An analytical solution to the bending problem of micro-plate ...scientiairanica.sharif.edu/article_21278_493cab9ad68837f...sical continuum theory of elasticity is a theoretical, idealized,

Scientia Iranica F (2019) 26(3), 1997{2006

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologyhttp://scientiairanica.sharif.edu

An analytical solution to the bending problem ofmicro-plate using a new displacement potential function

F. Yakhkeshi and B. Navayi Neya�

Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran.

Received 13 July 2018; received in revised form 1 January 2019; accepted 23 February 2019

KEYWORDSMicromechanics;Couple stress theory;Length-scaleparameter;Rectangular plate;Displacementpotential function.

Abstract. In this paper, to include small-scale e�ect, the augmented Love DisplacementPotential Functions (DPF) are developed for isotropic micro or nano-scale mediums basedon couple stress theory. By substituting the new DPF into the equilibrium equations, thegoverning equations were simpli�ed to two linear partial di�erential equations of sixth andfourth orders. Then, the governing di�erential equations were solved for simply supportedrectangular plate using the separation of variable method by satisfying exact boundaryconditions without any simpli�cation assumptions. Displacements, bending and torsionalmoments of rectangular plate were obtained for di�erent length-scale parameters and aspectand Poisson's ratios. The obtained results were compared with other studies, which showedexcellent agreement.© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, thin beams and plates are widelyused in nanoelectromechanical systems (NEMS),microelectromechanical systems (MEMS) and mi-cro/nanotechnologies. Hence, investigating the me-chanical behavior of such components has always beenan important issue among the researchers. The clas-sical continuum theory of elasticity is a theoretical,idealized, and simpli�ed model with limited ability topredict behaviors of solid body in micro scale. Forinstance, based on the classical theory, stress concen-tration factor around a circular hole in an in�nite plateis independent of the radius of the hole [1]. Also,in this theory, it is assumed that all material bodiesare continuous and therefore, the laws of motion andpostulates of constitution are valid for any part of

*. Corresponding author.E-mail addresses: [email protected] (F.Yakhkeshi); [email protected] (B. Navayi Neya)

doi: 10.24200/sci.2019.51264.2087

the body regardless of their dimensions [2]. On thecontrary, any material body consists of granular orporous materials. Experimental studies such as thoseof McFarland and Calton [3] have indicated that theclassical continuum mechanics is incapable of justifyingthe size dependency in micro or nano scales due tothe lack of material length-scale parameter. In addi-tion, micro structure e�ects become important whenthe thickness of the structure is extremely low andcomparable to internal length scales of its materials [4].As a result, a number of theories comprising materiallength-scale parameters have been proposed. Thesetheories are known to bridge between macroscopicand microscopic physics [2]. Voigt, who was one ofthe pioneers in formulating the theory of elasticity,introduced couple stresses mij in addition to the forcestresses �ij [5]. In 1909, Cosserat brothers developedthe �rst mathematical model using couple stress theory.Based on their theory, during loading on a body madeof elastic material, the body not only is displaced, butalso rotates independently, meaning that three extrafreedoms need to be considered for any element [5,6].Despite its novelty, Cosserat brothers' theory remained

1998 F. Yakhkeshi and B. Navayi Neya/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 1997{2006

unknown during their lifetime. Afterwards, in 1968,Eringen introduced the micropolar theory based onthe Cosserat theory. Another higher-order contin-uum theory is the couple stress theory developed byToupin [7], Mindlin and Tiersten [8], Koiter [9], andNowacki [10]. The couple stress theory is a specialform of the micropolar theory in which rotation vectorsfor microstructure and macrostructure are consideredto be equal [11]. Hence, rotation components areobtained based on displacements [8]. In couple stresstheory, in addition to two Lame constants, there aretwo material length-scale parameters for an isotropicelastic material [12]. Yang et al. [13] utilized anadditional equilibrium equation for moments of couplesas well as two classical equilibrium equations in orderto describe the behavior of the couples. Thus, thecouple stress tensor was constrained to be symmetricand the two material length scales were reduced to onlyone. Yang's studies are known as the modi�ed couplestress theory. The modi�ed couple stress theory hasbeen used in some researches for static and dynamicproblems. Park and Gao [14], Ma et al. [15], Kong etal. [12] and Ke and Wang [16] stated that in staticanalysis, the displacement obtained by the modi�edcouple stress theory was smaller than that by theclassical theory. This implies that the size dependencyreduces de ection and increases bending rigidity. Also,when size e�ect is applied to the dynamic behavior ofmicrostructures, natural frequencies of free vibrationpredicted by the modi�ed couple stress theory arehigher than those by the classical model [15,17-23].

