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Interlaminar hygrothermal stresses in laminated plates

Arash Bahrami, Asghar Nosier *

Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Avenue, Tehran, Iran

Received 17 March 2007; received in revised form 31 May 2007Available online 10 June 2007

Abstract

Within the elasticity formulation the most general displacement field for hygrothermal problems of long laminated com-posite plates is presented. The equivalent single-layer theories are then employed to determine the global deformationparameters appearing in the displacement fields of general cross-ply, symmetric, and antisymmetric angle-ply laminatesunder thermal and hygroscopic loadings. Reddys layerwise theory is subsequently used to determine the local deformationparameters of various displacement fields. An elasticity solution is also developed in order to validate the efficiency andaccuracy of the layerwise theory in predicting the interlaminar normal and shear stress distributions. Finally, variousnumerical results are presented for edge-effect problems of several cross-ply, symmetric, and antisymmetric angle-ply lam-inates subjected to uniform hygrothermal loads. All results indicate high stress gradients of interlaminar normal and shearstresses near the edges of laminates. 2007 Elsevier Ltd. All rights reserved.

Keywords: Laminated plates; Interlaminar stresses; Elasticity formulation; Equivalent single-layer theories; Layerwise theory; Hygro-thermal loading

1. Introduction

With the ever-increasing applications of laminated composites in severe environmental conditions hygro-thermal behavior of such laminates has attracted considerable attention. The interlaminar normal and shearstresses, on the other hand, are believed to play a significant role in prediction of dominant cause of failure incomposite laminates. These stresses that exhibit highly localized concentration near the edges of the laminate

are the basis for damage in the form of free-edge delamination and the subsequent delamination growth in theinterior region of laminates, leading to failure at loads below those corresponding to in plane failure. There-fore among the extensive research areas in the analysis of fiber-reinforced composites, the issue of interlam-inar-edge stresses has been subject of tremendous investigations and various theoretical and experimentalmethods are employed to study the edge-effect problem of composite laminates. Since, the present study isdevoted to thermal and hygroscopic problems, in the proceeding only the pertinent works will be referred.A relatively comprehensive review of the various techniques of evaluation of interlaminar stresses that have

0020-7683/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2007.06.004

* Corresponding author. Tel.: +98 21 6616 5543; fax: +98 21 6600 0021.E-mail address:[email protected](A. Nosier).

Available online at www.sciencedirect.com

International Journal of Solids and Structures 44 (2007) 81198142

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appeared in the literature is available in the survey paper by Kant and Swaminathan (2000). Examination ofthe literature survey reveals that the first paper considering the thermal loading in the edge-effect problem ofcomposite laminates is developed by Pagano (1974). He utilized a higher-order shear deformation theory(HSDT) to formulate the edge-effect problem of symmetric balanced laminates subjected to a constant axialstrain and a constant temperature change. Based on the formulation presented byPipes and Pagano (1970),

Wang and Crossman (1977a,b)studied the free-edge stresses in symmetric balanced laminates under a uniformthermal load using a finite element procedure. An approximate elasticity solution was presented byWang andChoi (1982) to determine the boundary-layer stresses due to hygroscopic effects. Wang and Chou (1989)obtained the transient interlaminar thermal stresses in symmetric balanced laminate by means of a zeroth-order perturbation analysis of the equilibrium equations.Webber and Morton (1993) and Morton and Web-ber (1993)by following a similar approach to that used byKassapoglou and Lagace (1986, 1987), attempted tocalculate the interlaminar free-edge stress distribution in symmetric laminates subjected to thermal loads. Byemploying the complementary energy approach together with polynomial expansion of stress functions, Yin(1997) proposed a variational method to evaluate the thermal interlaminar stresses. Also Kim and Atlury(1995) developed an stress-based variational method to obtain interlaminar stresses under combinedthermo-mechanical loading. In an effort to determine the interlaminar stresses, an iterative technique in con-

junction with the extended Kantorovich method is presented byCho and Kim (2000)for thermal and mechan-

ical loads.Rohwer et al. (2001)investigated the transverse shear and normal stresses in composite laminatessubjected to thermal loading by using the extended two-dimensional method and the first-order shear defor-mation plate theory (FSDT). A two-dimensional global higher-order deformation theory was proposed byMatsunaga (2004)for the calculation of out-of-plane stresses in cross-ply laminates subjected to thermal load-ing. Recently, a comprehensive examination of interlaminar stresses in general cross-ply laminates has beenpresented by Tahani and Nosier (2003, 2004). They used a layerwise theory to predict free-edge stresses infinite general cross-ply laminates under various mechanical, thermal, and hygroscopic loadings.

Despite the intensive investigations on the analysis of interlaminar stresses in composite laminates, docu-mented in the literature during the past three decades, few publications focused on the study of such stressesunder hygrothermal loading conditions. In addition, for the case of hygrothermal loading, reliable numericalresults are limited to cross-ply and symmetric balanced laminates. The purpose of the present work, however,

is to analyze the interlaminar stresses in symmetric and unsymmetric cross-ply laminates, symmetric lami-nates, and antisymmetric angle-ply laminates subjected to uniform thermal and/or hygroscopic loadings.Towards this goal, both equivalent single-layer (ESL) and layerwise theories (LWT) are utilized. Each ofthe ESL and layerwise models has its advantages and disadvantages in terms of solution accuracy, solutioneconomy, and simplicity. The ESL models which follow from an assumed global displacement field lead tothe definition of effective overall rigidities and are incapable of providing precise calculation of local 3-D stressfield. On the other hand, the layerwise models which possess the ability of accurately describing the three-dimensional effects, such as free-edge stresses, are computationally expensive. Thus, it is attempted here tointroduce a solution scheme which achieves maximum solution accuracy with minimal solution cost byemploying an efficient analytical procedure. This, on the other hand, is done by utilizing an appropriateESL theory to determine the unknown constant parameters appearing in various laminates displacementfields. Interlaminar stresses are then determined by using an analysis based on the layerwise theory of Reddy.Lastly, an analytical elasticity solution is presented for a specific set of boundary and loading conditions inorder to assess the effectiveness and validity of the present developments.

2. Theoretical formulation

The laminate considered in the present investigation is of thicknessh, width 2b, and assumed to be long inthex-direction so that the strain components are functions ofy and z only (Fig. 1). It is further assumed thatthe mechanical loads (if present) are applied at x=a and x=a only. Also the thermal and hygrothermalloadings are considered to be uniform everywhere in the laminate. Under these conditions the most generalform of the displacement field within the kth layer of the laminate has been shown to be as follows ( Nosierand Bahrami, 2007; Lekhnitskii, 1981):

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uk1 x;y;z B4xyB6xzB2xu

ky;z

uk2 x;y;z B1xzB3x

1

2

B4x2 vky;z

uk3 x;y;z B1xyB5x

1

2B6x

2 wky;z

1

where,uk1 x;y;z,u

k2 x;y;z, andu

k3 x;y;zrepresent the displacement components in the x-,y-, andz-direc-

tions, respectively, of a material point initially located at (x,y, z) in the kth lamina of the laminate. It is nextnoted that as long as the loading conditions at x =aand x =a are similar (so that the laminate inFig. 1isglobally in equilibrium), the following conditions will hold:

uk1 x;y;z u

k1 x;y;z

uk2 x;y;z u

k2 x;y;z

uk3 x;y;z u

k3 x;y;z:

2

Upon imposing these restrictions on (1) it is readily seen that the constants B4 and B5 must vanish and thedisplacement field in(1)is, therefore, reduced to what follows:

uk1 x;y;z B2xB6xzu

ky;z 3a

uk2 x;y;z B1xzB3xv

ky;z 3b

uk3 x;y;z B1xy

12B6x

2 wky;z: 3c

Furthermore, by replacingu(k)(y, z) appearing in(3a)by B3y +u(k)(y, z), it becomes apparent that the terms

involvingB3in(3)will generate no strains and, therefore, can be omitted (these terms, in fact, will correspond

to an infinitesimal rotation of the laminate about thez-axis inFig. 1). Thus, the most general form of an arbi-trary laminated composite laminate is given as follows:

uk1 x;y;z B2xB6xzu

ky;z 4a

uk2 x;y;z B1xzv

ky;z 4b

uk3 x;y;z B1xy

12B6x

2 wky;z: 4c

The displacement field in(4)may be used, in principle, for calculating the stress field in any composite lam-inate subjected to arbitrary combinations of self-equilibrating mechanical and uniform hygrothermal loads. Inthe present study, however, our attention is focused on symmetric and unsymmetric cross-ply laminates, sym-

metric laminates, and antisymmetric angle-ply laminates subjected to uniform hygroscopic and thermal

Fig. 1. Laminate geometry and coordinate system.

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loadings. Since the solutions developed here in the present study significantly depend on the laminationscheme, each of the aforementioned laminates is examined separately.

2.1. Cross-ply laminates

For symmetric and unsymmetric cross-ply laminates (e.g., seeJones, 1998) subjected to uniform hygrother-mal loadings, based on physical grounds, the following restrictions will, furthermore, hold (seeFig. 1):

uk1 x;y;z u

k1 x;y;z 5a

uk2 x;y;z u

k2 x;y;z: 5b

Upon imposing (5a)on (4a) it is concluded that u(k)(y, z) = 0. Also from (5b) and (4b) it is concluded thatB1= 0. Thus, for cross-ply laminates the most general form of the displacement field is given as follows:

uk1 x;y;z B2xB6xz

uk2 x;y;z v

ky;z

u

k

3 x;y;z

1

2B6x2

wk

y;z:

6

It is to be noted that for symmetric cross-ply laminates it can readily be shown that (see Eqs.(8)below)B6= 0.

2.2. Symmetric laminates

For symmetrically laminated composite plates (e.g., seeJones, 1998) under uniform hygrothermal loadingsthe deformational behavior of the material points within the laminates remains the same with respect to themiddle surface. Thus, the following restrictions must hold with respect to the middle surface of any symmetriclaminate with N layers:

uk1 x;y;z u

Nk11 x;y;z 7a

uk2 x;y;z uNk12 x;y;z: 7b

From(7a) and (4a), it is concluded thatB6= 0. Also from(7b) and (4b)it is concluded thatB1= 0. Therefore,for symmetric laminates the most general form of the displacement field is given as:

uk1 x;y;z B2xu

ky;z

uk2 x;y;z v

ky;z

uk3 x;y;z w

ky;z:

8

2.3. Antisymmetric angle-ply laminates

For antisymmetric angle-ply laminates withNlayers (e.g., seeJones, 1998) subjected to uniform hygrother-mal loadings the following condition must hold (see alsoNosier and Bahrami, 2007):

uk1 x;y;z u

Nk11 x;y; z: 9

From(9) and (4a)it is concluded thatB6= 0 and, therefore, the most general displacement field for such lam-inates is given as follows:

uk1 x;y;z B2xu

ky;z

uk2 x;y;z B1xzv

ky;z

uk3 x;y;z B1xyw

ky;z:

10

The displacement fields in(6), (8), and (10) can be represented in one place as:


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uk1 x;y;z B2xdCB6xzdASu

ky;z

uk2 x;y;z dAB1xzv

ky;z

uk3 x;y;z dAB1xy

1

2dCB6x

2 wky;z

11

where dA=dAS= 0 anddC= 1 for unsymmetric cross-ply laminates,dA=dC= 0 anddAS= 1 for symmetriclaminates, anddA=dAS= 1 anddC= 0 for antisymmetric angle-ply laminates. It is to be noted that the termsinvolving the constant parameters B1, B2, and B6 describe the global deformations within the laminateswhereas the functionsu(k)(y, z),v(k)(y, z), andw(k)(y, z) correspond to the local deformations ofkth layer withinthe laminate.

2.4. Equivalent single-layer theories

It is well known by now that ESL theories are adequate in predicting the global responses of the compositelaminates. On the other hand, these theories are simpler and computationally less time consuming than thelayerwise theories. The objective of the present section is to present the simplest ESL theory which, on theother hand, will provide sufficiently accurate results for the unknown constant parameters B1, B2, and B6appearing in(11). For unsymmetric cross-ply and antisymmetric angle-ply laminates numerical studies con-ducted by the authors indicate the first-order shear deformation theory of plates (FSDT), also known asMindlinReissner plate theory, is fairly accurate in predicting these parameters. For symmetric laminates,however, an improved first-order theory must be developed and used for determining the appropriate constantparameter (i.e.,B2) appearing in the displacement field(11).

In FSDT the components of the displacement vector at a material point in the laminate are expressed in thefollowing form (Reddy, 2003):

u1x;y;z ux;y zwxx;y

u2x;y;z vx;y zwyx;y

u3x;y;z wx;y

12

whereu,v, andwdenote the displacements of a point located on the middle plane of the laminate. Also wxandwyare the rotations of transverse normals about the y- andx-axes, respectively. The displacement field in(6)indicates that for hygrothermal problems of cross-ply laminates the displacement field within FSDT (i.e., Eq.(12)) must take the following simpler form:

u1x;y;z B2xB6xz

u2x;y;z Vy zWyy

u3x;y;z 12B6x

2 Wy

13

with B6 being equal to zero when the cross-ply laminate is symmetric. Similarly, based on the displacementfield in(10), it is concluded that for hygrothermal problems of antisymmetric angle-ply laminates the displace-

ment field within FSDT must take the following simpler form:

u1x;y;z B2xUy zWxy

u2x;y;z B1xzVy zWyy

u3x;y;z B1xyWy:

14

As it is pointed out earlier, for symmetric laminates FSDT is incapable of accurately determining the unknownparameterB2appearing in(8). This is attributed to the fact that, when such laminates are subjected to uniformhygrothermal loads, the unknown functionswx, wy, andw vanish and the remaining terms in (12)will not beadequate for accurate determination ofB2. On the other hand, in symmetric laminates under hygrothermalloadings the displacement componentsu1,u2must be even functions of thickness coordinatezwhereasu3mustbe an odd function ofz. For such laminates it is found that a more accurate result for B2 may be found by

introducing the following displacement field:


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u1x;y;z ux;y jzj~wxx;y

u2x;y;z vx;y jzj~wyx;y

u3x;y;z zwzx;y:

