Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations

Accepted Manuscript Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations M.H. Mansouri, M.Sc., M. Shariyat, Professor PII:

S1359-8368(15)00460-6

DOI:

10.1016/j.compositesb.2015.08.030

Reference:

JCOMB 3710

To appear in:

Composites Part B

Received Date: 2 March 2015 Revised Date:

8 May 2015

Accepted Date: 7 August 2015

Please cite this article as: Mansouri MH, Shariyat M, Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.08.030. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations M.H. Mansouri*, M. Shariyat†

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Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran 19991-43344, Iran. Center of Excellence in Smart Structures and Dynamical Systems.

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Abstract Thermo-mechanical buckling analysis of the orthotropic auxetic plates (with negative Poisson ratios) has not been performed so far, especially, in the hygrothermal environments. The complexity increases when the auxetic plate is fabricated from functionally graded orthotropic materials and surrounded by an elastic foundation. The aforementioned analyses are carried out in the present research, for the first time. The buckling loads may be uniaxial or biaxial ones. Moreover, temperature and moisture dependent material properties are considered. The pre-buckling effects are also considered in the paper. The high-order sheardeformation governing differential equations are solved based on a new differential quadrature method (DQM). The resulting solution may cover many practical simpler combinations. A comprehensive parameter study is accomplished for a wide range of geometric and material properties parameters and various boundary conditions. Results reveal that the hygrothermal conditions lead to degradations in the material properties and buckling strengths, especially for higher gradation exponents, the elastic foundation may enhance the buckling behavior through monitoring the buckling pattern, the buckling load decreases with the orthotropy angle, and the auxeticity reduces the buckling strength. Keywords: A. Plate; B. Buckling; B. Thermomechanical; B. Environmental degradation; Auxetic functionally graded orthotropic materials.

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1 Introduction The traditional orthotropic laminated composite structures suffer from the abrupt changes in the mechanical and thermal material properties in the transverse direction; evidence that may lead to severe interfacial stresses. Therefore, this phenomenon may be exaggerated when the load carrying component is placed in an environment with high temperature and/or high concentration moisture [1]. This phenomenon may significantly affect the local and even global behaviors (e.g., buckling strength [2]) of the structure. The discontinuity in the material properties may be resolved through using a large number of perfectly bonded thin orthotropic (composite) layers with slightly varying material properties [3-10]. Therefore, the resulting plate may be considered as an orthotropic functionally graded plate. The heterogeneity of the materials may also stem from effects of the humidity or moisture [11,12]. Mechanical/thermal

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M.Sc., E-mail addresses: [email protected], [email protected]. Tel.: +98 913 383 9396, Fax: +98 21 8131 3538. † Corresponding author, Professor, E-mail addresses: [email protected] and [email protected]. Tel.: +98 9122727199; Fax: +98 21 88674748, zip code: 19991-43344. Home page: http://wp.kntu.ac.ir/shariyat/publications.html

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buckling of the functionally graded orthotropic plates have been investigated by Asemi and Shariyat [8], Shariyat and Asemi [9], and Mansouri and Shariyat [10]. The elastic foundation changes the buckling strengths and modes of the functionally graded plates. Xiang et al. [13] investigated buckling of simply supported symmetric cross-ply rectangular laminates on Pasternak foundations, based on the first-order shear deformation plate theory and a Navier-type solution. Akavci [14] presented buckling analysis of simply supported composite plates on Winkler-Pasternak foundations, by a hyperbolic displacement model solved by using Navier technique. Kiani et al. [15] analytically determined buckling loads of clamped rectangular plates made of functionally graded materials on elastic foundations and under various types of thermal loadings, based on the classical plate theory. Alipour and Shariyat proposed semi-analytical solutions for buckling analysis of variable thickness transversely graded [16] and bidirectional functionally graded [17] viscoelastic circular plates on elastic foundations. A closed-form solution for buckling analysis of thick functionally graded plates resting on Pasternak foundations was presented by Thai and Kim [18] using the third-order shear deformation theory. Recently, mechanical and thermal postbuckling analyses of FGM rectangular plates resting on nonlinear elastic foundations were performed by Zhang and Zhou [19] using a high-order shear deformation theory and Ritz method. Non-linear buckling problem of functionally graded orthotropic plates surrounded by elastic foundations was treated by Shariyat and Asemi [9], employing the general threedimensional theory of elasticity and the complex B-spline finite element method. The fiber reinforced composite structures used in many aerospace, civil, and mechanical structures may be exposed to significant temperature rises and/or moisture absorption which have adverse effects on the material properties and strengths (e.g., the buckling strengths) of the mentioned structures and may lead to different failure modes such as moisture-induced distortions, swelling, and dimensional changes in humid and rainy weathers. Benkhedda et al. [20] proposed an approach for investigation of the mechanical characteristics changes and stresses of plates subjected to simultaneous variations of moisture and temperature. Sai Ram and Sinha [21] investigated the hygrothermal effects on the static instability of laminated composite plates, using the first-order shear-deformation theory and the geometric stiffness concept for determination of the buckling load. Chao and Shyu [22] studied biaxial buckling of laminated composite plates under hygrothermal environments, using the first-order shear deformation plate theory. The nonlinear governing equations were solved by an iterative numerical method. The hygrothermal effects on the postbuckling of simply supported composite plates subjected to a uniaxial compression was investigated by Shen [23] using Reddy's higher-order shear-deformation plate theory and including dependency of the material properties on the temperature and moisture. A perturbation technique was employed to determine buckling loads. Patel et al. [24] studied deflection, buckling and natural frequencies of thick composite laminates exposed to hygrothermal environment, using a plate theory with third-order global and linear zigzag local components. Borton [25] studied the moisture-related static buckling of the symmetric angle-ply laminates, based on the classical plate theory. Effects of the moisture and temperature on the postbuckling of the laminated composite plates were investigated by Pandey et al. [26] based on a high-order shear deformation theory and von Karman’s nonlinear kinematics. The mechanical loading consisted of uniaxial, biaxial, shear or their combinations. The degradation in material properties due to moisture and temperature was taken into account. Thermal buckling of transversely graded plates in hygrothermal environments were studied by Lee and Kim [27], using the first-order shear-deformation theory and von Karman strain–displacement relations. The effect of moisture concentration and thermal gradient on the buckling of the laminated composite plates with central cutouts were investigated by Natarajan et al. [28], using Mindlin plate theory. 2

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Auxetic materials are solids with negative Poisson’s ratios [29,30]. Unlike conventional solids, auxetic rods expand laterally when stretched axially while auxetic plates transform into synclastic domes when a bending moment is applied on two opposite sides. Lim [31] investigated effects of the negative Poisson’s ratios on buckling of the thick isotropic circular and square plates, using Mindlin plate theory and concluded that in contrast to the circular plates, buckling load of the square plate increases as the Poisson’s ratio becomes more negative. Through buckling and vibration analyses of circular plates under various boundary conditions within the entire range of Poisson’s ratios of the isotropic solids, i.e., from -1 to 0.5, Lim [32] deduced that as the Poisson’s ratio of the plate becomes more negative, the critical bucking loads and the natural frequencies reduce. In the present paper, biaxial buckling analysis of functionally graded orthotropic plates under hygrothermal effects is accomplished, for the first time. In addition to considering the hyrothermal effects on both the material properties and buckling strengths, auxeticity of the materials is considered. The displacement form of the governing equations that are derived based on Reddy’s high-order plate theory, solved by the new version of the DQ method. Novelties of the present research may be summarized as: o The hygrothermal effects are investigated for the functionally graded materials. o Considering the gradual transverse variations and orthotropy of the material properties in the present research, has increased the complexity further and led to remarkable bendingextension-shear couplings. o Temperature and moisture dependent material properties are considered. o The only one available article in literature on buckling of the rectangular auxetic plates, has treated isotropic plates. In the present research, the functionally graded orthotropic materials, elastic foundation, and the hygrothermal effects on both the material properties and buckling loads are considered as well. o The majority of the relevant buckling analyses have been accomplished considering thermal or uniaxial compressive loads. In addition to the biaxial analysis, different boundary conditions (simply supported and clamped conditions) are considered in the present research. o The pre-buckling effects are included. o The governing equations of buckling are solved by means of the new version of the differential quadrature method. o The special cases of the considered general problem, cover many practical problems.

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2 Governing equations of the auxetic functionally graded plate in the hygrothermal environment From the practical point of view, the functionally graded orthotropic plate may be constructed from a large number (NL) of perfectly bonded thin orthotropic layers with gradually varying material properties [8-12]. Fig. 1 shows a typical buckling pattern of the considered auxetic functionally graded plate that is surrounded by a Winkler-Pasternak elastic foundation. Length, width and thickness of the plate are denoted respectively by a, b and h. The z coordinate is measured from the mid-plane of the plate and is positive upward. Reddy’s third-order shear-deformation theory [33] is employed for description of the displacement field: ∂w   u ( x, y, z ) = u0 ( x, y ) + zϕ x ( x, y ) − c1 z 3  ϕ x ( x, y ) +  ∂x   (1)  ∂w  v( x, y, z ) = v0 ( x, y ) + zϕ y ( x, y ) − c1 z 3  ϕ y ( x, y ) +  ∂y   w( x, y, z ) = w( x, y ) 3

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Fig. 1 A typical biaxial buckling pattern of an auxetic functionally graded orthotropic plate surrounded by a Winkler-Pasternak elastic foundation, in a hygrothermal environment.

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where u, v, w, u0 , v0 , ϕ x , and ϕ y are the displacement components in the x, y, and z directions, in-plane displacements of the reference plate, and rotations of the section in the x-z 4 and y-z planes, respectively and c1 = 2 . Therefore, the strain components become: 3h 2  ∂ϕ ∂ w  ∂u ∂ϕ ε x = 0 + z x − c1 z 3  x + 2  ∂x ∂x  ∂x ∂x 

∂ϕ y ∂ϕ y ∂ 2 w  ∂v0 3 εy = +z − c1 z  + 2 ∂y ∂y  ∂y ∂y 

Q12 Q22 Q26

Q16   Q26  Q66 

(i)

 ε x − ε xHT   HT   εy −εy   ε xy − ε xyHT   

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(i )

 σx   Q11     σ y  =  Q12 σ  Q  xy   16

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 ∂ϕ ∂ϕ   ∂ϕ ∂ϕ ∂u ∂v ∂2w  ε xy = 0 + 0 + z  x + y  − c1 z 3  x + y + 2  ∂y ∂x ∂x  ∂x ∂x∂y   ∂y  ∂y  4z2    4 z 2  ∂w  ∂w  γ xz = 1 − 2   ϕ x +  γ yz = 1 − 2  ϕ y +  h  ∂x  h  ∂y    The stress components of the ith layer may be related to the strain components as:

(2)

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Q σ  ,  yz  =  44  σ xz   Q45 (i )

Q45   γ yz     Q55   γ xz  (i )

(i )

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ij

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( Q , i, j = 1,.., 6 ) are elements of the transformed stiffness matrix and the hygrothermal strains (denoted by the superscript HT) contain the thermal and moisture effects of the environment, as follows: (i )

(i )

(i )

(i )

(i )

 ε xHT   ε xT   ε xM   αx   βx   HT   T  M     (4)  ε y  =  ε y  +  ε y  = ∆T ( x, y )  α y  + ∆C ( x, y )  β y  α  β   γ xyHT   T  M  xy   xy     γ xy   γ xy  ∆T ( x, y ) and ∆C ( x, y ) are respectively, the temperature and moisture concentration rises with respect to their reference values. The superscripts T and M stand for the thermal and moisture effects, respectively. α and β are the thermal and moisture expansion coefficients, in the geometric coordinates. If number of the layers is large enough, the transverse variations of the material properties may be representable by a continuous function. Using of symmetric lamination schemes leads

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ACCEPTED MANUSCRIPT to higher buckling loads [34-38]. For this reason, variations of the elastic material properties of the heterogeneous orthotropic plate are chosen as follows: Qij = Q

ref ij

e

kz h

(5)

, i, j = 1, 2, 4,5, 6

(i )

 αx   cos 2 θ    2  α y  =  sin θ α   2 cos θ sin θ  xy  

sin 2 θ

  2 cos θ   −2 cos θ sin θ 

(i )

kz   α1e h  kz α e h  2

    