Microplates can be considered as important partsof various microsystems. Therefore, the static and dy-namic behaviors of the microplate have been studied bymany researchers [24-28]. Tsiatas [17] developed a newKirchho� plate method for static analysis of isotropicmicro-plates based on the modi�ed couple stress theorywith only one material-length scale parameter. Theproposed model was capable of analyzing plates withcomplex geometries and boundary conditions.

Even after the introduction of the modi�ed couplestress theory, the couple stress theory was being studiedin parallel. Asghari et al. [11] showed that from thetwo existing material length-scale parameters in theconstitutive equations of the couple stress theory, onlyone of them would appear. Fathalilou et al. [29]showed that bending rigidity of couple stress theorywas closer than that of modi�ed couple stress theoryto the experimental results.

Due to indeterminacy of the spherical part of thecouple-stress tensor and appearance of body couple inthe constitutive relations of force-stress tensor, formu-lation of the couple stress theory is more di�cult thanclassical theory [8,30]. Hadjesfandiari and Dargush [31]developed the consistent couple stress theory for solids.They introduced the skew-symmetric couple stress ten-

sor by utilizing the principle of virtual work and somekinematical considerations. It is necessary to pointout that Mindlin and Tiersten [8] derived the generalsolution to the displacement in isotropic elasticitywithin the context of the indeterminate couple stresstheory. Their formulation included two couple stressparameters, which limited its application. Hadjesfan-diari and Dargush [31] showed that these coe�cientswere not independent. They presented the governingequation with only one length scale parameter.

Utilizing potential functions is a suitable methodto simplify and solve elasticity problems. Hence,several potential functions have been developed forelasticity problems within both displacement and stressformulations. Displacement formulation is more com-mon for 3D elasticity problems. There are somedisplacement potential functions such as Helmholtz de-composition, Galerkin vector, Papkovich-Neuber func-tions, Love's strain function, and Lame's strain po-tential function that can be used for isotropic me-dia [1]. It is to be noted that Love [32] originallyderived his solutions expressively for the axisymmet-ric case. Love's displacement potential function wasthen extended to non-axisymmetric irrotational prob-lems [33]. In honor of Love, a non-axisymmetricDPF, named the non-axisymmetric Love solution, andits completeness were investigated by Tran-Cong [33]for a simply connected and z-convex smooth surface.In non-axisymmetric augmented Love DPF, irrota-tional limitation is omitted by adding the secondfunction in displacement vectors. Completeness ofaugmented Love DPF was investigated for elastostaticsand elastodynamics problems by Tran-Cong [33], Pakand Eskandari-Ghadi [34], and Eskandari-Ghadi andPak [35]. One of the useful displacement poten-tial functions is Lekhnitskii-Hu-Nowacki, proposed fortransversely isotropic materials. Lekhnitskii presentedparticular potential functions for the general solutionto axisymmetric problems in transversely isotropicmaterials that were the generalization forms of Lovesolution [1]. Hu (1953) and Nowacki (1954) developedthe Lekhnitskii solution to a general case of elastostaticproblems for transversely isotropic media [36]. Wangand Wang [37] proved that Lekhnitskii-Hu-Nowacki'ssolution was complete for elastostatic problems intransversely isotropic media. Afterwards, Eskandari-Ghadi [36] generalized the Lekhnitskii-Hu-Nowacki so-lution to dynamic problems. They successfully ap-plied augmented Love, Lekhnitskii-Hu-Nowacki, andEskandari-Ghadi displacement potential functions tobending, buckling, and vibration of isotropic and trans-versely isotropic simply supported thick rectangularplates problems [38,39].