15

Numerical studies reveal, furthermore, that the thickness stretching term (i.e., zwz) introduced in(15)has anegligible effect on the accuracy ofB2and, therefore, can be omitted when symmetric laminates are subjectedto uniform hygrothermal loadings. Based on the displacement field in (8), it is, thus, concluded that the sim-plest appropriate displacement field for symmetric laminates is given as follows:

u1x;y;z B2xUy jzj ~Wxy

u2x;y;z Vy jzj ~Wyy

u3x;y;z 0:

16

The theory that will be developed here using(16)will be referred to as the improved first-order shear defor-mation plate theory (IFSDT). It remains to find and, thereafter, solve the equilibrium equations correspond-

ing to the displacement fields in Eqs.(13), (14), and (16)in order to determine the constant parameters existingin these displacement fields. For brevity, however, here the detail of the procedure is demonstrated for the dis-placement field in(16)only and the results corresponding to the displacement fields in (13) and (14)are sum-marized. By using the displacement field (16)in the principle of minimum total potential energy ( Fung andTong, 2001) and treating the constant B2 as an unknown parameter five equilibrium equations are obtainedwhich may be presented as follows:

dU :N0xy0 17a

dV :N0y0 17b

d ~Wx : ~Qx ~M0xy0 17c

d ~Wy : ~Qy ~M0y0 17d

dB2 :

Z bb

Nx dy 0 18

where a prime indicates an ordinary differentiation with respect to variable y and the stress and moment resul-tants appearing in(17) and (18)are defined as follows:

Nx;Ny;Nxy

Z h=2h=2

rx;ry;rxy dz; ~Qx; ~Qy

Z h=2h=2

sgnzrxz;ryz dz

~My; ~Mxy

Z h=2

h=2

jzjry; rxy dz 19a

with

sgnz 1 for z< 0

1 for z> 0:

19b

Upon substitution of the displacement field (16)into Eq. (19a), through the linear strain-displacement rela-tions of elasticity and the plane-stress constitutive law (Herakovich, 1998) of a lamina, the stress and momentresultants may be expressed as:


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Nx

Ny

Nxy~My~Mxy

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

9>>>>>>=

>>>>>>;

A16 A12 ~B16 ~B12 A11

A26 A22 ~B26 ~B22 A12

A66 A26 ~B66 ~B26 A16~B26 ~B22 D26 D22 ~B12~B66

~B26 D66 D26

~B16

26666664

37777775

U0

V0

~W0x~W0y

B2

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

9>>>>>>=

>>>>>>;

NTx

NTy

NTxy

~MTy

~MTxy

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

9>>>>>>>=

>>>>>>>;

NHx

NHy

NHxy

~MHy

~MHxy

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

9>>>>>>>=

>>>>>>>;20a

and

~Qy~Qx

( ) k2

A44 A45

A45 A55

~Wy~Wx

( ) 20b

where the laminate rigidities Aij, ~Bij, and Dijare defined as:

Aij; ~Bij;Dij

Z h=2h=2

Qij1; jzj;z2 dz 21

with Qijbeing the transformed reduced stiffness (Herakovich, 1998) of an orthotropic lamina. AlsoNT, ~MT are

referred to as the thermal resultants and defined as:

NTx ;NTy;N

Txy

Z h=2h=2

hQ11; Q12; Q16ax Q12; Q22; Q26ay Q16; Q26; Q66axy

iDT dz 22a

~MTy; ~MTxy

Z h=2h=2

hQ12; Q16ax Q22; Q26ay Q26; Q66axy

iDTjzj dz 22b

where DTdenotes the temperature change and ax, ay, and axydenote the transformed coefficients of thermalexpansions (Herakovich, 1998). The hygroscopic resultants are, similarly, defined as:

NHx;NHy;N

Hxy Z

h=2

h=2 hQ11; Q12; Q16bx Q12; Q22; Q26by Q16; Q26; Q66bxyiDMdz 23a~MHy; ~M

Hxy

Z h=2h=2

hQ12; Q16bx Q22; Q26by Q26; Q66bxy

iDMjzj dz: 23b

InEqs. (23), DMis the percent moisture (by weight) absorbed by each layer in the laminate andbx,by, andbxyindicate the transformed coefficients of hygroscopic expansions (Herakovich, 1998). Also in(20b)the constantk2 is called the shear correction factor which is often introduced for in a first-order theory to modify the lam-inate transverse shear rigidities.

For free edges of the laminate at y= baccording to the principle of minimum total potential energy thefollowing traction-free boundary conditions must be imposed:

NyNxy0 at y b 24a

and~Mxy ~My0 at y b: 24b

Integrating the equilibrium equations in(17a) and (17b)and imposing the boundary conditions in(24a)yield:

Nyy Nxyy 0: 25

These conditions are used to express U0 andV0 appearing in(20a)in terms of ~Wxand ~Wy, andB2. This way theresultants Nx, ~My, and ~Mxyare all expressed in terms of ~Wxand ~Wy, andB2. Finally, upon substitution of ~Mxyand ~My, and(20b)into the equilibrium equations (17c) and (17d), the following results are obtained:

d ~Wx : D66~W00x k

2A55~Wx D26~W00yk

2A45~Wy0 26a

d~

Wy : D26

~

W00x k

2

A45~

Wx D22

~

W00yk

2

A44~

Wy0 26b


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with the coefficients D22, D26, and D66being displayed inAppendix A. Upon imposing the boundary conditionsin (24b) on the general solution of (26), the generalized displacement functions ~Wx and ~Wy are obtained interms of the unknown constant B2 which, on the other hand, may be presented as follows:

~Wxy X2

j1

Ajsinhkjy; ~Wyy X2

j1

DjAjsinhkjy 27

where the coefficients Ajand Dj j 1; 2 are given as:

A1 d1B2mTy m

Hy 28a

A2 d2B2mTxy m

Hxy 28b

Dj D66k

2j k

2A55

D26k2j k

2A45j 1; 2: 28c

The constant parameters appearing in(28a) and (28b)are presented inAppendix A. In addition, k2j (j= 1,2)appearing in(28c)are the roots of the following characteristic equation (associated with Eqs. (26)):

k2 D66k2A55k2 D22k2A44 k2 D26k2A452 0: 29

Finally, upon substituting(27)into the stress resultant Nx and the subsequent result into the global equilib-rium equation(18), the constant parameter B2 is determined which, on the other hand, may be presented asfollows:

B2 1

hnTx n

Hx 30

where the expressions for h, nTx , and nHx are also listed inAppendix A.

It is remarked earlier that a procedure similar to that outlined here for symmetric laminates may beemployed to determine the unknown parameters appearing in the displacement fields of unsymmetric cross-ply and antisymmetric angle-ply laminates. For brevity, however, only the appropriate results are presented

here. For cross-ply laminates the constants B2 and B6 appearing in(13)are found to be as follows:

B2 1

h1h4h2h3h4 n

Tx n

Hx

h2 m

Tx m

Hx

31a

B6 1

h1h4h2h3h1 m

Tx m

Hx

h3 n

Tx n

Hx

: 31b

The expressions for the constant parameters appearing in (31)are for clarity displayed inAppendix B. Forantisymmetric angle-ply laminates the constants parameters B1 and B2 appearing in the displacement field(14)are found to be as follows:

B1 1

h1h4h2h3

h4 nTx n

Hx h2 mTxy mHxy h i 32a

B2 1

h1h4h2h3h1 m

Txy m

Hxy

h3 n

Tx n

Hx

h i: 32b

The constant parameters in(32)are listed inAppendix C.

