Seediscussions,stats,andauthorprofilesforthispublicationat:http://www.researchgate.net/publication/228952249

Non-localelasticplatetheoriesARTICLEinPROCEEDINGSOFTHEROYALSOCIETYAMATHEMATICALPHYSICALANDENGINEERINGSCIENCESDECEMBER2007ImpactFactor:2DOI:10.1098/rspa.2007.1903

CITATIONS58

DOWNLOADS119

VIEWS202

6AUTHORS,INCLUDING:

HpLeeNationalUniversityofSingapore286PUBLICATIONS3,917CITATIONS

SEEPROFILE

JnReddyTexasA&MUniversity102PUBLICATIONS1,678CITATIONS

SEEPROFILE

Availablefrom:C.M.WangRetrievedon:03July2015

doi: 10.1098/rspa.2007.1903, 3225-3240463 2007 Proc. R. Soc. A

Pin Lu, P.Q Zhang, H.P Lee, C.M Wang and J.N Reddy

Non-local elastic plate theories

Referenceshtml#ref-list-1http://rspa.royalsocietypublishing.org/content/463/2088/3225.full.

This article cites 20 articles, 2 of which can be accessed free

Email alerting service herethe box at the top right-hand corner of the article or click

Receive free email alerts when new articles cite this article - sign up in

http://rspa.royalsocietypublishing.org/subscriptions go to: Proc. R. Soc. ATo subscribe to

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from

me tethe ch

Proc. R. Soc. A (2007) 463, 32253240

doi:10.1098/rspa.2007.1903

Published online 25 September 2007

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from becomes signicant when dealing with micro-/nanoscale plate-like structures. Asillustrative examples, the bending and free vibration problems of a rectangular platewith simply supported edges are solved and the exact non-local solutions are discussed inrelation to their corresponding local solutions.

Keywords: non-local continuum mechanics; Kirchhoff plate theory;Mindlin plate theory; size effects; microelectromechanical systems;

nanoelectromechanical systems

1. Introduction

Size-dependent theories of continuum mechanics have received increasingattention in recent years due to the need to model and analyse very small-sized mechanical structures and devices in the rapid developmentsof micro-/nanotechnologies. One of the well-known models is the non-localelasticity theory (Kroner 1967; Eringen 1983, 2002). This non-local theory hasbeen applied to solve wave propagation, dislocation and crack problems. Thetheory includes scale effects and long-range atomic interactions so that it canbe used as a continuum model for atomic lattice dynamics. Therefore, thiscontinuum theory on one hand is suitable for modelling submicro- or nanosizedstructures, while on the other hand it avoids enormous computational effortswhen compared with discrete atomistic or molecular dynamics simulations(Sun & Zhang 2003; Zhang & Sun 2004). Owing to the aforementioned

*Author and address for correspondence: Institute of High Performance Computing, 1 Science

Pa(lu

ReAcchanics. The basic equations for the non-local Kirchhoff and the Mindlin plaories are derived. These non-local plate theories allow for the small-scale effect whiNon-local elastic plate theories

BY PIN LU1,2,*, P. Q. ZHANG1, H. P. LEE2,3, C. M. WANG3

AND J. N. REDDY3

1Department of Modern Mechanics, University of Science and Technology ofChina, Hefei, Anhui 230027, Peoples Republic of China

2Institute of High Performance Computing, 1 Science Park Road,01-01 The Capricorn, Science Park II, Singapore 117528,

Republic of Singapore3Engineering Science Programme, National University of Singapore,

Block E3A, 04-17, 7 Engineering Drive 1, Singapore 117574,Republic of Singapore

A non-local plate model is proposed based on Eringens theory of non-local continuumre

typin@ihpc.a-star.edu.sg).

ceived 20 March 2007cepted 4 September 2007 3225 This journal is q 2007 The Royal Socierk Road, 01-01 The Capricorn, Science Park II, Singapore 117528, Republic of Singapo

plate with simply supported edges are solved in order to examine the effect ofsmall scale on the bending and vibration solutions.

P. Lu et al.3226

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from 2. Non-local elastic plate model

(a ) Review of non-local elasticity theory

For non-local linear elastic solids, the equations of motion have the form(Eringen 1983, 2002)

tij;jC fiZ rui; 2:1where r and fi are, respectively, the mass density and the body (and/or applied)forces; ui is the displacement vector; and tij is the stress tensor of the non-localelasticity dened by

tijxZVajx 0Kxjsijx 0dvx 0; 2:2

in which x is a reference point in the body; ajx 0Kxj is the non-local kernelfunction; and sij is the local stress tensor of classical elasticity theory at anypoint x 0 in the body and satises the constitutive relations

sij Z cijkl3kl and 3kl Z uk;lCul;k=2; 2:3for a general elastic material, in which cijkl are the elastic modulus componentswith the symmetry properties cijklZcjiklZcijlkZcklij, and 3kl is the strain tensor. Itshould be emphasized here that the boundary conditions involving tractions arebased on the non-local stress tensor tij and not on the local stress tensor sij.The properties of the non-local kernel ajx 0Kxj have been discussed in detail