In this paper, the augmented Love displacementpotential functions are developed to take into accountthe small e�ect of micro and nano mediums. By

F. Yakhkeshi and B. Navayi Neya/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 1997{2006 1999

substituting the new DPF in equilibrium equations, thesixth- and fourth-order partial di�erential equationsare obtained as the governing equations. Then, thegoverning equations are solved for simply supportedrectangular plate by separation of the variable method,satisfying exact boundary conditions. The presentedmethod is exact and applicable to all isotropic platesof the problem with any thickness ratio, becausethere is no simplifying assumption in the procedure,displacement or stress components, thickness ratio, andlinear elastic material properties.

2. The governing equations

In continuum mechanics, for an arbitrary part ofthe material continuum with volume, V , enclosed byboundary surface, S, the linear and angular equilibriumequations for the couple stress theory are written as [1]:

�ji;j + Fi = 0; (1)

�ji;j + "ijk�jk + Ci = 0; (2)

where �ji and �ji are force stress and couple stress ten-sors, respectively; Fi and Ci are body force and coupleper unit volume, respectively; and "ijk is permutationsymbol. Here, the subscript denotes di�erentiationwith respect to spatial variables.

Writing force-stress tensor in terms of displace-ments and substituting it into the linear equilibriumequation, Eq. (3) is obtained. In this paper, we use theliner equilibrium equation derived by Hadjesfandiariand Dargush [31], which is a di�erent version of theMindlin and Tiersten [8] equation:�

�+�+�r2�r (r:~u)+�� r2�r2~u+ F =0; (3)

where ~u is the displacement vector of the continuummaterial, � and � are the Lame constants, �ij is theKronecker delta, and � is an extra material constantde�ned in couple stress theory.

In Eq. (3), ~u is of class C4 on R3 if u is continuouson R3 and its position derivative up to the 4th order isat each point in R3, where R is the set of real numbers.A solution, u, to Eq. (3) of class C4 on R3 will bereferred to as an elastic response in micro scale of anisotropic material.

Using the energy and the constitutive relations,the force stress tensor and couple stress tensor can bewritten as [30]:

�ji =�ekk�ij + 2�eij + 2�r2!ji = �uk;k�ij

+ �(ui;j + uj;i)� �l2r2(ui;j � uj;i); (4)

�ji = 4�(!i;j � !j;i); (5)

where ui is displacement vector component of thecontinuum material, and eij and !ij are strain androtation tensors and can be written as:

eij =12

(ui;j + uj;i); (6)

!ij =12

(ui;j � uj;i): (7)

The operator r is de�ned as:

r =@@x~i+

@@y~j +

@@z~k: (8)

Also, � can be represented in terms of length scaleparameter, l, as [31]:��

= l2: (9)

3. Displacement potential functions

In non-axisymmetric Love DPF, only one function isused, i.e., F (see Eqs. (10)-(12)), and for rotation-free augmented Love DPF, function � is added. Thecompleteness of augmented Love potential functions, asmentioned before, has been investigated for many elas-todynamics and elastostatics mechanics problems [33-35], but due to the lack of length scale parameter,the augmented Love displacement potential functionsare not suitable for solving of the micromechanicsproblems [40]. Hence, in this section, by developingthe augmented Love potential functions for isotropicmaterials, we introduce new displacement potentialfunctions that include length scale parameter.

u = �(�1 + l2r2)@2F@x@z

� @�@y; (10)

v = �(�1 + l2r2)@2F@y@z

+@�@x

; (11)

w = �2

�@2F@x2 +

@2F@y2

�+ (1� l2r2)

@2F@z2 ; (12)

where u, v, and w are displacement components inx, y, and z directions, respectively; F and � aredisplacements potential functions, respectively; and �1and �2 are de�ned as:

�1 =�+ ��

; �2 =�+ 2��

: (13)

Substituting the displacement potential functions intothe governing equation (3) with some algebraic manip-ulations, the governing equations can be written as:

�(1� l2r2)@@yr2� = 0; (14)

(1� l2r2)@@xr2� = 0; (15)

(1� l2r2)r4F = 0: (16)

Based on the �rst two equations of this system, one


may de�ne a vector function, namely G, as G =((1 � l2r2)r2�)ez, where ez is the unite vector in z-direction. Then, the �rst two equations are translatedto curl G = 0. The solution to this equation is notunique [41] and, in this problem, it is enough to bezero, meaning that:

(1� l2r2)r2� = 0; (17)

where F (x; y; z) is a class C6 �eld and �(x; y; z) is aclass C4 �eld on R3.