2.5. Layerwise laminated plate theory of Reddy

Due to existence of local high stress gradient and the three-dimensional nature of the boundary-layer phe-nomenon, the interlaminar stresses in the boundary-layer regions can not be computed accurately by the ESLtheories. Thus, Reddys layerwise theory that is capable of modeling localized three-dimensional effects is uti-lized here to carry out the hygrothermal interlaminar stress analysis in laminates with free edges. The theory

assumes that the displacement components of a generic point in the laminate are given by ( Nosier et al., 1993):

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u1x;y;z ukx;yUkz

u2x;y;z vkx;yUkz k 1; 2; . . . ;N1

u3x;y;z wkx;yUkz

33

withkbeing a dummy index indicating summation from 1 toN+ 1. In(33)u1,u2, andu3denote the displace-

ment components in thex-,y-, andz-directions, respectively, of a material point located at (x,y, z) in the unde-formed state. Also,uk(x,y), vk(x,y), and wk(x,y) indicate the displacements of all points located, initially, onthe kth plane in the x-, y-, and z-directions, respectively. In addition, Ncorresponds to the total number ofnumerical layers considered in a laminate. Furthermore, Uk(z) are the global Lagrangian interpolation poly-nomials associated with the kth surface (seeTahani and Nosier, 2004; Reddy, 2003; Nosier et al., 1993). It is tobe noted that the layerwise concept introduced here is very general in a sense that the accuracy of the solutioncan be improved as close as desired by increasing the number of the subdivisions through the thickness orincreasing the order of interpolation polynomials through the thickness. However, in the present study, theinterpolation functions Uk(z) are assumed to have linear variation through the thickness of each numericallayer. The elasticity-based displacement field in (11)indicates that the displacement field of LWT appearingin(33)must be rewritten in a simpler form as follows:

u1x;y;z B2xdCB6xzdASUkyUkz

u2x;y;z dAB1xzVkyUkz k 1; 2; . . . ;N1

u3x;y;z dAB1xy12dCB6x

2 WkyUkz:

34

Substitution of(34)into the linear straindisplacement relations of elasticity (e.g., seeFung and Tong, 2001)yields the following results:

ex B2dCB6z; eyV0kUk; ez WkU

0k; cyz W

0kUk VkU

0k

cxz dASUkU0k dAB1y; cxydASU

0kUk dAB1z: 35

The equilibrium equations of a laminate within LWT are obtained employing(35)in the principle of minimumpotential energy (e.g., seeFung and Tong, 2001). The results are, in general, 3(N+ 1) local equilibrium equa-tions corresponding to 3(N+ 1) unknown functionsUk, Vk, and Wk, and three global equations correspond-ing to the three unknown constants B1, B2, and B6 as follows:

dUk :dAS Qkx

dMkxydy

! 0 36a

dVk :Qky

dMkydy

0 36b

dWk

:N

k

z

dRky

dy 0 36c

dB1 :dA

Z h=2h=2

Z bb

rxzyrxyz dydz 0 37a

dB2 :

Z h=2h=2

Z bb

rx dydz 0 37b

dB6 :dC

Z h=2h=2

Z bb

rxzdydz 0: 37c

Here, the generalized stress resultants, appearing in Eqs.(36a)through(36c), are defined as:


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Mky;Mkxy;N

kz

Z h=2h=2

ryUk;rxyUk;rzU0k dz

Qkx ;Qky;R

ky

Z h=2h=2

rxzU0k; ryzU

0k;ryzUk dz:

38

It is next noted that the three-dimensional stressstrain relations within the kth layer of composite laminateare given as (e.g., seeHerakovich, 1998):

frgk

Ck

feg feTg feHg k

39

with Cbeing the transformed stiffness matrix and {eT} and {eH} being the thermal and hygroscopic strains,respectively. By merely substituting relations(35)into(39)and the subsequent results into(38), the generalizedstress resultants are expressed in terms of the displacement functions:

Mky;Mkxy;N

kz dASD

kj

26;Dkj

66;Bjk

36U0j D

kj

22;Dkj

26;Bjk

23V0j B

kj

23;Bkj

36;Akj

33Wj

Bk12;Bk16;A

k13B2dAD

k26;D

k66; B

k36B1dCD

k12;D

k16; B

k13B6

MkTy ;MkTxy ;N

kTz M

kHy ;M

kHxy ;N

kHz 40

Qkx;Qky;R

ky dASA

kj55;A

kj45;B

kj45Uj A

kj45;A

kj44;B

kj44Vj B

jk45;B

jk44;D

kj44W

0j

dAAk55;A

k45;B

k45B1y

where the expressions for laminate rigidities and the thermal and hygroscopic resultants are listed inAppendixD. Lastly, upon substitution of Eqs. (40)into(36) and (37), the governing equations of equilibrium are ob-tained in the following form:

dUk :dAS Dkj

66U00j A

kj

55UjDkj

26V00j A

kj

45Vj Bkj

36Bjk

45W0j

h i dAB1A

k55y 41a

dVk :dASDkj

26U00j dASA

kj

45UjDkj

22V00j A

kj

44Vj Bkj

23Bjk

44W0j dAB1A

k45y 41b

dWk :dASBkj

45Bjk

36U0j B

kj

44Bjk

23V0jD

kj

44W00j A

kj

33Wj

B2Ak13dAB1B

k45 B

k36 dCB

k13B6N

kTz N

kHz 41c

dB1 :dA

Z bb

hdASA

j55UjA

j

45Vj Bj

45W0jy dASD

j

66U0jD

j

26V0j B

j

36Wj

dAB1A55y2 D66 B2B16dCB6D16M

Txy M

Hxy

idy 0 42a

dB2 :

Z bb

dASBj16U

0jB

j12V

0jA

j13Wj B2A11dCB6B11dAB1B16N

Tx N

Hx

h idy 0 42b

dB6 :dC

Z bb

dASDj

16U0jD

j

12V0j B

j

13Wj B2B11dCB6D11dAB1D16MTx M

Hxh i

dy 0: 42c

To investigate the free-edge-effect problem,Eqs. (41) and (42)must be solved subject to the following traction-free boundary conditions at y= b:

dASMkxyM

kyR

ky0 at y b: 43

As it is seen from(41)the local displacement equations comprise of 3(N+ 1) coupled second-order differentialequations with constant coefficients. It is clear that such a linear system may be solved analytically using, forexample, the state space approach. This way, the displacement components will be found, in general, in termsofB1,B2,B6, and 6(N+ 1) constants of integration. Three global displacement equations in (42)and 6(N+ 1)boundary conditions in(43)are, then, employed to determine integration constants and unknown coefficientsB1,B2, andB6. A detailed description of the solution scheme has been presented in Nosier and Bahrami (2006,2007)and, therefore, for brevity will not be repeated here. It is to be noted that the constant parameters B1,B2,

and B6 appearing in(34)may be considered to be known from an analysis based on the first-order theories

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developed here (see Eqs.(30)through(32)) since they represent global response quantities of laminates. In factthe accuracy of the first-order theories in predicting these parameters will be assessed within the present studywhen numerical results are discussed.