by Eringen (1983). When a(jxj) takes on a Greens function of a linearadvantages, several researchers have applied the non-local continuum theory forthe mechanical analysis of micro- and nanostructures in more recent years(Peddieson et al. 2003; Sudak 2003; Wang & Hu 2005; Zhang et al. 2005; Lu et al.2006a; Xu 2006; Wang et al. 2006; Reddy 2007). However, most of these studiesfocused on one-dimensional beam-like structures.In modelling micro- or nanoelectromechanical systems (MEMS or NEMS) and

devices, some mechanical componentssuch as thin lm elements (Freund &Suresh 2003), nanosheet resonators (Bunch et al. 2007), paddle-like resonators(Evoy et al. 1999; Lobontiu et al. 2006) and two-dimensional suspendednanostructures (Tighe et al. 1997; Zalalutdinov et al. 2006)have to be modelledas a two-dimensional plate-like structure. For this purpose, the non-local platetheories are studied herein. Based on the non-local elasticity model, pioneered byEringen (1983, 2002), the general governing equations for a thin plate can bederived by integrating the equations of motion for the non-local linear elasticitythrough the thickness. With the proper assumptions for displacementcomponents, specic plate theories can be further obtained. Considered hereinare two well-known plate theories: the Kirchhoff plate theory and the Mindlinplate theory. The Kirchhoff plate theory is a thin-plate theory that neglects theeffect of transverse shear deformation, whereas the Mindlin plate theory is a rst-order shear-deformable plate theory that incorporates this effect which becomessignicant in thick plates and shear-deformable plates. Based on these two non-local plate model versions, the bending and vibration problems of a rectangularProc. R. Soc. A (2007)

3227Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from (b ) Plate equations of non-local elastic model

The foregoing non-local elastic model can be extended to two-dimensionalthin-plate structures. Consider a thin plate with a constant thickness h. ACartesian coordinate system xi (iZ1, 2, 3) is introduced so that the axes x1 andx2 lie in the mid-plane of the plate. Since the thickness of the plate is very smallwhen compared with the other two dimensions, it is assumed that s33Z0 in theconsidered plate theories. The constitutive relations (2.3) can thus be reduced to

sabZ c^abur3ur and sa3Z 2ca3u33u3; 2:9where

c^aburZ caburK cab33c33ur=c3333 2:10are the reduced elastic modulus components.The non-local resultant forces Nij and the non-local resultant moments Mij are

dened as

Nij Z

h=2Kh=2

tij dx 3 and Mij Z

h=2Kh=2

tijx 3 dx3: 2:11

The global governing equations of the plate structures can be derived byintegrating the equations of motion (2.1) through the thickness (Lu et al. 2006b).By multiplying equation (2.1) by dx3, then integrating through the thickness andnoting (2.11)1, we have

Nia;aCpiZ

h=2Kh=2

rui dx3; 2:12differential operator L, i.e.Lajx 0KxjZ djx 0Kxj; 2:4

the non-local constitutive relation (2.2) is reduced to the differential equation

Ltij Zsij 2:5and the integro-partial differential equation (2.1) is correspondingly reduced tothe partial differential equation

sij;jCLfiKruiZ 0: 2:6By matching the dispersion curves with lattice models, Eringen (1983, 2002)proposed a non-local model with the linear differential operator L dened by

LZ 1Ke 0a2V2; 2:7where a is an internal characteristic length (lattice parameter, granular size ormolecular diameters) and e0 is a constant appropriate to each material foradjusting the model to match some reliable results by experiments or othertheories. Therefore, according to (2.3), (2.5) and (2.7), the constitutive relationswith this kernel function may be simplied to

1Ke0a2V2tij Z cijkl3kl : 2:8For simplicity and to avoid solving integro-partial differential equations,

the non-local elasticity model, dened by the relations (2.5)(2.8), hasbeen widely adopted for tackling various problems of linear elasticity andmicro-/nanostructural mechanics.Proc. R. Soc. A (2007)

P. Lu et al.3228

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from where piZ h=2Kh=2 fi dx3. Furthermore, multiplying equation (2.1) by x3 dx3

followed by integrating through the thickness and noting (2.11)2, we have

Mib;bKNi3Z

h=2Kh=2

ruix3 dx3: 2:13

Since the equation with iZ3 in equation (2.13) has no physical application, it isomitted in the remaining part of the derivations.By applying the linear differential operator (2.7) and the differential equations

(2.5) to equation (2.11), we have

1Ke 0a2V2Nij ZNLij and 1Ke 0a2V2Mij ZMLij ; 2:14where NLij and M

Lij are the local (classical) resultant forces and the local resultant

moments dened by

NLij Z

h=2Kh=2

sij dx3 and MLij Z

h=2Kh=2

sijx3 dx3: 2:15

Furthermore, by applying the operator to equations (2.12) and (2.13), we obtainthe general equations of motion for the non-local plate model as

NLib;bZK1Ke 0a2V2piC h=2Kh=2 rui dx 3Ke 0a2

h=2Kh=2 V

2ruidx3 andMLab;bKN

La3Z

h=2Kh=2 ruax3 dx 3Ke0a2

h=2Kh=2 V

2ruax3dx3:2:16

The differential operator V2 in (2.16) is the three-dimensional Laplace operatorin general. For thin-plate models, it may be reduced to the two-dimensionalLaplace operator by ignoring the differential component with respect to x3, i.e.