It can be seen that by eliminating the lengthscale parameter (l), Eqs. (16) and (17) are simpli�edto the classical form [39]. The proposed displacementpotential functions and governing equations are validfor any isotropic medium. In the following, theseequations are solved for a simple rectangular plate.

4. Application to plates

The governing di�erential equations (16) and (17) aresolved here for a simply supported rectangular platewith thickness of t, length of a, and width of b (seeFigure 1).

As shown in Figure 1, an x, y, z Cartesiancoordinates system is used, where the z-axis is alongthe plate thickness and q is a static lateral load appliedto upper surface of plate and written in the form of theFourier series as:

q(x; y) =Xm

Xn

qmn sin�m�ax�

sin�n�by�; (18)

where n and m are integers, and qmn is the Fourierloading coe�cient written as [42]:

qmn=4ab

aZ0

bZ0

qz(x; y): sin�m�ax�

sin�n�by�dxdy;

(19)

where qz(x; y) is static lateral load.

4.1. Boundary conditionsFor a simply supported rectangular plate, the boundaryconditions on edges can be written as:w = 0; Mx = 0 at x = 0; a; (20)

w = 0; My = 0 at y = 0; b; (21)

where bending moments Mx and My are de�ned as:

Mx =Z t

2

� t2�xzdz; My =

Z t2

� t2�yzdz: (22)

The boundary conditions on the upper (�t=2) andlower (+t=2) surfaces of the plate can be represented as:

At z = � t2 :

�zz = �q; (23a)

�zx = 0; (23b)

�zy = 0; (23c)

�zx = 0; (23d)

�zy = 0: (23e)

At z = + t2 :

�zz = 0; (24a)

�zx = 0; (24b)

�zy = 0; (24c)

�zx = 0; (24d)

�zy = 0: (24e)

5. Solution procedure

Using the method of separation of variables, Eq. (16)can be solved for the potential function, F , in terms ofthree scalar functions, f , g, and h, which are functionsof x, y, and z, respectively.

F (x; y; z) = f(x)� g(y)� h(z): (25)

Substituting Eq. (25) into Eq. (16), by satisfyingthe displacement boundary conditions, i.e., Eqs. (20)

Figure 1. Geometry and loading of a plate in xyz Cartesian coordinates.


and (21), f and g are determined as:

f = C 01 sinm�ax; (26)

g = C 02 sinn�by; (27)

where C 01 to C 02 are constant coe�cients. Parameters�m, �n, and �mn are de�ned as:

�m =m�a; �n =

n�b; (28)

�mn =p

(�m)2 + (�n)2: (29)

Substituting f and g into Eq. (16), the governingdi�erential equation is obtained in terms of h:

d6hdz6 (�l2) +

d4hdz4 (1 + 3�2

mnl2)

+d2hdz2 (�2�2

mn � 3�4mnl

2)

+ h(�4mn + �6

mnl2) = 0: (30)

The solution to Eq. (30) can be written as:

h =c03 sinh�mnz + c04 cosh�mnz + c05z sinh�mnz

+ c06z cosh�mnz + c07z sinh (p�1 � z)

+ c08z cosh (p�1 � z) ; (31)

where �1 = 1l2 + �2

mn.Introducing new coe�cients C1 to C6, the poten-

tial function F can be rewritten as:

F =Xm

Xn

"�C1 sinh�mnz + C2 cosh�mnz

+ C3z sinh�mnz + C4z cosh�mnz

+ C5 sinh (p�1 � z) + C6 cosh (

p�1 � z)�

� sinm�ax sin

n�by

#: (32)

Similar to F , the potential function � can be derivedas:�(x; y; z) = f1(x)� g1(y)� h1(z): (33)

Substituting Eq. (33) into Eq. (17) and using theseparation of variables technique, by satisfying themoment boundary conditions, i.e., Eqs. (20) and (21),f1 and g1 are speci�ed as

f1 = A01 cosm�ax; (34)

g1 = A02 cosn�by: (35)

Substituting f1, i.e., Eq. (34), and g1, i.e., Eq. (35), into

Eq. (17), the governing di�erential equation is obtainedin terms of h1 as:d4h1

dz4 (l2)� d2h1

dz2 (1+2�2mnl

2)+h1�2mn(1+�2

mnl2)=0:

(36)

