Composite Structures 139 (2016) 13–29

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Nonlocal third-order shear deformation theory for analysis of laminatedplates considering surface stress effects

http://dx.doi.org/10.1016/j.compstruct.2015.11.0680263-8223/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: rajagopal@iith.ac.in (A. Rajagopal).

Piska Raghu a, Kasirajan Preethi a, Amirtham Rajagopal a,⇑, Junuthula N. Reddy b

aDepartment of Civil Engineering, IIT Hyderabad, IndiabDepartment of Mechanical Engineering, Texas A&M University, College Station, TX, USA

a r t i c l e i n f o

Article history:Available online 12 December 2015

Keywords:Laminated compositesThird order shear deformationSurface stressNonlocal theoryAnalytical solutions

a b s t r a c t

In this work, analytical solutions are presented for laminated composite plates using a nonlocal third-order shear deformation theory considering the surface stress effects. The theory is based on Eringen’stheory of nonlocal continuum mechanics (Eringen and Edelen, 1972) and the third-order plate theoryof Reddy (1984, 2004). The mathematical formulation for surface stress is based on Gurtin andMurdoch’s work (Gurtin and Murdoch, 1975, 1978). Analytical solutions of bending and vibration of sim-ply supported laminated and isotropic plates are presented using new formulation to illustrate the effectsof nonlocality and surface stress on deflection and vibration frequencies for various span-to-thicknessratios (a=h).

� 2015 Elsevier Ltd. All rights reserved.

1. Background

In modeling micro and nano structures, where material sizeeffects are prominent (e.g., study of elastic waves when dispersioneffect is taken into account and the determination of stress at thecrack tip when the singularity of the solution is of concern), con-ventional theories cannot model the material behavior accurately.There has been considerable focus in recent yeas towards thedevelopment of generalized continuum theories that account forthe inherent micro-structure in natural and engineering materials(see [6–9]). The notion of generalized continua unifies severalextended continuum theories that account for such sizedependence due to the underlying micro-structure of the material.A systematic overview and detailed discussion of generalized con-tinuum theories has been given by Bazant and Jirasek [10]. Thesetheories can be categorized as gradient continuum theories (seethe works by Mindilin et al. [11–13], Toupin [14], Steinmannet al. [7,15–17], Casterzene et al. [18], Fleck et al. [19,20], Askeset al. [21–23]), micro continuum theories (see Eringen[24,25,6,26,1], Steinmann et al. [27,28]), and nonlocal continuumtheories (see works by Eringen [26], Jirasek [29,30], Reddy[31–34] and others [35]). Recently, the higher-order gradient the-ory for finite deformation has been elaborated (for instance see[36–38,18,39]) within classical continuum mechanics in thecontext of homogenization approaches. A comparison of various

higher-order gradient theories can be found in [19]. A moredetailed formulation of gradient approach in spatial and materialsetting has been presented in [27].

Nonlocality of the stress–strain relationship introduces lengthscale at which classical elasticity theories are inadequate in model-ing the response. Classical theory is inherently size independent.The nonlocal formulations can be of integral-type formulationswith weighted spatial averaging or by implicit gradient modelswhich are categorized as strongly nonlocal, while weakly nonlocaltheories include, for instance, explicit gradient models [10]. Thenonlocality arises due to the discrete structure of matter and thefluctuations in the inter-atomic forces. The two dominant physicalmechanisms that lead to size dependency of elastic behavior at thenanoscale are surface energy effects and nonlocal interactions [40].Recently, various beam theories (e.g., Euler–Bernoulli, Timoshenko,Reddy, and Levinson beam theories) were reformulated usingEringen’s nonlocal differential constitutive model by Reddy [31],and analytical solutions for bending, buckling, and natural vibra-tions for isotropic plates were also presented. Various shear defor-mation beam theories were also reformulated by Reddy [31] usingnonlocal differential constitutive relations of Eringen. Subse-quently, similar works have been carried out by Aydogdu [32]and Civalek [33].

Nonlocal elastic rod models have been developed to investigatethe small-scale effect on axial vibrations of the nanorods byAydogdu [41] and Adhikari et al. [42]. Free vibration analysis ofmicrotubules based on nonlocal theory and Euler–Bernoulli beamtheory was carried out by Civalek et al. [33]. Free vibration analysis

14 P. Raghu et al. / Composite Structures 139 (2016) 13–29

of functionally graded carbon nanotubes using the Timoshenkobeam theory has been studied, and numerical solutions wereobtained using the Differential Quadrature Method (DQM) byJanghorban et al. [43] and others (see [44–46]). Eringen’s nonlocalelasticity theory has also been applied to study bending, buckling,and vibration of nanobeams using the Timoshenko beam theory(see [47–50]). Numerical solutions were obtained using two differ-ent collocation techniques, global (RDF) and local (RDF-FD), withmulti-quadrics radial basis functions (see Roque et al. [51]). Staticdeformation of micro- and nano-structures were studied usingnonlocal Euler–Bernoulli and Timoshenko beam theories andexplicit solutions were derived for displacements for standardboundary conditions by Wang et al. [52–54]. Iterative nonlocalelasticity for classical plates has been presented in [55]. Thaiet al. [56] developed a nonlocal shear deformation beam theorywith a higher-order displacement field that does not require shearcorrection factors [57].

Analytical study on the nonlinear free vibration of functionallygraded nanobeams incorporating surface effects has beenpresented in [58–60]. The effect of nonlocal parameter,surface elasticity modulus, and residual surface stress on the vibra-tional frequencies of Timoshenko beam has been studied in[61,62]. The coupling between nonlocal effect and surface stresseffect for the nonlinear free vibration case of nanobeams has beenstudied in [63].

Some explicit solutions involving trigonometric expansions arealso presented recently for nonlocal analysis of beams [64]. A finiteelement framework for nonlocal analysis of beams is presented in arecent work by Sciarra et al. [65]. Size effects on elastic moduli ofplate like nanomaterials has been studied in [66]. Studies to under-stand thermal vibration of single wall carbon nanotube embeddedin an elastic medium using DQM has also been reported in [67].The recent studies has been towards the application of nonlocalnonlinear formulations for the vibration analysis of functionallygraded beams [68]. The effect of surface stresses on bending prop-erties of metal nanowires is presented in [69]. There has been someworks on transforming nonlocal approaches to gradient type for-mulations [70]. Semi analytical approach for large amplitude freevibration and buckling of nonlocal functionally graded beams hasbeen reported in [71]. Barretta et al. [72] derived a new variationalframe work following the gradient type nonlocal constitutive lawand a thermodynamic approach. Wang et al. [52] presented thescale effect on static deformation of micro- and nano-rods or tubesthrough nonlocal EulerBernoulli beam theory and Timoshenkobeam theory. Explicit solutions for static deformation of suchstructures with standard boundary conditions are derived. Huuet al. [56] based on the modified couple stress theory andTimoshenko beam theory examined static bending, buckling andfree vibration behaviors of size dependent functionally gradedsandwich micro beams [57].

These nonlocal laminated plate theories allow for the small-scale effect which becomes significant when dealing with microand nano laminated plate-like structures [31]. The nonlocalityarises due to the discrete structure of matter and the fluctuationsin the inter atomic forces [73]. In the case of plate like structures,when the width to thickness ratio of the plate becomes less than20, transverse shear stresses play a key role on the behavior ofthe plate. Various theories have been developed to take care ofshear strains into account such as first order shear deformationtheory (FSDT) by Mindlin, Third order shear deformation theory(TSDT) by Reddy [3], and other generalized higher order sheardeformation theories (e.g., see [74–80]). Lu et al. [81] proposed anon-local plate model based on Eringens theory of nonlocal contin-uum mechanics. The basic equations for the non-local Kirchhoffand the Mindlin plate theories are derived. Maranganti et al. [40]estimated nonlocal elasticity length scales for various classes of

materials like semiconductors, metals, amorphous solids, and poly-mers using a combination of empirical molecular dynamics andlattice dynamics. The effect of inter atomic forces is also studied.Farajpoura et al. [82] investigated the buckling response of ortho-tropic single layered graphene sheet subjected to linearly varyingnormal stresses using the nonlocal elasticity theory. The nonlocaltheory of Eringen and the equilibrium equations of a rectangularplate are employed to derive the governing equations. Differentialquadrature method (DQM) has been used to solve the governingequations for various boundary conditions.