2.6. Theory of elasticity

In order to verify the validity of the solutions obtained in the previous sections, an analytical elasticity solu-tion is presented here for particular boundary and loading conditions. A generally stacked laminate subjectedto a uniform hygrothermal loading is considered. It is, moreover, assumed that the ends of the laminate inFig. 1 are gripped so that the line AB (EF) is not allowed to have rotations about the x-axis and the linecc(the x-axis and the line dd). Under such assumptions the constantsB1and B6vanish and the displacementfield of elasticity in(4)is simplified to what follows:

uk1 x;y;z B2xu

ky;z

uk2 x;y;z v

ky;z

uk3 x;y;z w

ky;z:

44

The constantB2appearing in(44)represents the uniform axial strain in thex-direction due to a hygrothermalloading and will be determined here within the elasticity theory. By employing the displacement field (44)inthe principle of minimum total potential energy (e.g., see Fung and Tong, 2001) the local and global equilib-rium equations are readily found to be:

du :orkxy

oy

orkxzoz

0 45a

dv :orky

oy

orkyz

oz 0 45b

dw :or

k

yz

oy or

kz

oz 0 45c

and

dB2 :

Z bb

Z h=2h=2

rx dzdy 0; 46

respectively. For traction-free boundary conditions at y= b, no analytical solution seems to exist for Eqs.(45a)through(45c). It is, however, noted thatEqs. (45)admit an analytical solution for the following bound-ary conditions at y= b:

rky uk3 r

kxy 0 at y b: 47

Using the three-dimensional stressstrain relations in (39)together with strain-displacement relationships ofelasticity (e.g., seeFung and Tong, 2001), the local equilibrium equations in (45)may be expressed in termsof the displacement components as follows:

Ck66u

k;yy

Ck55u

k;zz

Ck26v

k;yy

Ck45v

k;zz

Ck45

Ck36w

k;yz 0

Ck26u

k;yy

Ck45u

k;zz

Ck22v

k;yy

Ck44v

k;zz

Ck23

Ck44w

k;yz 0

Ck45

Ck36u

k;yz

Ck23

Ck44v

k;yz

Ck44w

k;yy

Ck33w

k;zz 0

48

where a coma followed by a variable denotes partial differentiation with respect to that variable. Similarly, theboundary conditions in(47)are expressed in terms of the displacement components as follows:


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Ck26u

k;y

Ck22v

k;y

Ck23w

k;z

Ck12e

kTx e

kHx B2

Ck22e

kTy e

kHy

Ck23e

kTz e

kHz

Ck26c

kTxy c

kHxy

Ck66u

k;y

Ck26v

k;y

Ck36w

k;z

Ck16e

kTx e

kHx B2 at y b

Ck

26ekTy e

kHy

Ck36e

kTz e

kHz

Ck66c

kTxy c

kHxy

wk 0:

49

It remains next to obtain the solution of Eqs.(48)satisfying the boundary conditions in(49)and the followingconditions:

The traction-free conditions at the top surface of the first layer:

r1z r1yz r

1xz 0 at the top surface of the 1st layer: 50a

The traction-free conditions at the bottom surface of the Nth layer:

rNz rNyz r

Nxz 0 at the bottom surface of the Nth layer: 50b

The displacement continuity conditions at every interface within the laminate:

uk1 u

k11 ; u

k2 u

k12 ; and u

k3 u

k13 at interfaces: 50c

The stress equilibrium conditions at every interface within the laminate:

rkz rk1z ;r

kyz r

k1yz ; and r

kxz r

k1xz at interfaces: 50d

Eqs.(48)with the boundary conditions in(49)may be solved by means of the Fourier series technique. A com-plete description of the solution procedure is discussed inNosier and Bahrami (2007)and, for the sake of brev-ity, is not repeated here. The displacement components within the kth layer of the laminate are found to be:

uk1 x;y;z B2x X

1

m0X

6

i1

Akmiekkmizsinamy X

1

m1

Akmsinamy

uk2 x;y;z

X1m0

X6i1

BkmiAkmiekkmizsinamy

X1m1

Bkmsinamy

uk3 x;y;z

X1m0

X6i1

CkmiAkmiekkmizcosamy

51

where am= (2m+ 1)p/2b and kkmi (i= 1,2, . . ., 6) are the roots of the following sixth-order polynomialequation:

a2mC

k66

Ck55k

2km a

2mC

k26 k

2km

Ck45 amkkm

Ck45

Ck36

a2m

Ck

26

Ck

45

k2

km

a2m

Ck

22

k2km

Ck

44

amkkmCk

44

Ck

23

amkkmCk45

Ck36 amkkm

Ck44

Ck23 a

2mC

k44 k

2km

Ck33

0: 52

Also the parameters Akmand Bkm appearing in(51)are given by:

Akm C

k12

Ck26 C

k16

Ck22

Ck66

Ck22

Ck2

26

B2am and Bkm C

k16

Ck26 C

k12

Ck66

Ck66

Ck22

Ck2

26

B2am 53a

with

am 8b

p2

1m

2m12: 53b

Moreover, the coefficients Bkmi and Ckmi appearing in(51)are determined from the following relations:


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BkmiCkmi

a2mC

k26 k

2kmi

Ck45 amkkmi

Ck45

Ck36

a2mC

k22 k

2kmi

Ck44 amkkmi

Ck44

Ck23

" #1a2m

Ck66

Ck55k

2kmi

a2mC

k26

Ck45k

2kmi

( ): 54

The next step of the analysis is to determine the 6Nunknowns, namely,Akmi(k= 1,2, . . ., Nand i= 1, . . ., 6)for each Fourier integer m and the unknown parameter B2. These unknowns will be determined in a try and

error process by imposing the conditions in(50a)through(50d)and Eq.(46). For this purpose, an initial valueis assumed forB2. This value is then substituted into conditions(50a)through(50d)to obtain a system of 6Nalgebraic equations in terms of Akmi (for each m) which upon solving yields the unknown coefficients Akmi(k= 1,2, . . ., Nand i= 1, . . .,6). Next, upon carrying out the integration in (46)a new B2 is found which,on the other hand, may be presented as:

B2 1

bPN

k1C

k11hk

bNTx NHx

PNk1

P1m1

hkCk12Bkm

Ck16Akm1

m

2PNk1

P1m0

P6i1

Bkmi Ck

12C

k

16

kkmi

Ckmi Ck

13

am

Akmi1

msinhkkmihk

2

26664

37775 55

where NTx and NHx appearing in(55)are defined as:

NTx XNk1

Ck11a

kx

Ck12a

ky

Ck13a

kz

Ck16a

kxyhkDT

NHx XNk1

Ck11b

kx

Ck12b

ky

Ck13b

kz

Ck16b

kxyhkDM:

56

It is reminded here that for convenience the z coordinate is located at the middle surface of each ply. The newvalue ofB2is then compared with that assumed initially. If the difference between two values is significant, thenew value ofB2in (55)is used as the initial value and the procedure is repeated until B2is obtained with anydesirable degree of accuracy. It is to be noted that by substituting B2 into(53a)and subsequent results into(51)the displacement components will be obtained. Upon substitution of the displacement field into strain-dis-

placement relations of linear elasticity (e.g., seeFung and Tong, 2001) and using the three-dimensional Hookelaw(39)the stress components are finally determined.It is pointed out that the elasticity solution presented here is, although analytical, not an exact solution

since the Gibbs phenomenon appears in the Fourier expansions introduced in the solution procedure (seeNos-ier and Bahrami, 2007). In fact, according to the solution obtained here, the interlaminar normal stressrzwillvanish at points located on the edges of the laminate at y= bwhich is not a correct result. The exact value ofrz on these edges may, however, be determined by considering the following three-dimensional strainstressrelations (e.g., seeHerakovich, 1998):

ekx S

k11r

kx

Sk12r

ky

Sk13r

kz

Sk16r

kxy a

kx DTb

kx DM

ekz S

k13r

kx

Sk23r

ky

Sk33r

kz

Sk36r

kxy a

kz DTb

kz DM

57

where Sk

ij s are the off-axis compliances of thekth layer. At the edges of the laminateuk

3 is specified to vanish(see(47)). Therefore, at all points on these edges (except for points located at the intersections of these edgeswith interfaces, bottom surface, and top surface of the laminate) it can be concluded:

ezou

k3

oz 0 at y b: 58

Next, substitution of(47) and (58)(and ekx B2) into(57)results in:

B2 Sk11r

kx

Sk13r

kz a

kx DTb

kx DM 59a

0 Sk13r

kx

Sk33r

kz a

kz DTb

kz DM: 59b

Upon solvingEqs. (59)the exact value ofrkz is obtained to be as follows:


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rkz S

k11a

kz DTb

kz DM

Sk13B2a

kx DTb

kx DM

Sk2

13 Sk11

Sk33

: 60

Relation(60)indicates that the interlaminar normal stress rkz has a constant value at the edges of each lamina(aty = b) and, in addition, this constant value changes from one layer to another (adjacent) layer because of

changes in fiber orientations.

3. Numerical results and discussions

In this section the validity and accuracy of the theories and procedures introduced here in the present studyare first assessed trough several numerical examples by considering cross-ply laminates with special boundaryconditions(47)aty= b. For this purpose numerical results of ESL and layerwise theories for a uniform tem-perature change are compared with those of elasticity theory. The interlaminar stresses within various cross-ply, symmetric, and antisymmetric angle-ply laminates with free edges at y= b are then closely examined.The material layers within any laminate are assumed to have identical thickness (hk), density, and on-axisproperties. The mechanical and thermal properties of each lamina are, furthermore, assumed to be the same

as those of graphite/epoxy T300/5208 (Herakovich, 1998):

E1 132 GPa; E2 E3 10:8 GPa; G12 G13 5:65 GPa; G23 3:38 GPa;

m12 m13 0:24; m23 0:59; a1 0:77106=C; a2 a3 0:2510

6=C:61

Moreover, the thickness of each physical lamina is assumed to be 1 mm (i.e., hk= 1 mm). In addition, since theresponses of various laminates due to thermal and hygroscopic loads are similar, only thermal results (due to auniform temperature change DT= 1 C) are presented here. Furthermore, in the numerical examples the value5/6 is used for the shear correction factor, k2, in the first-order plate theories and the interlaminar stress com-ponents according to LWT are computed by integrating the local equilibrium equations of elasticity (see, e.g.,Reddy, 2003).

Numerical values ofB2 according to FSDT, IFSDT, LWT, and elasticity theory are listed in Table 1forcross-ply [90/0/0/90] and [0/90/0/90] laminates with boundary conditions in (47) under the uniformthermal load DT= 1 C. The ratio of the laminate width to its thickness is assumed to be 5 (i.e., 2 b/h= 5)for the numerical results inTable 1. It is observed that the layerwise theory overestimates the numerical valuesofB2 and by, however, increasing the number of numerical layers (p) in each physical layer (see Tahani andNosier, 2004; Nosier et al., 1993) the results of LWT approach those of the elasticity theory. It is to be notedthat the numerical values ofB2 as predicted by FSDT and IFSDT are sufficiently accurate as compared tothose of elasticity theory. In addition, the slight differences observed between the first-order theories andthe elasticity theory in predictingB2have no considerable effects on the interlaminar stress distributions withinvarious laminates. The interlaminar stress components are depicted inFigs. 2 and 3for thermal problems ofcross-ply [90/0/0/90] and [0/90/0/90] laminates subjected to the boundary conditions(47)at y = b. It

Table 1Numerical value of the constant parameter B2 10

6 according to FSDT, IFSDT, LWT, and elasticity theory for special boundaryconditions in Eqs.(47)

P= 1 P= 2 P= 3 P= 4 P= 5 P= 6 P= 7 P= 8

[90/0/0/90]Layerwise theory 1.7296 1.6980 1.6906 1.6871 1.6852 1.6840 1.6833 1.6828Elasticity theory 1.6817IFSDT 1.7876

[0/90/0/90]Layerwise theory 1.6633 1.6234 1.6252 1.6113 1.6093 1.6080 1.6072 1.6067Elasticity theory 1.6051

FSDT 1.5389


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is observed that the results of the layerwise theory are in excellent agreements with those of the elasticitytheory.

The free-edge-effect problems are now examined by using several numerical examples. The numerical valuesof the unknown constant parameters appearing in various displacement fields are presented for cross-ply, sym-metric, and antisymmetric angle-ply laminates inTable 2. The results are generated for various width to thick-ness ratios. Close examination ofTable 2reveals that the FSDT and IFSDT results are sufficiently accurate

Fig. 2. Distribution of interlaminar stresses near the middle plane of [0/90/0/90] laminate and in the top 90 layer.

Fig. 3. Distribution of interlaminar stresses near 90/0 interface of [90/0/0/90] laminate and in the top 90layer.

Table 2

Unknown constants of displacement field according to FSDT and LWTLaminate Theory Constants 2b/h= 5 2b/h= 10 2b/h= 20

[90/0/0/0] FSDT B2 7.7625e7 7.7625e7 7.7625e7B6 1.7317e3 1.7317e3 1.7317e3

LWT B2 7.6235e7 7.7038e7 7.7697e7B6 1.7197e3 1.7274e3 1.7358e3

[90/0/0/90] IFSDT B2 1.5323e6 1.5472e6 1.5546e6LWT B2 1.5141e6 1.5386e6 1.5511e6

[45/10/10/45] IFSDT B2 1.8309e6 1.9222e6 1.9672e6LWT B2 1.7793e6 1.8970e6 1.9577e6