V2Zv2=vx 21Cv2=vx22. With this approximation, the equations of motion (2.16)

become

NLib;bZK1Ke 0a2V2 piK h=2Kh=2 rui dx3

and

MLab;bKNLa3Z 1Ke 0a2V2

h=2Kh=2 ruax 3 dx3

2:17

and the non-local resultant force and moment tensors, Nij and Mij, respectively,in (2.11) can be simplied as

NijxZAajx 0KxjNLij x 0dAx 0 and

MijxZAajx 0KxjMLij x 0dAx 0;

2:18

where the integrals are taken along the mid-plane A of the plate, NLij and MLij are

given in (2.15). The two-dimensional non-local kernel ajx 0Kxj in equation(2.18) can be dened to satisfy the relation (2.4), in which the differentialoperator is as given in equation (2.7) instead of a two-dimensional Laplace

operator, i.e. LZ1Ke 0a2v2=vx21Cv2=vx22. This approximation is acceptablefor plates with very small thicknessspan ratios. For thick-plate models, theexact expressions (2.11) and (2.16) may be required.The later derivations for the thin-plate models are based on the simplied

equations (2.17) and (2.18). Beginning from equations (2.11) and (2.16), thederivations can be shown to arrive at the same formulations, but the non-localresultant force and moment tensors are dened by equation (2.11) and not byequation (2.18).Proc. R. Soc. A (2007)

3229Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from in which Qa is the effective shear forces as dened by Q1ZM1b,bCM12,2 andQ2ZM2b,bCM21,1.The local resultant forces NLab and the local resultant moments M

Lab for the

Kirchhoff plate theory can be obtained by substituting equations (2.9), (2.10),(3.2) and (3.3) into equation (2.15) as

NLabZAabur30urCBaburkur and

MLabZBabur30urCDaburkur;

3:73. Basic equations for two plate theories

Equations (2.9)(2.18) are the general equations of the non-local plate model.For different plate theories, the related equations of motion can be obtained bysubstituting the assumed displacement components ui into these equations.There are a number of plate theories, of which the most commonly used are theKirchhoff and the Mindlin plate theories. The basic equations of these two platetheories are derived in this section based on the foregoing non-local relations.

(a ) Kirchhoff plate theory

In the Kirchhoff plate theory, the displacement components are assumed tohave the form

uaZ u0aK x 3u

03;a and u3Z u

03; 3:1

where u0iZu0i xb; t is the displacement components of the mid-plane at time t.

The strain components for the plate theory can be obtained by substitutingequation (3.1) into equation (2.3)2 as

3abZ 30abCx3kab and 33aZ 0; 3:2

with

30abZ1

2u0a;bCu

0b;a

and kabZKu

03;ab: 3:3

The equations of motion for the plate theory can be obtained by substitutingequation (3.1) into equations (2.12) and (2.13), i.e.

Nab;bCpaZ I0u0a and

Mab;abCp3Z I0u03K I2u

03;aa;

3:4

where

I0Z

h=2Kh=2

r dx3Z rh and I2Z

h=2Kh=2

rx23 dx3Zrh3

12: 3:5

The boundary conditions are given by either one of each of the following pairs ofconditions being specied:

Nab or u0a; Qa or u

03 and Mab or u

03;a; 3:6Proc. R. Soc. A (2007)

a a 3 a 3 3

P. Lu et al.3230

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from where jaZjaxb; t are independent variables. The strain components forthe plate theory can be obtained by substituting equation (3.10) into equation(2.3)2 as

3abZ 30abCx33

1ab and 33aZ

1

2u03;aCja

; 3:11with

30abZ1

2u0a;bCu

0b;a

and 31abZ

1

2ja;bCjb;a: 3:12

The equations of motion of the Mindlin plate theory can be obtained bysubstituting equation (3.10) into equations (2.12) and (2.13), thus yielding

Nib;bCpiZ I0u0i and

Mab;bKNa3Z I2 ja:3:13

The boundary conditions are given by either one of each of the following pairs ofconditions being specied:

Nib or u0i and Mab or ja: 3:14

The local resultant forces NLab and the local resultant moments MLab for the

Mindlin plate theory can be obtained by substituting equations (2.9), (2.10),where

AaburZ

h=2Kh=2

c^abur dx 3; BaburZ

h=2Kh=2

c^aburx3 dx 3 and

DaburZ

h=2Kh=2

c^aburx23 dx3

3:8

are the extensional, the coupling and the bending stiffnesses, respectively. For asymmetric composite plate, BaburZ0.By substituting equations (3.7) and (2.14) into equation (3.4), the equations of

motion for the non-local Kirchhoff plate theory can be expressed in terms of thedisplacements as

Aaburu0u;rbKBaburu

03;urbC 1Ke0a2V2 paK I0u0a

Z 0 and

Baburu0u;rabKDaburu

03;urabC 1Ke0a2V2 p3K I0u03

Z 0;

3:9

in which the mass inertia I2 dened in equation (3.5) is neglected for theKirchhoff plate theory. Using the Voigt notation, the plate constants Aabur,Babur and Dabur can be converted to the conventional form expressed by twoindices as AIJ, BIJ and DIJ.

