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International Journal of Engineering Science 90 (2015) 44–57

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International Journal of Engineering Science

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Modified nonlocal elasticity theory for functionally gradedmaterials

http://dx.doi.org/10.1016/j.ijengsci.2015.01.0050020-7225/� 2015 Elsevier Ltd. All rights reserved.

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

H. Salehipour, A.R. Shahidi, H. Nahvi ⇑Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran

a r t i c l e i n f o

Article history:Received 15 December 2014Received in revised form 5 January 2015Accepted 30 January 2015

Keywords:Functionally graded materialsModified nonlocal elasticityMicro/nanoplatesFree vibration

a b s t r a c t

In this paper, it will be shown that the nonlocal theory of Eringen is not generally suitablefor analysis of functionally graded (FG) materials at micro/nano scale and should be mod-ified. In the current work, an imaginary nonlocal strain tensor is introduced and used todirectly obtain the nonlocal stress tensor. Similar to the stress tensor in Eringen’s nonlocaltheory, the imaginary nonlocal strain tensor at a point is assumed to be a function of localstrain tensor at all neighbor points. To compare the new modified nonlocal theory withEringen’s theory, free vibration of FG rectangular micro/nanoplates with simply supportedboundary conditions are investigated based on the first-order plate theory and three-di-mensional (3-D) elasticity theory. The material properties are assumed to be functionallygraded only along the plate thickness. The effects of nonlocal parameter and material gra-dient index on the natural frequencies of FG micro/nano plates are discussed. The presentdeveloped nonlocal theory can be used in conjunction with different analytical and numer-ical methods to analyze mechanical response of micro/nano structures made of FGmaterials.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials are an advanced class of composite materials where the volume fractions of the constitu-tive materials are varied continuously as a function of position from one point to the other. This continuity providescontinuous distribution of material properties and results in eliminating common interface difficulties of laminate compositematerials (Koizumi, 1993). The advanced physical and mechanical characteristics of FG materials make them to be researchmatter for diverse disciplines as tribology, geology, optoelectronics, biomechanics, fracture mechanics and micro- andnanotechnology (Suresh & Mortensen, 1998; Suresh, 2001). Introducing of FG materials to micro- and nanotechnologyhas led to easily achieving the micro/nano devices, with better physical properties, such as micro/nano electromechanicalsystems (Witvrouw & Mehta, 2005; Lee et al., 2006), shape memory alloy thin films (Fu, Du, & Zhang, 2003) and atomic forcemicroscopes (Rahaeifard, Kahrobaiyan, & Ahmadian, 2009).

The size dependence of mechanical behavior in micro/nano scale makes the applicability of classic continuum theorysomewhat questionable. The classic continuum theory which is scale independent does not capture small size effect, andso, cannot correctly predict mechanical behavior of micro/nano structures. Hence, non-classical continuum theories suchas classical couple stress (Toupin, 1962; Mindlin & Tiersten, 1962; Mindlin, 1963; Koiter, 1964), strain gradient (Aifantis,

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H. Salehipour et al. / International Journal of Engineering Science 90 (2015) 44–57 45

1999), nonlocal elasticity (Eringen, 1972, 1972) and modified couple stress (Yang, Chong, Lam, & Tong, 2002) have beenextended to accommodate the size effect of micro/nanomaterials. In the classical couple stress theories (Toupin, 1962;Mindlin & Tiersten, 1962; Mindlin, 1963; Koiter, 1964), the material particle can be under applied loads including not onlya force to translate the material particle but also a couple to rotate it. Based on this concept, strain energy is a function ofboth strain and curvature tensors. Recently, the modified couple stress theory has been developed by Yang et al. (2002).The two main differences between the modified couple stress theory and the classical couple stress theory are the involve-ment of only one material length scale parameter and that the strain energy is a function of the strain and only the symmet-ric part of the curvature tensor. In the strain gradient theory (Aifantis, 1999), it is assumed that the strain energy functiondepends on the gradient of the strain tensor in addition to its strain tensor. The strain gradient theory includes three lengthscale parameters for the dilatation gradient vector, the curvature tensor and the deviatoric stretch gradient tensor. In thenonlocal theory of Eringen (Eringen, 1972, 1972), the stress at a point in a continuum body is assumed to be a functionof the strain at all neighbor points of the domain, because of atomic forces and micro/nano scale effect.

The nonlocal theory of Eringen has been developed for isotropic and homogeneous materials. But, it has been used inmany works to study different mechanical behavior of micro/nano structures made of FG materials such as micro/nanobeams and plates, see for example (Natarajan, Chakraborty, Thangavel, Bordas, & Rabczuk, 2012; Eltaher, Emam Samir, &Mahmoud, 2012; Simsek, 2012; Ke, Wang, Yang, & Kitipornchai, 2012; Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian,2012; Lei, He, Zhang, Gan, & Zeng, 2012; Jung & Han, 2013; Hosseini-Hashemi, Bedroud, & Nazemnezhad, 2013; Eltaher,Emam Samir, & Mahmoud, 2013; Uymaz, 2013; Simsek & Yurtcu, 2013; Rahaeifard, Kahrobaiyan, Ahmadian, &Firoozbakhsh, 2013; Simsek & Reddy, 2013; Rahim Nami & Janghorban, 2014; Nazemnezhad & Hosseini-Hashemi, 2014;Rahmani & Pedram, 2014; Hosseini-Hashemi, Nazemnezhad, & Bedroud, 2014; Daneshmehr, Rajabpoor, & pourdavood,2014; Eltaher, Khairy, Sadoun, & Omar, 2014; Daneshmehr, Rajabpoor, & Pourdavood, 2014; KE, Yang, Kitipornchai, &Wang, 2014; Akgoz & Civalek, 2014; Salehipour, Nahvi, & Shahidi, 2015). In this paper, it will be illustrated that the theoryof Eringen cannot generally be applied to analyze mechanical behavior of FG micro/nano structures and must be modified forsuch analysis. For this purpose, a nonlocal strain tensor is defined which is similar to the stress tensor in Eringen’s theory.The nonlocal strain tensor is directly used to obtain the nonlocal stress tensor. Based on the presented modified theory, freevibration of FG rectangular micro/nanoplates is analytically carried out using the first-order plate theory and three-dimen-sional elasticity. The plate is assumed to be simply supported at all edges and the material properties are functionally gradedonly along the plate thickness. Numerical results are presented to compare the results of the modified nonlocal theory withthose of Eringen’s nonlocal theory. Contrary to that observed in the Eringen’s theory, the modified nonlocal theory predictsthat by increasing nonlocal parameter and material gradient index, the results of first-order plate theory and 3-D elasticitytheory are very close.