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where, Qijref are the reference elastic coefficients of the orthotropic material of the reference plane, i.e., the mid-plane, layer. k is the material properties exponent. Similarly, the thermal and moisture expansion coefficients may be defined as:

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(6) (i ) (i ) kz   cos 2 θ    βx  sin 2 θ h    β1e    2 2 cos θ  β y  =  sin θ   kz  β e h  β    2 cos sin − 2 cos sin θ θ θ θ  xy     2  where the subscripts 1 and 2 denote directions parallel and normal to the fibers, respectively and θ denotes orientation of the fiber with respect to the x axis. Determination of the equivalent mechanical and hygrothermal material properties of the fiber-matrix mixture, requires using appropriate micromechanical homogenization techniques. Denoting volume fractions of the fiber and matrix by V f and Vm , respectively, so that: V f + Vm = 1 (7) the following equivalent material properties may be defined according to Shen [23] (Voigt and Reuss models [39,40]): E1 = V f E f + Vm Em (8)

V f Vm 1 = + G12 G f Gm

, Gm =

Em 2(1 + ν m )

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(10) (11)

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ν 12 = V fν f + Vmν m V E α + Vm Emα m α1 = f f f

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ν 2f Em E f +ν m2 E f Em − 2ν fν m 1 V f Vm = + − V f Vm E2 E f Em V f E f + Vm Em

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V f E f + Vm Em

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α 2 = (1 + ν f )V f α f + (1 + ν m )Vmα m −ν 12α1 (13) Vm Em β m ρ β1 = (14) (V f E f + Vm Em )Vm ρ m (1 + ν m ) β m ρ β2 = −ν 12 β1 (15) ρm ρ = V f ρ f + Vm ρ m (16) ρ is the mass density. The governing equations associated with Reddy’s third-order theory are [33]:

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ACCEPTED MANUSCRIPT N x , x + N xy , y = 0 N xy , x + N y , y = 0

M x , x + M xy , y − c1 ( Px , x + Pxy , y ) − Qx + c2 Rx = 0

(17)

M xy , x + M y , y − c1 ( Pxy , x + Py , y ) − Qy + c2 Ry = 0 4 and h2 A12 A22

A16 A26 A66

B11 B12 B16 D11

B12 B22 B26 D12 D22

B16 B26 B66 D16 D26

E11 E12 E16 F11 F12

E12 E 22 E 26 F12 F22

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F16 H 11

F26 H 12 H 22

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u 0, x   N xHT     HT  v0, y   Ny     N HT u0, y + v0, x    xy  ϕ x x ,    0  −  ϕ y, y   0  + ϕ ϕ   0 F66   x, y y,y    − c1 (ϕ x , x + w, xx ) H 16     0  − c1 (ϕ y , y + w, yy )   0 H 26       H 66   − c1 (ϕ x , y + ϕ y , x + 2 w, xy )   0 E16 E 26 E66 F16 F26

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 N x   A11     Ny    N xy       Mx    My =     M xy    P    x    Py    P    xy  

) +F e +F =0

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where c2 =

HT

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Qx , x + Qy , y − c2 ( Rx , x + Ry , y ) + c1 ( Px , x + 2 Pxy , xy + Py , yy ) + F

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 Qy   A44 A45 D44 D45   ϕ y + w, y       A55 D45 D55   ϕ x + w, x   Qx  =   Ry   Sym. F44 F45   −c1 (ϕ y + w, y )       F55  −c1 (ϕ x + w, x )   Rx   N HT are the in-plane hygrothermal forces

 αx     α y  dz α   xy 

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 βx     β y  dz    β xy 

Q16   Q26  Q66 

 N xM  M  Ny  M  N xy

Q16   Q26  Q66 

 Q11   Q12   Q16

Q12

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 h /2   = ∆C ( x, y ) ∫ −h/ 2   Aij , Bij , etc., are:

Q22

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Q26

(18)

(19)

(i )

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 N xT   Q11 Q12 h /2  T   N y  = ∆T ( x, y ) ∫  Q12 Q22 −h/ 2   N xyT   Q16 Q26  

             

(i )

(20)

h /2

( Aij , Bij , Dij , Eij , Fij , H ij ) =

∫

Qij (1, z , z 2 , z 3 , z 4 , z 6 )dz

(21)

−h/ 2

) F HT , F e and F are the imposed forces due to the hygrothermal effects, Winkler-Pasternak elastic foundation, and the externally applied edge loads per unit length: F HT = N xHT w, xx + 2 N xyHT w, xy + N yHT w, yy F e = − kw w + k p ( w, xx + w, yy ) ) ) ) ) F = N x w, xx + 2 N xy w, xy + N y w, yy

(22)

kw and k p are the Winkler and Pasternak coefficients of the elastic foundation. Eq. (17) may be rewritten in terms of the displacement parameters, based on Eqs. (18) to (22). Hereafter, u0 and v0 will be denoted by u and v for the sake of simplicity. 6

ACCEPTED MANUSCRIPT A11u, xx + 2 A16u, xy + A66u, yy + A16 v, xx + ( A12 + A66 )v, xy + A26 v, yy + ( B11 − c1 E11 )ϕ x , xx +2( B16 − c1 E16 )ϕ x , xy + ( B66 − c1 E66 )ϕ x , yy + ( B16 − c1 E16 )ϕ y , xx + ( B12 − c1 E12 + B66 − c1 E66 )

ϕ y , xy + ( B26 − c1E26 )ϕ y , yy − c1E11w, xxx − 3c1E16 w, xxy − c1 ( E12 + 2 E66 ) w, xyy − c1 E26 w, yyy = 0 A16u, xx + ( A12 + A66 )u, xy + A26u, yy + A66 v, xx + 2 A26 v, xy + A22 v, yy + ( B16 − c1 E16 )ϕ x , xx

(23)

+ ( B66 − c1 E66 + B12 − c1 E12 )ϕ x , xy + ( B26 − c1 E26 )ϕ x , yy + ( B66 − c1 E66 )ϕ y , xx + 2( B26 − c1 E26 )

ϕ y , xy + ( B22 − c1E22 )ϕ y , yy − c1E16 w, xxx − c1 ( E12 + 2 E66 ) w, xxy − 3c1 E26 w, xyy − c1E22 w, yyy = 0

(24)

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( B11 − c1 E11 )u, xx + 2( B16 − c1 E16 )u, xy + ( B66 − c1 E66 )u, yy + ( B16 − c1 E16 )v, xx

+ ( B12 − c1 E12 + B66 − c1 E66 )v, xy + ( B26 − c1 E26 )v, yy + ( D11 − 2c1 F11 + c12 H11 ) ϕ x , xx

+2 ( D16 − 2c1 F16 + c12 H16 ) ϕ x , xy + ( D66 − 2c1 F66 + c12 H 66 ) ϕ x , yy + ( D16 − 2c1 F16 + c12 H16 )

ϕ y , xx + ( D12 − 2c1 F12 + c12 H12 + D66 − 2c1 F66 + c12 H 66 ) ϕ y , xy + ( D26 − 2c1F26 + c12 H 26 ) ϕ y , yy

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−c1 ( F11 − c1 H11 ) w, xxx − 3c1 ( F16 − c1 H16 ) w, xxy − c1 ( F12 − c1 H12 + 2 F66 − 2c1 H 66 ) w, xyy −c1 ( F26 − c1 H 26 ) w, yyy − ( A55 − 2c2 D55 + c22 F55 ) w, x − ( A45 − 2c2 D45 + c22 F45 ) w, y

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− ( A55 − 2c2 D55 + c22 F55 ) ϕ x − ( A45 − 2c2 D45 + c22 F45 ) ϕ y = 0

(25)

( B16 − c1 E16 )u, xx + ( B12 − c1 E12 + B66 − c1 E66 )u, xy + ( B26 − c1 E26 )u, yy + ( B66 − c1 E66 ) v, xx +2( B26 − c1 E26 )v, xy + ( B22 − c1 E22 )v, yy + ( D16 − 2c1 F16 + c12 H16 )ϕ x , xx

+ ( D12 − 2c1 F12 + c12 H12 + D66 − 2c1 F66 + c12 H 66 ) ϕ x , xy + ( D26 − 2c1 F26 + c12 H 26 ) ϕ x , yy

+ ( D66 − 2c1 F66 + c12 H 66 ) ϕ y , xx + 2 ( D26 − 2c1 F26 + c12 H 26 ) ϕ y , xy + ( D22 − 2c1 F22 + c12 H 22 )

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ϕ y , yy − c1 ( F16 − c1 H16 ) w, xxx − c1 ( F12 − c1 H12 + 2 F66 − 2c1H 66 ) w, xxy − 3c1 ( F26 − c1H 26 ) w, xyy − c1 ( F22 − c1 H 22 ) w, yyy − ( A45 − 2c2 D45 + c22 F45 ) w, x − ( A44 − 2c2 D44 + c22 F44 ) w, y

− ( A45 − 2c2 D45 + c22 F45 ) ϕ x − ( A44 − 2c2 D44 + c22 F44 )ϕ y = 0

(26)

c1  E11u, xxx + 3E16u, xxy + ( E12 + 2 E66 )u, xyy + E26u, yyy + E16 v, xxx + ( E12 + 2 E66 )v, xxy +3E26v, xyy

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+ E22 v, yyy + ( F11 − c1 H11 )ϕ x , xxxx + 3( F16 − c1 H16 )ϕ x , xxy + ( F12 − c1 H12 + 2 F66 − 2c1 H 66 )ϕ x , xyy + ( F26 − c1 H 26 )ϕ x , yyy + ( F16 − c1 H16 )ϕ y , xxx + ( F12 − c1 H12 + 2 F66 − 2c1 H 66 )ϕ y , xxy

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+3( F26 − c1 H 26 )ϕ y , xyy + ( F22 − c1 H 22 ) ϕ y , yyy  + ( A55 − 2c2 D55 + c22 F55 ) ϕ x , x

+ ( A45 − 2c2 D45 + c22 F45 ) ϕ x , y + ( A45 − 2c2 D45 + c22 F45 ) ϕ y , x + ( A44 − 2c2 D44 + c22 F44 ) ϕ y , y −c12  H11w, xxxx + 4 H16 w, xxxy + 2( H12 + H 66 ) w, xxyy + 4 H 26 w, xyyy + H 22 w, yyyy 

+ ( A55 − 2c2 D55 + c22 F55 ) w, xx + 2 ( A45 − 2c2 D45 + c22 F45 ) w, xy + ( A44 − 2c2 D44 + c22 F44 ) w, yy + ( N xHT w, xx + 2 N xyHT w, xy + N yHT w, yy ) − kw w + k p ( w, xx + w, yy ) ) ) ) + N x w, xx + 2 N xy w, xy + N y w, yy = 0

(

)

(27)

3 The two-step buckling load determination procedure While the transversely symmetric material properties distribution increases the plate strength, it also provides the necessary conditions for the buckling and bifurcation occurrence; otherwise, increasing the loads leads to increased bending rather than buckling. Since in this 7

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situation, the plate remains flat prior to the buckling onset, determination of the bifurcation buckling point may be accomplished through an eigenvalue analysis [8,9,38]. Determination of the buckling load may be accomplished through a two-step procedure. In the first stage, distribution of the in-plane stresses associated with the pre-buckling state is determined to be used in the second stage that is devoted to determination of the buckling load through an eigenvalue solution. Since in the pre-buckling state, the temperature and moisture distributions with respect to the stress-free state are uniform; lateral deflections and rotations of the plate are zero in that state and Eq. (17) reduces to: N x , x + N xy , y = 0 (28) N xy , x + N y , y = 0 where  N x   A11 A12 A16   u, x   N xHT       HT   (29)  N y  =  A12 A22 A26   v, y  −  N y  HT          N xy   A16 A26 A66   u, y + v, x   N xy  Substituting Eq. (29) into (28), the governing equations of the pre-buckling state may be expressed in terms of the in-plane displacements, as follows: A11u, xx + 2 A16u, xy + A66u, yy + A16 v, xx + ( A12 + A66 )v, xy + A26 v, yy = 0 (30) A16u, xx + ( A12 + A66 )u, xy + A26u, yy + A66 v, xx + 2 A26 v, xy + A22 v, yy = 0 Solving Eq. (30) gives the distributions of the in-plane pre-buckling hygrothermal forces per unit length ( F HT ) that may be used in the full form of Eq. (17) in determination of the mechanical buckling load.