(b ) Mindlin plate theory

In the Mindlin plate theory, the displacement components are assumed tohave the form

u Z u0Cx j and u Z u0; 3:10Proc. R. Soc. A (2007)

For a symmetrical orthotropic plate, the coupling stiffnesses Babur in equation

3231Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from (3.7) are zero. The constitutive relations for the local bending moments are thusreduced to

ML11

ML22

ML12

8>>>:

9>>=>>;Z

D11 D12 0

D12 D22 0

0 0 D66

264

375

k11

k22

2k12

8>:

9>=>;

ZK

D11 D12 0

D12 D22 0

0 0 D66

264

375

u03;11

u03;22

2u03;12

8>>>:

9>>=>>;; 4:1

in which the subscripts of the bending stiffness components have been writtenwith two-index Voigt notation. The equation of motion (3.9)2 for bendingbecomes

D11u03;1111C2D12C2D66u03;1122CD22u03;2222Z 1Ke 0a2V2 p3K I0u03

: 4:2(3.11) and (3.12) into equation (2.15) as

NLabZAabur30urCBabur3

1ur; N

L3bZ 2A3b3r33r and

MLabZBabur30urCDabur3

1ur;

3:15

where the constants Aabur, Babur and Dabur are given in equation (3.8), and

A3b3rZ h=2Kh=2 c^3b3r dx3.

By substituting equations (3.15) and (2.14) into equation (3.13), the equationsof motion for the non-local Mindlin plate theory can be expressed in terms ofdisplacements as

Aaburu0u;rbCBaburju;rbC 1Ke 0a2V2 paKI u0a

Z 0;

A3b3r u03;rbCjr;b

C 1Ke 0a2V2 p3KI u03

Z 0 and

Baburu0u;rbCDaburju;rbKA3a3r u

03;rCjr

K1Ke 0a2V2I2 jaZ 0:

3:16

4. Bending and free vibrations of symmetrically orthotropic plates

In order to illustrate the applications of the foregoing non-local plate theories, weconsider the case of symmetrically orthotropic plates for Kirchhoff and Mindlinplate models. For such plates, the in-plane and the out-of-plane variables areuncoupled, and only exural deformations are considered in the examples for thesake of simplicity. The bending and free vibration solutions of a simplysupported, rectangular plate based on both Kirchhoff and Mindlin non-local platemodels are then derived, and are compared with the results based on local(classical) plate theories.

(a ) Solutions based on Kirchhoff plate theoryProc. R. Soc. A (2007)

P. Lu et al.3232

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from In the case of cylindrical bending, equation (4.2) reduces to Euler beam-typeequations (Lu et al. 2006a)

D11u03;1111Z 1Ke0a2V2 p3K I0u03

: 4:3

The boundary conditions for the simply supported edges of the rectangularplate are dened by

u03Z 0; M11Z 0 along edges x 1Z 0; l 1 and

u03Z 0; M22Z 0 along edges x2Z 0; l 2:4:4

From (2.14), it follows that these conditions are equivalent to:

u03Z 0; ML11Z 0 along edges x 1Z 0; l 1 and

u03Z 0; ML22Z 0 along edges x2Z 0; l 2:

4:5

Consider the static bending problem of a simply supported plate subjected to atransverse sinusoidally distributed load given by

p3ZP3nm sin 2nx1 sin hmx 2; 4:6where P3nm is a known constant, and

2nZnp=l 1 and hmZmp=l 2; 4:7with n and m being positive integers. The deection solution that satises theboundary conditions (4.4) or (4.5) can be assumed to take the form

u03ZU3nm sin 2nx 1 sin hmx 2; 4:8in which 2n and hm are dened in equation (4.7), and U3nm is the constant to bedetermined. By substituting equations (4.8) and (4.6) into equation (4.2), oneobtains U3nm as

U3nmZ Hnm2ULnmK ; 4:9where

HnmZ1Ce 0a222nCh2m

q4:10

is the non-local effect-related parameter, and

ULnm

K ZP3nm

D1124nC2D12C2D6622nh2mCD22h4m

4:11

is the value of the maximum transverse displacement based on the localKirchhoff plate theory. Since HnmO1, it is clear that the transversedisplacements predicted by the non-local plate theories are generally largerthan those predicted by the classical plate theories as the non-local effect makesthe plate models more exible.For the free transverse vibration problem of the simply supported, rectangular

plate, the time-dependent displacement solution satisfying the boundaryconditions (4.4) or (4.5) can be assumed to take the form

u03ZU3nm sin 2nx1 sin hmx 2 sin unmt; 4:12where unm is the related order natural frequency of the transverse vibration, and2n and hm are dened in equation (4.7). By substituting equation (4.12) intoProc. R. Soc. A (2007)