Wang et al. [80] presented a large-deflection mathematicalanalysis of rectangular plates under uniform lateral loading. Theanalysis is based on solving two fourth-order, second-degree, par-tial differential von Krmn equations relating the lateral deflectionsto the applied load. Plates with two boundary conditions, namely,simply supported edges and held edges, are considered. Neveset al. [83] derived higher-order shear deformation theory for mod-eling functionally graded plates to account for extensibility in thethickness direction. Arash et al. [84] studied the application ofthe nonlocal continuum theory in modeling of carbon nano tubesand graphene sheets. A variety of nonlocal continuum models inmodeling of the two materials under static and dynamic loadingsare introduced and reviewed. The superiority of nonlocal contin-uum models to their local counterparts, the necessity of the cali-bration of the small-scale parameter and the applicability ofnonlocal continuum models are discussed. Yan et al. [85] appliednonlocal continuum mechanics to derive complete and asymptoticrepresentation of the infinite higher-order governing differentialequations for nano-beam and nano-plate models.

Wang et al. [86] presented elastic buckling analysis of micro-and nano-rods/tubes based on Eringens nonlocal elasticity theoryand the Timoshenko beam theory. Sun et al. [66] presented a semicontinuum model for nano structured materials that possess aplate like geometry such as ultra-thin films. This model accountsfor the discrete nature in the thickness direction. In-plane Youngsmodulus, and in-plane and out-plane Poissons ratios are investi-gated with this model. It is found that the values of the Youngsmodulus and Poissons ratios depend on the number of atomic lay-ers in the thickness direction and approach the respective bulk val-ues as the number of atom layers increases. Murmu et al. [87]solved vibration of double-nano beam-systems which are impor-tant in nano-optomechanical systems and sensor applications.Expressions for free bending-vibration of double-nano beam-system are established within the framework of Eringens nonlocalelasticity theory. The increase in the stiffness of the couplingsprings in double-nano beam-system reduces the nonlocal effectsduring the out-of-phase modes of vibration. Wang et al. [88] con-ducted study of the mechanisms of nonlocal effect on the trans-verse vibration of two-dimensional 2D nano plates, for example,mono layer graphene and boron-nitride sheets. It is found thatsuch a nonlocal effect stems from a distributed transverse forcedue to the curvature change in the nanoplates and the surfacestress due to the nonlocal atom-to-atom interaction. Using theprinciple of virtual work the governing equations are derived forrectangular nanoplates. Solutions for buckling loads are computedusing differential quadrature method (DQM). It is shown that thenonlocal effect is quite significant in graphene sheets and has adecreasing effect on the buckling loads.

Murmu et al. [89–91] have studied small-scale effects on thefree in-plane vibration of nano plates employing nonlocal contin-uum mechanics. Equations of motion of the nonlocal plate modelfor this study are derived and presented. Explicit relations fornatural frequencies are obtained through direct separation of vari-ables. It has been shown that nonlocal effects are quite significantin in-plane vibration studies and need to be included in thecontinuum model of nanoplates such as in graphene sheets.

P. Raghu et al. / Composite Structures 139 (2016) 13–29 15

Han et al. [92] studied influence of the molecular structure onindentation size effect in polymers. The indentation size effect inpolymers is examined which manifests itself in increased hardnessat decreasing indentation depths. Nikolov et al. [93] applied themicro-mechanical origin of size effects in elasticity of solid poly-mers. It was shown that size effects related to rotational gradientscan be interpreted in terms of Frank elasticity arising from thefinite bending stiffness of the polymer chains and their interac-tions. A relationship between the gradient of the nematic directorfield, related to the orientation of the polymer segments, and thecurvature tensor associated with rotational gradients was derived.

The focus of the present work is to develop analytical solutionsfor bending and free vibration of laminates composite plates usingthe nonlocal third-order shear deformation theory which accountsfor surface stress effects. The nonlocal theory used here is that ofEringen [1], which considers the size effect by assuming that stressat a point depends not only on the strain at that point but also onstrains at the neighboring points. Navier solutions of bending andvibration of simply supported rectangular laminates are presentedusing this nonlocal theory to illustrate the effect of nonlocality ondeflections and natural frequencies for various side-to-thicknessratios a=h and plate aspect ratios a=b.

The paper is organized as follows. Section 2 contains an intro-duction to the nonlocal theories. In Section 3 the third-order sheardeformation theory of Reddy [3,2] is extended to include the non-local and surface effects. The surface stresses for plates and lami-nates are discussed in detail in this section. The equilibriumequations are also presented in the section. In Section 4 the Naviersolutions for antisymmetric cross-ply and angle-ply laminates arepresented. Section 5 is devoted to numerical examples. Conclu-sions and remarks are presented in 6.

2. Nonlocal theories

Nonlocal elasticity theory invokes the length scale parameter inorder to account for the size effects [1]. Neglecting the size effects,when dealing with micro and nano scale fields, may result ininaccurate solutions. So one must consider the small scale effectsand atomic forces to obtain solutions with acceptable accuracy.In nonlocal elasticity theory, it is assumed that the stress at a pointin a continuum body is function of the strain at all neighbor pointsof the continuum. The effects of small scale and atomic forces areconsidered as material parameters in the constitutive equation.Following experimental observations, Eringen proposed aconstitutive model that expresses the nonlocal stress tensor rnl

at point x as

rnl ¼Z

Kðjx0 � xj; sÞrðx0Þdx0 ð1Þ

where, rðxÞ is the classical macroscopic stress tensor at point x andKðjx0 � xj; sÞ is the Kernel function which is normalized over the vol-ume of the body represents the nonlocal modulus. jx0 � xj is thenonlocal distance and s is the material constant that depends onthe internal and external characteristic length.

As per Hooke’s law we have

rðxÞ ¼ CðxÞ : �ðxÞ ð2Þwhere e is the strain tensor and C is the fourth-order elasticitytensor. Eqs. (1) and (2) together form the nonlocal constitutiveequation for Hookean solid. Eq. (1) can be represented equivalentlyin differential form as

1� s2l2r2� �

rnl ¼ r ð3Þ

where s ¼ ðe0aÞ2l2

; e0 is a material constant and a and l are the internal

and external characteristic lengths respectively. In general,r2 is the

three-dimensional Laplace operator. The nonlocal parameter l can

be taken as l ¼ s2l2.

3. Third-order shear deformation theory

In the third-order shear deformation theory (TSDT) ofReddy [2,3] the assumptions of straightness and normalityof the transverse normal after deformation are relaxed byexpanding the displacements as cubic functions of thicknesscoordinate. Consequently, the transverse shear strains andtransverse shear stresses vary quadratically through the thicknessof the laminate and avoids the need for shear correction factors.Here, the Reddy third-order shear deformation theory is reformu-lated to account for the Eringen’s nonlocal model and surface stresseffect.

3.1. Displacement field

The displacement field is based on a quadratic variation oftransverse shear strains (and hence stresses) and vanishing oftransverse shear stresses on top and bottom of a general laminatecomposed of different layers. The displacement field of the Reddythird-order theory [2,3] is

uðx; y; zÞ ¼ u0ðx; yÞ þ z/x �4z3

3h2 /x þ@w0

@x

� �

vðx; y; zÞ ¼ v0ðx; yÞ þ z/y �4z3

3h2 /y þ@w0

@y

� �ð4Þ

wðx; y; zÞ ¼ w0ðx; yÞ

where u0;v0;w0 are in-plane displacements of a point on the mid-plane (i.e., z ¼ 0). /x and /y denote the rotations of a transverse nor-mal line at the mid-plane (/x ¼ @u

@z and /y ¼ @u@z). The total thickness

of the laminate is given by h.

3.2. Strain–displacement relations

The strain fields of the TSDT is

exxeyycxy

8><>:

9>=>; ¼

eð0Þxx

eð0Þyy

cð0Þxy

8><>:

9>=>;þ z

eð1Þxx

eð1Þyy

cð1Þxy

8><>:

9>=>;þ z3

eð3Þxx

eð3Þyy

cð3Þxy

8><>:

9>=>; ð5Þ

cyzcxz

� �¼ cð0Þyz

cð0Þxz

( )þ z2

cð2Þyz

cð2Þxz

( )ð6Þ

where

eð0Þxx

eð0Þyy

cð0Þxy

8><>:

9>=>; ¼

@u0@x@v0@y

@u0@y þ @v0

@x

8>><>>:

9>>=>>;;

eð1Þxx

eð1Þyy

cð1Þxy

8><>:

9>=>; ¼

@/x@x@/y

@y

@/x@y þ @/y

@x

8>><>>:

9>>=>>; ð7Þ

eð3Þxx

eð3Þyy

cð3Þxy

8><>:

9>=>; ¼ �c1

@/x@x þ @2w0

@x2

@/y

@y þ @2w0@y2

@/x@y þ @/y

@x þ 2 @2w0@x@y

8>>><>>>:

9>>>=>>>;

ð8Þ

cð0Þyz

cð0Þxz

( )¼

/y þ @w0@y

/x þ @w0@x

( );

cð2Þyz

cð2Þxz

( )¼ �c2

/y þ @w0@y

/x þ @w0@x

( )ð9Þ

where c1 ¼ 43h2

and c2 ¼ 3c1.