Eq. (36) can be solved as:

h1 =A03 sinh�mnz +A04 cosh�mnz

+A05 sinh (p�1 � z) +A06 cosh (

p�1 � z) : (37)

By introducing new coe�cients A1 to A4, the potentialfunction � is expressed as:

� =Xm

Xn

"�A1 sinh�mnz +A2 cosh�mnz

+A3 sinh (p�1 � z) +A4 cosh(

p�1 � z)�

� cosm�ax cos

n�by

#: (38)

To �nd the unknown coe�cients C1-C6 and A1-A4,the remaining boundary conditions must be satis�ed.Hence, the displacements components and boundaryconditions, Eqs. (23) and (24), are written in terms off , g, h and f1, g1, h1 as:

u = ��1f 0gh0 � l2r2f 0gh0 � f1g01h1; (39)

v = ��1fg0h0 � l2r2fg0h0 � f 01g1h1; (40)

w = �2f 00gh+ �2fg00h+ fgh00 � l2r2fgh00; (41)

�zx=��+

k�2�k�f 0gh00+ k

2

�� 1l2

+�2mn

�f 0gh

+k

�2mn

�� 1�2

+12

�f 0gh(4)

+�� k

2�2

�f1g01h01 +

�k

2�2�2mn

�f1g01h0001 ;

(42)

�zy=��+

k�2�k�fg0h00+ k

2

�� 1l2

+�2mn

�fg0h

+k

�2mn

�� 1�2

+12

�fg0h(4)

+��+

k2�2

�f 01g1h01+

�� k

2�2�2mn

�f 01g1h0001 ;

(43)

�zy =k�h0 � 1

�2mn

h000�f 0g

+k�2

�� 1�2mn

h001 + h1

�f1g01; (44)


�zx =k��h0 + 1

�2mn

h000�fg0

+k�2

�� 1�2mn

h001 + h1

�f 01g1; (45)

�zz =��2

mn � l2��4mn � k

l2

�fgh0

+��2 + l2��2

mn +k2

�fgh000

+�� k

2�2mn

�fgh(5); (46)

where k = 2�l2�2�mn2.Satisfying boundary conditions (23b), (23c) and

(24b), (24c) lead to the zero value of �. It is noticeablethat the length scale parameter remains although po-tential function � is eliminated. Thus, the coe�cientsC1 to C6 should be obtained based on boundaryconditions, i.e., Eqs. (23) and (24). Eqs. (23b), (23c)and (24b), (24c) can be rewritten as:�� k +

k�2

�h00 + k

2

�� 1l2

+ �2mn

�h

+k

�2mn

�� 1�2

+12

�h(4) = 0 at z = � t

2:(47)

Similarly, �zx and �zy can be written as:

At z = � t2 :

�zx = k��h0 + 1

�2mn

h000�fg0 = 0; (48)

�zy = k�h0 � 1

�2mn

h000�f 0g = 0: (49)

It can be seen that Eqs. (48) and (49) are identical andcan be rewritten as:��kh0 + k

�2mn

h000�

= 0 at z = � t2: (50)

Thus, the previous boundary conditions are reducedto 6 boundary conditions, namely force stresses �zz,

Eqs. (23a) and (24a); shear stresses, Eqs. (23c)and (24c); and couple stresses, Eqs. (23e) and (24e).Therefore, there is a system of 6 independent equationsin which 6 constants C1 to C6 remain unknown.

Using MATLAB software, the 6 constant coef-�cients and, consequently, the displacement potentialfunction F can be determined. Having F , the displace-ments, strains, force stresses, couple stress tensors, andshear forces, bending and twisting moments can bedetermined.

6. Numerical examples

In this section, it is shown that how the new displace-ment potential functions can be employed in order tosolve the real-case problems. The results obtained byutilizing the new displacement potential functions arecompared to the classical data as well as the outcomesof the previous study.

6.1. Veri�cationsVeri�cation of the proposed displacement potentialfunctions is shown in the following by eliminatinglength scale parameter (l) in the governing equations(Eqs. (16) and (17)). Therefore, the equations aresimpli�ed to the classical form [39], demonstrating thevalidity of the new displacement potential functions.