EP

TE D

4 Mathematical descriptions of the boundary conditions A variety of the boundary conditions are considered in the present research (results section) due to considering various buckling situations. In this regard, the following two types of the simply supported (SS) and two types of the clamped (C) edge conditions are considered: at x = 0, a  v = N x = w = M x − c1 Px = Px = ϕ y = 0 SS1:  at y = 0, b u = N y = w = M y − c1 Py = Py = ϕ x = 0 (31) at x = 0, a u = N xy = w = M x − c1 Px = Px = ϕ y = 0 SS 2 :  at y = 0, b v = N xy = w = M y − c1 Py = Py = ϕ x = 0

AC C

u = N xy = w = w, x = ϕ x = ϕ y = 0 C1 :  v = N xy = w = w, y = ϕ x = ϕ y = 0 u = v = w = w, x = ϕ x = ϕ y = 0 C2 :  u = v = w = w, y = ϕ x = ϕ y = 0

at

x = 0, a

at

y = 0, b

at

x = 0, a

at

y = 0, b

(32)

5 DQ Discretization of the governing equations Several DQ methods have been proposed so far. Wang et al. [41] and Tornabene et al. [42] have reviewed these methods. The traditional methods have limitations in treating different types of the boundary conditions. The main problem of these techniques is that the discretized system matrix may become ill-conditioned [43-45]. The method introduced by Wang et al. [46,47] leads to simpler computations. Indeed, based on Eqs. (31) and (32), six boundary conditions have to be imposed for the either simply supported or clamped plates while Reddy’s higher-order theory has only five unknown displacement parameters. In this method, first-order derivatives of the quantities at the edges are defined as extra degrees of freedom. 8

ACCEPTED MANUSCRIPT Therefore, the main difference between the new version of the DQM and the traditional one appears only in treating the lateral deflections. Furthermore, the weighting coefficient matrix of the first-order derivatives is identical with that of the common DQM. Weighting coefficients of the second-order derivatives are defined at the points located on the edges as follows: n −1

n+ 2

j =1 k = 2

j =1

n

(1) (1) (2) ′ f1′′= C11(1) f1′ + C1(1) n f n + ∑∑ C1k Ckj f j = ∑ C1 j f j n −1

(33)

n+2

f n′′ = C f ′+ C f ′ + ∑∑ C C f j = ∑ C (1) nn n

(1) nk

j =1 k = 2

(1) kj

j =1

(2) nj

fj

RI PT

n

(1) n1 1

where ( f )T = ( f1 , f1′, f 2 , f3 ,..., f n , f n′) . The weighting coefficients of the second-order derivatives at the inner points may be computed from: n

n+ 2

n

f i′′= ∑∑ Cik(1)Ckj(1) f j = ∑ Cij(2) f j

i = 2,3,..., n − 1

,

j =1

(34)

SC

j =1 k =1

In Eqs. (33) and (34), Cij( 2 ) are the weighting coefficients of the second-order derivatives in the

M AN U

new version of DQM while Cij(1) are the weighting coefficients of the first-order derivative of the common DQM and are already given by Shu and Richards [48]. The weighting coefficients of the third- and fourth-order derivatives have been given as: n

Cij(3) = ∑ Cik(1) Ckj(2) k =1 n

(4) ij

C

= ∑C

(2) ik

k =1

C

(2) kj

i = 1, 2,..., n ,

,

(35)

j = 1, 2,..., n + 2

EP

TE D

Based on this method, the inner points, points located at the edges, and the corner points have one, two, and three degrees of freedom, related to the lateral deflection per point of the rectangular plate. Imposing of the new version of the DQM to both the governing equations and the relevant edge conditions constitutes the eigenvalue problem that leads to determination of the buckling loads. The resulting equations of the eigenvalue problem may be solved through the standard techniques and the resulting eigenvalues may be regarded as the buckling loads. Therefore, based on the new differential quadrature approach of Wang [46], the discretized form of Eqs. (23) to (27) becomes (i=2,3,...,nx-1, j=2,3,...,ny-1): nx

ny

nx

A11 ∑ C u + 2 A16 ∑ C (2) ik kj

ny

∑C ∑C k =1 nx

k =1

AC C

k =1

nx

(1) ik

(1) ik

l =1

ny

nx

∑ C u kl + A66 ∑ C (2)jl uil + A16 ∑ Cik(2)vkj + ( A12 + A66 ) (1) jl

l =1

l =1

k =1

ny

(1) jl

nx

(2) v k l + A26 ∑ C (2) j l vil + ( B11 − c1 E11 )∑ Cik ϕ xkj + 2( B16 − c1 E16 ) l =1

k =1

ny

ny

nx

l =1

k =1

∑ Cik(1) ∑ C (1)jl ϕ x kl + ( B66 − c1E66 )∑ C (2)jl ϕ xil + ( B16 − c1E16 )∑ Cik(2)ϕ y kj k =1

l =1

ny

nx

+( B12 − c1 E12 + B66 − c1 E66 )∑ C

(1) ik

k =1

nx + 2

nx + 2

−c1 E11 ∑ C wkj − 3c1 E16 ∑ C k =1

(3) ik

k =1

(2) ik

∑C l =1

ny

ny

∑C l =1

ϕ y k l + ( B26 − c1E26 )∑ C (2) j l ϕ yil

(1) jl

l =1 nx

(1) jl

w k l − c1 ( E12 + 2 E66 )∑ C k =1

(1) ik

ny + 2

∑C l =1

(2) jl

w kl

ny + 2

−c1 E26 ∑ C (3) j l wil = 0

(36)

l =1

9

ACCEPTED MANUSCRIPT nx

ny

nx

A16 ∑ C u + ( A12 + A66 )∑ C (2) ik kj

k =1

(1) ik

k =1

nx

ny

k =1

l =1

∑C l =1

ny

(1) jl

nx

u k l + A26 ∑ C u + A66 ∑ Cik(2) vkj (2) jl il

l =1

ny

k =1

nx

(2) (2) +2 A26 ∑ Cik(1) ∑ C (1) j l v k l + A22 ∑ C j l vil + ( B16 − c1 E16 ) ∑ Cik ϕ x kj l =1

k =1

nx

ny

ny

k =1

l =1

(2) +( B66 − c1 E66 + B12 − c1 E12 )∑ Cik(1) ∑ C (1) j l ϕ x k l + ( B26 − c1 E26 )∑ C j l ϕ xil l =1

nx

nx

ny

k =1

l =1

k =1

RI PT

+( B66 − c1 E66 )∑ Cik(2)ϕ y kj + 2( B26 − c1 E26 )∑ Cik(1) ∑ C (1) jl ϕ y k l nx + 2

ny

(3) +( B22 − c1 E22 )∑ C (2) j l ϕ yil − c1 E16 ∑ Cik wkj − c1 ( E12 + 2 E66 ) ny

∑ C ∑C (2) ik

k =1

l =1

k =1

ny + 2

nx

(1) jl

w k l − 3c1 E26 ∑ C

(1) ik

k =1

∑C l =1

nx

ny + 2

(2) jl

w k l − c1 E22 ∑ C (3) j l wil = 0 l =1

ny

nx

( B11 − c1 E11 )∑ C u + 2( B16 − c1 E16 )∑ C (2) ik kj

k =1

nx

∑C l =1

ny

(1) jl

u k l + ( B66 − c1 E66 )∑ C (2) j l ui l l =1

M AN U

k =1

(1) ik

(37)

SC

l =1

nx + 2

nx

ny

k =1

l =1

+ ( B16 − c1 E16 )∑ Cik(2) vkj + ( B12 − c1 E12 + B66 − c1 E66 )∑ Cik(1) ∑ C (1) jl v k l k =1 ny

nx

2 (2) + ( B26 − c1 E26 )∑ C (2) jl vil + ( D11 − 2c1 F11 + c1 H 11 ) ∑ Cik ϕ xkj l =1

k =1

nx

+2 ( D16 − 2c1 F16 + c H16 ) ∑ C 2 1

k =1

(1) ik

ny

∑C l =1

ny

ϕ x k l + ( D66 − 2c1F66 + c H 66 ) ∑ C (2) j l ϕ xil

(1) jl

2 1

l =1

nx

TE D

+ ( D16 − 2c1 F16 + c12 H16 ) ∑ Cik(2)ϕ y kj + ( D12 − 2c1 F12 + c12 H12 + D66 − 2c1 F66 + c12 H 66 ) k =1

nx

∑C k =1

(1) ik

ny

ny

nx + 2

l =1

k =1

∑ C ϕ y kl + ( D26 − 2c1F26 + c H 26 ) ∑ C (2)jl ϕ yil − c1 ( F11 − c1H11 ) ∑ Cik(3) wkj l =1

2 1

(1) jl

nx + 2

ny

EP

−3c1 ( F16 − c1 H16 ) ∑ Cik(2) ∑ C (1) j l w k l − c1 ( F12 − c1 H 12 + 2 F66 − 2c1 H 66 ) k =1

nx

ny + 2

l =1

ny + 2

nx

l =1

k =1

k =1

AC C

∑ Cik(1) ∑ C (2)jl w kl − c1 ( F26 − c1H 26 ) ∑ C (3)jl wil − ( A55 − 2c2 D55 + c22 F55 ) ∑ Cik(1) wkj l =1

ny

2 − ( A45 − 2c2 D45 + c22 F45 ) ∑ C (1) j l wil − ( A55 − 2c2 D55 + c2 F55 ) ϕ xij l =1

− ( A45 − 2c2 D45 + c F45 ) ϕ yij = 0

(38)

2 2

10

ACCEPTED MANUSCRIPT nx

ny

nx

( B16 − c1 E16 )∑ C u + ( B12 − c1 E12 + B66 − c1 E66 )∑ C k =1

ny

∑C l =1

(2) ik kj

(1) ik

k =1

nx

∑C

k =1

(2) ik kj

ny

k =1

u k l + ( B26 − c1 E26 )

ny

nx

u + ( B66 − c1 E66 )∑ C v + 2( B26 − c1 E26 )∑ C

(2) jl il

l =1

(1) jl

(1) ik

∑C l =1

(1) jl

v k l + ( B22 − c1 E22 )

nx

∑ C (2)jl vil + ( D16 − 2c1F16 + c12 H16 ) ∑ Cik(2)ϕ x kj + ( D12 − 2c1F12 + c12 H12 + D66 − 2c1F66 l =1

k =1

ny

ny

k =1

l =1

2 (2) + c12 H 66 ) ∑ Cik(1) ∑ C (1) j l ϕ x k l + ( D26 − 2c1 F26 + c1 H 26 ) ∑ C j l ϕ xil l =1

RI PT

nx

+ ( D66 − 2c1 F66 + c H 66 ) ∑ C ϕ y kj + 2 ( D26 − 2c1 F26 + c H 26 ) ∑ C nx

2 1

k =1

nx

(2) ik

2 1

nx + 2

ny

k =1

l =1

k =1

nx + 2

ny

ny

∑C l =1

ϕ y kl

(1) jl

SC

(3) + ( D22 − 2c1 F22 + c H 22 ) ∑ C (2) j l ϕ yil − c1 ( F16 − c1 H16 ) ∑ Cik wkj 2 1

(1) ik

−c1 ( F12 − c1 H12 + 2 F66 − 2c1 H 66 ) ∑ Cik(2) ∑ C (1) jl w k l − 3c1 ( F26 − c1 H 26 )

∑C ∑ C k =1

(1) ik

nx

∑C k =1

(1) ik

l =1

l =1

ny + 2

(2) jl

M AN U

k =1

ny + 2

nx

2 w k l − c1 ( F22 − c1 H 22 ) ∑ C (3) j l wil − ( A45 − 2c2 D45 + c2 F45 ) l =1

ny

2 wkj − ( A44 − 2c2 D44 + c22 F44 ) ∑ C (1) jl wil − ( A45 − 2c2 D45 + c2 F45 ) ϕ xij

− ( A44 − 2c2 D44 + c F44 ) ϕ yij = 0

l =1

AC C

EP

TE D

2 2

11

(39)