16 P. Raghu et al. / Composite Structures 139 (2016) 13–29

3.3. Stress–strain relationships

The constitutive equations for each layer in the global coordi-nates are given by [3]

rxx

ryy

rxy

8>>><>>>:

9>>>=>>>;

¼

�Q11�Q12

�Q16

�Q12�Q22

�Q26

�Q16�Q26

�Q66

26664

37775

exx

eyy

exy

8>>><>>>:

9>>>=>>>;;

ryz

rxz

( )¼

�Q44�Q45

�Q45�Q55

" # cyz

cxz

( ) ð10Þ

where

�Q11 ¼ Q11 cos4 hþ 2 Q12 þ 2Q66ð Þ sin2 h cos2 hh i

þ Q22 sin4 h

�Q12 ¼ Q11 þ Q22 � 4Q66ð Þ sin2 h cos2 hþ Q12 sin4 hþ cos4 h� �

�Q16 ¼ Q11 � Q12 � 2Q66ð Þ sin h cos3 hþ Q12 � Q22 þ 2Q66ð Þ sin3 h cos h�Q22 ¼ Q11 sin

4 hþ 2 Q16 þ 2Q66ð Þ sin2 h cos2 hþ Q22 cos4 h ð11Þ

�Q26 ¼ Q11 � Q12 � 2Q66ð Þ sin3 h cos hþ Q12 � Q22 þ 2Q66ð Þ sin h cos3 h�Q66 ¼ Q11 þ Q22 � 2Q12 � 2Q66ð Þ sin2 h cos2 hþ Q66 sin

4 h cos4 h�Q44 ¼ Q44 cos2 hþ Q55 sin

2 h�Q45 ¼ Q55 � Q44ð Þ cos h sin h ð12Þ�Q55 ¼ Q44 sin

2 hþ Q55 cos2 h

Q11 ¼E1

1�m12m21; Q12 ¼

m12E1

1�m12m21; Q22 ¼

E2

1�m12m21; Q66 ¼G12;

ð13Þ

Q16 ¼ Q26 ¼ 0; Q44 ¼ G23; Q45 ¼ G12; Q55 ¼ G13 ð14Þwhere h is the orientation, measured counterclockwise, from thefiber direction to the positive x-axis, E1 and E2 are elastic moduli,m12 and m21 are Poisson’s ratios, and G12;G13 and G23 are the shearmoduli.

3.3.1. Surface stressBecause of interaction between the elastic surface and bulk

material, in-plane forces in different directions act on the plate.The resulting in-plane loads lead to surface stresses. The generalexpression for surface stresses as given by Gurtin and Murdoch(see [4,5]) is given by

Table 1Comparison of dimensionless maximum deflection, stresses, and fundamental frequency c

a=b a=h l ss (N/m) �x �w

1 10 0 0.0 3.12158 4.6730 1.7 3.12158 4.6730 3.4 3.12157 4.6730 6.8 3.12157 4.6731 0.0 2.33344 5.5073 0.0 1.70051 7.1765 0.0 1.40336 8.8451 1.7 2.33344 5.5073 3.4 1.70051 7.1765 6.8 1.40335 8.845

1 20 0 0.0 6.02405 4.4960 1.7 6.02405 4.4960 3.4 6.02405 4.4960 6.8 6.02406 4.4961 0.0 4.50309 5.3083 0.0 3.28214 6.9325 0.0 2.70821 8.5551 1.7 4.50308 5.3083 3.4 3.28214 6.9325 6.8 2.70821 8.556

rsab ¼ ssdab þ 2ðls � ssÞeab þ ðks þ ssÞuc;cdab þ ssua;b ð15aÞ

rs3b ¼ ssu3;b ð15bÞ

where a;b; c ¼ 1;2 and where ks and ls are the Lame’s constantsand ss is the surface stress parameter. From Eqs. (15), we can writeindividual surface stresses as

rsxx ¼ 2ls þ ks � ssð Þexx þ ks þ ssð Þeyy þ 1þ @u

@x

� �ss ð16Þ

rsyy ¼ 2ls þ ks � ssð Þeyy þ ks þ ssð Þexx þ 1þ @v

@y

� �ss ð17Þ

rsxy ¼ 2 ls � ssð Þexy þ ss @u

@yð18Þ

rsxz ¼ ss @w

@xð19Þ

rsyz ¼ ss @w

@yð20Þ

Gurtin and Murdoch [4,5] also gave the surface equilibrium equa-tions as

rsia;a þ ri3 ¼ qs€ui ð21Þ

where i ¼ 1;2;3 and a ¼ 1;2. The equilibrium Eq. (21) will not besatisfied since we have taken rzz to be zero. To satisfy the equilib-rium condition (21), we assume rzz to vary linearly through thethickness and is given by the equation

rzz ¼@rs

xz@x þ @rs

yz

@y � qs @w@t2

� ����at top

þ @rsxz

@x þ @rsyz

@y � qs @w@t2

� ����at bottom

2

264

375

þ@rs

xz@x þ @rs

yz

@y � qs @w@t2

� ����at top

� @rsxz

@x þ @rsyz

@y � qs @w@t2

� ����at bottom

h

264

375zð22Þ

The superscript s is used to denote the quantities corresponding tothe surface.

onsidering nonlocal and surface effects.

�rxx �ryy �syz �sxz

11 0.28817 0.28817 0.45601 0.4560111 0.28817 0.28817 0.45600 0.4560012 0.28816 0.28816 0.45600 0.4560012 0.28816 0.28816 0.45600 0.4560067 0.32446 0.32446 0.99054 0.9905479 0.39105 0.39105 2.05960 2.0596091 0.45954 0.45954 3.12866 3.1286665 0.32447 0.32447 0.99054 0.9905478 0.39108 0.39108 2.05960 2.0596092 0.45955 0.45955 3.12866 3.12866

14 0.28749 0.288171 0.46252 0.4625215 0.28749 0.287488 0.45252 0.4525216 0.28749 0.287489 0.46252 0.4625217 0.28749 0.28749 0.46252 0.4625211 0.32345 0.32346 1.00854 1.0085405 0.39539 0.39539 2.10057 2.1005799 0.46733 0.46733 3.12606 3.1260612 0.32346 0.32346 1.00854 1.0085406 0.39539 0.39539 2.10057 2.1005700 0.46733 0.46733 3.12605 3.12605

Fig. 1. (a) Deflection ratio �wnl=�w versus a=h for various values of l and ss ¼ 0. (b)Deflection ratio versus a=h for various values of ss and l ¼ 0.

Fig. 2. (a) Deflection ratio versus a=h for various values of l and ss , (b) frequencyratio versus a=h for various values of l and ss ¼ 0.

P. Raghu et al. / Composite Structures 139 (2016) 13–29 17

3.4. Stress resultants

The stress resultants for TSDT including surface stress effectsare given as

Nxx

Nyy

Nxy

8>>><>>>:

9>>>=>>>;

¼ZA

rkxx

rkyy

rxy

8>>><>>>:

9>>>=>>>;

dAþIC

rsxx

rsyy

0

8>>><>>>:

9>>>=>>>;

dS ð23Þ

Mxx

Myy

Mxy

8>>><>>>:

9>>>=>>>;

¼ZA

rkxx

rkyy

rxy

8>>><>>>:

9>>>=>>>;zdAþ

IC

rsxx

rsyy

0

8>>><>>>:

9>>>=>>>;zdS ð24Þ

Pxx

Pyy

Pxy

8>>><>>>:

9>>>=>>>;

¼ZA

rkxx

rkyy

rxy

8>>><>>>:

9>>>=>>>;z3 dAþ

IC

rsxx

rsyy

0

8>>><>>>:

9>>>=>>>;z3 dS ð25Þ

Qxz

Qyz

Rxz

Ryz

8>>><>>>:

9>>>=>>>;

¼ZA

rxz

ryz

rxzz2

ryzz2

8>>><>>>:

9>>>=>>>;

dAþIC

rsxz

rsyz

rsxzz

2

rsyzz

2

8>>>><>>>>:

9>>>>=>>>>;

dS ð26Þ

where rkxx ¼ rxx þ m

ð1�mÞrzz and rkyy ¼ ryy þ m

ð1�mÞrzz.