2. Nonlocal elasticity of Eringen

According to Eringen’s nonlocal elasticity, the stress tensor at a point in a continuum body not only depends on the straintensor at that point but also on the strains at all neighbor points of the continuum body. The constitutive equations for anisotropic homogeneous nonlocal continuum body are expressed as

rij ¼Z

Va x0 � xj jð Þtijðx0ÞdVðx0Þ; 8x 2 V

tij ¼ Cijklekl

ð1Þ

where rij and tij are the nonlocal and local stress tensors, respectively; eij is the strain tensor and Cijkl is the fourth order elas-ticity tensor. The nonlocal kernel function a j x0 � x jð Þ incorporates into the constitutive equations the nonlocal effects oflocal stress at the source point x0. Function a j x0 � x jð Þ depends on s ¼ e0a=l, in which e0 is material constant, and a and lare internal and external characteristic lengths. Following experimental observations, Eringen (1972) proposed the followingforms of function a j x0 � x jð Þ for 2-D and 3-D problems, respectively:

a x0 � xj jð Þ ¼ 2pl2s2� ��1

K0 x0 � xj j=lsð Þ ð2aÞ

a x0 � xj jð Þ ¼ pl2s� ��3=2

exp � x0 � xj j2=l2s� �

ð2bÞ

in which K0 is the modified Bessel function. The integral form of Eq. (1) cannot be solved easily. Hence, a differential form ofthe constitutive equations for homogenous materials is proposed by Eringen (1972) as

1� lr2� �

rij ¼ tij ð3Þ

where l ¼ e0að Þ2 is nonlocal parameter and r2 is Laplacian operator. Based on Eq. (3), stress at a point of continuum bodydepends on both strain and second gradient of strain at that point. The convenient differential form of the nonlocal Eq. (3) isbroadly used in different 1, 2 and 3 dimensional mechanical problems.

46 H. Salehipour et al. / International Journal of Engineering Science 90 (2015) 44–57

3. Modified nonlocal elasticity for FG materials

Based on the concept of nonlocal elasticity, stress at a point of continuum body depends on strain at all neighbor points.But, for FG materials, Eringen’s nonlocal constitutive equations incorporate material properties, elasticity modulus andPoison’s ratio, of neighbor points in the definition of nonlocal stress. Hence, the nonlocal theory should be modified byremoving the effects of material properties at neighbor points from nonlocal constitutive equations. In view of this, the fol-lowing imaginary nonlocal strain tensor is introduced:

�eij ¼Z

Va x0 � xj jð Þeijðx0ÞdVðx0Þ; 8x 2 V ð4Þ

and nonlocal stress rij is defined directly by

rij ¼ Cijkl�ekl ð5Þ

in which Cijkl is the classic fourth order elasticity tensor. The differential form of the constitutive Eq. (4) is expressed as

1� lr2� �

�eij ¼ eij ð6Þ

Nonlocal parameter l ¼ e0að Þ2 depends on the type of material. Thus, for FG materials, l varies from one point to the other.For the homogeneous materials, the constitutive equations of the new modified nonlocal theory, Eqs. (4)–(6), are convertedto the constitutive equations of Eringen’s theory, Eqs. (1)–(3). It is obvious from Eqs. (5) and (6) that the nonlocal stress at apoint is independent of the gradient of material properties at that point.

4. Free vibration analysis of FG micro/nanoplate based on modified nonlocal theory

Consider a rectangular FG micro/nanoplate with a length a, width b and thickness h located along the x; y and z coordi-nates, respectively. The material properties including elasticity modulus E and mass density q are assumed to vary along thethickness direction based on the exponential law as

EðzÞ ¼ E0 expð/zÞ ð7aÞqðzÞ ¼ q0 expð/zÞ ð7bÞ

where / is the material gradient index, and E0 and q0 are the material properties at the bottom surface of plate. The nonlocalparameter l and Poisson’s ratio should be functions of coordinate z, but for simplicity and to obtain analytical solution, theyare assumed to be constant. Although the exponential variation of FG materials is unlikely satisfied in practice, this distri-bution rule is applied to obtain analytical solution for three-dimensional elasto-dynamic equations.

Based on the nonlocal balance law, the nonlocal linear elasticity equations of motion are expressed in the absence of bodyforces as

rij; j ¼ qðzÞ€ui; ði ¼ x; y; zÞ ð8Þ

where ðux;uy;uzÞ ¼ ðu;v ;wÞ are components of the displacement vector in the Cartesian coordinate system.