ACCEPTED MANUSCRIPT ny ny nx nx nx  (3) (2) (1) (1) c1  E11 ∑ Cik ukj + 3E16 ∑ Cik ∑ C jl u k l + ( E12 + 2 E66 )∑ Cik ∑ C (2) jl u k l l =1 l =1 k =1 k =1  k =1 ny

nx

l =1

k =1

nx

ny

k =1

l =1

nx

ny

k =1

l =1

(3) (2) (1) (1) (2) + E23 ∑ C (3) j l uil + E16 ∑ Cik vkj + ( E12 + 2 E66 )∑ Cik ∑ C j l v k l + 3 E26 ∑ Cik ∑ C j l v k l ny

nx

l =1

k =1

nx

ny

k =1

l =1

(3) (2) (1) + E22 ∑ C (3) j l vil + ( F11 − c1 H11 )∑ Cik ϕ x kj + 3( F16 − c1 H16 )∑ Cik ∑ C jl ϕ x k l

+ ( F12 − c1 H12 + 2 F66 − 2c1 H 66 )∑ C k =1

(1) ik

∑C l =1

ny

ϕ x k l + ( F26 − c1H 26 )∑ C (3) j l ϕ xil

(2) jl

RI PT

ny

nx

l =1

nx

nx

ny

k =1

l =1

+ ( F16 − c1 H16 )∑ Cik(3)ϕ y kj + ( F12 − c1 H12 + 2 F66 − 2c1 H 66 ) ∑ Cik(2) ∑ C (1) jl ϕ y k l k =1

 (3) 2 +3( F26 − c1 H 26 )∑ Cik(1) ∑ C (2) j l ϕ y k l + ( F22 − c1 H 22 )∑ C jl ϕ yil  + ( A55 − 2c2 D55 + c2 F55 ) k =1 l =1 l =1 

∑C k =1

ny

nx

ϕ x kj + ( A45 − 2c2 D45 + c F45 ) ∑ C ϕ xil + ( A45 − 2c2 D45 + c F45 ) ∑ Cik(1)ϕ y kj

(1) ik

2 2

(1) jl

l =1

2 2

M AN U

nx

ny

SC

ny

nx

k =1

ny nx + 2  nx + 2 (4) (3) + ( A44 − 2c2 D44 + c F44 ) ∑ C ϕ yil − c  H11 ∑ Cik wkj + 4 H16 ∑ Cik ∑ C (1) jl w k l l =1 l =1 k =1 k =1  ny + 2 ny + 2 ny + 2 nx + 2 nx  (2) (2) (1) (3) +2( H12 + 2 H 66 ) ∑ Cik ∑ C jl w k l + 4 H 26 ∑ Cik ∑ C jl w k l − H 22 ∑ C (4) w il  jl l =1 l =1 l =1 k =1 k =1  ny

2 2

(1) jl

2 1

nx + 2

nx

+ ( A55 − 2c2 D55 + c F55 ) ∑ C wkj + 2 ( A45 − 2c2 D45 + c F45 ) ∑ C 2 2

2 2

TE D

k =1

(2) ik

k =1

(1) ik

ny

∑C l =1

(1) jl

w kl

ny + 2

nx + 2  HT (2) HT + ( A44 − 2c2 D44 + c F44 ) ∑ C (2) + ( , ) w N x y jl il i j ∑ Cik wkj + 2 N xy ( xi , y j )  x l =1 k =1  2 2

ny + 2

ny + 2   nx + 2 (2) C ∑ C w k l + N ( xi , y j ) ∑ C wil  +  k p ∑ Cik wkj + k p ∑ C (2) ∑ j l wil k =1 l =1 l =1 l =1   k =1 ny ) nx ) n y + 2 (2)   ) nx + 2 − k w w( xi , y j )  +  N x ∑ Cik(2) wkj + 2 N xy ∑ Cik(1) ∑ C (1) w + N jl kl y ∑ C j l wil  = 0 l =1 l =1 k =1  k =1  ny

nx

(1) jl

HT y

(2) jl

AC C

EP

(1) ik

(40)

6 Numerical results and discussions 6.1 Verification of the results Since present problem has not been investigated by other researchers, two simpler examples are considered for verification purposes. Simultaneously, a convergence study regarding choosing the appropriate sampling points is carried out. The sampling points are chosen based on roots of the Chebyshev-Gauss-Lobatto polynomials: 1 i −1  xi = 1 − cos π , i = 1, 2,..., nx (41) 2 nx − 1  A similar equation may be used for the y coordinate. As a first verification example, uniaxial and biaxial buckling loads of a three-layer cross-ply [0/90/0] relatively thick (b/h=10) rectangular composite plate on an elastic foundation are 12

ACCEPTED MANUSCRIPT determined. The plate is simply supported by edges experience SS1 type conditions. The dimensionless material properties and stiffness of the plate and the Pasternak-Winkler elastic foundation are: G23 E1 G12 G13 = 40 , = = 0.6 , = 0.5 , ν 12 = 0.25 E2 E2 E2 E2 k b4 k w = w 3 = 100 , E2 h

kp =

k pb 2 E2 h 3

= 10

RI PT

The dimensionless buckling loads ( N cr b 2 / ( E2 h 3 ) ) of the present research are extracted for

different mesh sizes and aspect ratios and compared with results reported by Xiang et al. [13], Akavci [14], and Setoodeh and Karami [49], in Table 1. Xiang et al. used a first-order while Akavci [14] employed a hyperbolic shear-deformation theory and both presented Navier-type solutions. Setoodeh and Karami [49] used a 3D finite element model.

Present

Akavci [14] Xiang et al. [13] Setoodeh et al. [49]

7×7 9×9 11×11 13×13 15×15 17×17 -

a/b = 1 Uniaxial Biaxial 50.7705 21.9571 50.7675 22.0059 50.7673 22.0017 50.7673 22.0018 50.7673 22.0018 50.7673 22.0018 50.813 21.980 50.751 22.228 49.226 21.866

M AN U

Mesh Size

TE D

Approach

SC

Table 1 A comparison among the dimensionless uniaxial and biaxial buckling loads predicted by various approaches ( kw = 100, k p = 10 ). a/b = 2 Uniaxial Biaxial 49.2391 19.3201 49.0676 19.3792 49.0484 19.3741 49.0492 19.3743 49.0491 19.3743 49.0491 19.3743 49.058 19.350 49.266 19.590 49.039 19.140

AC C

EP

As may be readily noted from results of Table 1, due to using a high-order shear-deformation theory in the present research, there is a good agreement among the results, even for the 7×7 mesh size. The second verification example evaluates the hygrothermal effects on the buckling of the laminated composite plates. In this regard, a four-layer asymmetric [0/90/0/90] laminated graphite/epoxy thick rectangular (a/b=0.5, a/h=10) plate with C2 type edge conditions previously considered by Ram and Sinha [21] is re-examined. Results are expressed in terms of the buckling ratio λ = ( N x )cr / ( N x )cr  . The dependency of the material ∆C = 0, T = 300 K

properties on temperature and moisture is also considered as Ram and Sinha [21] who used a first-order shear-deformation theory (FSDT). The base material properties are: ( E1 )∆C ,∆T =0 = 130GPa, ( E2 )∆C ,∆T =0 = 9.5GPa, ( G12 )∆C ,∆T =0 = 6GPa, G13 = G12 ,

G23 = 0.5G12 , ν 12 = 0.3, α1 = −0.3 × 10 −6 / K, α 2 = 28.1× 10 −6 / K, β1 = 0, β 2 = 0.44 Patterns of variations of the material properties with temperature and moisture can be found in reference [21]. The same problem was examined by Pandey et al. [26] who used a high-order shear-deformation theory (HSDT). For this reason present results are obtained based on the employed high-order theory and a simplified first-order theory [by substitution c1 = 0 in Eq. (1) and choosing a 5/6 shear correction factor] and compared in Table 2 with those of references [21] and [26].

13

ACCEPTED MANUSCRIPT Table 2 A comparison among the FSDT and HSDT buckling results (in terms of the buckling ratio λ = N cr / ( N cr ) ∆C =0,T =300 K ) of a clamped laminated [0/90/0/90] graphite/epoxy plate under hygrothermal effects (a/b=0.5, a/h=10). Buckling ratio N cr ( N cr ) ∆T ,∆C =0 HSDT [26]

FSDT [21]

∆C=0.5 % ∆C=1.0 % ∆T=50 °c ∆T=100 °c

0.987 0.971 0.945 0.876

0.984 0.970 0.951 0.888

Present DQ FSDT HSDT 0.9848 0.9844 0.9707 0.9701 0.9508 0.9498 0.8867 0.8848

RI PT

Environment conditions

M AN U

SC

The treated two verification examples reveal that present approach can accurately predict the uniaxial and biaxial buckling loads, considering the hygrothermal and elastic foundation effects, in addition to the higher convergence rate (in comparison to the finite element or double series solutions). Results of Table 1 shows that a mesh size of 17×17 may lead to higher accuracies; for this reason, this mesh size is adopted for derivation of the remaining results.

6.2 General specifications of the considered plates In the present research, the material properties are assumed to be both temperature and moisture dependent. Thickness of the thick plate is h=5mm (h/a=0.2). The plate basic material properties correspond to graphite/epoxy materials [23]: E f E0 = 230, G f E0 = 9, ν f = 0.203, α f = −0.54 × 10 −6 C −1 , ρ f = 1750 kg m3

TE D

E0 = 1 GPa , ν m = 0.34, α m = 45 × 10−6 C −1 ,

ρ m = 1200 kg m3 , V f = 0.6 ,

AC C

EP

G13 = G12 , G23 = 0.5G12 , β m = 2.68 × 10−3 / wt % H 2 O The hygrothermal effects on the elasticity modulus may be defined trough the following linear relation [23]: Em E0 = 3.51 − 0.003∆T − 0.142∆C (42) The following dimensionless buckling load and elastic coefficients of the elastic foundation are defined to present more general results: k p b2 103 N cr k b4 N cr = , Kw = w 3 , K p = E0 b E0 h E0 h 3 On the other hand, it is assumed that the plate is placed in a hydrostatic compression in the ) ) biaxial buckling analyses, i.e., N y = N x as the first verification example.

6.3 Effects of the orthotropy angle and transverse gradation of the material properties The sensitivity analysis presented in the present section includes investigation of simultaneous effects of the orthotropy angle ( θ ), the material properties gradation exponent (k), loading type (uniaxial or biaxial), boundary condition (simply supported and clamped), elastic coefficients of the elastic foundation (Kw, Kp), and temperature and moisture concentration. Results associated with the uniaxial compression are presented in Tables 3 and 4 for plates with simply supported and clamped edges, respectively, whereas results of the biaxial compression are given in Tables 5 and 6. In addition to the buckling loads, the 14

ACCEPTED MANUSCRIPT buckling load ratio with respect to the zero temperature and moisture rises condition, i.e., λ = N cr / ( N cr )∆C =0,T =300 K = N cr ( N cr ) ∆T ,∆C =0 are indicated in these tables.

SC

RI PT

During extraction of the results of the biaxial buckling, it was observed that the results are symmetric with respect to the 45 degrees orthotropy angle. For this reason, results associated with higher orthotropy angles are not reported here. However, results of the uniaxial compression show that the trends appeared in Tables 3 and 4, continue; so that the buckling load decrease with the orthotropy angle. Results of Tables 3 to 6 reveal that the degradations in the buckling strengths due to the hygrothermal effects are more remarkable for higher gradation exponents and the elastic foundation may enhance the buckling strengths. Simultaneous effects of the temperature and moisture rises on the uniaxial and biaxial thermo-mechanical buckling loads of the orthotropic FGM plate with elastic foundation are illustrated graphically in Figs. 2 and 4 and 3 and 5 for the SS2 and C1 edge conditions, respectively ( a / b = 1, h / a = 0.1, θ = 45, k = 0.5 , K w = 500 , K p = 50 ).

N cr

k=0 383.9282 331.0002 338.6629 289.0210 392.0339 339.1059 346.7686 297.1267 543.9282 491.0002 498.6629 449.0210 552.0339 499.1059 506.7686 457.1267 375.7045 318.7516 326.1194 273.0379 383.0849 326.0390 333.4664 280.2800 527.0950 467.2461 484.8415 426.5120 529.0249 469.1718

0.5 436.2277 373.7589 380.3268 322.5631 444.3333 381.8646 388.4325 330.6688 569.2277 533.7589 540.3268 482.5631 604.3333 541.8646 548.4325 490.6688 426.8355 359.3200 365.4233 303.3424 434.1866 366.5818 372.7457 310.5634 576.8160 508.5265 524.4171 460.3856 578.7462 510.4532

AC C

EP

TE D

θ (Deg.) Kw, Kp ∆T (K), ∆C (%) 0 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0 200,0 0,2 200,2 15 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0 200,0

M AN U

Table 3 The buckling load and buckling load ratio for an orthotropic FGM simply supported (SS2) plate resting on an elastic foundation and subjected to uniaxial compression, in hygrothermal environment (a/b=1, a/h=5).