After substituting the values of stresses from Eqs. (10), and(16)–(22) into Eqs. (23)–(26), we obtain stress resultants in termsof strains as,

Nxx

Nyy

Nxy

8>><>>:

9>>=>>; ¼

A11 A12 A16

A12 A22 A26

A16 A26 A66

2664

3775

eð0Þxx

eð0Þyy

cð0Þxy

8>><>>:

9>>=>>;þ

B11 B12 B16

B12 B22 B26

B16 B26 B66

2664

3775

eð1Þxx

eð1Þyy

cð1Þxy

8>><>>:

9>>=>>;

þE11 E12 E16

E12 E22 E26

E16 E26 E66

2664

3775

eð3Þxx

eð3Þyy

cð3Þxy

2664

3775þ ð2ls þ ksÞ

Z11

Z21

Z31

2664

3775

ð27Þ

Fig. 3. (a) Frequency ratio versus a=h for various values of ss , (b) frequency ratioversus a=h for various values of l and ss .

Fig. 4. (a) Distribution of �rxx predicted by both local and nonlocal TSDT (b)distribution of �ryy predicted by both local and nonlocal TSDT.

18 P. Raghu et al. / Composite Structures 139 (2016) 13–29

Mxx

Myy

Mxy

8><>:

9>=>; ¼

B11 B12 B16

B12 B22 B26

B16 B26 B66

264

375

eð0Þxx

eð0Þyy

cð0Þxy

2664

3775þ

D11 D12 D16

D12 D22 D26

D16 D26 D66

264

375

eð1Þxx

eð1Þyy

cð1Þxy

8>><>>:

9>>=>>;

þF11 F12 F16

F12 F22 F26

F16 F26 F66

264

375

eð3Þxx

eð3Þyy

cð3Þxy

8>><>>:

9>>=>>;þ

L11L21L31

8><>:

9>=>;

ð28Þ

Pxx

Pyy

Pxy

8><>:

9>=>; ¼

E11 E12 E16

E12 E22 E26

E16 E26 E66

264

375

eð0Þxx

eð0Þyy

cð0Þxy

8>><>>:

9>>=>>;þ

F11 F12 F16

F12 F22 F26

F16 F26 F66

264

375

eð1Þxx

eð1Þyy

cð1Þxy

8>><>>:

9>>=>>;

þH11 H12 H16

H12 H22 H26

H16 H26 H66

264

375

eð3Þxx

eð3Þyy

cð3Þxy

2664

3775þ

O11

O21

O31

8><>:

9>=>;

ð29Þ

Qxz

Qyz

Rxz

Ryz

8>>><>>>:

9>>>=>>>;

¼

ss cð0Þxz ð2bþ hÞ þ cð2Þxzbh2

2 þ h3

12

� �h iss cð0Þyz ð2aþ hÞ þ cð2Þyz

ah2

2 þ h3

12

� �h iss cð0Þxz

bh2

2 þ h3

12

� �þ cð2Þxz

bh4

8 þ h5

80

� �h iss cð0Þyz

ah2

2 þ h3

12

� �þ cð2Þyz

ah4

8 þ h5

80

� �h i

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

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

ð30Þ

where Zi1; Li1 and Oi1 ði ¼ 1;2;3Þ are given in the Appendix.

Aij;Bij;Dij;Eij;Fij;Hij ¼XN

k¼1

Z zkþ1

zk

�Q ðkÞij 1;z;z2;z3;z4;z6� �

dz; ði;j¼1;2;6Þ

ð31Þ

Fig. 5. (a) Distribution of �syz predicted by both local and nonlocal TSDT (b)distribution of �sxz predicted by both local and nonlocal TSDT.

P. Raghu et al. / Composite Structures 139 (2016) 13–29 19

Using Eq. (3), the nonlocal stress resultants can be written as

1� lr2� � Nnl

xx

Nnlyy

Nnlxy

8>>>><>>>>:

9>>>>=>>>>;

¼Nxx

Nyy

Nxy

8>>><>>>:

9>>>=>>>;

ð32Þ

1� lr2� � Mnl

xx

Mnlyy

Mnlxy

8>>>><>>>>:

9>>>>=>>>>;

¼Mxx

Myy

Mxy

8>>><>>>:

9>>>=>>>;

ð33Þ

1� lr2� � Pnl

xx

Pnlyy

Pnlxy

8>>>><>>>>:

9>>>>=>>>>;

¼Pxx

Pyy

Pxy

8>>><>>>:

9>>>=>>>;

ð34Þ

1� lr2� � Qnl

xz

Qnlyz

Rnlxz

Rnlyz

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

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

¼

Qxz

Qyz

Rxz

Ryz

8>>><>>>:

9>>>=>>>;

ð35Þ

3.5. Equations of equilibrium

The governing equilibrium equations for the third-order sheardeformation theory for laminated plates can be derived using theprinciple of virtual displacements (Hamilton’s principle). By substi-tuting the nonlocal stress resultants in terms of displacements intothe statement of principle of virtual displacements and usingintegration-by-parts, the equations of motion can be obtained as[73]

@Nnlxx

@xþ @Nnl

xy

@y¼ 1� lr2� �

I0€u0 þ J1€/x � c1I3@ €w0

@x

� �ð36Þ

@Nnlxy

@xþ @Nnl

yy

@y¼ 1� lr2� �

I0€v0 þ J1€/y � c1I3@ €w0

@y

� �ð37Þ

@ �Qnlx

@xþ @ �Qnl

y

@yþ @

@xNnl

xx@w0

@xþ Nnl

yy@w0

@y

� �þ @

@yNnl

xy@w0

@xþ Nnl

yy@w0

@y

� �

þ c1@2Pnl

xx

@x2þ 2

@2Pnlxy

@x@yþ @2Pnl

yy

@y2

!þ q

¼ 1� lr2� �

I0 €w0 � c21I6@ €w0

@x2þ @2 €w0

@y2

!" #

þ 1� lr2� �

c1 I3@€u0

@xþ @€v0

@y

� �þ J4

@€/x

@xþ @€/y

@y

!" #( )ð38Þ

@ �Mnlxx

@xþ @ �Mnl

xy

@y� �Qnl

x ¼ 1� lr2� �

J1€u0 þ K2€/x � c1J4

@ €w0

@x

� �ð39Þ

@ �Mnlxy

@xþ @ �Mnl

yy

@y� �Qnl

y ¼ 1� lr2� �

J1€v0 þ K2€/y � c1J4

@ €w0

@y

� �ð40Þ

where

�Mnlab ¼ Mnl

ab � c1Pnlab ða;b ¼ 1;2;6Þ : �Qa ¼ Qnl

a � c2Rnla ða ¼ 4;5Þ

ð41Þ

Ii ¼XNk¼1

Z zkþ1

zk

qðkÞðzÞi dz ði ¼ 0;1;2; . . . ;6Þ ð42Þ

Ji ¼ Ii � c1Iiþ2; K2 ¼ I2 � 2c1I4 þ c21I6; c1 ¼ 4

3h2 ; c2 ¼ 3c1

ð43Þ

4. Navier’s solution procedure

In Navier’s method, the generalized displacements areexpanded in a double Fourier series in terms of unknown parame-ters. The choice of the functions in the series is restricted to thosewhich satisfy the boundary conditions of the problem. Substitutionof the displacement expansions into the governing equations

Table 2Dimensionless maximum deflections and stresses in simply supported antisymmetric cross-ply laminate (0/90/0/90) under sinusoidally distributed transverse load.

a=b a=h l ss (N/m) �w �x �rxx �ryy �syz �sxz

1 10 0 0.0 1.05268 0.02440 0.74527 0.69058 0.62303 0.623030 1.7 1.05271 0.02440 0.74529 0.69060 0.62303 0.623030 3.4 1.05273 0.02440 0.7453 0.69062 0.62303 0.623020 6.8 1.05276 0.02440 0.74534 0.69066 0.623022 0.623021 0.0 1.05314 0.01824 0.74546 0.69086 0.62471 0.624713 0.0 1.05405 0.01329 0.74585 0.69141 0.62808 0.628085 0.0 1.05496 0.01097 0.74624 0.69196 0.62145 0.621451 1.7 1.05316 0.01824 0.74548 0.69088 0.62470 0.624703 3.4 1.05409 0.01330 0.74589 0.69144 0.62807 0.628085 6.8 1.05504 0.01097 0.74632 0.69203 0.62144 0.62143