4:16

3233Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from For the case of cylindrical bending, the equations in (4.16) reduce to theequations of the non-local Timoshenko beam model (Lu et al. 2007).The boundary conditions for the simply supported edges of the rectangular

plate are dened by

u03Z 0; j2Z 0; M11Z 0 along edges x1Z 0; l 1 and

u03Z 0; j1Z 0; M22Z 0 along edges x2Z 0; l 2:4:17

In view of equation (2.14), it follows that these conditions are equivalent to

u03Z 0; j2Z 0; ML11Z 0 along edges x1Z 0; l 1 and

u03Z 0; j1Z 0; ML22Z 0 along edges x2Z 0; l 2:

4:18

For the free transverse vibration problem, the solutions satisfying theequation (4.2) with p3Z0, unm can be obtained as

unmZuLnm

K

Hnm; 4:13

where

uLnm

K Z

D112

4nC2D12C2D6622nh2mCD22h4m

I0

s4:14

is the natural frequency based on the Kirchhoff classical plate theory, and Hnm isthe non-local effect-related parameter dened in equation (4.10). The freevibration and natural frequencies of rectangular plates based on the localKirchhoff plate theory were discussed in detail by Leissa (1973).

(b ) Solutions based on Mindlin plate theory

For the symmetrical orthotropic plate, the coupling stiffnesses Babur inequation (3.15) are also zero. The constitutive relations for the uncoupled localbending components are thus reduced to

ML11

ML22

ML12

NL23

NL13

8>>>>>>>>>>>>>>>:

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

D11 D12 0 0 0

D12 D22 0 0 0

0 0 D66 0 0

0 0 0 A44 0

0 0 0 0 A55

266666664

377777775

j1;1

j2;2

j1;2Cj2;1

u03;2Cj2

u03;1Cj1

8>>>>>>>>>>>>>>>:

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

; 4:15

in which the strain components 31ab and 33a have been written in the displacementcomponents according to equations (3.11) and (3.12). The equations of motionfor bending can be obtained from equation (3.16) as

A55u03;11Cj1;1CA44u03;22Cj2;2Z 1Ke0a2V2 I0u03Kp3

;

D11j1;11CD12CD66j2;12CD66j1;22KA55u03;1Cj1Z 1Ke0a2V2I2j1 andD66j2;11CD66CD12j1;12CD22j2;22KA44u03;2Cj2Z 1Ke0a2V2I2j2:Proc. R. Soc. A (2007)

P. Lu et al.3234

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from boundary conditions (4.17) or (4.18) can be assumed to take the form

u03ZU3nm sin 2nx1 sin hmx 2 sin unmt;

j1ZJ1nm cos 2nx1 sin hmx 2 sin unmt and

j2ZJ2nm sin 2nx1 cos hmx2 sin unmt;

4:19

where 2n and hn are dened in equation (4.7).By substituting equation (4.19) into equation (4.16) with p3Z0, we have

k11KI0Hnmunm2 k12 k13k12 k22KI2Hnmunm2 k23k13 k23 k33KI2Hnmunm2

2664

3775

U 03nm

J1nm

J2nm

8>:

9>=>;Z 0;

4:20

where

k11ZA55x2nCA44h

2m; k12ZA552n; k13ZA44hm;

k23Z D12CD662nhm; k22ZD11x2nCD66h2mCA55 and

k33ZD166x2nCD22h

2mCA44

4:21

and Hnm is as given in equation (4.10). By setting the determinant of thecoefcient matrix in equation (4.20) to be zero, one obtains the correspondingcharacteristic equation as

Hnmunm6Ca1Hnmunm4Ca2Hnmunm2Ca3Z 0; 4:22

where

a1ZKk11I0K

k22Ck33I2

;

a2Zk11k22Ck11k33Kk

212Kk

213

I0I2C

k22k33Kk223

I 22and a3ZK

D

I0I22

;

DZ k11k22k33C2k12k13k23K k11k223K k22k

213K k33k

212:

4:23

By solving the characteristic equation (4.22), the frequencies for the xed valuesn and m are obtained as

u1nmZuL1nm

M

Hnm; u2nmZ

uL2nm

M

Hnmand u3nmZ

uL3nm

M

Hnm; 4:24Proc. R. Soc. A (2007)

u2nm M ZK 3a1K3a2 cos 3

K3

;

3235Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from uL3nm 2

M ZK2

3

a21K3a2

qcos

gC4p

3K

a13

and

gZ cosK127a3C2a

31K9a1a2

2a21K3a23

q ;

4:25

are the natural frequencies based on the local Mindlin plate theory. It can be seenthat, for each combination of n and m, we obtain three natural frequencies. Thelowest of these corresponds to the mode where the transverse deectiondominates, whereas the other two frequencies are much higher and correspondto shear modes (Soedel 1993).The static bending problem of a simply supported rectangular plate under a

sinusoidally distributed transverse load (4.6), based on the non-local Mindlinplate theory, can be solved similarly. Assume the static displacementcomponents to take the forms as shown in equation (4.19), but omitting thetime-dependent terms, i.e. by letting sin unmtZ1. By substituting the staticdisplacement components into the governing equations (4.16), one obtains themaximum values of the displacement components as

U3nmZ Hnm2ULnmM ; J1nmZ Hnm2JL1nmM and

J2nmZ Hnm2JL2nmM ; 4:26where

ULnmM Zk22k33Kk

223

DP3nm; JL1nmM Z

k13k23K k12k33D

P3nm and

JL2nmM Zk12k23K k22k13

DP3nm

4:27

are the values of the maximal generalized displacement components based on thelocal Mindlin plate theory, and kij and D are dened in equations (4.21) and(4.23), respectively. Again, it can be seen that the displacements predicted by thenon-local Mindlin plate theory are larger than those predicted by the localMindlin plate theory.