15

1 496.0261 421.8973 425.9264 358.5944 504.1318 430.0029 434.0321 366.7001 656.0261 581.8973 585.9264 518.5944 664.1318 590.0029 594.0321 526.7001 485.2444 404.6156 407.9447 335.0364 492.5640 411.8512 415.2418 342.2362 633.3955 554.9424 566.9953 494.1458 635.3260 556.8703

N cr ( N cr ) ∆T , ∆C =0 k=0 1.0000 0.8621 0.8821 0.7528 1.0000 0.8650 0.8845 0.7579 1.0000 0.9027 0.9168 0.8255 1.0000 0.9041 0.9180 0.8281 1.0000 0.8484 0.8680 0.7267 1.0000 0.8511 0.8705 0.7316 1.0000 0.8865 0.9198 0.8092 1.0000 0.8869

0.5 1.0000 0.8568 0.8719 0.7394 1.0000 0.8594 0.8742 0.7442 1.0000 0.9377 0.9492 0.8478 1.0000 0.8966 0.9075 0.8119 1.0000 0.8418 0.8561 0.7107 1.0000 0.8443 0.8585 0.7153 1.0000 0.8816 0.9092 0.7981 1.0000 0.8820

1 1.0000 0.8506 0.8587 0.7229 1.0000 0.8530 0.8609 0.7274 1.0000 0.8870 0.8931 0.7905 1.0000 0.8884 0.8944 0.7931 1.0000 0.8338 0.8407 0.6904 1.0000 0.8361 0.8430 0.6948 1.0000 0.8761 0.8952 0.7802 1.0000 0.8765

ACCEPTED MANUSCRIPT

0,10

45

0,0

10,0

0,10

0.9127 0.7988 1.0000 0.8347 0.8635 0.7012 1.0000 0.8354 0.8669 0.7088 1.0000 0.8732 0.8987 0.7795 1.0000 0.8736 0.8991 0.7802 1.0000 0.8242 0.8509 0.6886 1.0000 0.8249 0.8516 0.6900 1.0000 0.8678 0.8915 0.7699 1.0000 0.8680 0.8930 0.7704

0.9033 0.7856 1.0000 0.8287 0.8532 0.6860 1.0000 0.8294 0.8538 0.6926 1.0000 0.8655 0.8861 0.7613 1.0000 0.8659 0.8864 0.7620 1.0000 0.8172 0.8357 0.6695 1.0000 0.8179 0.8364 0.6708 1.0000 0.8595 0.8778 0.7504 1.0000 0.8598 0.8791 0.7509

EP

10,10

0.9201 0.8099 1.0000 0.8393 0.8681 0.7109 1.0000 0.8401 0.8765 0.7196 1.0000 0.8799 0.9087 0.7944 1.0000 0.8803 0.9090 0.7951 1.0000 0.8295 0.8623 0.7031 1.0000 0.8302 0.8630 0.7045 1.0000 0.8748 0.9021 0.7857 1.0000 0.8750 0.9036 0.7862

RI PT

0,10

573.8864 499.1046 436.7626 361.9598 372.6485 299.6200 438.4448 363.6361 374.3374 303.6789 545.4217 472.0563 483.2884 415.2488 547.0935 473.7181 484.9540 416.9071 361.0641 295.0743 301.7536 241.7251 362.5303 296.5116 303.2365 243.1781 464.9105 399.6065 408.1140 348.8870 465.7313 400.4255 409.4465 349.7200

SC

10,0

528.2097 462.3119 389.1904 324.8607 336.0778 272.9192 390.8797 326.5424 338.8398 277.0601 497.7846 434.6800 447.3786 388.0273 499.4580 436.3414 449.0455 389.6851 320.9613 264.5287 273.0919 221.0177 322.4573 265.9907 274.5960 222.4862 425.3609 369.1195 379.2242 327.4921 426.1845 369.9387 380.5648 328.3197

M AN U

0,0

486.7696 428.4362 347.3185 291.5154 301.5011 246.9005 349.0165 293.2037 305.8976 251.1663 455.8606 401.1112 414.2373 362.1513 457.5362 402.7720 415.9058 363.8077 285.6376 236.9239 246.2945 200.8192 287.1627 238.4118 247.8215 202.3067 390.5410 341.6394 352.2906 306.8672 391.3675 342.4592 353.6432 307.6914

TE D

30

0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2

AC C

Table 4 The buckling load and buckling load ratio for an orthotropic FGM clamped (C1) plate resting on an elastic foundation and subjected to uniaxial compression, in hygrothermal environment (a/b=1, a/h=5).

θ

(Deg.) Kw, Kp ∆T (K), ∆C (%) 0 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2

N cr ( N cr ) ∆T , ∆C =0

N cr k=0 526.4623 461.1754 483.5280 413.2655 528.1058 464.9049 487.9000 418.1728 623.4763 560.3020 583.3784

0.5 588.9685 514.8977 537.5432 458.5039 590.6091 519.4601 542.1400 463.5041 685.9770 615.4026 639.2168

16

1 660.5928 575.7063 597.5493 508.3160 662.2309 580.3489 602.2490 513.4222 757.5991 678.4387 702.2871

k=0 1.0000 0.8760 0.9184 0.7850 1.0000 0.8803 0.9239 0.7918 1.0000 0.8987 0.9357

0.5 1.0000 0.8742 0.9127 0.7785 1.0000 0.8795 0.9179 0.7848 1.0000 0.8971 0.9318

1 1.0000 0.8715 0.9046 0.7695 1.0000 0.8764 0.9094 0.7753 1.0000 0.8955 0.9270

ACCEPTED MANUSCRIPT

10,10

30

0,0

10,0

0,10

10,10

0,0

AC C

45

10,0

0,10

10,10

17

0.8352 1.0000 0.8989 0.9359 0.8356 1.0000 0.8581 0.8981 0.7539 1.0000 0.8616 0.9015 0.7582 1.0000 0.8891 0.9236 0.8154 1.0000 0.8894 0.9238 0.8158 1.0000 0.8430 0.8821 0.7318 1.0000 0.8435 0.8825 0.7327 1.0000 0.8802 0.9113 0.7970 1.0000 0.8804 0.9115 0.7974 1.0000 0.8307 0.8681 0.7086 1.0000 0.8313 0.8686 0.7096 1.0000 0.8751 0.9041 0.7865 1.0000 0.8754 0.9043 0.7870

0.8306 1.0000 0.8974 0.9320 0.8310 1.0000 0.8555 0.8907 0.7446 1.0000 0.8587 0.8939 0.7486 1.0000 0.8855 0.9172 0.8066 1.0000 0.8858 0.9174 0.8070 1.0000 0.8393 0.8732 0.7209 1.0000 0.8397 0.8736 0.7217 1.0000 0.8745 0.9023 0.7839 1.0000 0.8748 0.9025 0.7843 1.0000 0.8260 0.8575 0.6954 1.0000 0.8265 0.8580 0.6964 1.0000 0.8683 0.8934 0.7709 1.0000 0.8685 0.8937 0.7714

0.8252 1.0000 0.8957 0.9272 0.8256 1.0000 0.8515 0.8805 0.7319 1.0000 0.8544 0.8834 0.7356 1.0000 0.8814 0.9091 0.7957 1.0000 0.8816 0.9093 0.7962 1.0000 0.8343 0.8613 0.7062 1.0000 0.8347 0.8617 0.7070 1.0000 0.8678 0.8908 0.7677 1.0000 0.8681 0.8910 0.7682 1.0000 0.8198 0.8434 0.6778 1.0000 0.8203 0.8438 0.6787 1.0000 0.8602 0.8798 0.7517 1.0000 0.8604 0.8801 0.7522

RI PT

0,10

625.1812 759.2183 680.0607 703.9129 626.8085 589.3388 501.8307 518.9087 431.3429 590.8151 504.8177 521.9407 434.6147 692.6493 610.4901 629.7207 551.1538 694.1018 611.9442 631.1777 552.6133 467.8408 390.3337 402.9697 330.4086 469.0341 391.5211 404.1702 331.6055 573.7678 497.9267 511.0993 440.4920 574.9335 499.0844 512.2666 441.6485 379.9652 311.4878 320.4483 257.5519 380.9565 312.4810 321.4646 258.5644 480.9557 413.7003 423.1638 361.5224 481.8433 414.5846 424.0780 362.4259

SC

10,0

569.7673 687.5952 617.0226 640.8406 571.3918 525.2179 449.3218 467.8288 391.1032 526.6944 452.2593 470.8003 394.3008 628.4096 556.4785 576.3973 506.8501 629.8590 557.9281 577.8499 508.3033 417.1087 350.0737 364.2168 300.6994 418.3091 351.2674 365.4231 301.9001 522.8287 457.2076 471.7482 409.8290 523.9951 458.3653 472.9157 410.9853 338.8329 279.8845 290.5437 235.6297 339.8388 280.8900 291.5702 236.6508 439.6783 381.7595 392.8262 338.9511 440.5658 382.6392 393.7349 339.8453

M AN U

0,0

520.7131 625.0935 561.9200 585.0004 522.3347 469.1738 402.6200 421.3555 353.6995 470.6512 405.5178 424.2812 356.8349 572.2640 508.8063 528.5221 466.6142 573.7105 510.2515 529.9706 468.0617 372.6138 314.0996 328.6844 272.6896 373.8222 315.3005 329.8976 273.8959 478.1432 420.8395 435.7344 381.0688 479.3101 421.9969 436.9019 382.2243 302.6628 251.4275 262.7492 214.4722 303.6849 252.4469 263.7881 215.5042 403.3793 353.0113 364.6795 317.2635 404.2673 353.8867 365.5837 318.1489

TE D

15

200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2

EP

10,10

ACCEPTED MANUSCRIPT Table 5 The buckling load and buckling load ratio for an orthotropic FGM simply supported (SS2) plate resting on an elastic foundation and subjected to biaxial compression, in ) ) hygrothermal environment (a/b=1, a/h=5, N y = N x ).

θ

N cr ( N cr ) ∆T , ∆C =0

EP

AC C

1 218.4500 177.0077 178.8166 142.2352 220.0711 178.6289 180.4377 143.8564 298.4500 257.0077 258.8166 222.2352 300.0711 258.6289 260.4377 223.8564 223.6355 180.2604 182.4067 143.9687 225.1470 181.7602 183.9118 145.4598 303.6355 260.2604 262.4067 223.9687 305.1470 261.7602 263.9118 225.4598 227.3394 184.2069 186.1586 147.6497 229.7429 186.3823 188.4326 149.6890 307.3394 264.2069 266.1586 227.6497 309.7429

18

k=0 1.0000 0.8237 0.8481 0.6874 1.0000 0.8254 0.8495 0.6904 1.0000 0.8803 0.8968 0.7877 1.0000 0.8810 0.8975 0.7890 1.0000 0.8206 0.8462 0.6822 1.0000 0.8221 0.8475 0.6849 1.0000 0.8770 0.8946 0.7822 1.0000 0.8777 0.8952 0.7834 1.0000 0.8265 0.8501 0.6902 1.0000 0.8278 0.8516 0.6926 1.0000 0.8806 0.8968 0.7867 1.0000

SC

0.5 192.3221 157.2896 160.6545 129.1750 193.9432 158.9108 162.2756 130.7962 272.3221 237.2896 240.6545 209.1750 273.9432 238.9108 242.2756 210.7962 197.3417 160.6828 164.3935 131.3403 198.8583 162.1886 165.9045 132.8385 277.3417 240.6828 244.3935 211.3403 278.8583 242.1886 245.9045 212.8385 200.4778 164.2569 167.7154 134.7983 202.9545 166.5092 170.0699 136.9178 280.4778 244.2569 247.7154 214.7983 282.9545

M AN U

k=0 169.4059 139.5423 143.6659 116.4500 171.0271 141.1635 145.2870 118.0711 249.4059 219.5423 223.6659 196.4500 251.0271 221.1635 225.2870 198.0711 174.2482 142.9843 147.4541 118.8738 175.7695 144.4954 148.9702 120.3781 254.2482 222.9843 227.4541 198.8738 255.7695 224.4954 228.9702 200.3781 176.8625 146.1828 150.3511 122.0673 179.4081 148.5069 152.7801 124.2610 256.8625 226.1828 230.3511 202.0673 259.4081