20 0 0.0 0.86949 0.01235 0.73810 0.69693 0.63228 0.632280 1.7 0.86963 0.01235 0.73830 0.69705 0.63227 0.632270 3.4 0.86976 0.01235 0.73840 0.69718 0.63227 0.632260 6.8 0.86003 0.01235 0.73870 0.69742 0.63225 0.632251 0.0 0.86959 0.00923 0.73825 0.69700 0.63273 0.632733 0.0 0.86979 0.00673 0.73837 0.69713 0.63364 0.633645 0.0 0.86999 0.00555 0.73850 0.69727 0.63454 0.634541 1.7 0.86972 0.00923 0.73838 0.69712 0.63273 0.632723 3.4 0.87006 0.00672 0.73863 0.69726 0.63345 0.633625 6.8 0.87053 0.00550 0.73902 0.69776 0.63451 0.63450

20 P. Raghu et al. / Composite Structures 139 (2016) 13–29

should result in a unique, invertible, set of algebraic equationsamong the parameters of the expansion (see Reddy [3]). Navier’ssolution can be developed for rectangular laminates with two setsof simply supported boundary conditions (SS-1 and SS-2). It turnsout that the type of simply supported boundary conditions forantisymmetric cross-ply (SS-1) and antisymmetric angle-ply (SS-2) are different, as will be shown shortly. In the following subsec-tions, Navier solutions of cross-ply laminates for the SS-1 boundaryconditions and anti-symmetric angle-ply laminates for the SS-2boundary conditions including nonlocal and surface stress effectsare presented.

4.1. Boundary conditions and displacement expansions for SS-1

The SS-1 boundary conditions for the third-order sheardeformation plate theory are

u0ðx;0; tÞ ¼ 0; u0ðx;b; tÞ ¼ 0; v0ð0;y; tÞ ¼ 0; v0ða;y; tÞ ¼ 0/xðx;0; tÞ ¼ 0; /xðx;b; tÞ ¼ 0; /yð0;y; tÞ ¼ 0; /yða;y; tÞ ¼ 0 ð44Þw0ðx;0; tÞ ¼ 0; w0ðx;b; tÞ ¼ 0; w0ð0;y; tÞ ¼ 0; w0ða;y; tÞ ¼ 0

Nxxð0;y;tÞ¼0; Nxxða;y;tÞ¼0; Nyyðx;0;tÞ¼0; Nyyðx;b;tÞ¼0

�Mxxð0;y;tÞ¼0; �Mxxða;y;tÞ¼0; �Myyðx;0;tÞ¼0; �Myyðx;b;tÞ¼0ð45Þ

The following displacement expansions that satisfy SS-1 boundaryconditions are used:

u0ðx; y; tÞ ¼X1n¼1

X1m¼1

UmnðtÞ cosax sin by

v0ðx; y; tÞ ¼X1n¼1

X1m¼1

VmnðtÞ sinax cos by

w0ðx; y; tÞ ¼X1n¼1

X1m¼1

WmnðtÞ sinax sinby

/xðx; y; tÞ ¼X1n¼1

X1m¼1

XmnðtÞ cosax sin by

/yðx; y; tÞ ¼X1n¼1

X1m¼1

YmnðtÞ sinax cos by

ð46Þ

where a ¼ mpa and b ¼ np

b . Umn; Vmn; Wmn; Xmn and Ymn are coeffi-cients that are to be determined.

4.2. Boundary conditions and displacement expansions for SS-2

The SS-2 boundary conditions for the third-order shear defor-mation plate theory are

u0ð0;y;tÞ¼0; u0ða;y;tÞ¼0; v0ðx;0;tÞ¼0; v0ðx;b;tÞ¼0

/xðx;0;tÞ¼0; /xðx;b;tÞ¼0; /yð0;y;tÞ¼0; /yða;y;tÞ¼0 ð47Þw0ðx;0;tÞ¼0; w0ðx;b;tÞ¼0; w0ð0;y;tÞ¼0; w0ða;y;tÞ¼0

Nxyð0;y;tÞ¼0; Nxyða;y;tÞ¼0; Nxyðx;0;tÞ¼0; Nxyðx;b;tÞ¼0

�Mxxð0;y;tÞ¼0; �Mxxða;y;tÞ¼0; �Myyðx;0;tÞ¼0; �Myyðx;b;tÞ¼0

The following displacement expansions that satisfy SS-2 boundaryconditions are used:

u0ðx; y; tÞ ¼X1n¼1

X1m¼1

UmnðtÞ sinax cosby

v0ðx; y; tÞ ¼X1n¼1

X1m¼1

VmnðtÞ cosax sinby

w0ðx; y; tÞ ¼X1n¼1

X1m¼1

WmnðtÞ sinax sinby

/xðx; y; tÞ ¼X1n¼1

X1m¼1

XmnðtÞ cosax sinby

/yðx; y; tÞ ¼X1n¼1

X1m¼1

YmnðtÞ sinax cosby

ð48Þ

4.3. The Navier solutions

For both SS-1 and SS-2, the transverse load qzðx; y; tÞ isexpressed in double Fourier sine series as

qzðx; y; tÞ ¼X1n¼1

X1m¼1

QmnðtÞ sinax sin by

QmnðtÞ ¼4ab

Z a

0

Z b

0qzðx; y; tÞ sinax sin bydxdy

ð49Þ

Substituting the displacement expansions in Eq. (48) into the gov-erning equations of equilibrium yields the following ordinary differ-ential equations in time:

SDþM€D ¼ F ð50Þ

Fig. 6. Distribution of normal stress predicted by both local and nonlocal TSDT fora=h ¼ 10 (a) �rxx (b) �ryy .

Fig. 7. Distribution of shear stress predicted by both local and nonlocal TSDT fora=h ¼ 10 (a) �syz (b) �sxz .

P. Raghu et al. / Composite Structures 139 (2016) 13–29 21

where D is the displacement vector, €D is the acceleration vector, Sand M are the stiffness and mass matrices, respectively. The dis-placement D and force vector F are given as

D¼

Umn

Vmn

Wmn

Xmn

Ymn

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

9>>>>>>>>>>=>>>>>>>>>>;; F¼ð1�lr2Þ

0

0

Qmn

0

0

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

9>>>>>>>>>>=>>>>>>>>>>;; Qmn¼

16q0

p2mnfor uniform load

ð51Þ

The coefficients of the stiffness matrix S and mass matrix M for thetwo types of boundary conditions are given in the subsections tofollow.

For static bending analysis we set €D to zero and obtain

SD ¼ F ð52Þ

For the natural vibration, we assume that the solution is periodicDðtÞ ¼ D0eixt , where x is the frequency of natural vibration andi ¼

ffiffiffiffiffiffiffi�1

p. Thus, the free vibration problem consists of solving the

eigenvalue problem

S�x2M� �

D ¼ 0 ð53Þ

22 P. Raghu et al. / Composite Structures 139 (2016) 13–29

4.3.1. Anti-symmetric cross-ply laminateThe coefficients of the stiffness matrix Sij for anti-symmetric

cross-ply laminate are

S11 ¼ A11a2 þ A66b2 þ ð2ls þ ksÞ a2hþ 2a2b

� �S12 ¼ ðA12 þ A66Þab

S13 ¼ �c1½E11a2 þ ðE12 þ 2E66Þb2�a� c1ð2ls þ ksÞ h4

32a3 þ bh3

4a3

!

S14 ¼ B11a2 þ B33b2 þ ð2ls þ ksÞ �c1h

4

32a2 þ bha2 � c1bh

3

4a2

!

S15 ¼ ðB12 þ B66ÞabS21 ¼ S12

S22 ¼ A66a2 þ A22b2 þ ð2ls þ ksÞ b2hþ 2b2a

� �S23 ¼ �c1½E22b

2 þ ðE12 þ 2E66Þa2�b� c1ð2ls þ ksÞ h4

32b3 þ ah3

4b3

!

S24 ¼ ðB12 þ B66Þab

S25 ¼ B66a2 þ B22b2 þ ð2ls þ ksÞ �c1h

4

32b2 þ ahb2 � c1ah

3

4b2

!