(c ) Discussions

For a simply supported rectangular plate, it can be seen from equations (4.13)and (4.24) that, for given n and m, the ratio between the non-local and the localfrequencies is 1/Hnm for both Kirchhoff and Mindlin plate theories. By deninguNnm and u

Lnm to be the non-local and the local natural frequencies obtained inwhere

uL1nm 2

M ZK2

3

a21K3a2

qcos

g

3K

a13

;

L 2 2 2q gC2p a1Proc. R. Soc. A (2007)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.20 0.2 0.4

0.20.4

0.60.8

1.0

0.6 0.8 1.0

R11

e0 a /l1 l1/l2

Figure 1. Variations of frequency ratio, R11, with respect to non-local parameter e0a=l 1 and aspectratio l1/l2.

1.0(a) (b)

0.9

0.8

0.7

0.6

0.5R nm

0.4

0.3

0.2

0.1

0 0.2 0.4 0.6 0.8 1.0

l1/l2 = 1.0

e0a/l1

n=1, m=1n=1, m=2n=2, m=1n=2, m=2n= 2, m=3n=3, m=2n=3, m=3

n=1, m=1n=1, m=2n=2, m=1n=2, m=2n= 2, m=3n=3, m=2n=3, m=3

0.20 0.4 0.6 0.8 1.0

l1/l2 = 0.4

e0a/l1

Figure 2. (a,b) Variations of frequency ratios Rnm with respect to non-local parameter e 0a=l 1 fordifferent aspect ratios l1/l 2.

P. Lu et al.3236

Proc. R. Soc. A (2007)

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from

3237Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from Table 1. Frequency ratios Rnm in terms of non-local parameter e 0a=l 1 and aspect ratio l1/l2.

e 0a=l 1

(n, m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

l 1/l2Z1.0(1, 1) 1.0000 0.9139 0.7475 0.6001 0.4904 0.4105 0.3512 0.3061 0.2708(1, 2) 1.0000 0.8183 0.5799 0.4287 0.3353 0.2738 0.2308 0.1993 0.1752(2, 1) 1.0000 0.8183 0.5799 0.4287 0.3353 0.2738 0.2308 0.1993 0.1752(2, 2) 1.0000 0.7475 0.4904 0.3912 0.2708 0.2196 0.1844 0.1587 0.1393(2, 3) 1.0000 0.6618 0.4038 0.2823 0.2155 0.1739 0.1456 0.1251 0.1097equations (4.13), (4.14), (4.24) and (4.25), the ratios can be written as

RnmZ1

HnmZ

uNnm

uLnmZ

11Cp2e 0a=l 12n2Cm2l 1=l 22

q ; 4:28in which e 0a=l 1 is a non-dimensional non-local parameter, and l1/l2 is the aspectratio of the rectangular plate.The properties of the natural frequencies of the simply supported rectangular

plates based on the local Kirchhoff and Mindlin plate theories have been wellstudied (see, for instance, Leissa (1973) and Soedel (1993)). The correspondingnon-local Kirchhoff and Mindlin plate models modify the frequency results by the

(3, 2) 1.0000 0.6618 0.4038 0.2823 0.2155 0.1739 0.1456 0.1251 0.1097(3, 3) 1.0000 0.6001 0.3512 0.2426 0.1844 0.1484 0.1241 0.1066 0.0934

l 1/l2Z0.8(1, 1) 1.0000 0.9277 0.7719 0.6380 0.5278 0.4451 0.3827 0.3346 0.2967(1, 2) 1.0000 0.8602 0.6448 0.4902 0.3886 0.3197 0.2707 0.2343 0.2063(2, 1) 1.0000 0.8282 0.5942 0.4419 0.3465 0.2834 0.2391 0.2066 0.1816(2, 2) 1.0000 0.7791 0.5278 0.3827 0.2967 0.2412 0.2028 0.1748 0.1535(2, 3) 1.0000 0.7137 0.4539 0.3216 0.2468 0.1997 0.1674 0.1440 0.1263(3, 2) 1.0000 0.6834 0.4240 0.2979 0.2279 0.1840 0.1542 0.1326 0.1162(3, 3) 1.0000 0.6380 0.3827 0.2662 0.2028 0.1635 0.1368 0.1175 0.1030