TE D

(Deg.) Kw, Kp ∆T (K), ∆C (%) 0 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0 200,0 0,2 200,2 15 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0 200,0 0,2 200,2 30 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0

0.5 1.0000 0.8178 0.8353 0.6717 1.0000 0.8194 0.8367 0.6744 1.0000 0.8714 0.8837 0.7681 1.0000 0.8721 0.8844 0.7695 1.0000 0.8142 0.8330 0.6655 1.0000 0.8156 0.8343 0.6680 1.0000 0.8678 0.8812 0.7620 1.0000 0.8685 0.8818 0.7632 1.0000 0.8193 0.8366 0.6724 1.0000 0.8204 0.8380 0.6746 1.0000 0.8709 0.8832 0.7658 1.0000

1 1.0000 0.8103 0.8186 0.6511 1.0000 0.8117 0.8199 0.6537 1.0000 0.8611 0.8672 0.7446 1.0000 0.8619 0.8679 0.7460 1.0000 0.8060 0.8156 0.6438 1.0000 0.8073 0.8169 0.6461 1.0000 0.8571 0.8642 0.7376 1.0000 0.8578 0.8649 0.7389 1.0000 0.8103 0.8189 0.6495 1.0000 0.8113 0.8202 0.6516 1.0000 0.8597 0.8660 0.7407 1.0000

RI PT

N cr

ACCEPTED MANUSCRIPT

0,10

10,10

266.3823 268.4326 229.6890 224.9489 181.3213 183.5879 144.7596 227.2339 183.4490 185.7835 146.7951 304.9489 261.3213 263.5879 224.7596 307.2339 263.4490 265.7835 226.7951

0.8809 0.8974 0.7874 1.0000 0.8214 0.8472 0.6831 1.0000 0.8229 0.8488 0.6859 1.0000 0.8774 0.8951 0.7824 1.0000 0.8779 0.8957 0.7834

0.8712 0.8838 0.7666 1.0000 0.8147 0.8338 0.6659 1.0000 0.8160 0.8353 0.6685 1.0000 0.8679 0.8815 0.7618 1.0000 0.8684 0.8822 0.7629

0.8600 0.8666 0.7415 1.0000 0.8061 0.8161 0.6435 1.0000 0.8073 0.8176 0.6460 1.0000 0.8569 0.8644 0.7370 1.0000 0.8575 0.8651 0.7382

RI PT

10,0

246.5092 250.0699 216.9178 198.5297 161.7350 165.5272 132.1928 200.8753 163.9227 167.7867 134.2898 278.5297 241.7350 245.5272 212.1928 280.8753 243.9227 247.7867 214.2898

SC

0,0

228.5069 232.7801 204.2610 175.2946 143.9923 148.5062 119.7377 177.6995 146.2376 150.8269 121.8929 255.2946 223.9923 228.5062 199.7377 257.6995 226.2376 230.8269 201.8929

M AN U

45

200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2

Table 6 The buckling load and buckling load ratio for an orthotropic FGM clamped (C1) plate resting on an elastic foundation and subjected to biaxial compression, in hygrothermal ) ) environment (a/b=1, a/h=5, N y = N x ).

N cr

k=0 208.4908 171.6962 179.0879 144.9838 209.2925 172.4730 179.8728 145.7441 288.4908 251.6962 259.0879 224.9838 289.2925 252.4730 259.8728 225.7441 206.8993 170.2540 177.4188 143.4416 207.8902 171.2038 178.3856 144.3636 286.8993 250.2540

0.5 233.5720 191.1374 197.8878 159.0919 234.3608 191.9014 198.6594 159.8389 313.5720 271.1374 277.8878 239.0919 314.3608 271.9014 278.6594 239.8389 232.0420 189.6869 196.2062 157.4581 233.0180 190.6184 197.1552 158.3591 312.0420 269.6869

AC C

EP

TE D

θ (Deg.) Kw, Kp ∆T (K), ∆C (%) 0 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0 0,2 200,2 10,10 0,0 200,0 0,2 200,2 15 0,0 0,0 200,0 0,2 200,2 10,0 0,0 200,0 0,2 200,2 0,10 0,0 200,0

19

1 262.0616 212.6463 217.9544 173.5458 262.8378 213.3974 218.7124 174.2790 342.0616 292.6463 297.9544 253.5458 342.8378 293.3974 298.7124 254.2790 260.5997 211.1626 216.2345 171.7605 261.5602 212.0745 217.1645 172.5507 340.5997 291.1626

N cr ( N cr ) ∆T , ∆C =0 k=0 1.0000 0.8235 0.8590 0.6954 1.0000 0.8241 0.8594 0.6964 1.0000 0.8725 0.8981 0.7799 1.0000 0.8727 0.8983 0.7803 1.0000 0.8229 0.8575 0.6933 1.0000 0.8235 0.8581 0.6944 1.0000 0.8723

0.5 1.0000 0.8183 0.8472 0.6811 1.0000 0.8188 0.8477 0.6820 1.0000 0.8647 0.8862 0.7625 1.0000 0.8649 0.8864 0.7629 1.0000 0.8175 0.8456 0.6786 1.0000 0.8180 0.8461 0.6796 1.0000 0.8643

1 1.0000 0.8114 0.8317 0.6622 1.0000 0.8119 0.8321 0.6631 1.0000 0.8555 0.8711 0.7412 1.0000 0.8558 0.8713 0.7417 1.0000 0.8103 0.8298 0.6591 1.0000 0.8108 0.8303 0.6597 1.0000 0.8549

ACCEPTED MANUSCRIPT

0,10

10,10

45

0,0

10,0

0,10

AC C

EP

10,10

20

0.8972 0.7788 1.0000 0.8726 0.8975 0.7793 1.0000 0.8201 0.8548 0.6885 1.0000 0.8208 0.8554 0.6897 1.0000 0.8708 0.8957 0.7762 1.0000 0.8711 0.8960 0.7768 1.0000 0.8206 0.8538 0.6880 1.0000 0.8215 0.8546 0.6896 1.0000 0.8722 0.8958 0.7777 1.0000 0.8726 0.8962 0.7784

0.8852 0.7610 1.0000 0.8645 0.8854 0.7615 1.0000 0.8145 0.8425 0.6733 1.0000 0.8151 0.8431 0.6744 1.0000 0.8626 0.8833 0.7580 1.0000 0.8629 0.8836 0.7586 1.0000 0.8148 0.8413 0.6725 1.0000 0.8156 0.8421 0.6740 1.0000 0.8637 0.8833 0.7591 1.0000 0.8641 0.8836 0.7598

0.8697 0.7392 1.0000 0.8551 0.8700 0.7394 1.0000 0.8071 0.8263 0.6533 1.0000 0.8077 0.8268 0.6544 1.0000 0.8529 0.8676 0.7357 1.0000 0.8532 0.8679 0.7362 1.0000 0.8072 0.8249 0.6521 1.0000 0.8079 0.8256 0.6535 1.0000 0.8539 0.8673 0.7364 1.0000 0.8543 0.8677 0.7371

RI PT

10,0

296.2345 251.7605 341.5602 292.0745 297.1645 252.5507 256.7688 207.2357 212.1678 167.7510 257.6560 208.0981 213.0412 168.5978 336.7688 287.2357 292.1678 247.7510 337.6560 288.0981 293.0412 248.5978 250.3301 202.0634 206.4888 163.2503 251.6338 203.3014 207.7505 164.4461 330.3301 282.0634 286.4888 243.2503 331.6338 283.3014 287.7505 244.4461

SC

0,0

276.2062 237.4581 313.0180 270.6184 277.1552 238.3591 228.7333 186.3066 192.7140 154.0142 229.6320 187.1815 193.6000 154.8743 308.7333 266.3066 272.7140 234.0142 309.6320 267.1815 273.6000 234.8743 222.6219 181.3884 187.2938 149.7155 223.9534 182.6553 188.5858 150.9421 302.6219 261.3884 267.2938 229.7155 303.9534 262.6553 268.5858 230.9421

M AN U

30

257.4188 223.4416 287.8902 251.2038 258.3856 224.3636 204.0461 167.3433 174.4189 140.4801 204.9564 168.2306 175.3172 141.3533 284.0461 247.3433 254.4189 220.4801 284.9564 248.2306 255.3172 221.3533 198.2311 162.6634 169.2429 136.3789 199.5904 163.9586 170.5645 137.6349 278.2311 242.6634 249.2429 216.3789 279.5904 243.9586 250.5645 217.6349

TE D

10,10

0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2 0,0 200,0 0,2 200,2

SC

RI PT

ACCEPTED MANUSCRIPT

EP

TE D

M AN U

Fig. 2 Variations of the uniaxial and biaxial buckling loads with moisture rises of the orthotropic FGM plate with SS2 edge conditions, for various temperature rises ( a / b = 1, h / a = 0.1, θ = 45, k = 0.5 , K w = 500 , K p = 50 ).

AC C

Fig. 3 Variations of the uniaxial and biaxial buckling loads with moisture rises of the orthotropic FGM plate with C1 edge conditions, for various temperature rises ( a / b = 1, h / a = 0.1, θ = 45, k = 0.5 , K w = 500 , K p = 50 ).

21

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Fig. 4 Variations of the uniaxial and biaxial buckling loads with temperature rises of the orthotropic FGM plate with SS2 edge conditions, for various moisture rises ( a / b = 1, h / a = 0.1, θ = 45, k = 0.5 , K w = 500 , K p = 50 ).

Fig. 5 Variations of the uniaxial and biaxial buckling loads with temperature rises of the orthotropic FGM plate with C1 edge conditions, for various moisture rises ( a / b = 1, h / a = 0.1, θ = 45, k = 0.5 , K w = 500 , K p = 50 ). Figs. 2 to 5 imply that variations of the dimensionless buckling loads with the temperature rise or moisture concentration are linear. Indeed, this fact has occurred due to the linear elastic behavior of the structure; so that the buckling load is proportional to the elastic modulus of the material (or its integrals through thickness, e.g., the bending or extensional rigidities), as may be noted from the numerical or closed-form solutions available in the text books [50,51] and papers [52-57]. For this reason, the first power of the elastic modulus has commonly been employed in definition of the dimensionless buckling load per unit load. If the elasticity modulus is a linear function of the temperature rise and moisture concentration (as in the present research), a linear relation may exist between the resulting dimensionless buckling 22

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loads and the temperature rise or moisture. This evident has been reported either for values of the buckling loads (in explicit [21,28] or implicit forms [58-60]) or for the hygrothermal effects on the buckling load [61]. Since in Figs. 2 to 5, plates with quite identical geometric and material specifications are considered, effects of the linear dependency of the temperature and moisture rises of the elasticity modulus are illustrated. To present a sensitivity analysis regarding, e.g., the type of the temperature-dependency of the elasticity modulus on the resulting buckling loads, the following dependency function is adopted temporary in addition to the main linear dependency function described by Eq. (42): Em E0 = 3.51 − 0.003∆T 1.2 − 0.142∆C (43) Results of the linear [Eq. (42)] and nonlinear [Eq. (43)] temperature-dependency of the elasticity modulus on the dimensionless buckling loads are compared in Fig. 6 ( a / b = 1, h / a = 0.1, θ = 45o , k = 0.5, K w = 500, K p = 50, ∆C = 3% ). These results confirm that the dimensionless buckling load is significantly affected by the type of the temperaturedependency of the elasticity modulus. A similar discussion may be presented for the moisturedependency of the elasticity modulus. Therefore, the linear relations observed in Figs. 2 to 5 are due to the linear dependency of the elastic modulus of the material on the temperature rise and moisture concentration.

Fig. 6- Influence of the temperature-dependency of the material properties on the resulting buckling loads of the orthotropic FGM plate with SS2 edge conditions.