S31 ¼ S13 � c1bh3

8a3ð4ls þ ksÞ

S32 ¼ S23 � c1ah3

8b3ð4ls þ ksÞ

S33 ¼ �A55a2 þ �A44b2 þ c21½H11a4 þ 2ðH12 þ 2H66Þa2b2 þ H22b

4�

þ c1mh4ss

40ð1� mÞ ða4 þ b4 þ a2b2Þ þ ð2ls

þ ksÞ h7

448ðc21a4 þ c21b

4Þ þ h6

32ðbc21a4 þ ac21b

4Þ" #

S34 ¼ A55a� c1½F11a3 þ ðF12 þ 2F66Þab2�

þ ð2ls þ ksÞ �c1h5

80a3 þ c21h

7

448a3 � c1h

4

8ba3 þ c21h

6

32ba3

!ð54Þ

Table 3Comparison of dimensionless maximum deflection, stresses, and fundamental frequency c

a=b a=h l ss (N/m) �w �x

1 10 0 0.0 1.04791 0.6390 1.7 1.04793 0.6390 3.4 1.04795 0.6390 6.8 1.04799 0.6391 0.0 1.22372 0.4773 0.0 1.57532 0.3485 0.0 1.92690 0.2871 1.7 1.22374 0.4473 3.4 1.57539 0.3485 6.8 1.92710 0.287

20 0 0.0 0.77599 0.3210 1.7 0.77610 0.3210 3.4 0.77621 0.3210 6.8 0.77642 0.3211 0.0 0.80988 0.2403 0.0 0.87760 0.1755 0.0 0.94545 0.1441 1.7 0.81000 0.2403 3.4 0.87790 0.1755 6.8 0.94600 0.144

S35 ¼ �A44b� c1½F22b3 þ ðF12 þ 2F66Þa2b� þ ð2ls

þ ksÞ �c1h5

80b3 þ c21h

7

448b3 � c1h

4

8ab3 þ c21h

6

32ab3

!

S41 ¼ S14 þ ð2ls þ ksÞ bha2 � c1bh3

8a2

!

S42 ¼ S24

S43 ¼ S34 þ mh2ss

6ð1� mÞ �c1h

4mss

40ð1� mÞ

!ða3 þ ab2Þ

þ ð2ls þ ksÞ �c1h5

80a3 þ c21h

7

448a3 � c1h

4

8ba3 þ c21h

6

32ba3

!

S44 ¼ �A55 þ �D11a2 þ �D66b2

þ ð2ls þ ksÞ h3

12a2 � c1h

5

40a2 � c1bh

4

4a2 þ c21h

7

448a2 þ c21bh

6

32a2

!

S45 ¼ ð�D12 þ �D66ÞabS51 ¼ S15

S52 ¼ S25 þ ð2ls þ ksÞ ahb2 � c1ah3

8b2

!

S53 ¼ S35 þ mh2ss

6ð1� mÞ �c1h

4mss

40ð1� mÞ

!ðb3 þ ba2Þ

þ ð2ls þ ksÞ �c1h5

80b3 þ c21h

7

448b3 � c1h

4

8ab3 þ c21h

6

32ab3

!

S54 ¼ S45

S55 ¼ �A44 þ �D33a2 þ �D22b2

þ ð2ls þ ksÞ h3

12b2 � c1h

5

40b2 � c1ah

4

4a2 þ c21h

7

448b2 þ c21bh

6

32b2

!

where

onsidering nonlocal and surface effects.

�rxx �ryy �syz �sxz

04 0.78113 0.51637 0.35436 0.8321004 0.78115 0.51639 0.35435 0.8321504 0.78117 0.51640 0.35430 0.8321604 0.78121 0.51643 0.35425 0.8321770 0.87500 0.56740 0.98390 1.4864012 1.06275 0.66960 2.24299 2.79510291 1.25050 0.77180 3.50208 4.1037069 0.87503 0.56750 0.98380 1.4865011 1.06282 0.66970 2.24297 2.7952029 1.25066 0.77190 3.50203 4.10380

40 0.81068 0.40153 0.32469 0.8752045 0.81081 0.40160 0.32464 0.8753042 0.81094 0.40167 0.32460 0.8753439 0.8112 0.4018 0.32450 0.8754025 0.84070 0.40867 0.48547 1.0572009 0.90000 0.42295 0.80704 1.4201049 0.96070 0.43720 1.12860 1.7849024 0.84080 0.40874 0.48543 1.0572308 0.90100 0.42311 0.80694 1.4211048 0.96316 0.43756 1.1284 1.78510

P. Raghu et al. / Composite Structures 139 (2016) 13–29 23

Aij ¼ Aij � c1Dij; Bij ¼ Bij � c1Eij;

Dij ¼ Dij � c1Fij ði; j ¼ 1;2;6Þ

Fij ¼ Fij � c1Hij;

�Aij ¼ Aij � c1Dij ¼ Aij � 2c1Dij þ c21Fij ði; j ¼ 1;2;6Þ ð55Þ�Dij ¼ Dij � c1Fij ¼ Dij � 2c1Fij þ c21Hij ði; j ¼ 1;2;6ÞThe coefficients of the mass matrix Mij are

Fig. 8. Distribution of stresses predicted by both local and nonlocal TSDT fora=h ¼ 10 (a) �rxx (b) �ryy .

M11 ¼ I0; M22 ¼ I0

M33 ¼ I0 þ c21I6ða2 þ b2Þ þ 2aqs c1h4mqs

160ð1� mÞ ða2 þ b2Þ; M34 ¼�c1J4a

M35 ¼�c1J4b; M43 ¼� mh2qs

6ð1� mÞaþ c1h4mqs

160ð1� mÞb

M44 ¼ K2; M53 ¼� mh2qs

6ð1� mÞbþc1h

4mqs

160ð1� mÞa

M55 ¼ K2

ð56Þ

Fig. 9. Distribution of shear stresses predicted by both local and nonlocal TSDT fora=h ¼ 10 (a) �syz (b) �sxz .

24 P. Raghu et al. / Composite Structures 139 (2016) 13–29

where qs is the surface density.The in-plane stresses in each laminate layer can be computed

using Eq. (10), where the strains are given as

exx

eyy

exy

8>>><>>>:

9>>>=>>>;

¼X1m¼1

X1n¼1

�Rxxmnð1;1Þ þ z�Sxxmnð1;1Þ þ c1z3�Txx

mnð1;1Þ� �

sinax sin by

�Rxxmnð2;1Þ þ z�Sxxmnð2;1Þ þ c1z3�Txx

mnð2;1Þ� �

sinax sin by

�Rxxmnð3;1Þ þ z�Sxxmnð3;1Þ þ c1z3�Txx

mnð3;1Þ� �

sinax sin by

8>>>><>>>>:

9>>>>=>>>>;ð57Þ

where

�Rxxmn

�Ryymn

�Rxymn

8>>><>>>:

9>>>=>>>;

¼�aUmn

�bVmn

bUmn þ aVmn

8>>><>>>:

9>>>=>>>;;

�Sxxmn

�Syymn

�Sxymn

8>>><>>>:

9>>>=>>>;

¼�aXmn

�bYmn

bXmn þ aYmn

8>>><>>>:

9>>>=>>>;

ð58Þ

�Txxmn

�Tyymn

�Txymn

8>>><>>>:

9>>>=>>>;

¼aXmn þ a2Wmn

bYmn þ b2Wmn

�ðbXmn þ aYmn þ 2abWmnÞ

8>>><>>>:

9>>>=>>>;

ð59Þ

The transverse stresses are determined from the followingequations

ryz

rxz

( )¼ð1�c2z2Þ

X1m¼1

X1n¼1

Q44 0

0 Q55

" # ðYmnþbWmnÞsinaxcosbyðXmnþaWmnÞcosaxsinby

( )

ð60Þ

4.3.2. Anti-symmetric angle-ply laminateThe stiffness coefficients Sij for the SS-2 case are given by

S11 ¼ A11a2 þ A66b2 þ ð2ls þ ksÞ a2hþ 2a2b

� �S12 ¼ ðA12 þ A66Þab

S13 ¼ �c1ð3E16a2 þ E26b2Þb� c1ð2ls þ ksÞ h4

32a3 þ bh3

4a3

!

S14 ¼ 2B16abþ ð2ls þ ksÞ �c1h4

32a2 þ bha2 � c1bh

3

4a2

!

S15 ¼ B16a2 þ B26b2; S21 ¼ S12

S22 ¼ A66a2 þ A22b2 þ ð2ls þ ksÞ b2hþ 2b2a

� �S23 ¼ �c1ðE16a2 þ 3E26b

2Þa� c1ð2ls þ ksÞ h4

32b3 þ ah3

4b3

!

S24 ¼ S15

S25 ¼ 2B26abþ ð2ls þ ksÞ �c1h4

32b2 þ ahb2 � c1ah

3

4b2

!

