2 12   11  1  l 2 mn   11  1  F1 exp  1z    4 2 2  4

CR IP T

l 2 22   2 2  1 l 2 mn   2 2  1  F2 exp  2 z    4 33  4  l 232  33  1

l 2 mn   33  1  F3 exp  3z    4 44  4  l 242   4 4  1  l 2 mn  44  1  

(19a)

F4 exp  4 z    4 55  4  l 2 52   55  1  l 2 mn   55  1  F5 exp  5 z  z  0, h

0

AN US

  4 6 6  4  l 2 62   6 6  1  l 2 mn   6 6  1  F6 exp  6 z  

E (z )  12 1  1  F1 exp  1z    22 2  2  F2 exp  2 z    32 3  3  F3 exp  3 z  

 

 4  F4 exp  4 z    52 5  5  F5 exp  5 z    62 6  6  F6 exp  6 z  

z  0, h

0

(19b)

M

2 4 4

ED

 E ( z )   1  2  1   1  mn  1   1   k w  k P  mn   F1 exp  1z      E (z ) 1   2  mn  2   1   k w  k P  mn   F2 exp  2 z      1  2 

CE

PT

 E (z )  1   3  mn  3   1   k w  k P  mn   F3 exp  3z    1  2     E (z ) 1   4  mn  4   1   k w  k P  mn   F4 exp  4 z      1  2 

AC

 E (z ) 1   5  mn  5   1   k w  k P  mn   F5 exp  5 z      1  2    E (z ) 1   6  mn  6   1   k w  k P  mn   F6 exp  6 z   0    1  2   z 0

15

(19c)

ACCEPTED MANUSCRIPT



 E (z )  1   1  mn  1  F1 exp  1z    1   2  mn  2  F2 exp  2 z     1   3  mn  3  F3 exp  3z    1   4  mn  4  F4 exp  4 z  

 1   

5



 mn  5  F5 exp  5 z    1   6  mn  6  F6 exp  6 z   1   q mn 

(19d) z h

0

q mn

4  ab

b a

 Q (x , y )sin(

m

CR IP T

where x )sin(  n y )dxdy

0 0

(20)

From the solution of Eqs. (19a-d), the constant coefficients Fi (i  1, 2,...,6) are obtained. For

AN US

complex values of i (i  1, 2,...,6) , the solution procedure is the same as above-described, but Eqs. (15) and (19) should be changed accordingly. By obtaining the coefficients

ED

M

Fi (i  1, 2,...,6) , the complete solution is achieved.

PT

3. Numerical results and discussion

For numerical verification, only square micro/nanoplates are considered under the following

AC

CE

sinusoidal transverse load:

Q (x , y )  q sin( x a)sin( y b )

(21)

For generalization of numerical results, the following non-dimensional forms of parameters are used:

16

ACCEPTED MANUSCRIPT

u 

10uE 0 h 3 10vE 0 h 3 10wE 0 h 3 , v  , w  qa 4 qa 4 qa 4

 ij 

(22a)

 ij h

(22b)

qa

KW  k w a 4 D ,

K p  k pa2 D

(22c)

l l h

CR IP T

(22d)

  E h E 0  exp( h )

(22e)

Young's modulus of plate top surface.

AN US

where D  E 0 h 3 12(1  2 ) is a reference for the plate bending rigidity, and E h is the

To the best of authors’ knowledge, since they could not find any result in the literature for bending of FG micro/nanoplates with the exponential variation of material properties, a

M

comparative study is made for central transverse displacement of a homogeneous

ED

micro/nanoplate between the results of present approach and those obtained from 2D plate theories. These are shown in Table 1. It can be seen that the difference between the results of

PT

this 3D elasticity solution and those of 2D theories seems very small when the plate is very thin ( a h  20 ). Furthermore, it is observed that the length scale parameter has a decreasing

CE

effect on the transverse displacement. In other words, the inclusion of length scale parameter

AC

leads to an increase in the bending rigidity of plate. The normalized transverse displacements at the center of a square FG micro/nanoplate

resting on elastic foundation are tabulated in Table 2 for the length-to-thickness ratio equal to five and different values of length scale parameter, material gradient index, and Winkler and Pasternak foundation rigidities. Similar results are listed in Table 3 for the length-to-thickness ratio equal to ten. It is obvious that as the material gradient index and/or length scale 17