6.4 Effects of the temperature and moisture concentration dependency of the material properties In the present section, influence of the temperature and moisture concentration dependency of the material properties on the resulting uniaxial and biaxial buckling loads is evaluated through comparing results of the temperature and moisture dependent (TMD) and temperature and moisture independent (TMID) material properties. At the same time, effects of the thickness and aspect ratios on the results are studied, in addition to effects of the orthotropy angle and the elastic foundation. Effects of the thickness ratio are evaluated in Tables 7 and 8 23

ACCEPTED MANUSCRIPT for SS2 and C1 boundary conditions, respectively, whereas, the aspect ratio effects are reported in Tables 9 and 10. Since the thin plate (h/a=0.05) buckles in lower temperatures, the relevant buckling loads are determined for lower temperature and moisture values (Tables 7 and 8). Table 7 Influence of the thickness ratio on uniaxial and biaxial buckling loads of orthotropic FGM plates with SS2 type edge conditions ( a / b = 1, θ = 45, K w = 100, K p = 10 ).

50,0.5 (TMD) 50,0.5 (TMID) 0.1

0,0 200,2 (TMD) 200,2 (TMID)

0.15

0,0

200,2 (TMID) 0.2

0,0

EP

200,2 (TMD)

k=0 8.7626 (1.0000) 7.0382 (0.8032) 7.2086 (0.8227) 56.2060 (1.0000) 38.7160 (0.6888) 43.7163 (0.7778) 147.3249 (1.0000) 111.3164 (0.7556) 128.4142 (0.8716) 275.7404 (1.0000) 217.1372 (0.7875) 250.1648 (0.9072)

TE D

200,2 (TMD)

Uniaxial 0.5 1 19.9611 23.1901 (1.0000) (1.0000) 15.6348 17.5393 (0.7833) (0.7563) 16.0377 17.9912 (0.8034) (0.7758) 109.9763 124.3981 (1.0000) (1.0000) 74.1126 79.8356 (0.6739) (0.6418) 86.8354 93.6864 (0.7896) (0.7531) 253.2476 280.1997 (1.0000) (1.0000) 187.0002 200.1597 (0.7384) (0.7143) 220.2658 236.4371 (0.8698) (0.8438) 433.3979 472.9369 (1.0000) (1.0000) 335.4740 356.9214 (0.7741) (0.7547) 393.5412 419.8727 (0.9080) (0.8878)

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200,2 (TMID)

Biaxial 0.5 10.1230 (1.0000) 7.9122 (0.7816) 8.1068 (0.8008) 63.3609 (1.0000) 41.6455 (0.6573) 47.1537 (0.7442) 162.3215 (1.0000) 118.8433 (0.7321) 137.7785 (0.8488) 298.6689 (1.0000) 229.3539 (0.7679) 265.4856 (0.8889)

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( N cr ) ∆T , ∆C =0 )

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0.05

k=0 17.2755 (1.0000) 13.9005 (0.8046) 14.2540 (0.8251) 97.5001 (1.0000) 68.2288 (0.6998) 79.6895 (0.8173) 229.6054 (1.0000) 174.0421 (0.7580) 204.2200 (0.8894) 398.5894 (1.0000) 314.8132 (0.7898) 368.0123 (0.9233)

cr

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(N

h/a ∆T (K), ∆C (%)

N cr

1 11.7607 (1.0000) 8.8720 (0.7544) 9.0907 (0.7730) 71.7370 (1.0000) 44.3627 (0.6184) 50.2676 (0.7007) 179.5398 (1.0000) 126.3157 (0.7036) 147.0214 (0.8189) 324.7558 (1.0000) 241.6357 (0.7441) 280.8024 (0.8647)

Table 8 Influence of the thickness ratio on uniaxial and biaxial buckling loads of orthotropic FGM plates with C1 type edge conditions ( a / b = 1, θ = 45, K w = 100, K p = 10 ).

(N

h/a ∆T (K), ∆C (%) k=0 23.6509 (1.0000) 50,0.5 (TMD) 20.3867 (0.8620) 50,0.5 (TMID) 21.1564 (0.8945) 0.1 0,0 110.9333

0.05

0,0

N cr

cr

( N cr ) ∆T , ∆C =0 )

Uniaxial 0.5 1 27.3660 31.7470 (1.0000) (1.0000) 23.2379 26.4366 (0.8492) (0.8327) 24.1335 27.4722 (0.8819) (0.8653) 124.4106 139.8623

24

k=0 13.4994 (1.0000) 11.4887 (0.8511) 11.9067 (0.8820) 70.2748

Biaxial 0.5 15.6438 (1.0000) 13.0902 (0.8368) 13.5772 (0.8679) 78.9391

1 18.1841 (1.0000) 14.8834 (0.8185) 15.4470 (0.8495) 88.9288

ACCEPTED MANUSCRIPT (1.0000) (1.0000) (1.0000) 48.5035 52.4260 56.1841 (0.6902) (0.6641) (0.6318) 57.4586 62.3095 66.9020 (0.8176) (0.7893) (0.7523) 163.6371 179.9569 198.5626 (1.0000) (1.0000) (1.0000) 122.2523 130.6888 139.1063 (0.7471) (0.7262) (0.7006) 144.2915 154.8554 165.3139 (0.8818) (0.8605) (0.8326) 290.0893 314.3602 341.9424 (1.0000) (1.0000) (1.0000) 227.0366 240.2567 253.6554 (0.7826) (0.7643) (0.7418) 264.1160 280.6648 297.3197 (0.9105) (0.8928) (0.8695)

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(1.0000) (1.0000) 85.2732 91.9812 (0.6854) (0.6577) 102.4110 110.6643 (0.8232) (0.7912) 265.9448 293.6365 (1.0000) (1.0000) 196.6863 210.7518 (0.7396) (0.7177) 234.5002 251.8385 (0.8818) (0.8577) 446.8446 488.2913 (1.0000) (1.0000) 346.5692 369.6839 (0.7756) (0.7571) 407.0521 435.0794 (0.9109) (0.8910)

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(1.0000) 78.5008 (0.7076) 200,2 (TMID) 94.0012 (0.8474) 0.15 0,0 241.6020 (1.0000) 200,2 (TMD) 182.9471 (0.7572) 200,2 (TMID) 217.4405 (0.9000) 0.2 0,0 410.4261 (1.0000) 200,2 (TMD) 324.4786 (0.7906) 200,2 (TMID) 379.9359 (0.9257) 200,2 (TMD)

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200,2 (TMD) 200,2 (TMID) 0,0

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200,2 (TMD)

200,2 (TMID)

3

k=0 487.6872 (1.0000) 392.0138 (0.8038) 455.5435 (0.9341) 472.0681 (1.0000) 378.0284 (0.8008) 438.9965 (0.9299) 465.3176 (1.0000) 371.7248 (0.7989) 433.1417 (0.9309) 463.5336 (1.0000) 369.9007 (0.7980) 430.0390 (0.9277)

0,0

200,2 (TMD) 200,2 (TMID)

Uniaxial 0.5 531.7641 (1.0000) 420.3129 (0.7904) 490.0201 (0.9215) 514.0318 (1.0000) 404.0020 (0.7859) 470.9254 (0.9161) 506.8472 (1.0000) 397.5954 (0.7844) 464.3390 (0.9161) 504.7348 (1.0000) 395.1689 (0.7829) 461.0028 (0.9134)

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Table 9 Influence of the aspect ratio on uniaxial and biaxial buckling loads of orthotropic FGM plates with SS2 type edge conditions ( h / b = 0.2, θ = 30, K w = 100, K p = 10 ).

25

N cr

cr

( N cr ) ∆T , ∆C =0 )

1 581.8654 (1.0000) 450.4659 (0.7742) 526.5266 (0.9049) 561.7090 (1.0000) 431.2966 (0.7678) 504.3308 (0.8979) 554.0498 (1.0000) 424.8214 (0.7668) 496.7244 (0.8965) 551.5541 (1.0000) 421.6207 (0.7644) 493.2215 (0.8942)

k=0 327.2356 (1.0000) 255.3565 (0.7803) 301.2705 (0.9207) 276.4864 (1.0000) 217.4131 (0.7863) 250.5560 (0.9062) 246.8423 (1.0000) 197.8277 (0.8014) 220.9996 (0.8953) 239.4723 (1.0000) 193.1681 (0.8066) 213.2815 (0.8906)

Biaxial 0.5 355.3199 (1.0000) 271.5871 (0.7643) 321.6313 (0.9052) 299.9051 (1.0000) 230.0171 (0.7670) 266.2547 (0.8878) 265.5775 (1.0000) 207.3639 (0.7808) 232.0844 (0.8739) 256.6049 (1.0000) 202.0209 (0.7873) 222.6416 (0.8676)

1 387.2615 (1.0000) 288.4387 (0.7448) 342.6459 (0.8848) 326.5294 (1.0000) 242.7082 (0.7433) 281.9536 (0.8635) 286.9825 (1.0000) 216.5600 (0.7546) 242.6642 (0.8456) 276.0890 (1.0000) 210.2914 (0.7617) 231.1325 (0.8372)

ACCEPTED MANUSCRIPT Table 10 Influence of the aspect ratio on uniaxial and biaxial buckling loads of orthotropic FGM plates with C1 type edge conditions ( h / b = 0.2, θ = 30, K w = 100, K p = 10 ).

200,2 (TMD) 200,2 (TMID) 1

0,0 200,2 (TMD) 200,2 (TMID)

2

0,0 200,2 (TMD) 200,2 (TMID)

3

0,0

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k=0 560.1947 (1.0000) 456.9936 (0.8158) 529.0565 (0.9444) 489.3683 (1.0000) 392.1171 (0.8013) 457.2510 (0.9344) 472.7876 (1.0000) 378.0720 (0.7997) 440.4607 (0.9316) 469.5425 (1.0000) 375.0681 (0.7988) 436.3092 (0.9292)

200,2 (TMID)

1 680.4474 (1.0000) 541.1640 (0.7953) 626.5723 (0.9208) 585.0819 (1.0000) 451.6591 (0.7720) 529.0844 (0.9043) 563.8879 (1.0000) 432.8927 (0.7677) 506.8680 (0.8989) 559.9835 (1.0000) 428.7061 (0.7656) 501.7510 (0.8960)

k=0 350.8689 (1.0000) 274.9749 (0.7837) 324.7894 (0.9257) 292.7141 (1.0000) 228.6693 (0.7812) 266.5844 (0.9107) 265.3471 (1.0000) 208.4576 (0.7856) 239.2390 (0.9016) 260.0143 (1.0000) 204.4282 (0.7862) 233.9665 (0.8998)

Biaxial 0.5 382.4696 (1.0000) 294.2849 (0.7694) 348.6337 (0.9115) 317.3214 (1.0000) 242.1186 (0.7630) 283.4185 (0.8932) 286.9629 (1.0000) 219.5916 (0.7652) 253.0975 (0.8820) 281.0896 (1.0000) 215.1280 (0.7653) 247.2962 (0.8798)

1 418.6426 (1.0000) 314.8321 (0.7520) 373.8330 (0.8930) 345.2767 (1.0000) 255.7659 (0.7408) 300.3748 (0.8700) 311.5496 (1.0000) 230.6134 (0.7402) 266.7094 (0.8561) 305.0799 (1.0000) 225.6701 (0.7397) 260.3275 (0.8533)

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0.5

Uniaxial 0.5 616.1604 (1.0000) 497.0220 (0.8066) 575.6259 (0.9342) 534.1017 (1.0000) 420.9431 (0.7881) 492.1387 (0.9214) 515.4469 (1.0000) 404.8777 (0.7855) 472.9135 (0.9175) 512.0262 (1.0000) 400.8932 (0.7830) 468.4187 (0.9148)

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6.5 Auxeticity effects on the buckling results As mentioned before, auxeticity (negative Poisson ratio) may be established through proper manufacturing techniques. Since extent of the resulting auxeticity depends on the employed fabrication technique, a sensitivity analysis that covers the possible Poisson ratio ranging from -1 to 0.5 (for the fiber, so that Poisson’s ratio becomes about -0.4 to 0.4 for the entire plate) is accomplished and the relevant effects on the uniaxial and biaxial buckling loads are studied for various rises in the temperature and moisture. Results are given in Table 11. From results of Table 11, one may deduce that the auxeticity generally reduces the buckling strength, in contrast to what has been reported by Lim [31] for the isotropic rectangular plates (without elastic foundations and temperature or moisture effects) but in agreement with the results reported by Lim in Ref. [32] for the circular plates. Therefore, it can be concluded that using auxetic materials is not only not recommended for circular plates [32] but also rectangular plates. However, presence of the elastic foundation has to somewhat, monitored effect of the auxeticity on the buckling loads.