S31 ¼ S13 þ c1bh3

8a3ð4ls þ ksÞ

S32 ¼ S23 þ c1ah3

8b3ð4ls þ ksÞ

S33 ¼ �A55a2 þ �A44b2 þ c21½H11a4 þ 2ðH12 þ 2H66Þa2b2

þ H22b4� þ c1mh

4ss

40ð1� mÞ ða4 þ b4 þ a2b2Þ þ ð2ls

þ ksÞ h7

448ðc21a4 þ c21b

4Þ þ h6

32ðbc21a4 þ ac21b

4Þ" #

ð61Þ

S34 ¼ A55a� c1½F11a3 þ ðF12 þ 2F66Þab2� þ ð2ls

þ ksÞ �c1h5

80a3 þ c21h

7

448a3 � c1h

4

8ba3 þ c21h

6

32ba3

!

S35 ¼ �A44a� c1½F22b3 þ ðF12 þ 2F66Þa2b2� þ ð2ls

þ ksÞ �c1h5

80b3 þ c21h

7

448b3 � c1h

4

8ab3 þ c21h

6

32ab3

!

S41 ¼ S14 þ ð2ls þ ksÞ bha2 � c1bh3

8a2

!

S42 ¼ S24

S43 ¼ S34 þ mh2ss

6ð1� mÞ �c1h

4mss

40ð1� mÞ

!ða3 þ ab2Þ

S44 ¼ �A55 þ �D11a2 þ �D66b2 þ ð2ls

þ ksÞ h3

12a2 � c1h

5

40a2 � c1bh

4

4a2 þ c21h

7

448a2 þ c21bh

6

32a2

!

S45 ¼ ð�D12 þ �D66Þab; S51 ¼ S15

S52 ¼ S25 þ ð2ls þ ksÞ ahb2 � c1ah3

8b2

!

S53 ¼ S35 þ mh2ss

6ð1� mÞ �c1h

4mss

40ð1� mÞ

!ðb3 þ ba2Þ þ ð2ls

þ ksÞ �c1h5

80b3 þ c21h

7

448b3 � c1h

4

8ab3 þ c21h

6

32ab3

!

S54 ¼ S45

S55 ¼ �A44 þ �D33a2 þ �D22b2 þ ð2ls

þ ksÞ h3

12b2 � c1h

5

40b2 � c1ah

4

4a2 þ c21h

7

448b2 þ c21bh

6

32b2

!

The coefficients of the mass matrix for anti-symmetric angle-plylaminates are same as those in Eq. (56). The in-plane stresses ineach layer can be computed using the Eq. (10) where the strainsare given as

eð0Þxx

eð0Þyy

eð0Þxy

8><>:

9>=>; ¼

X1m¼1

X1n¼1

aUmn cosax cos bybVmn cosax cos by

�ðbUmn þ aVmnÞ sinax sinby

8><>:

9>=>; ð62Þ

eð1Þxx

eð1Þyy

eð1Þxy

8><>:

9>=>; ¼ �

X1m¼1

X1n¼1

aXmn sinax sinby

bYmn cosax cos by�ðbXmn þ aYmnÞ cosax cos by

8><>:

9>=>; ð63Þ

eð3Þxx

eð3Þyy

eð3Þxy

8>>><>>>:

9>>>=>>>;

¼ c1X1m¼1

X1n¼1

ðaXmn þ a2WmnÞ sinax sin by

ðbYmn þ b2WmnÞ sinax sin by

�ðbXmn þ aYmn þ 2abWmnÞ cosax cos by

8>>><>>>:

9>>>=>>>;ð64Þ

P. Raghu et al. / Composite Structures 139 (2016) 13–29 25

5. Numerical examples

5.1. Preliminary comments

In this section we present several examples of the analyticalsolutions obtained in this study. Navier’s solution is obtained using100 terms in the series for uniformly distributed load. Both bend-ing and free vibration solutions are presented for each problem.The following four examples are considered:

(1) Isotropic plates with following material propertiesE ¼ 30� 106 Mpa, m ¼ 0:3; a ¼ 10 mm, q0 ¼ 1, and q ¼ 1,and subjected to a uniformly distributed transverse load ofintensity q0.

(2) Antisymmetric cross-ply (0�=90�=0�=90�) laminated plates.(3) Symmetric cross-ply (0�=90�=90�=0�) laminated plates.(4) Antisymmetric angle-ply (30�=� 30�=30�=� 30�) laminated

plates.

For all examples the thickness of all layers are equal and eachlayer is orthotropic with following material properties: E1 ¼175� 103 MPa, E2 ¼ 7� 103 MPa, G12 ¼ 3:5� 103 MPa, G13 ¼3:5� 103 MPa, G23 ¼ 1:4� 103 MPa, m12 ¼ 0:25; m13 ¼ 0:25; m21 ¼ðE2=E1Þm12, and a ¼ 20 mm. The following notations are used fordeflection and frequency namely: �wnl is the dimensionless maxi-mum deflection with nonlocal effect; �ws is the dimensionless max-imum deflection with surface effect; �wnls dimensionless maximumdeflection with nonlocal and surface effect. �xnl dimensionlessfundamental frequency with nonlocal effect; �xs dimensionlessfirst mode frequency with surface effect. �xnls dimensionlessfundamental frequency with nonlocal and surface effect. For theisotropic plate example, the dimensionless maximum centerdeflection, fundamental frequency, respectively, are obtained as�w ¼ w0 � ðEh3

=q0a4Þ � 102; �x ¼ xh

ffiffiffiffiffiffiffiffiffiq=G

p. For all the laminated

plate examples, the maximum center deflection, and fundamental

frequency are dimensionless as �w ¼ w� ðE2h3=q0a

4Þ � 102 and�x ¼ xh

ffiffiffiffiffiffiffiffiffiffiffiffiffiq=G13

p, where a; b, and h are the length, width and thick-

ness of the plate, respectively; q0 is the intensity of the uniformlydistributed transverse load and q is the material density; E and Gare Young’s and shear moduli, respectively. The dimensionlessstress measures used are

Table 4Dimensionless maximum deflections, fundamental frequencies, and stresses in simplysinusoidally distributed transverse load.

a=b a=h l ss (N/m) �w �x

1 10 0 0.0 0.74203 0.16980 1.7 0.74204 0.16980 3.4 0.74205 0.16980 6.8 0.74206 0.16981 0.0 0.74956 0.12693 0.0 0.76463 0.09255 0.0 0.77970 0.07631 1.7 0.74957 0.12693 3.4 0.76465 0.09255 6.8 0.77973 0.0763

20 0 0.0 0.55072 0.08550 1.7 0.55077 0.08550 3.4 0.55083 0.08550 6.8 0.55094 0.08551 0.0 0.55213 0.06393 0.0 0.55494 0.04655 0.0 0.55775 0.03841 1.7 0.55218 0.06393 3.4 0.55505 0.04655 6.8 0.55796 0.0384

�rxx ¼ rxxa2;b2;h2

� �h2

b2q0

!; �ryy ¼ ryy

a2;b2;h2

� �h2

b2q0

!

�syz ¼ syza2;0;0

� � hbq0

� �; �sxz ¼ sxz 0;

b2;0

� �hbq0

� � ð65Þ

5.2. Isotropic plates

A simply supported isotropic square plate subjected to auniformly distributed transverse load is considered. Both staticbending and free vibration analysis has been performed. Table 1shows the nondimensional maximum values of deflections, stres-ses and fundamental frequency. Two different aspect ratios ofa=h ¼ 10 and a=h ¼ 20 are considered. The nonlocal parameter land surface effect parameter ss are varied. It is observed that themaximum values of the dimensionless deflection increases withincrease in nonlocal parameters l and ss. The dimensionless fre-quency decreases with an increase in nonlocal parameters. For afixed nonlocal parameter an increase in surface parameter has astiffening effect and there is a decrease in the maximum deflectionand increase in the frequency as given in Table 1. The rate ofchange in the solutions for the surface effect is quite small; thechange is not seen in some cases unless additional decimal placesare reported.

Fig. 1(a) shows the variation of deflection ratio �wnl=�w (ratio ofdimensionless maximum deflection with nonlocal effect to thedimensionless maximum deflection with out nonlocal effect) withincreasing values of a=h. Fig. 1(b) shows variation of deflectionratio �ws= �w with increasing values of a=h where �ws= �w is the ratioof dimensionless maximum deflection with surface effect to thedimensionless maximum deflection with out surface effect. It isobserved that the ratio increases as the value of a=h increases.

Fig. 2(a) shows the variation of deflection ratio �wnl= �w withincreasing values of a=h. Fig. 2(b) shows the variation of frequencyratio i.e., �xnls= �x with increasing values of a=h where �xnls= �x is theratio of dimensionless fundamental frequency with nonlocal effectto the dimensionless fundamental frequency without the nonlocaleffect.