[45/10/10/45] FSDT B1 5.5921e3 5.5921e3 5.5921e3B2 8.6592e7 8.6592e7 8.6592e7

LWT B1 5.5468e3 5.5725e3 5.5826e3

B2 7.9602e7 8.3327e7 8.5260e7


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for thin to moderately thick laminates. For thick laminates, however, numerical investigation indicates thatslight inaccuracy in global terms (i.e., terms involving the unknown parameters B1, B2, and B6 appearing in(4)) has insignificant effects on the accuracy of stress distributions within various laminates. It is, therefore,concluded here that the explicit expressions obtained for these parameters according to first-order theories(i.e., relations(30), (31), and (32)for symmetric, cross-ply, and antisymmetric angle-ply laminates) may con-

veniently be used within other theories such as LWT and elasticity theory (see alsoNosier and Bahrami, 2006,2007). This, in fact, is done here in the remaining of the present study. In order to obtain accurate results forinterlaminar stresses, each physical layer is subdivided into, unless otherwise mentioned, 20 numerical layers(i.e., p = 20). Moreover, the ratio of width to thickness is assumed to be 10 (i.e., 2b/h= 10). It is to be notedthat for symmetric cross-ply laminates either relation(30) or (31)may be used to determine the single constantparameter B2(noting thatB6= 0 for symmetric laminates).Fig. 4shows the distributions of the interlaminarnormal stressrzalong the two 0/90interfaces in the [0/90/0/90] laminate. It is observed that the rzexhib-its different behavior at these interfaces. More explicitly, it is seen that the maximum numerical value ofrzisquite larger in the top 0/90interface. The distributions of the interlaminar stressesrzandryzalong the upperand middle interfaces of the symmetric cross-ply [90/0/0/90] laminate are shown inFig. 5. Both stresses areseen to grow abruptly in the vicinity of the free edge, while being zero in the interior region of the laminate. Itis also noted that the interlaminar shear stress ryzrises toward the free edge and decreases rather suddenly to

zero at the free edge. By increasing the number of numerical layers in each lamina ryzbecomes slightly closerto zero but it may never become zero. This is, most likely, due to the fact that within LWT the generalizedstress resultantRky, rather than ryz, is forced to vanish at the free edge (see Eq.(43)). It is reminded here that

Fig. 4. Interlaminar normal stress along the two 0/90 interfaces of [0/90/0/90] laminate.

Fig. 5. Distributions of interlaminar stresses along the top and middle interfaces of [90/0/0/90] laminate.


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the interlaminar shear stress rxzis identically zero everywhere in cross-ply laminates. Also, due to symmetry,the interlaminar shear stress ryz vanishes at the middle surface of symmetric cross-ply laminates. The varia-tions of interlaminar normal stress rz through the thickness of the unsymmetric cross-ply laminate [90/0/90/0] are depicted inFig. 6as the free edge is approached. It is observed that the maximum negative value

ofrzoccurs within the top 0layer and the maximum positive value ofrzoccurs within the bottom 90layer

Fig. 6. Interlaminar normal stressrz through the thickness of [90/0/90/0] laminate.

Fig. 7. Distribution of interlaminar shear stressryz through the thickness of bottom 90 layer of [0/90/90/0] laminate.

Fig. 8. Distribution of interlaminar stresses along the 25/80 interface of [25/80/80/25] laminate.


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both near the 90/0interfaces at the free edge (i.e.,y=b). It is also seen thatrzdiminishes away from the freeedge as the interior region of the laminate is approached.Fig. 7shows the variations of transverse shear stressryzat the free edge (i.e., y =b) and through the thickness of the bottom 90layer of the [0/90/90/0] lam-inate. It is clear from the figure that ryzhas a nonzero value at the interface-edge junction (i.e., interface cor-ner) of the laminate. A close examination ofFig. 7reveals that, except for the interface corner point, the value

ofryzalong the free edge of the laminate approaches zero as the number of numerical layers in each physicalply,p, is increased. The variations of the interlaminar normal and shear stresses along the upper interface (i.e.,the 25/80interface) of the antisymmetric angle-ply laminate [25/80/80/25] are plotted inFig. 8. It isseen that the magnitude ofryz is quite smaller than those ofrzand rxz(with rxzand rz surprisingly havingsimilar magnitudes). It is reminded here that extensive numerical studies indicate that the interlaminar shearstress ryzis equal to zero at the middle surface of all antisymmetric angle-ply laminates subjected to uniformhygrothermal loads. The distributions of interlaminar stresses rzandrxzacross the 0/60interfaces of [60/0/0/60] and [0/60/60/0] laminates are compared in Fig. 9a and b. It is observed that both transversenormal stress rzand transverse shear stress rxzexhibit similar behavior in the two laminates. The maximuminterfacial values for both rz and rxzoccur, however, in the [0/60/60/0] laminate. The distributions ofthrough-the-thickness interlaminar normal stressrz at y=b for the antisymmetric angle-ply laminate [45/60/60/45] and the symmetric laminate [45/60/60/45] are shown in Fig. 10. It is observed that the

value ofrzat the free edge is noticeably larger in the antisymmetric angle-ply laminate. The maximum valueofrzin the [45/60/60/45] laminate occurs at the middle surface while the maximum value ofrzin [45/60/60/45] occurs in the 60 layers near the top and bottom interfaces. It is to be noted that, in both sym-metric and antisymmetric angle-ply laminates, the interlaminar normal stress rzis an even function of thick-ness coordinate. Similarly, the variations of interlaminar shear stress rxzthrough the thickness and at the freeedge of the [(45/45)2] and [45/45]slaminates are presented inFig. 11. It is noticed from this figure that

Fig. 9. (a) Interlaminar normal stressrz along the 0/60 interfaces of [60/0/0/60] and [0/60/60/0] laminates. (b) Interlaminar

shear stress rxz along the 0/60interfaces of [60/0/0/60] and [0/60/60/0] laminates.


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in the antisymmetric angle-ply laminate the maximum positive values ofrxzoccur near the 45/45interfaceswhereas the maximum negative value ofrxz occurs at the middle surface of the laminate. In the symmetriclaminate, however, the maximum positive and negative values ofrxzoccur near the 45/45 and 45/45interfaces, respectively. It is significant to note that the magnitudes of maximum values ofrxzfor both lam-inates are approximately equal, with rxzbeing an odd (even) function of thickness coordinate in symmetric(antisymmetric angle-ply) laminates.

Finally, the effect of fiber orientation is examined in Fig. 12a and b by comparing the variations of inter-

laminar normal stressrzat the top interface-edge and middle surface-edge junctions of the [0/h/h/0] and [0/h/h/0] laminates as a function ofh. It is observed that at the top interface-edge (i.e., 0/h interface-edge)

junctions the two laminates display very similar behavior, with the numerical values ofrzbeing approximatelyidentical. At the middle surface-edge junction, however, the distributions ofrzin the two laminates are, exceptfor small hs, quite different. The maximum values ofrzat the top interface-edge junction occur at h = 90inboth laminates whereas at the middle surface-edge junction rzbecomes maximum when h55 in the anti-symmetric angle-ply laminate and when h= 90 in the symmetric laminate.

4. Conclusions

In the present investigation an elasticity formulation is presented for the displacement field of a long gen-

erally stacked laminate subjected to hygrothermal loads. It is found that the components of the displacements

Fig. 10. Distributions of interlaminar normal stress rz through the thickness of [45/60/60/45] and [45/60/60/45] laminates.

Fig. 11. Distributions of interlaminar shear stressrxz through the thickness of [(45/45)2] and [45/45]s laminates.