ACCEPTED MANUSCRIPT parameter increase, the transverse displacement decreases. In addition, Winkler and Pasternak foundation rigidities have similar decreasing effect on the transverse displacement. Figs. 2 and 3 depict the distribution of longitudinal displacement, u (0, a 2, z ) , and transverse displacement, w (a 2, a 2, z ) , along the thickness of a FG square micro/nanoplate resting on elastic foundation. The length-to-thickness ratio of plate is considered to be five

CR IP T

( a h  5 ). The material gradient index and length scale parameter are considered in four different cases: (a)   1, l  0 (i.e., homogeneous and macro), (b)   5, l  0 (i.e., FG and macro), (c)   1, l  1 (i.e., homogeneous and micro/nano), and (d)   5, l  1 (i.e., FG and

AN US

micro/nano). Results are presented for Winkler foundation stiffnesses equal to zero and 100; and Pasternak foundation stiffnesses equal to zero and 10. On the contrary to what observed from Fig. 2a (for homogeneous plate), the position of neutral plane for a FG macroplate is not at the mid-surface of plate. Moreover, from Figs. 2 and 3, one can see that by increasing

M

foundation stiffness, the longitudinal and transverse displacements decrease. The rate of this

ED

decrease is lower than those for FG ones. Also, by including length scale parameter, the effect of foundation stiffness on the plate reduces. It is deduced from Figs. 3a and 3b that the

PT

variation of transverse displacement through the plate thickness is small when the small scale effect is neglected ( l  0 ). However, from Figs. 3c and 3d, it is observed that this trend is not

CE

dominant for a nonzero value of length scale parameter ( l  1 ), and the absolute value of

AC

transverse displacement increases continuously from the top to the bottom surface attached to the elastic foundation. Figs. 4-7 present through-thickness variations of the normal and shear stress

components,  ij , for a FG square micro/nanoplate ( a h  5 ), resting on Winkler-Pasternak foundation. It can be found that the effect of foundation stiffnesses on the stress components is remarkable for homogeneous plates, as compared with those of FG ones. It is seen from 18

ACCEPTED MANUSCRIPT Figs. 4 and 5 that the distribution of in-plane stresses,  xy and  xx , along the plate thickness, are nonlinear for FG plates. This is on the contrary to what is observed for homogeneous plates. In the sequel, it is observed that the inclusion of small scale effect yields a reduction in the values of in-plane stress components. From Figs. 6c and 6d, it is observed that for a nonzero value of length scale parameter (e.g., l  1 ( the out-of-plane shear stress  xz is not

CR IP T

zero at the top and bottom surfaces of plate. This is attributed to the boundary conditions of modified couple stress theory as appeared in Eqs. (17a-b). In other words, at the lateral plate surfaces, the classical continuum boundary conditions of  xz  0 and  yz  0 are converted

AN US

to Eqs. (17a-b) for the modified couple stress theory. Figs. 6c and 6d illustrate that for a

nonzero value of length scale parameter ( e.g., l  1 ), the absolute value of out-of-plane shear stress is maximum at the top surface of plate. It is deduced from Figs. 7a-d that the absolute

M

value of out-of-plane normal stress  zz increases from the bottom to the top surface of plate. The slope of these curves at the top and bottom surfaces of plate is minute whenever the

ED

small scale effect is neglected (cases 1 and 2). This is not true in other cases. In case 3 (homogeneous plate with l  1 ), the through-thickness variation of stress  zz is nearly linear,

PT

and in case 4 (FG plate with l  1 (, the slope of curves increases by increasing the coordinate

AC

CE

z h.

4. Conclusion

A 3D elasticity analytical solution based on modified couple stress theory was proposed for static bending response of both homogeneous and FG micro/nanoplates subjected to transverse load and resting on Winkler-Pasternak elastic foundation. The Young’s modulus of 19

ACCEPTED MANUSCRIPT plate was assumed to vary exponentially from the bottom to top surface of plate, while Poisson’s ratio was considered to be constant. The present solution was verified by its comparison with the results from 2D plate theories. The through-thickness variations of displacement and stress fields were studied for a thick plate. The results were presented for

Some of important results could be listed as: 

The absolute values of displacement components are higher for FG plates as compared with homogeneous ones.