4.1. First-order plate theory

The first-order plate theory is based on the following displacement field:

uðx; y; z; tÞ ¼ u0ðx; y; tÞ þ zwxðx; y; tÞ ð9aÞvðx; y; z; tÞ ¼ v0ðx; y; tÞ þ zwyðx; y; tÞ ð9bÞwðx; y; z; tÞ ¼ w0ðx; y; tÞ ð9cÞ

where ðu0;v0;w0Þ are the displacements of a material point on the midplane of plate in the ðx; y; zÞ coordinate directions. Thelocal stress components associated with the displacement field in Eq. (9) are defined as

txx

tyy

txy

txz

tyz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ ½C�

exx

eyy

exy

exz

eyz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ ½C�

u0;x þ zwx;x

v0;y þ zwy;y

12 u0;y þ v0;x þ zwx;y þ zwy;x

� �12 w0;x þ wxð Þ12 w0;y þ wy

� �

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð10Þ


where

C½ � ¼ EðzÞ1� m2

1 m 0 0 0m 1 0 0 00 0 1� mð Þ 0 00 0 0 1� mð Þ 00 0 0 0 1� mð Þ

26666664

37777775

ð11aÞ

eij ¼12ðui; j þ uj;iÞ; ði; j ¼ x; y; zÞ ð11bÞ

The global equations of plate can be extracted by integrating elasto-dynamic Eqs. (8) through the plate thickness as

Nxx;x þ Nxy;y ¼ I0€u0 þ I1€wx ð12aÞ

Nyy;y þ Nxy;x ¼ I0 €v0 þ I1€wy ð12bÞ

Q x;x þ Q y;y ¼ I0 €w0 ð12cÞ

where

Ii ¼Z h

2

�h2

qðzÞzidz; ði ¼ 0;1;2Þ ð13aÞ

Nij ¼Z h

2

�h2

rijdz; ði; j ¼ x; yÞ ð13bÞ

Qx;Q y

� �¼ ks

Z h2

�h2

rxz;ryz� �

dz ð13cÞ

Further, multiplying Eq. (8) by z then integrating through the plate thickness and employing integration-by-part, we have

Mxx;x þMxy;y � Q x ¼ I1€u0 þ I2€wx ð14aÞ

Myy;y þMxy;x � Q y ¼ I1 €v0 þ I2€wy ð14bÞ

where

Mij ¼Z h

2

�h2

rijzdz; ði; j ¼ x; yÞ ð15Þ

Eqs. (12) and (14) are equations of motion in terms of nonlocal stress components. The transverse shear correction factor ks isintroduced in the first-order theory to provide non-uniform shear rigidities in the transverse sections and is often taken 5=6for the isotropic materials. For 2-D problems, Laplacian operator r2 reduces to 2-D one, r2 ¼ @2=@x2 þ @2=@y2. Since thematerial properties of plate are only functions of thickness coordinate z, thus for the present 2-D problem, the modified non-local theory is converted to Eringen’s nonlocal theory.

In order to obtain equations of motion in terms of displacement components, linear operator ð1� lr2Þ is used in Eqs.(12) and (14) as

NLxx;x þ NL

xy;y ¼ 1� lr2� �

I0€u0 þ I1€wx

h ið16aÞ

NLyy;y þ NL

xy;x ¼ 1� lr2� �

I0 €v0 þ I1€wy

h ið16bÞ

Q Lx;x þ Q L

y;y ¼ 1� lr2� �

I0 €w0½ � ð16cÞ

MLxx;x þML

xy;y � Q Lx ¼ 1� lr2

� �I1€u0 þ I2

€wx

h ið16dÞ

MLyy;y þML

xy;x � Q Ly ¼ 1� lr2

� �I1 €v0 þ I2

€wy

h ið16eÞ

here NLij;Q

Li and ML

ij are local force and moment components defined as

NLij ¼

Z h2

�h2

tijdz; ði; j ¼ x; yÞ ð17aÞ

Q Lx;Q

Ly

� �¼ ks

Z h2

�h2

txz; tyz� �

dz ð17bÞ

MLij ¼

Z h2

�h2

tijzdz; ði; j ¼ x; yÞ ð17cÞ


From Eqs. (10), (16) and (17), the nonlocal equations of motion can be obtained in terms of the displacement components as