To compare the results, non-dimensional displace-ment of the mid-plane is introduced as �w = wD=qa4,in which D is rigidity of plate (D = (Et3=12(1� v2))),and non-dimensional bending moments in x and ydirections are introduced as �Mx = Mx=qa2 and �My =My=qa2.

To verify the accuracy of the Presented Work(PW), the central non-dimensional de ection ( �wc) andcentral non-dimensional bending moment in x and ydirections ( �Mxc and �Myc) are compared with theirexact classical solutions, namely �wcc, �M c

xc, and �M cyc.

Considering t = 0:01 in contrast to l, which is in microrange, the l=t ratio becomes approximately zero. Withthe elimination of the l=t ratio, the solutions becomesimilar to those to the classical method for thin plate.Thus, �wc, �Mxc, and �Myc are adapted as �wcc, �M c

xc,and �M c

yc, demonstrating the accuracy of the modeldeveloped in this study [43]. According to Table 1,the results of the central de ection in rectangular platefor 4 di�erent aspect ratios are in excellent agreement

Table 1. Comparison of �wcc and �wc, �Mcxc and �Mxc , and �Mc

yc and �Myc in the rectangular plate for various aspect ratios.

b=a �wcc �wc �Mcxc

�Mxc�Mcyc �Myc

1 0.00406 0.00406 0.047900 0.048236 0.047900 0.0482362 0.01013 0.01014 0.101700 0.102293 0.046400 0.0474493 0.01223 0.01227 0.118900 0.119985 0.0406000 0.0427264 0.01282 0.01292 0.123500 0.125377 0.038400 0.041458


Table 2. Comparison of normalized central de ections of square micro-plate in the PW and Tsiatas results versus (l=t)2

for 3 di�erent Poisson's ratios.

(l=t)2wc=wcc

v = 0:25 v = 0:3 v = 0:35PW Tsiatas PW Tsiatas PW Tsiatas

0 1 1 1 1 1 10.05 0.648 0. 817 0. 655 0.825 0.684 0.850.10 0.516 0.69 0.53 0.7 0.547 0.720.20 0.415 0.53 0.43 0.545 0.45 0.565

with those obtained by the analytical solutions ofTimoshenko and Krieger [43]. Also, the comparisonof �Mxc and �Myc with �M c

xc and �M cyc is summarized in

Table 1.Also, the presented results in Table 1 show that

the central non-dimensional bending moments in rect-angular plate for the 4 aspect ratios are consistentwith the solutions for the classical method. The slightdi�erence is due to the length scale parameter.

Table 2 shows the normalized central de ection,wc=wcc, for a square plate versus (l=t)2 for 3 di�erentPoisson's ratios (v = 0:25; 0:3; 0:35) of PW and Tsiatasresult.

According to Table 2, the de ection of the platedecreases by the increment of (l=t). This is similarto the results of Tsiatas, in which the normalizedde ection decreased with the increase in (l=t). Also,it is observed that the de ection of the micro-plateslightly decreases with increase in the Poisson's ratio.

6.2. Results and discussionThe results for the analysis of micro plates underuniformly distributed load have been presented inFigures 2-6. Here, the following assumptions have beenmade: Young's modulus: E = 2:1 � 1011 N/m2 andPoisson's ratio: v = 0:3.

Figure 2 shows non-dimensional displacements of

Figure 2. Non-dimensional displacement of the squarenano-plate at y = (1=2)b for various values of l=t.

the simply supported square plate (b=a = 1) on the liney = (1=2)b for various length-to-thickness, l=t, ratios.

According to Figure 2, displacement of platedecreases with increase in l=t and it is always smallerthan the classical de ection. This can be justi�ed withrigidity of microplate de�ned as D = (Et3=12(1�v2))+�tl2. Rigidity of micro plate is the sum of bendingrigidity of the plate and the contribution of rotationgradients to the bending rigidity [17]. By consideringlength scale parameter, the rigidity of couple stresstheory increases. Hence, the displacement of platedecreases. It is noticeable that the displacementsestimated here are less than those estimated by theclassical theory and are adapted to Tsiatas results.In both Tsiatas model and PW, the displacementtreatments are equal and for the same value of l=t , thedisplacement for PW is less than that for the Tsiatasmodel. It is noticeable that in the couple stress theory,size e�ect is applied using the l=t ratio.

The e�ect of the aspect ratio, b=a, on the centraldisplacement of the rectangular microplates for variousl=t ratios is illustrated in Figure 3.