The inclusion of small scale effect in modified couple stress theory has a decreasing

AN US

effect on the static bending response of plate. 

CR IP T

different values of length scale parameter, material gradient index, and foundation stiffnesses.

By increasing foundation stiffnesses, the absolute values of displacements and stresses decrease, except out-of-plane normal stress.

The inclusion of small scale effect decreases the influence of foundation stiffnesses on

M



the bending response of plate.

In the modified couple stress theory, the out-of-plane shear stresses are not

ED



CE

References

PT

necessarily zero at the lateral plate surfaces.

AC

[1] Ichikawa K, "Functionally Graded Materials in the 21st Century: A Workshop on Trends and Forecasts", Springer, 2001.

[2] Witvrouw A, Mehta A. The use of functionally graded poly-SiGe layers for MEMS applications. Mater Sci Forum 2005;492-493:255-60.

20

ACCEPTED MANUSCRIPT [3] Lee Z, Ophus C, Fischer LM, Nelson-Fitzpatrick N, Westra KL, Evoy S, et al. Metallic NEMS components fabricated from nanocomposite Al–Mo films. Nanotechnol 2006;17:3063-70. [4] Fu Y, Du H, Zhang S. Functionally graded TiN/TiNi shape memory alloy films. Mater Lett 2003;57;2995–99.

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[5] Rahaeifard M, Kahrobaiyan MH, Ahmadian MT. Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials. 3rd international conference on micro- and nanosystems. DETC2009-86254;2009;539-44.

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[10] Aifantis EC. Strain gradient interpretation of size effects. Int J Fractures 1999;95;1-4. [11] Eringen AC. Nonlocal polar elastic continua. Int J Eng Sci 1972;10:1–16.

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[12] Gurtin ME, Weissmuller J, Larche F. The general theory of curved deformable interfaces in solids at equilibrium. Philis Mag A 1998:1093–109.

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[13] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002;39;2731-43.

[14] Salehipour H, Shahidi AR, Nahvi H. Modified nonlocal elasticity theory for functionally graded materials. Int J Eng Sci 2015; 90:44-57. [15] Reddy JN, Berry Jessica. Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Compos Struct 2012;94(12):3664–8. 21

ACCEPTED MANUSCRIPT [16] Reddy JN, Kim J. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 2012;94;1128-43. [17] Kim J, Reddy JN. Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Compos Struct 2013;103;86-98. [18] Sharafkhani N, Rezazadeh GH, Shabani R. Study of mechanical behavior of circular

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FGM micro-plates under nonlinear electrostatic and mechanical shock loadings. Acta Mech 2012;223;579-91.

[19] Natarajan S, Chakraborty S, Thangavel M, Bordas S, Rabczuk T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci

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2012;65;74-80.

[20] Ke LL, Yang J, Kitipornchai S, Bradford MA. Bending, buckling and vibration of sizedependent functionally graded annular microplates. Compos Struct 2012;94;3250-57.

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[21] Thai H-T, Choi D-H. Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos Struct 2013;95;142-53.

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[22] Thai H-T, Kim S-E. A size-dependent functionally graded Reddy plate model based on a

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modified couple stress theory. Compos Part B 2013;50;1636-45. [23] Thai H-T, Vo TP. A size-dependent functionally graded sinusoidal plate model based on

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a modified couple stress theory. Compos Struct 2013;96;376–83. [24] Sahmani S, Ansari R. On the free vibration response of functionally graded higher-order

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shear deformable microplates based on the strain gradient elasticity theory. Compos Struct 2013;95;430–42.

[25] Jung WY, Han SC, Park WT. A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium. Compos Part B 2014;60;746-756.

22

ACCEPTED MANUSCRIPT [26] Jung WY, Park WT, Han SCH. Bending and vibration analysis of S-FGM microplates embedded in Pasternak elastic medium using the modified couple stress theory. Int J Mech Sci 2014;87;150-62. [27] Hosseini-Hashemi Sh, Bedroud M, Nazemnezhad R. An exact analytical solution for

elasticity. Compos Struct 2013;103;108–18.