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field are composed of two distinct parts, signifying the global and local deformations within laminates. Basedon physical arguments regarding the behavior of symmetric, cross-ply, and antisymmetric angle-ply laminatesspecial displacement fields are obtained for such laminates. The ESL theories are then employed to determinethe unknown constant parameters appearing in the global deformation part of various displacement fields. Itis also found that for each lamination scheme an appropriate ESL theory must be employed for the efficientand accurate prediction of these parameters. For cross-ply and antisymmetric angle-ply laminates it is foundthat the usual first-order theory (MindlinReissner plate theory) yields accurate results for the unknown con-stant parameters appearing in various displacement fields. Numerical investigations, however, reveal that forsymmetric laminates the usual first-order theory (FSDT) is inadequate in predicting these parameters. There-

fore, an improved first-order theory (IFSDT) is introduced to obtain these parameters for symmetric lami-nates. Next, Reddys layerwise theory (LWT) is utilized to calculate the interlaminar stresses. Theunknown constants appearing in various displacement fields are also determined within LWT. For specialboundary and loading conditions an analytical elasticity solution is presented to verify the accuracy ofLWT, in predicting the interlaminar stresses andB2, and that of FSDT and IFSDT in predictingB2. Excellentagreements are seen to exist between the results of LWT and the elasticity theory. Several numerical resultsaccording to LWT are then developed for the free-edge interlaminar stresses through the thickness and acrossthe interfaces of various cross-ply, symmetric, and antisymmetric angle-ply laminates.

Appendix A

The coefficients D22, D26, and D66 appearing in(26)are given as follows:

Fig. 12. (a) Interlaminar normal stressrzat the top interface-edge junctions (i.e., 0/hinterface-edge junctions) of the [0/h/h/0] and [0/h/h/0] laminates as a function ofh. (b) Interlaminar normal stressrzat the middle interface-edge junctions of the [0/h/h/0] and [0/h/h/0] laminates as a function ofh.


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D22 D22~B26A26~B22A22~B26 ~B22A26~B26A66~B22

A22A66A226

D26 D26~B26A26~B26A22~B66 ~B22A26~B66A66~B26

A22A66A226

D66 D66~B66A26~B26A22~B66 ~B26A26~B66A66~B26

A22A66A226

:

The constant parameters in Eqs.(28a) and (28b)are defined as:

d1; d2 1

a1a4a2a3a2B16a4B12; a3B12a1B16

mTy;mTxy

1

a1a4a2a3a4 M

Ty a2 M

Txy; a1 M

Txy a3 M

Ty

mHy;mHxy

1

a1a4a2a3a4 M

Hy a2 M

Hxy; a1 M

Hxy a3 M

Hy

where

a1 D26 D1 D22k1coshk1b; a2 D26 D2 D22k2coshk2b

a3 D66 D1 D26k1coshk1b; a4 D66 D2 D26k2coshk2b

B12 ~B12~B26A12A26A16A22 ~B22A16A26A12A66

A22A66A226

B16 ~B16~B66A12A26A16A22 ~B26A16A26A12A66

A22A66A226

and

MTy ~MTy

~B26A22NTxy A26N

Ty ~B22A66N

Ty A26N

Txy

A22A66A226

MTxy ~MTxy

~B66A22NTxy A26N

Ty ~B26A66N

Ty A26N

Txy

A22A66A226

MHy ~MHy

~B26A22NHxy A26N

Hy ~B22A66N

Hy A26N

Hxy

A22A66A226

MHxy ~MHxy

~B66A22NHxy A26N

Hy ~B26A66N

Hy A26N

Hxy

A22A66A226

:

In addition, the constant coefficients appearing in(30)are defined as follows:

h A11bd1B16 B12 D1 sinhk1b d2B16 B12 D2 sinhk2b

nTx b NTx m

TyB16 B12 D1 sinhk1b m

TxyB16 B12 D2 sinhk2b

nHx b NHx m

HyB16 B12 D1 sinhk1b m

HxyB16 B12 D2 sinhk2b

where

A11 A11A16A12A26A16A22 A12A16A26A12A66

A22A66A226

and


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NTx NTx

A16A22NTxy A26N

Ty A12A66N

Ty A26N

Txy

A22A66A226

NHx NHx

A16A22NHxy A26N

Hy A12A66N

Hy A26N

Hxy

A22A66A226

:

Appendix B

The constant coefficients appearing in(31)are given as follows:

h1 A11a1A12a2B12a3; h2 B11a1A12a4B12a5

h3 B11a1B12a2D12a3; h4 D11a1B12a4D12a5

and

nTx NTx a1N

Tya2M

Tya3; m

Tx M

Tx a1N

Tya4M

Tya5

nHx NHxa1NHya2MHya3; mHx MHxa1NHya4MHya5

where

a1 A22D22B222; a2 B12B22A12D22; a3 A12B22A22B12

a4 B22D12B12D22; a5 B12B22A22D12:

It is reminded here that the laminate rigidities and stress, thermal, and hygroscopic resultants are the same asthose defined inEqs. (19), (21), (22), and (23). Furthermore, here:

MTx ;MTy Z

h=2

h=2

Q11; Q12ax Q12; Q22ayDTzdz

MHx;MHy

Z h=2h=2

Q11; Q12bx Q12; Q22byDMzdz

and

Bij

Z h=2h=2

Qijzdz:

It is observed that the constant parameters B2 and B6 appearing in(31)are independent of shear correctionfactor. In other words, the results presented in(31)for B2 and B6 are also obtainable from the classical lam-ination theory.

Appendix C

The constant coefficients appearing in(32)are given by:

h1 2A12B26A22B16

h2 A11A22A212

h3 2B226A22D66

h4 A22B16A12B26

and


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nTx A22NTx A12N

Ty

mTxyA22MTxy B26N

Ty

nHx A22NHx A12N

Hy

mHxyA22MHxy B26N

Hy

where

MTxy

Z h=2h=2

Q16ax Q26ay Q66axyDTzdz

MHxy

Z h=2h=2

Q16bx Q26by Q66bxyDMzdz

with the remaining thermal and hygroscopic resultants being the same as those defined in (22a) and (23a). Forantisymmetric angle-ply laminates the results in(32)may also be arrived at by using the classical laminationtheory.

Appendix D

The laminate rigidities, introduced in(40), are given as follows (also see Nosier and Bahrami, 2006, 2007):

Akjpq;Bkjpq;D

kjpq

C

k1pq

hk1;

Ck1pq

2 ;

hk1 Ck1pq

6

if j k1

Ck1pq

hk1

Ckpq

hk;C

k1pq

2

Ckpq

2 ;

hk1 Ck1pq

3

hkCkpq

3

if j k

C

kpq

hk; C

kpq

2 ;

hkCkpq

6

if j k1

0; 0; 0 if j< k1 or j> k1

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

and

Akpq;Bkpq; B

kpq;D

kpq

C1pq;h1 C

1pq

2 ; C1pq

z21

z22

2h1; C

1pq

h1

z31

z32

3 z2

z21

z22

2

h i if k 1

Ck1pq ;hk1 C

k1pq

2 ; Ck1pq

z2k

z2k1

2hk1;C

k1pq

hk1

z3k

z3k1

3 zk1

z2k

z2k1

2

h i if k N1

Ck1pq Ckpq;

hk1 Ck1pq

2

hkCkpq

2 ; Ck1pq

z2k

z2k1

2hk1 Ckpq

z2k

z2k1

2hk;

C

k1pq

hk1

z3k

z3k1

3 zk1

z2k

z2k1

2

h i

Ckpq

hk

z3k

z3k1

3 zk1

z2k

z2k1

2

h i if 1< k< N1:

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

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