The results indicate that by the increasing the as-pect ratio (b=a), the central displacement will increasefor di�erent l=t ratios. Also, for the ratios of b=a =3 and 4, the displacements are approximately equal.

Figure 3. Non-dimensional maximum displacement forvarious values of l=t and b=a.


Figure 4. Non-dimensional bending moment ( �Mxc) fordi�erence values of l=t and b=a.

Figure 5. Non-dimensional bending moment ( �Myc) forvarious values of l=t and b=a.

Such behavior can be justi�ed for an in�nite lengthstripe. Figure 3 also shows that by increasing l=t,the central displacement of rectangular plate alwaysdecreases nonlinearly. This response is similar to thatobtained by Tsiatas [17], and Papargyri-Beskou andBeskos [4].

Maximum bending moment of the rectangularplate in x direction (Mx) depends on the b=a ratio.In this regard, �Mxc for di�erent l=t and b=a ratiosunder uniform distributed load is determined. Figure 4shows the variations of �Mxc with respect to di�erentratios of l=t and di�erent plate dimensions of b=a. Itis observed that, unlike for l=t > 0:5, for small l=tratios, with increase in b=a, the maximum bendingmoments increase. In addition, it is observed that as l=tincreases, thickness of the plate becomes smaller thanlength scale parameter and �Mxc shows convergencebehavior for di�erent b=a ratios.

The variation of non-dimensional maximum mo-ment in y direction ( �Myc) is depicted in Figure 5. Thebehavior of �Myc is similar to that of �Mxc with respect

Figure 6. Distribution of non-dimensional torsionalmoment �Mxy at x = a for various values of l=t.

to the variations of l=t. In contrast, when b=a increases,�Myc slightly decreases and the convergence between the

results improves for all the values of b=a. The twistingmoment Mxy can be written as:

Mxy =Z t

2

� t2�xyzdz: (51)

In order to investigate the in uence of twisting mo-ment, distribution of non-dimensional twisting mo-ment, �Mxy = Mxy=qa2, in x = a for the square plate isshown in Figure 6. The �gure shows that maximum�Mxy occurs at the edges of the plate and increases

by decreasing l=t. Also, the di�erences between thetwisting moments are reduced as the l=t ratio increases.

7. Conclusions

In this paper, the augmented Love displacement poten-tial functions were developed for mediums of isotropicmicro and nano scales based on couple stress theory.Using the new DPF for an arbitrary isotropic andhomogeneous medium in micro or nano scale led tothe formation of di�erential equations in the form oftwo independent partial di�erential equations of sixthand fourth orders. These equations were solved formicro rectangular plates using the method of sepa-ration of variables. In contrast with the results ofthe previous researches, the proposed solution did notrequire simplifying assumptions, especially for shearstress or strain in thickness of plate. The numericalresults of this study showed that with the reduction inplate thickness and by considering length parameter,the sti�ness of the plate increased and the plate dis-placement decreased, compared with the classic theory.Moreover, it was observed that in the couple stresstheory for rectangular plates, with increase in b=a, theplate maximum displacement, wc, and the maximummoment, Mxc, increased, while the maximum momentMyc of the plate decreased. In addition, wc, Mxc, andMyc decreased by increasing l=t. In the investigationinto torsional moment, it was concluded that when the


length parameter increased, Mxy on a simply supportededge decreased and the convergence of the values ofMxy improved.

Acknowledgment

This work was supported by Babol Noshirvani Univer-sity of Technology through Grant No. 91/13150490.This support is gratefully acknowledged.

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Biographies

Fatemeh Yakhkeshi is a PhD candidate in theDepartment of Civil Engineering at Babol NoshirvaniUniversity of Technology (NIT). She received her BSand MSc degrees in Civil Engineering from BabolNoshirvani University of Technology in 2012 and 2014,respectively. Her MSc thesis was entitled \Analysis ofplates in micro and nano scales based of couple stresstheory." Her research interests include micromechan-ics, fracture mechanics, and plates analysis.

Bahram Navayi Neya is an Associate Professor inthe Department of Civil Engineering at Babol Noshir-vani University of Technology (NIT). His researchinterests include theory of plate and shell, theory ofelasticity, dynamic analysis, and concrete dams.

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