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free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal

[28] Daneshmehr A, Rajabpoor A, pourdavood M. Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions. Int J Eng Sci 2014;82:84-100.

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[29] Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Darabi MA. Nonlinear

vibrations of functionally graded Mindlin microplates based on the modified couple stress

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CE

PT

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theory. Compos Struct 2014;114;124–34.

23

ACCEPTED MANUSCRIPT [30] Salehipour H, Nahvi H, Shahidi AR. Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity. Physica E 2015;66:350–58. [31] Salehipour H, Nahvi H, Shahidi AR. Exact closed-form free vibration analysis for functionally graded micro/nano plates based on modified couple stress and three-

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dimensional elasticity theories. Compos Struct 2015;124:283-91.

[32] Salehipour H, Nahvi H, Shahidi AR. Closed-form elasticity solution for three-

dimensional deformation of functionally graded micro/nano plates on elastic foundation. LAJSS 2015;12:747-62.

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[33] Jung W-Y, Han S-C. Static and eigenvalue problems of Sigmoid Functionally Graded Materials (S-FGM) micro-scale plates using the modified couple stress theory. Appl Math Modell 2015;39:3506-24.

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[34] Akgoz B, Civalek O. A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech 2015; 226: 2277-94.

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[35] Akgoz B, Civalek O. Bending analysis of FG microbeams resting on Winkler elastic

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foundation via strain gradient elasticity. Compos Struct 2015;134: 294-301. [36] Akgoz B, Civalek O. Modeling and analysis of micro-sized plates resting on elastic

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medium using the modified couple stress theory. Meccanica 2013;48:863-73.

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[37] Akgoz B, Civalek O. Shear deformation beam models for functionally graded microbeams with new shear correction factors. Compos Struct 2014;112: 214-25.

[38] Wang Y, Ding H, Xu R. Three-dimensional analytical solutions for the axisymmetric bending of functionally graded annular plates. Appl Math Modell 2016;40:5393420.

24

ACCEPTED MANUSCRIPT [39] Zhang B, He Y, Liu D, Shen L, Lei J. An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl Math Modell 2015;39:3814-45. [40] He L, Lou J, Zhang E, Wang Y, Bai Y. A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory.

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Compos Struct 2015;130:107-15.

[41] Gurses M, Akgoz B, Civalek O. Mathematical modeling of vibration problem of nanosized annular sector plates using the nonlocal continuum theory via eight-node

Computation 2012;219: 3226-40.

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discrete singular convolution transformation. Applied Mathematics and

[42] Nateghi A, Salamat-talab M, Rezapour J, Daneshian B. Size dependent buckling analysis of functionally graded micro beams based on modified couple stress

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theory. Appl Math Modell 2012;36: 4971-87.

[43] Bedroud M, Nazemnezhad R, Hosseini-Hashemi S, Valixani M. Buckling of FG

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circular/annular Mindlin nanoplates with an internal ring support using nonlocal

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elasticity. Appl Math Modell 2016;40:3185-210. [44] Kashtalyan M. Three-dimensional elasticity solution for bending of functionally

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graded rectangular plates. Eur J Mech A Solids 2004;23:853-64.

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[45] Kashtalyan M, Menshykova M. Three-dimensional elastic deformation of a functionally graded coating/substrate system. Int J Solids Struct 2007;44:5272-88.

[46] Kashtalyan M, Menshykova M. Effect of a functionally graded interlayer on threedimensional elastic deformation of coated plates subjected to transverse loading. Compos Struct 2009;89:167-76.

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ACCEPTED MANUSCRIPT [47] Huang ZY, Lu CF, Chen WQ. Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations. Compos Struct 2008;85:95-104. [48] Woodward B, Menshykova M. Three-dimensional elasticity solution for bending of transversely isotropic functionally graded plates. Eur J Mech A Solids 2011;30:705-

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18.

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Fig. 1: Geometry and coordinate system of a FG micro/nano plate resting on an elastic

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foundation.