K1 2u0;xx þ 1� mð Þu0;yy þ 1þ mð Þv0;xy� �

þ K2 2wx;xx þ 1� mð Þwx;yy þ 1þ mð Þwy;xy

� �¼ 1� lr2� �

I0€u0 þ I1€wx

h ið18aÞ

K1 2v0;yy þ 1� mð Þv0;xx þ 1þ mð Þu0;xy� �

þ K2 2wy;yy þ 1� mð Þwy;xx þ 1þ mð Þwx;xy

� �¼ 1� lr2� �

I0 €v0 þ I1€wy

h ið18bÞ

ksK1 1� mð Þ r2w0 þ wx;x þ wy;y

� �¼ 1� lr2� �

I0 €w0½ � ð18cÞ

K2 2u0;xx þ 1� mð Þu0;yy þ 1þ mð Þv0;xy� �

þ K3 2wx;xx þ 1� mð Þwx;yy þ 1þ mð Þwy;xy

� �� ksK1 1� mð Þ w0;x þ wxð Þ

¼ 1� lr2� �

I1€u0 þ I2€wx

h ið18dÞ

K2 2v0;yy þ 1� mð Þv0;xx þ 1þ mð Þu0;xy� �

þ K3 2wy;yy þ 1� mð Þwy;xx þ 1þ mð Þwx;xy

� �� ksK1 1� mð Þ w0;y þ wy

� �¼ 1� lr2� �

I1 €v0 þ I2€wy

h ið18eÞ

where

Ki ¼Z h

2

�h2

EðzÞ2ð1� m2Þ z

i�1ð Þdz; ði ¼ 1;2;3Þ ð19Þ

The simply-supported boundary conditions in the nonlocal field are in the form

v0 ¼ wy ¼ w0 ¼ Nxx ¼ Mxx ¼ 0; ðx ¼ 0; aÞ ð20aÞ

u0 ¼ wx ¼ w0 ¼ Nyy ¼ Myy ¼ 0; ðy ¼ 0; bÞ ð20bÞ

According to the Navier approach, the solutions are found as

u0ðx; y; tÞ ¼X1n¼1

X1m¼1

Umn cosðamxÞ sinðbnyÞejxm;nt ð21aÞ

v0ðx; y; tÞ ¼X1n¼1

X1m¼1

Vmn sinðamxÞ cosðbnyÞejxm;nt ð21bÞ

w0ðx; y; tÞ ¼X1n¼1

X1m¼1

Wmn sinðamxÞ sinðbnyÞejxm;nt ð21cÞ

wxðx; y; tÞ ¼X1n¼1

X1m¼1

w xmn cosðamxÞ sinðbnyÞejxm;nt ð21dÞ

wyðx; y; tÞ ¼X1n¼1

X1m¼1

w ymn sinðamxÞ cosðbnyÞejxm;nt ð21eÞ

here j ¼ffiffiffiffiffiffiffi�1p

;am ¼ mp=a; bn ¼ np=b and xm;n is the natural frequency. Substitution of displacements from Eq. (21) into Eqs.(18a)–(18e) and observing non-trivial solution leads to the eigenvalue problem, Eq. (22). The natural frequencies are extract-ed from this equation.

K11 �M11x2m;n K12 0 K14 �M14x2

m;n K15

K12 K22 �M22x2m;n 0 K24 K25 �M25x2

m;n

0 0 K33 �M33x2m;n K34 K35

K14 �M14x2m;n K24 K34 K44 �M44x2

m;n K45

K15 K25 �M25x2m;n K35 K45 K55 �M55x2

m;n

��

��¼ 0 ð22Þ


where

K11 ¼ �K1 2a2m þ 1� mð Þb2

n

� �K12 ¼ �K1 1þ mð Þambn

K14 ¼ �K2 2a2m þ 1� mð Þb2

n

� �K15 ¼ K24 ¼ �K2 1þ mð Þambn

K22 ¼ �K1 2b2n þ 1� mð Þa2

m

� �K25 ¼ �K2 2b2

n þ 1� mð Þa2m

� �K33 ¼ �ksK1 1� mð Þ a2

m þ b2n

� �K34 ¼ �ksK1am 1� mð ÞK35 ¼ �ksK1bn 1� mð ÞK44 ¼ �K3 2a2

m þ 1� mð Þb2n

� �� ksK1 1� mð Þ

K45 ¼ �K3 1þ mð Þambn

K55 ¼ �K3 2b2n þ 1� mð Þa2

m

� �� ksK1 1� mð Þ

M11 ¼ M22 ¼ M33 ¼ �I0 1þ l a2m þ b2

n

� �� M14 ¼ M25 ¼ �I1 1þ l a2

m þ b2n

� �� M44 ¼ M55 ¼ �I2 1þ l a2

m þ b2n

� ��

ð23Þ

4.2. Three-dimensional elasticity

Based on the constitutive equations of modified nonlocal theory, the equations of motion (8) can be expressed as

Cijklekl� �

;j

qðzÞ ¼ €ui; ði ¼ x; y; zÞ ð24Þ

where the tensor Cijkl in three-dimensional elasticity is given as

C½ � ¼ EðzÞ1þ m

1�m1�2m

m1�2m

m1�2m 0 0 0

m1�2m

1�m1�2m

m1�2m 0 0 0

m1�2m

m1�2m

1�m1�2m 0 0 0

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

2666666664

3777777775

ð25Þ

By applying operator ð1� lr2Þ on Eq. (24), we get

1� lr2� � Cijkl;jekl þ Cijkl�ekl;j

qðzÞ

� ¼ 1� lr2� �

€ui; ði ¼ x; y; zÞ ð26Þ

Using Eqs. (6), (25) and material properties from Eq. (7) in Eq. (26), yields the following equations of motion:

E0

q0 1þ mð Þ1� m

1� 2mexx;x þ

m1� 2m

eyy;x þm

1� 2mezz;x þ exy;y þ exz;z þ /exz

� ¼ 1� lr2� �

€u ð27aÞ

E0

q0 1þ mð Þm

1� 2mexx;y þ

1� m1� 2m

eyy;y þm

1� 2mezz;y þ exy;x þ eyz;z þ /eyz

� ¼ 1� lr2� �

€v ð27bÞ

E0

q0 1þ mð Þm

1� 2mexx;z þ

m1� 2m

eyy;z þ1� m

1� 2mezz;z þ /

m1� 2m

exx þm

1� 2meyy þ

1� m1� 2m

ezz

�þ exz;x þ eyz;y

� ¼ 1� lr2� �

€w

ð27cÞ

Substituting strain components from Eq. (11b) into Eqs. (27a)–(27c), the 3-D modified nonlocal equations of motion for anisotropic FG material are obtained in terms of displacements as:

E0

q0 1þmð Þ1�m

1�2mu;xxþ

m1�2m

v ;xyþm

1�2mw;xzþ

u;yyþv ;xy

2þu;zzþw;xz

2þ/

u;zþw;x

2

� �� ¼ 1�lr2� �

€u ð28aÞ

E0

q0 1þmð Þm

1�2mu;xyþ

1�m1�2m

v ;yyþm

1�2mw;yzþ

u;xyþv ;xx

2þv ;zzþw;yz

2þ/

v ;zþw;y

2

� �� ¼ 1�lr2� �

€v ð28bÞ

E0

q0 1þmð Þm

1�2mu;xzþ

m1�2m

v ;yzþ1�m

1�2mw;zzþ/

m1�2m

u;xþm

1�2mv ;yþ

1�m1�2m

w;z

�þu;xzþw;xx

2þv ;yzþw;yy

2

� ¼ 1�lr2� �

€w

ð28cÞ


In order to solve the governing vibration equations, two different displacement fields are used for the in-plane and out-of-plane vibration modes.

4.2.1. In-plane vibration analysisFor the in-plane free vibration, the following expressions for the displacement components are used:

u ¼ f ðzÞX1n¼1

X1m¼1

am cosðamxÞ sinðbnyÞe jxqm;nt ð29aÞ

v ¼ gðzÞX1n¼1

X1m¼1

bn sinðamxÞ cosðbnyÞe jxqm;nt ð29bÞ

w ¼ 0 ð29cÞ
where unknown functions f ðzÞ and gðzÞ determine the through thickness variations of in-plane displacements. The displace-ment components in Eq. (29) satisfy simply supported boundary conditions given by the three-dimensional elasticity as
rxx ¼ v ¼ w ¼ 0 x ¼ 0; að Þ ð30aÞ

ryy ¼ u ¼ w ¼ 0 y ¼ 0; bð Þ ð30bÞ

Substituting the displacement field of Eqs. (29) into Eqs. (28), reduces the partial differential equations of motion to the fol-lowing ordinary differential equations:

lXqmn � G0

� �f 00ðzÞ þ �G0/ð Þf 0ðzÞ þ � lcmn þ 1ð ÞXq

mn þ G0cmn þG0a2

m

1� 2m

�f ðzÞ þ G0b

2n

1� 2m

!gðzÞ ¼ 0 ð31aÞ

lXqmn � G0

� �g00ðzÞ þ �G0/ð Þg0ðzÞ þ � lcmn þ 1ð ÞXq

mn þ G0cmn þG0b

2n

1� 2m

!gðzÞ þ G0a2

m

1� 2m

�f ðzÞ ¼ 0 ð31bÞ

a2m 2m/f ðzÞ þ f 0ðzÞ� �

¼ �b2n 2m/gðzÞ þ g0ðzÞð Þ ð31cÞ

where

G0 ¼E0

2ð1þ mÞ ; cmn ¼ a2m þ b2

n; Xqmn ¼ q0 xq

m;n

� �2ð32Þ

In general, the solutions of Eqs. (31) are in the following form

f ðzÞ ¼ C exp kzð Þ; gðzÞ ¼ D exp kzð Þ ð33Þ

From Eqs. (31c) and (33), it is concluded that

D ¼ �a2m

b2n

C ð34Þ

Using Eqs. (33) and (34) in either of Eqs. (31a) or (31b), the characteristic equation is derived as

G0 � lXqmn

� �k2 þ G0/ð Þkþ Xq

mn þ lXqmn � G0

� �cmn

� �¼ 0 ð35Þ

The roots of Eq. (35) are obtained as

k1

k2

� ¼ 1

2 G0 � lXqmn

� � �G0/�ffiffiffiffiUp

�G0/þffiffiffiffiUp

( )ð36Þ

where

U ¼ G20/

2 � 4 G0 � lXqmn

� �Xq

mn þ lXqmn � G0

� �cmn

� �ð37Þ

Thus for U > 0:

f ðzÞ ¼ C1 exp k1zð Þ þ C2 exp k2zð Þ ð38aÞ

gðzÞ ¼ �a2m

b2n

ðC1 exp k1zð Þ þ C2 exp k2zð ÞÞ ð38bÞ

and for U < 0:

gðzÞ ¼ exp gzð Þ C1 cos fzð Þ þ C2 sin fzð Þð Þ ð39aÞ

f ðzÞ ¼ �a2m

b2n

exp gzð Þ C1 cos fzð Þ þ C2 sin fzð Þð Þð Þ ð39bÞ


in which

g ¼ �G0/

2 G0 � lXqmn

� � ; f ¼ffiffiffiffiffiffiffiffi�Up

2 G0 � lXqmn

� � ð40Þ

The local strain components eij are obtained by substituting the displacement field, Eqs. (29), and also Eqs. (38) and (39) intoEq. (11b). By using the local strain eij in Eq. (6), the following expressions for the strain components �eij can be derived: forU > 0:

exx ¼ �eyy ¼�C1 expðk1zÞ

1þ l cmn � k21

� �� C2 expðk2zÞ1þ l cmn � k2

2

� � !X1

n¼1

X1m¼1

a2m sinðamxÞ sinðbnyÞejxq

m;nt ð41aÞ

exy ¼C1 expðk1zÞ

1þ l cmn � k21

� �þ C2 expðk2zÞ1þ l cmn � k2

2

� � !X1

n¼1

X1m¼1

b2n � a2

m

2bnam cosðamxÞ cosðbnyÞejxq

m;nt ð41bÞ

exz ¼12

C1k1 expðk1zÞ1þ l cmn � k2

1

� �þ C2k2 expðk2zÞ1þ l cmn � k2

2

� � !X1

n¼1

X1m¼1

am cosðamxÞ sinðbnyÞejxqm;nt ð41cÞ

eyz ¼ �12

C1k1 expðk1zÞ1þ l cmn � k2

1


2

� � !X1

n¼1

X1m¼1

a2m

bnsinðamxÞ cosðbnyÞejxq

m;nt ð41dÞ

ezz ¼ 0 ð41eÞ

and for U < 0:

exx ¼ �eyy ¼ �R1expðgzÞcosðfzÞ � R2expðgzÞsinðfzÞð ÞX1n¼1

X1m¼1

a2m sinðamxÞ sinðbnyÞejxq

m;nt ð42aÞ

exy ¼ R1expðgzÞcosðfzÞ þ R2expðgzÞsinðfzÞð ÞX1n¼1

X1m¼1

b2n � a2

m

2bnam cosðamxÞ cosðbnyÞejxq

m;nt ð42bÞ

exz ¼12

gR1 þ fR2ð ÞexpðgzÞcosðfzÞ þ gR2 � fR1ð ÞexpðgzÞsinðfzÞð ÞX1n¼1

X1m¼1

am cosðamxÞ sinðbnyÞejxqm;nt ð42cÞ

eyz ¼ �12

gR1 þ fR2ð ÞexpðgzÞcosðfzÞ þ gR2 � fR1ð ÞexpðgzÞsinðfzÞð ÞX1n¼1

X1m¼1

a2m

bnsinðamxÞ cosðbnyÞejxq

m;nt ð42dÞ

ezz ¼ 0 ð42eÞ

where

R1 ¼1þ lcmn � lðg2 � f2Þ� �

C1 þ 2lgfð ÞC2

1þ lcmn � lðg2 � f2Þ� �2 þ 2lgfð Þ2

; R2 ¼1þ lcmn � lðg2 � f2Þ� �

C2 � 2lgfð ÞC1

1þ lcmn � lðg2 � f2Þ� �2 þ 2lgfð Þ2

ð43Þ

The bottom and top surfaces of plate are free of traction, therefore:

rxz ¼ ryz ¼ rzz ¼ 0 z ¼ � h2;h2

�ð44Þ

By substituting Eqs. (41) and (42) into Eq. (5) and using Eq. (25), the lateral boundary conditions, Eq. (44), are converted totwo independent algebraic equations for the cases U > 0 and U < 0 as follows:

for U > 0:

EðzÞ C1k1 expðk1zÞ1þ l cmn � k2

1


2

� � !��

z¼�h2

¼ 0

EðzÞ C1k1 expðk1zÞ1þ l cmn � k2

1


2

� � !��

z¼h2

¼ 0

ð45Þ


and for U < 0:

EðzÞ gR1 þ fR2ð ÞexpðgzÞcosðfzÞ þ gR2 � fR1ð ÞexpðgzÞsinðfzÞð Þjz¼�h2¼ 0

EðzÞ gR1 þ fR2ð ÞexpðgzÞcosðfzÞ þ gR2 � fR1ð ÞexpðgzÞsinðfzÞð Þjz¼h2¼ 0

ð46Þ

To obtain nontrivial solution, Ci – 0, and extract the natural frequencies of in-plane modes, the determinant of coefficientmatrices are set to zero. The roots of characteristic equations yield the natural frequencies of in-plane modes.

4.2.2. Out-of-plane vibration analysisIn order to solve equations of motion for the out-of-plane free vibration modes, the following displacement field is used:

u ¼ f ðzÞX1n¼1

X1m¼1

am cosðamxÞ sinðbnyÞejxqm;nt ð47aÞ

v ¼ f ðzÞX1n¼1

X1m¼1

bn sinðamxÞ cosðbnyÞejxqm;nt ð47bÞ

w ¼ gðzÞX1n¼1

X1m¼1

sinðamxÞ sinðbnyÞejxqm;nt ð47cÞ

Employing Eqs. (47a)–(47c) in Eqs. (28a)–(28c) and simplifying the consequent results give only two independent ordinarydifferential equations in the following form

A1f 00ðzÞ þ A2f 0ðzÞ þ A3f ðzÞ þ A4g0ðzÞ þ A5gðzÞ ¼ 0 ð48aÞ

B1g00ðzÞ þ B2g0ðzÞ þ B3gðzÞ þ B4f 0ðzÞ þ B5f ðzÞ ¼ 0 ð48bÞ

in which

A1 ¼ G0 � lXqmn; A2 ¼ A5 ¼ G0/; A3 ¼ Xq

mn þ lXqmn �

2G0 1� mð Þ1� 2m

�cmn; A4 ¼

G0

1� 2m

B1 ¼2G0 1� mð Þ

1� 2m� lXq

mn; B2 ¼2G0/ 1� mð Þ

1� 2m; B3 ¼ Xq

mn þ lXqmn � G0

� �cmn

B4 ¼ �G0

1� 2mcmn; B5 ¼ �

2G0m/1� 2m

cmn

ð49Þ

The general solutions of Eqs. (48a) and (48b) have the following exponential form

f ðzÞ ¼ K exp kzð Þ; gðzÞ ¼ L exp kzð Þ ð50Þ

Substituting Eq. (50) into Eqs. (48a) and (48b) and simplifying results in a matrix equation as

A1k2 þ A2kþ A3 A4kþ A5

B4kþ B5 B1k2 þ B2kþ B3

" #KL

� ¼

00

� ð51Þ

For nontrivial solution, determinant of the matrix should be set to zero, thus:

A1k2 þ A2kþ A3 A4kþ A5

B4kþ B5 B1k2 þ B2kþ B3

�� ¼ 0 ð52Þ

The roots of the resulted characteristic equation are obtained as

k1

k2

k3

k4

8>>><>>>:

9>>>=>>>;¼ � M2

4M1� 1

2

ffiffiffiffiffiffiU1

p� 1

2


pð53Þ

where

U1 ¼M2

2

4M21

� 2M3

3M1þ

U3 þffiffiffiffiffiffiU4p� �1=3

321=3M1

þ21=3 M2

3 � 3M2M4 þ 12M1M5

� �3M1 U3 þ

ffiffiffiffiffiffiU4p� �1=3

U2 ¼3M2

2

4M21

� 2M3

M1�U1 þ �M3

2

M31

þ 4M2M3

M21

� 8M4

M1

!=4


p ð54Þ


and

M1 ¼ A1B1

M2 ¼ A1B2 þ A2B1

M3 ¼ A1B3 þ A2B2 þ A3B1 � A4B4

M4 ¼ A2B3 þ A3B2 � A4B5 � A5B4

M5 ¼ A3B3 � A5B5

U3 ¼ 2M33 � 9M2M3M4 � 72M1M3M5 þ 27 M1M2

4 þM5M22

� �U4 ¼ U2

3 � 4 M23 � 3M2M4 þ 12M1M5

� �3

ð55Þ

The roots in Eq. (53) are real or complex. When the roots are real, we get

f ðzÞ ¼ K1 exp k1zð Þ þ K2 exp k2zð Þ þ K3 exp k3zð Þ þ K4 exp k4zð Þ ð56aÞ

gðzÞ ¼ L1 exp k1zð Þ þ L2 exp k2zð Þ þ L3 exp k3zð Þ þ L4 exp k4zð Þ ð56bÞ

By substituting Eqs. (56) into either of Eqs. (48a) or (48b), the coefficient Li can be obtained in terms of Ki as

L1 ¼ �A1k

21 þ A2k1 þ A3

A4k1 þ A5K1

L2 ¼ �A1k

22 þ A2k2 þ A3

A4k2 þ A5K2

L3 ¼ �A1k

23 þ A2k3 þ A3

A4k3 þ A5K3

L4 ¼ �A1k

24 þ A2k4 þ A3

A4k4 þ A5K4

ð57Þ

For the roots with complex values, functions f ðzÞ and gðzÞmust be changed accordingly. But, to be concise, the new forms ofthese functions are not presented here.

Using Eqs. (6), (11b), (47) and (56), the strain components eij are obtained as

exx

eyy

exy

8><>:

9>=>; ¼ K1v1ðzÞ þ K2v2ðzÞ þ K3v3ðzÞþ

�K4v4ðzÞ

��X1n¼1

X1m¼1

�a2m sinðamxÞ sinðbnyÞ

�b2n sinðamxÞ sinðbnyÞ

ambn cosðamxÞ cosðbnyÞ

8><>:

9>=>;ejxq

m;nt

0B@

1CA ð58aÞ

�exz

�eyz

� ¼1

2K1k1þL1ð Þv1ðzÞþ K2k2þL2ð Þv2ðzÞþ K3k3þL3ð Þv3ðzÞþ K4k4þL4ð Þv4ðzÞ

� ��X1n¼1

X1m¼1

am cosðamxÞsinðbnyÞbn sinðamxÞcosðbnyÞ

� ejxq

m;nt

�

ð58bÞ

ezz ¼ L1k1v1ðzÞ þ L2k2v2ðzÞ þ L3k3v3ðzÞ þ L4k4v4ðzÞ� �X1

n¼1

X1m¼1

sinðamxÞ sinðbnyÞejxqm;nt ð58cÞ

where

viðzÞ ¼expðkizÞ

1þ l cmn � k2i

� � ; ði ¼ 1;2;3;4Þ ð59Þ

It should be noted that according to Eq. (44) the shear and normal tractions at the lateral surfaces are zero. By inserting Eqs.(58) into Eq. (5) and using Eq. (25), the six boundary conditions of Eq. (44) are changed to the following four independentalgebraic equations:

EðzÞ K1k1 þ L1ð Þv1ðzÞ þ K2k2 þ L2ð Þv2ðzÞ þ K3k3 þ L3ð Þv3ðzÞ þ K4k4 þ L4ð Þv4ðzÞ� �

¼ 0; z ¼ � h2;h2

�ð60aÞ

EðzÞ L1k1 1� mð Þ � mcmnK1ð Þv1ðzÞ þ L2k2 1� mð Þ � mcmnK2ð Þv2ðzÞ þ L3k3 1� mð Þ � mcmnK3ð Þv3ðzÞ�þ L4k4 1� mð Þ � mcmnK4ð Þv4ðzÞ

�¼ 0; z ¼ � h

2;h2

�ð60bÞ


Eqs. (60) are four algebraic equations in terms of four coefficients Ki. In order to obtain the natural frequencies of the out-of-plane modes, the determinant of coefficient matrix must set to zero.