l 1/l2Z0.6(1, 1) 1.0000 0.9390 0.8066 0.6730 0.5636 0.4792 0.4141 0.3633 0.3229(1, 2) 1.0000 0.8977 0.7237 0.5619 0.4539 0.3774 0.3216 0.2795 0.2468(2, 1) 1.0000 0.8361 0.6062 0.4530 0.3561 0.2916 0.2462 0.2128 0.1872(2, 2) 1.0000 0.8066 0.5636 0.4141 0.3229 0.2633 0.2218 0.1914 0.1682(2, 3) 1.0000 0.7637 0.5091 0.3668 0.2836 0.2302 0.1934 0.1666 0.1463(3, 2) 1.0000 0.7018 0.4419 0.3120 0.2391 0.1933 0.1628 0.1394 0.1222(3, 3) 1.0000 0.6730 0.4141 0.2902 0.2218 0.1790 0.1499 0.1289 0.1130

l 1/l2Z0.4(1, 1) 1.0000 0.9472 0.8282 0.7018 0.5942 0.5088 0.4419 0.3890 0.3465(1, 2) 1.0000 0.9277 0.7719 0.6380 0.5278 0.4451 0.3827 0.3346 0.2967(2, 1) 1.0000 0.8420 0.6152 0.4615 0.3635 0.2980 0.2517 0.2176 0.1915(2, 2) 1.0000 0.8282 0.5942 0.4419 0.3465 0.2834 0.2391 0.2066 0.1816(2, 3) 1.0000 0.8066 0.5636 0.4141 0.3229 0.2633 0.2218 0.1914 0.1682(3, 2) 1.0000 0.7159 0.4562 0.3234 0.2483 0.2009 0.1684 0.1449 0.1271(3, 3) 1.0000 0.7018 0.4419 0.3120 0.2391 0.1933 0.1620 0.1394 0.1222

Proc. R. Soc. A (2007)

components.

P. Lu et al.3238

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from The foregoing simple examples show that one can apply the non-local platemodels to predict the mechanical properties of micro- and nanoscale plate-likestructures. For complex boundary value problems, analytical solutions aregenerally not available and numerical treatments are required.

5. Concluding remarks

In this paper, the general equations and relations of non-local elastic plate modelshave been presented, and the governing equations of two non-local plate theoriesmodied from their corresponding local Kirchhoff plate theory and local Mindlinplate theory have been derived. The non-local theories can be applied for theanalysis of micro-and nanoscale plate-like structures, in which the small-scaleeffects become signicant. As illustrative examples, the bending and freevibration problems of a simply supported rectangular plate based on both thenon-local Kirchhoff and Mindlin plate models have been studied. The resultsshow that, for very small-sized plates, the inuences of the non-local effects onthe mechanical properties are considerable.For general boundary value problems, like non-zero boundary force conditions,

the method for solution of the non-local plate theories will be more complicatedthan those of the local plate theories. It is known that the force boundaryconditions for the non-local plate models are based on the non-local componentsNij and Mij dened in equation (2.11) or equation (2.18). Since these generalizedforce components are coupled by a set of the second-order differential equations(2.14), it is very difcult to obtain their explicit expressions as in the case of one-dimensional non-local beam models (Lu et al. 2006a). Therefore, the non-localgeneralized force components of the non-local plate theories for general boundaryfactor Rnm. Therefore, the properties of Rnm are of interest for the examplespresented herein. Figure 1 shows the variations of R11 with respect to e0a=l 1 andl1/l2. It can be seen that R11 decreases rapidly with increasing e0a=l 1 for allaspect ratios l1/l2. This means that, for very small-sized plate-like structures inMEMS or NEMS, in which the size effect becomes signicant, the frequencyproperties predicted using the local plate theories are considerably over-estimated. On the other hand, it can be seen from gure 2 that the decreasingrate of R11 is slightly increased with increasing aspect ratios l1/l2. For higherorder frequencies, the changes of the corresponding parameters Rnm have similartrends as shown in gure 1, and are plotted in gure 2 for aspect ratios l1/l2Z1and 0.4. Some numerical values are given in table 1. It can be observed that thenon-local effects have more signicant inuences on the higher order frequencies.For instance, for e 0a=l 1Z0:2, the frequencyu

N11 drops by approximately 20%while

uN33 drops by approximately 60% when compared with the frequencies obtainedfrom the local plate theories.On the other hand, the solutions for the simply supported plates given in

equations (4.9) and (4.26) show that the displacements obtained by the non-localplate models are larger than those predicated by the local plate theories. Thisimplies that the non-local effects soften the structures, and make them moreexible. These mechanical properties for the structures in micro- and nanoscalesshould be taken into consideration in design and fabrication of MEMS/NEMSProc. R. Soc. A (2007)

Reddy, J. N. 2007 Nonlocal theories for bending, buckling and vibration of beams. Int. J. Engng.

3239Non-local elastic plate theories

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from Sci. 45, 288307. (doi:10.1016/j.ijengsci.2007.04.004)Soedel, W. 1993 Vibrations of shells and plates, 2nd edn. New York, NY: Marcel Dekker.Sudak, L. J. 2003 Column buckling of multiwalled carbon nanotubes using nonlocal continuum

mechanics. J. Appl. Phys. 94, 72817287. (doi:10.1063/1.1625437)Sun, C. T. & Zhang, H. T. 2003 Size-dependent elastic moduli of platelike nanomaterials. J. Appl.