Fig. 3(a) shows the variation of frequency ratio �xs= �x withincreasing values of a=h where �xs= �x is the ratio of dimensionlessfundamental frequency with surface effect to the dimensionlessfundamental frequency without the surface effect. Fig. 3(b) shows

supported (SS-2) antisymmetric angle-ply laminates (30=� 30=30=� 30) under

�rxx �ryy �syz �sxz

5 0.36808 0.14231 0.64644 0.8870655 0.368085 0.14231 0.64644 0.8870645 0.36809 0.14231 0.64644 0.8870615 0.3681 0.14232 0.64644 0.8870596 0.36951 0.14280 0.75265 1.015364 0.37238 0.14379 0.96506 1.271965 0.37525 0.14477 1.17746 1.528566 0.36952 0.14280 0.75265 1.015354 0.37239 0.14379 0.96505 1.271956 0.37528 0.14478 1.17745 1.52855

2 0.32459 0.12745 0.58790 0.952532 0.32463 0.12746 0.58789 0.952532 0.32466 0.12748 0.58787 0.952542 0.32473 0.12750 0.58784 0.952543 0.32471 0.12752 0.61420 0.988019 0.32496 0.12768 0.66679 1.058975 0.32521 0.12783 0.71938 1.129933 0.32475 0.12754 0.61419 0.988029 0.32503 0.12771 0.66676 1.058985 0.32535 0.12789 0.71931 1.12995

26 P. Raghu et al. / Composite Structures 139 (2016) 13–29

the variation of frequency ratio �xnl= �xwith increasing values of a=hwhere �xnl= �x is the ratio of dimensionless fundamental frequencywith surface and nonlocal effect to the dimensionless fundamentalfrequencywith no surface and nonlocal effects. As stated earlier, therate of change in the solutions for the surface effect is quite small.

Figs. 4(a), (b) and 5(a), (b) respectively shows the variation of�rxx; �ryy; �syz; �sxz with thickness coordinate z=h for various values ofnonlocal parameter l. The plot clearly indicates that the nonlocalparameter has a significant effect on the stresses.

5.3. Antisymmetric cross-ply (0�=90�=0�=90�) laminated plates

A simply supported square antisymmetric cross ply laminatedplate subjected to a uniformly distributed transverse load isconsidered. Both static bending and free vibration analysis

Fig. 10. Distribution of normal stresses predicted by both local and nonlocal TSDTfor a=h ¼ 10 (a) �rxx (b) �ryy .

has been performed. Table 2 shows the dimensionlessmaximum deflections, stresses and first mode frequency for aspectratio of a=h ¼ 10 and a=h ¼ 20 with nonlocal and surface effects.The dimensionless stresses are computed as before except �ryy isnow computed at h=4. The nonlocal parameter l and surface effectparameter ss are varied. It is observed that the maximum values ofthe dimensionless deflection increases with increase in nonlocalparameter. The dimensionless frequency decreases with increasein nonlocal parameter. For a fixed nonlocal parameter an increasein surface parameter there is a increase in the maximum deflectionand decrease in the frequency as given in Table 2.

Figs. 6(a), (b) and 7(a), (b) respectively shows the variation of�rxx; �rxx; �syz, and �sxz with thickness coordinate z=h for various valuesof nonlocal parameter l clearly indicating the dependence of stres-ses on nonlocal parameter.

Fig. 11. Distribution of shear stresses predicted by both local and nonlocal TSDT fora=h ¼ 10 (a) �syz (b) �sxz .

P. Raghu et al. / Composite Structures 139 (2016) 13–29 27

5.4. Symmetric cross-ply (0�=90�=90�=0�) laminated plates

A simply supported, symmetric cross-ply, square laminatedplate subjected to a uniformly distributed transverse load is con-sidered. Both static bending and free vibration analysis has beenperformed. Table 3 shows the nondimensional maximum deflec-tions, stresses, and fundamental frequency for aspect ratios ofa=h ¼ 10 and a=h ¼ 20, and with nonlocal and surface effects.The dimensionless stresses are computed as before except �ryy

and �sxz and are computed at h=4. The nonlocal parameter l andsurface effect parameter ss are varied. It is observed that the max-imum values of the dimensionless deflection increases withincrease in nonlocal parameter. The dimensionless frequencydecreases with increase in nonlocal parameter. For a fixed nonlocalparameter an increase in surface parameter there is a increase inthe maximum deflection and decrease in the frequency as givenin Table 3.

Figs. 8(a), (b) and 9(a), (b) respectively shows the variation of�rxx; �ryy; �syz, and �sxz with thickness coordinate z=h for various valuesof nonlocal parameter l.

5.5. Antisymmetric angle-ply (30�=� 30�=30�=� 30�) plates

An simply supported square antisymmetric angle-ply laminatedplate subjected to a uniformly distributed load is considered forstatic bending and free vibration analysis. Table 4 shows the

L11L21L31

8><>:

9>=>; ¼

mh6ss6ð1�mÞ

@2w@x2 þ @2w

@y2

� �þ ð2ls þ ksÞ eð0Þxx bhþ eð1Þxx

h3

12 þ bh2

2

� �þ eð3Þxx

h5

80 þ bh4

8

� �h iþ ssðbþ hÞ

mh6ss6ð1�mÞ

@2w@x2 þ @2w

@y2

� �þ ð2ls þ ksÞ eð0Þyy ahþ eð1Þyy

h3

12 þ ah2

2

� �þ eð3Þyy

h5

80 þ ah4

8

� �h iþ ssðaþ hÞ

0

8>>><>>>:

9>>>=>>>;

ð67Þ

O11

O21

O31

8><>:

9>=>; ¼

mh4ss40ð1�mÞ

@2w@x2 þ @2w

@y2

� �þ ð2ls þ ksÞ eð0Þxx

bh3

4 þ eð1Þxxh5

80 þ bh4

8

� �þ eð3Þxx

h7

448 þ bh6

32

� �h iþ ssðbþ hÞ

mh4ss40ð1�mÞ

@2w@x2 þ @2w

@y2

� �þ ð2ls þ ksÞ eð0Þyy

ah3

4 þ eð1Þyyh5

80 þ ah4

8

� �þ eð3Þyy

h7

448 þ ah6

32

� �h iþ ssðaþ hÞ

0

8>>><>>>:

9>>>=>>>;

ð68Þ

dimensionless maximum deflections, stresses and first modefrequency for a=h ¼ 10;20 with nonlocal and surface effects. Thenonlocal parameter l and surface effect parameter ss are varied.It is observed that the maximum values of the dimensionlessdeflection increases with increase in nonlocal parameter. Thedimensionless frequency decreases with increase in nonlocalparameter. For a fixed nonlocal parameter an increase in surfaceparameter there is a increase in the maximum deflection anddecrease in the frequency as given in Table 4.

Figs. 10(a), (b) and 11(a), (b), respectively, show the variation of�rxx; �rxx; �syz, and �sxz with thickness coordinate z=h for various valuesof the nonlocal parameter l.

6. Conclusions and remarks

In this work, analytical solutions are presented for laminatedcomposite plates using the Reddy nonlocal third-order shear defor-mation theory considering the surface stress effects. The nonlocaltheory considers the size effect by assuming that stress at a pointdepends on the strain at that point as well as on strains at theneighboring points. Analytical (Navier’s) solutions of bending andvibration of a simply supported composite laminated and isotropic

plates are developed using this theory to illustrate the effect ofnonlocality and surface stress on deflection and vibration frequen-cies for various span-to-thickness (a=h) ratios. The results indicatethat the maximum center deflections increase with an increase inthe nonlocal parameter l and surface stress parameter ss, latterhaving relatively less effect. The opposite is observed for frequen-cies. The difference in solutions between the two theoriesdecreases as the value of a=h increases. Thus, the parameters asso-ciated with the nonlocal formulation have the softening effect.

The nonlocal third-order theory can be extended to include thevon Kármán nonlinearity and finite element models of the theorycan be developed. Bending and vibration solutions of compositelaminates for other types of boundary conditions can be developedusing the finite element method. The effect of nonlocal parameteron the bending and free vibration response of plates with non-rectangular geometries and for boundary conditions that do notadmit analytical solutions can be determined with the help ofthe finite element model.

Appendix A

Z11

Z21

Z31

264

375 ¼

eð0Þxx ðbþ hÞ þ eð1Þxx bhþ eð3Þxx ðh432 þ bh3

4 Þ þ ssðbþ hÞeð0Þyy ðaþ hÞ þ eð1Þyy ahþ eð3Þyy ðh432 þ ah3

4 Þ þ ssðaþ hÞ0

2664

3775 ð66Þ

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