5. Numerical examples and discussion

For the sake of generality and convenience, the following non-dimensional parameters are used:

Table 1Compar

�l

0

.1

.3

/ ¼ E h=2ð Þ=E �h=2ð Þ ¼ expð/hÞ ð61aÞ

l ¼ l=h2 ð61bÞ

xqm;n ¼ xq

m;na2ffiffiffiffiffiffiffiffiffiq=E

p=h ð61cÞ

where E �h=2ð Þ and E h=2ð Þ are the elasticity modulus at the bottom and top surfaces of plate, respectively.To the best of author’s knowledge, the results of FG micro/nanoplates based on the experimental data or atomistic meth-

ods are not available in the literature. Hence, the comparison studies are carried out between the results corresponding totheory of Eringen, 3-D elasticity solution, and the present modified nonlocal theory. Based on the modified nonlocal theory,the non-dimensional natural frequencies xq

1;1 of a square FG micro/nanoplate are tabulated in Table 1 and compared withthe results given by Salehipour et al. (2015). The bold-face values denote the natural frequencies related to the in-planemodes. It can be observed that for �l ¼ 0, the results of present 3-D elasticity solution are the same as that presented bySalehipour et al. (2015). As expected, the natural frequency of the in-plane mode for the present 2-D (first order) and 3-Dsolutions are the same and does not change with the material gradient index. But, Eringen’s nonlocal theory (3-D solution)predicts higher frequency values for in-plane mode rather than present solutions. As shown in the table, for all three modes,

ison study of the non-dimensional natural frequencies �xq1;1 of a FG micro/nanoplate.

�/ a=h Method q (mode number)

1 2 3

2 5 Salehipour et al. (2015) 5.2476 13.777 23.140Present (First-order) 5.2306 13.777 23.287Present (3-D elasticity) 5.2476 13.777 23.140

10 Salehipour et al. (2015) 5.7108 27.554 46.504Present (First-order) 5.7054 27.554 46.574Present (3-D elasticity) 5.7108 27.554 46.504












the differences between the results of present study and those of Eringen’s theory (3-D solution) increase when the nonlocalparameter or gradient index increase.

Figs. 1 and 2 are drawn to display variation of non-dimensional fundamental frequency of square FG micro/nanoplatesversus the material gradient index based on the present 2-D (first-order) nonlocal theory, 3-D modified nonlocal theory(elasticity solution) and also Eringen’s theory (3-D elasticity solution). The curves are plotted for the nonlocal parameter val-ues l ¼ 0:1 and 0.2, and the length-to-thickness ratios of 5 and 10. It is clearly seen that based on the Eringen’s theory (3-Delasticity solution), the fundamental frequency rises when the gradient index increases. But, the other theories predict com-pletely different behavior. In other words, the 2-D nonlocal theory (first-order) and 3-D modified nonlocal theory illustratethat the frequency slowly decreases when the gradient index increases. The unexpected results of the Eringen’s theory (3-Delasticity solution) which have large differences with the results of other theories, originate from the fact that the operatorr2 in Eringen’s nonlocal theory is applied on both material properties and strain tensor.

Figs. 3 and 4 are plotted to show the variation of fundamental frequency of FG micro/nanoplates versus nonlocal para-meter, using aforementioned theories for the length-to-thickness ratios of 5 and 10, respectively. The material gradient indexvalues are set to be 2 and 3. It is evident that by increasing the nonlocal parameter, the differences between the results

Fig. 1. Variation of non-dimensional fundamental natural frequency for a square FG micro/nanoplate ða=h ¼ 5Þ versus the gradient index �/: (a) �l ¼ 0:1; (b)�l ¼ 0:2.

Fig. 2. Variation of non-dimensional fundamental natural frequency for a square FG micro/nanoplate ða=h ¼ 10Þ versus the gradient index �/: (a) �l ¼ 0:1;(b) �l ¼ 0:2.

Fig. 3. Variation of non-dimensional fundamental natural frequency for a square FG micro/nanoplate ða=h ¼ 5Þ versus the nonlocal parameter �l: (a) �/ ¼ 2;(b) �/ ¼ 3.

Fig. 4. Variation of non-dimensional fundamental natural frequency for a square FG micro/nanoplate ða=h ¼ 10Þ versus the nonlocal parameter �l: (a)�/ ¼ 2; (b) �/ ¼ 3.


obtained by different aforementioned theories increase. But, the differences between the results of Eringen’s theory (3-Delasticity solution) and that of other two theories are more significant. It is seen from the figures that for higher value ofthe gradient index or for thinner plates, the variation of frequency with nonlocal parameter based on the Eringen’s theory(3-D elasticity solution) is completely different from that predicted by the other two theories. It is concluded from Figs. 1–4 that for sufficiently large values of gradient index and nonlocal parameter, the fundamental frequency corresponding toEringen’s theory (3-D elasticity solution) takes very higher values. This is attributed to the important role of the materialgradient index in the Eringen’s nonlocal theory.

6. Conclusion

The present study serves to propose a size-dependent modified theory based on the nonlocal Eringen’s theory for analysisof structures made of FG materials. The modified theory eliminates the couple effect of nonlocal parameter and gradient ofmaterial properties. In view of this, the nonlocal stress components are directly obtained from a new nonlocal strain tensorwhich is a function of the neighbor strain tensors in the continuum body. Based on the modified nonlocal theory and utilizingfirst-order plate theory and three-dimensional elasticity theory, analytical solutions are obtained for free vibration of FGmico/nanoplates. Numerical examples are carried out to compare the results of present study with the results obtained from


three-dimensional Eringen’s nonlocal elasticity theory. It is observed that for sufficiently large values of the nonlocal para-meter and gradient index, the results of three-dimensional Eringen’s theory are much higher than that of present solutions.The presented modified nonlocal theory can be applied as a useful size-dependent theory in the future researches to studymechanical behavior of micro/nano structures made of FG materials.

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