Phys. 93, 12121218. (doi:10.1063/1.1530365)Tighe, T. S., Worlock, J. M. & Roukes, M. L. 1997 Direct thermal conductance measurements on

suspendedmonocrystalline nanostructures.Appl. Phys. Lett. 70, 26872689. (doi:10.1063/1.118994)Wang, L. F. & Hu, H. 2005 Flexural wave propagation in single-walled carbon nanotubes. Phys.

Rev. B 71, 195412. (doi:10.1103/PhysRevB.71.195412)Wang, C. M., Zhang, Y. Y., Sai, S. R. & Kitipornchai, S. 2006 Buckling analysis of micro- and

nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D 39, 39043909. (doi:10.1088/0022-3727/39/17/029)value problems should be determined by the relations (2.11) or (2.18), in whichthe three-dimensional non-local kernel satisfying the relations (2.4) and (2.7) isgiven by

ajxjZ 4pl 2t2K1jxjK1expKjxj=lt; tZ e 0a=l 5:1and the two-dimensional non-local kernel satisfying the relations is given by

ajxjZ 2pl 2t2K1K0jxj=lt; tZ e 0a=l; 5:2where K0 is the modied Bessel function and l is a characteristic length of theconsidered structure. For more examples of different boundary value problemsbased on the non-local elasticity models, refer to Eringen (2002).

References

Bunch, J. S., van der Zande, A. M., Verbridge, S. S., Frank, I. W., Tanenbsum, D. M., Parpia,J. M., Craighead, H. G. & McEuen, P. L. 2007 Electromechanical resonators from graphenesheets. Science 315, 490493. (doi:10.1126/science.1136836)

Eringen, A. C. 1983 On differential equations of nonlocal elasticity and solutions of screwdislocation and surface waves. J. Appl. Phys. 54, 47034710. (doi:10.1063/1.332803)

Eringen, A. C. 2002 Nonlocal continuum eld theories. New York, NY: Springer.Evoy, S., Carr, D. W., Sekaric, L., Olkhovets, A., Parpia, J. M. & Craighead, H. G. 1999

Nanofabrication and electrostatic operation of single-crystal silicon paddle oscillations. J. Appl.Phys. 86, 60726077. (doi:10.1063/1.371656)

Freund, L. B. & Suresh, S. 2003 Thin lm materials. Cambridge, UK: Cambridge University Press.Kroner, E. 1967 Elasticity theory of materials with long-range cohesive forces. Int. J. Solids Struct.

3, 731742. (doi:10.1016/0020-7683(67)90049-2)Leissa, A. W. 1973 The free vibration of rectangular plates. J. Sound Vib. 31, 257293.Lobontiu, N., Ilic, B., Garcia, E., Reissman, T. & Craighead, H. G. 2006 Modeling of

nanofabricated paddle bridges for resonant mass sensing. Rev. Sci. Instrum. 77, 073301.(doi:10.1063/1.2221560)

Lu, P., Lee, H. P., Lu, C. & Zhang, P. Q. 2006a Dynamic properties of exural beams using anonlocal elasticity model. J. Appl. Phys. 99, 073510. (doi:10.1063/1.2189213)

Lu, P., He, L. H., Lee, H. P. & Lu, C. 2006b Thin plate theory including surface effects. Int.J. Solids Struct. 43, 46314647. (doi:10.1016/j.ijsolstr.2005.07.036)

Lu, P., Lee, H. P., Lu, C. & Zhang, P. Q. 2007 Application of nonlocal beam models for carbonnanotubes. Int. J. Solids Struct. 44, 52895300. (doi:10.1016/j.ijsolstr.2006.12.034)

Peddieson, J., Buchanan, G. R. & McNitt, R. P. 2003 Application of nonlocal continuum models tonanotechnology. Int. J. Eng. Sci. 41, 305312. (doi:10.1016/S0020-7225(02)00210-0)Proc. R. Soc. A (2007)

Xu, M. 2006 Free transverse vibrations of nano-to-micron scale beams. Proc. R. Soc. A 462,29772995. (doi:10.1098/rspa.2006.1712)

Zalalutdinov, M. K., Baldwin, J. W., Marcus, M. H., Reichenbach, R. B., Parpia, J. M. & Houston,B. H. 2006 Two-dimensional array of coupled nanomechanical resonators. Appl. Phys. Lett. 88,143504. (doi:10.1063/1.2190448)

Zhang, H. T. & Sun, C. T. 2004 Nanoplate model for platelike nanomaterials. AIAA J. 42,20022009.

Zhang, Y. Q., Liu, G. R. & Xie, X. Y. 2005 Free transverse vibration of double-walled carbonnanotubes using a theory of nonlocal elasticity. Phys. Rev. B 71, 195404. (doi:10.1103/PhysRevB.71.195404)

P. Lu et al.3240

on April 22, 2012rspa.royalsocietypublishing.orgDownloaded from Proc. R. Soc. A (2007)