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Fig. 2: Through-thickness distribution of longitudinal displacement u (0, a 2, z ) of a square

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plate ( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 , (b)

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  5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Fig. 3: Through-thickness distribution of transverse displacement w (a 2, a 2, z ) of a square

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plate ( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 , (b)

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  5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Fig. 4: Through-thickness distribution of in-plane shear stress  xy (0,0, z ) of a square plate

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( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 , (b)

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  5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Fig. 5: Through-thickness distribution of longitudinal normal stress  xx (a 2, a 2, z ) of a

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square plate ( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 ,

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(b)   5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Fig. 6: Through-thickness distribution of transverse shear stress  xz (0, a 2, z ) of a square

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plate ( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 , (b)

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  5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Fig. 7: Through-thickness distribution of out-of-plane normal stress  zz (a 2, a 2, z ) of a

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square plate ( a h  5 ) resting on elastic foundation, for four different cases: (a)   1, l  0 ,

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(b)   5, l  0 , (c)   1, l  1 , (d)   5, l  1.

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Table 1  

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a b h  

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Comparison of normalized out-of-plane displacement  w ( , , )  of a square homogeneous 2 2 2 micro/nanoplate.

0 0.2 0.4 0.6 0.8 1

10

0 0.2 0.4 0.6 0.8 1

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AC

20

0 0.2 0.4 0.6 0.8 1

HSDT [22] ( =0.3) 0.3433 0.2875 0.1934 0.1251 0.0838 0.0588

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5

Present ( =0.3) 0.3357 0.2851 0.1991 0.1351 0.0953 0.0709

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l

Present ( =0.38) 0.3204 0.2769 0.1993 0.1383 0.0990 0.0744

FSDT [21] ( =0.38) 0.3306 0.2876 0.2086 0.1456 0.1049 0.0793

CPT [21] ( =0.38) 0.2635 0.2294 0.1652 0.1126 0.0779 0.0558

0.2942 0.2514 0.1757 0.1177 0.0810 0.0583

0.2961 0.2520 0.1742 0.1150 0.0780 0.0522

0.2779 0.2414 0.1739 0.1193 0.0834 0.0606

0.2803 0.2440 0.1762 0.1211 0.0849 0.0619

0.2635 0.2294 0.1652 0.1126 0.0779 0.0558

0.2838 0.2428 0.1697 0.1131 0.0773 0.0550

0.2842 0.2430 0.1693 0.1124 0.0765 0.0542

0.2671 0.2324 0.1674 0.1143 0.0793 0.0570

0.2677 0.2330 0.1680 0.1148 0.0797 0.0574

0.2635 0.2294 0.1652 0.1126 0.0779 0.0558

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a h

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Table 2 a b h 2 2 2

l 0

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Central out-of-plane displacement w ( , , ) of a square FG micro/nanoplate with a h  5.



KW

KP

1

0

0 10 0 10 0 10 0 10

5

0

0.5

1

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100

0

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100

5

0

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100

AC

1

1

5

M

100

0 100 0 100

Central out-of-plane displacement -0.3357 -0.2131 -0.2595 -0.1804 -0.1531 -0.1225 -0.1359 -0.1114

0 10 0 10 0 10 0 10

-0.1636 -0.1279 -0.1432 -0.1154 -0.0693 -0.0622 -0.0655 -0.0592

0 10 0 10 0 10 0 10

-0.0709 -0.0633 -0.0668 -0.0601 -0.0289 -0.0276 -0.0282 -0.0269

35

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Table 3 a b h 2 2 2

Central out-of-plane displacement w ( , , ) of a square FG micro/nanoplate with a h  10.



KW

KP

1

0

0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10

100 0

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5

100

1

0

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0.5

100

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5

AC

1

1

0

100 0 100

5

M

l 0

0 100

36

Central out-of-plane displacement -0.2942 -0.1928 -0.2323 -0.1642 -0.1342 -0.1085 -0.1198 -0.0990 -0.1437 -0.1143 -0.1271 -0.1036 -0.0612 -0.0553 -0.0580 -0.0527 -0.0583 -0.0528 -0.0554 -0.0504 -0.0240 -0.0230 -0.0235 -0.0225