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ACCEPTED MANUSCRIPT Table 11 Influence of the auxeticity on the uniaxial and biaxial buckling loads, for various temperature and moisture rises ( a / b = 1, h / a = 0.2, k = 0.5, θ = 45, K w = 100 , K p = 10 ).

ν 12 (ν f )

(N

∆T (K), ∆C (%)

N cr

cr

( N cr ) ∆T , ∆C =0 )

Uniaxial

-0.314 ( -0.75 )

0,0 200,2

-0.224 ( -0.6 )

0,0 200,2

-0.134 ( -0.45 )

0,0 200,2

-0.044 ( -0.3 )

0,0 200,2 0,0

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0.046 ( -0.15 )

200,2 0.106 ( -0.05 )

0,0

0.226 ( 0.15 )

EP

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200,2

0.316 ( 0.3 )

0,0

200,2

0.406 ( 0.45 )

0,0

200,2

C1 439.8895 (1.0000) 343.7459 (0.7814) 440.6620 (1.0000) 344.0464 (0.7807) 441.4873 (1.0000) 344.3726 (0.7800) 442.3659 (1.0000) 344.7248 (0.7793) 443.2991 (1.0000) 345.1032 (0.7785) 444.2880 (1.0000) 345.5082 (0.7777) 444.9790 (1.0000) 345.7932 (0.7771) 446.4395 (1.0000) 346.4000 (0.7759) 447.6061 (1.0000) 346.8881 (0.7750) 448.8364 (1.0000) 347.4055 (0.7740)

SS2 295.9158 (1.0000) 228.5803 (0.7725) 296.1477 (1.0000) 228.6326 (0.7720) 296.4236 (1.0000) 228.7014 (0.7715) 296.7438 (1.0000) 228.7869 (0.7710) 297.1089 (1.0000) 228.8890 (0.7704) 297.5194 (1.0000) 229.0078 (0.7697) 297.8187 (1.0000) 229.0963 (0.7692) 298.4796 (1.0000) 229.2959 (0.7682) 299.0309 (1.0000) 229.4655 (0.7674) 299.6310 (1.0000) 229.6524 (0.7665)

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C1 312.1931 (1.0000) 240.0039 (0.7688) 312.3256 (1.0000) 239.9602 (0.7683) 312.5099 (1.0000) 239.9416 (0.7678) 312.7453 (1.0000) 239.9477 (0.7672) 313.0134 (1.0000) 239.9783 (0.7667) 313.3681 (1.0000) 240.0331 (0.7660) 313.6206 (1.0000) 240.0830 (0.7655) 314.1933 (1.0000) 240.2147 (0.7645) 314.6824 (1.0000) 240.3414 (0.7638) 315.2231 (1.0000) 240.4920 (0.7629)

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200,2

SS2 426.4683 (1.0000) 332.7577 (0.7803) 427.2671 (1.0000) 333.0654 (0.7795) 428.1099 (1.0000) 333.3924 (0.7788) 428.9972 (1.0000) 333.7388 (0.7780) 429.9297 (1.0000) 334.1045 (0.7771) 430.9080 (1.0000) 334.4897 (0.7762) 431.5862 (1.0000) 334.7574 (0.7756) 433.0068 (1.0000) 335.3192 (0.7744) 434.1299 (1.0000) 335.7641 (0.7734) 435.3045 (1.0000) 336.2296 (0.7724)

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7 Conclusions Thermo-mechanical uniaxial and biaxial buckling analyses of orthotropic auxetic plates with temperature and moisture dependent material properties resting on Pasternak-Winkler elastic foundations in hygrothermal environments is performed in the present paper. Even the special cases of the present analysis have not been treated before. Novelties of the presented research are mentioned at the end of the introduction section. The major results may be highlighted as: o The presented formulations in the form of the new version of the DQM converge with a high rate, even for small mesh sizes. o Higher orthotropy angles reduce the buckling loads. o Higher gradation exponents lead to higher buckling strengths. o The hygrothermal degradations in the buckling strengths are higher in higher gradation exponents. o The elastic foundations may monitor some other effects and generally lead to higher buckling strengths. o In contrast to the only available paper in literature on the isotropic auxetic rectangular plates (without orthotropy, material gradation, hygrothermal or elastic foundation effects), the auxeticity reduces the buckling strength. o The proposed formulation covers many simpler practical problems.

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References [1] Kundu CK, Han J-H. Nonlinear buckling analysis of hygrothermoelastic composite shell panels using finite element method. Composites: Part B 2009; 40: 313–328. [2] Abediokhchi J, Kouchakzadeh MA, Shakouri M. Buckling analysis of cross-ply laminated conical panels using GDQ method. Composites: Part B 2013; 55: 440–446. [3] Na K-S, Kim J-H. Three-dimensional thermal buckling analysis of functionally graded materials. Composites: Part B 2004; 35: 429–437. [4] Sofiyev AH, Najafov AM, Kuruoglu N. The effect of non-homogeneity on the nonlinear buckling behavior of laminated orthotropic conical shells. Composites: Part B 2012; 43: 1196–1206. [5] Torabi J, Kiani Y, Eslami MR. Linear thermal buckling analysis of truncated hybrid FGM conical shells. Composites: Part B 2013; 50: 265–272. [6] Sofiyev AH. On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation. Composites: Part B 2014; 58: 524–533. [7] Satouri S, Kargarnovin MH, Allahkarami F, Asanjarani A. Application of Third Order Shear Deformation Theory in Buckling Analysis of 2D-Functionally Graded Cylindrical Shell Reinforced by Axial Stiffeners. DOI: 10.1016/j.compositesb.2015.04.036. [8] Asemi K, Shariyat M. Highly accurate nonlinear three-dimensional finite element elasticity approach for biaxial buckling of rectangular anisotropic FGM plates with general orthotropy directions. Composite Structures 2013; 106: 235-249. [9] Shariyat M, Asemi K. Three-dimensional non-linear elasticity-based 3D cubic B-spline finite element shear buckling analysis of rectangular orthotropic FGM plates surrounded by elastic foundations. Composites Part B 2014; 56: 934-947. [10] Mansouri MH, Shariyat M. Thermal buckling predictions of three types of high-order theories for the heterogeneous orthotropic plates, using the new version of DQM. Composite Structures 2014; 113(1): 40-55.

28

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[11] Sofiyev AH, Karaca Z. The vibration and buckling of laminated non-homogeneous orthotropic conical shells subjected to external pressure. Europ J Mech A/Solids 2009; 28: 317–328. [12] Sofiyev AH, Kuruoglu N. Buckling analysis of nonhomogeneous orthotropic thinwalled truncated conical shells in large deformation. Thin-Walled Struct 2013; 62: 131– 141. [13] Xiang Y, Kitipornchai S, Liew KM. Buckling and vibration of thick laminates on Pasternak foundations. Journal of Engineering Mechanics 1996; 122: 54-63. [14] Akavci SS. Buckling and free vibration analysis of symmetric and antisymmetric laminated composite plates on an elastic foundation. Journal of Reinforced Plastics and Composites 2007; 26; 1907-1919. [15] Kiani Y, Bagherizadeh E, Eslami MR. Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (Three approximate analytical solutions). ZAMM 2011; 91: 581- 593. [16] Alipour MM, Shariyat M. Semi-analytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations. Mechanics Research Communications 2011; 38(8): 594-601. [17] Alipour MM, Shariyat M. Semianalytical solution for buckling analysis of variable thickness two-directional functionally graded circular plates with nonuniform elastic foundations. ASCE Journal of Engineering Mechanics 2013; 139(5): 664-676. [18] Thai H-T, Kim S-E. Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation. International Journal of Mechanical Sciences 2013; 75: 34–44. [19] Zhang D-G, Zhou H-M. Mechanical and thermal post-buckling analysis of FGM rectangular plates with various supported boundaries resting on nonlinear elastic foundations. Thin-Walled Structures 2015; 89: 142–151. [20] Benkhedda A, Tounsi A, Bedia EAA. Effect of temperature and humidity on transient hygrothermal stresses during moisture desorption in laminated composite plates. Compos Struct 2008;82:623–35. [21] Sai Ram KS, Sinha PK. Hygrothermal effects on the buckling of laminated composite plates. Composite Structures 1992; 21: 233-247. [22] Chao L-P, Shyu S-L. Nonlinear buckling of fiber‐reinforced composite plates under hygrothermal effects. Journal of the Chinese Institute of Engineers 1996; 19: 657-667. [23] Shen H-S. Hygrothermal effects on the postbuckling of shear deformable laminated plates. International Journal of Mechanical Sciences 2001; 43: 1259-1281. [24] Patel BP, Ganapathi M, Makhecha DP. Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory. Composite Structures 2002; 56: 25–34. [25] Barton Jr O. Eigensensitivity analysis of moisture-related buckling of marine composite panels. Ocean Engineering 2007; 34: 1543–1551. [26] Pandey R, Upadhyay AK, Shukla KK. Hygrothermoelastic Postbuckling Response of Laminated Composite Plates. Journal of Aerospace Engineering, 23, 2010, 1-1–13. [27] Lee C-Y, Kim J-H. Hygrothermal postbuckling behavior of functionally graded plates. Composite Structures 2013; 95: 278–282. [28] Natarajan S, Deogekar PS, Manickam G, Belouettar S. Hygrothermal effects on the free vibration and buckling of laminated composites with cutouts. Composite Structures 2014; 108: 848–855. [29] Mir M, Najabat Ali M, Sami J, Ansari U. Review of Mechanics and Applications of Auxetic Structures. Advances in Materials Science and Engineering 2014, Article ID 753496, 17 pages. 29

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[30] Lim T-C. Auxetic Materials and Structures. Springer, 2015. [31] Lim T-C. Elastic stability of thick auxetic plates. Smart Mater. Struct. 2014; 23: 045004 (7pp). [32] Lim T-C. Buckling and Vibration of Circular Auxetic Plates. Journal of Engineering Materials and Technology 2014; 136: 021007-1-6. [33] Reddy JN. Mechanics of laminated composite plates and shells: Theory and analysis. 2nd edition, Boca Raton, CRC Press, 2004. [34] Thangaratnam KR, Palaninathan, Ramachandran J. Thermal buckling of composite laminated plates. Comput Struct 1989; 32: 1117-1124. [35] Nath Y, Shukla KK. Post-bucking of angle-ply laminated plates under thermal loading. Commun Nonlinear Sci Numer Sim 2001; 6: 1-16. [36] Shariyat M, Eslami MR. On thermal dynamic buckling analysis of imperfect laminated cylindrical shells. ZAMM 2000; 80: 171-182. [37] Shariyat M. A nonlinear double-superposition global–local theory for dynamic buckling of imperfect viscoelastic composite/sandwich plates: A hierarchical constitutive model. Composite Structures 2011; 93(7): 1890-1899. [38] Asemi K, Shariyat M, Salehi M, Ashrafi H. A full compatible three-dimensional elasticity element for buckling analysis of FGM rectangular plates subjected to various combinations of biaxial normal and shear loads. Finite Elements in Analysis and Design 2013; 74: 9–21. [39] Shen H.-S., Functionally graded materials: nonlinear analysis of plates and shells. CRC Press, Boca Raton, 2009. [40] Shariyat M, Moradi M. Enhanced algorithm for nonlinear impact of rectangular composite plates with SMA wires, accurately tracing the instantaneous and local phase changes. Composite Structures 2014; 108(1): 834–847. [41] Wang X, Liu F, Wang X, Gan L. New approaches in application of differential quadrature method to fourth-order differential equations. Commun Numer Meth Eng 2005; 21: 61-71. [42] Tornabene F, Fantuzzi N, Ubertini F, Viola E. Strong formulation finite element method based on differential quadrature: A Survey. Applied Mechanics Reviews 2015; 67(2), 020801-1-55. [43] Civan F, Sliepcevich CM. Differential quadrature for multi-dimensional problems. J Math Ana Appl 1984; 101: 423-443. [44] Jang SK. Application of differential quadrature to the analysis of structural components. PhD Thesis, University of Oklahoma, 1987. [45] Jang SK, Bert CW, Striz AG. Application of differential quadrature to static analysis of structural components. Int J Numer Meth Eng 1989; 28: 561-577. [46] Wang X, Tan M, Zhou Y. Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method. Thin-Walled Struct 2003; 41: 15–29. [47] Wang X, Wang Y, Yuan Z. Accurate vibration analysis of skew plates by the new version of the differential quadrature method. Appl Math Model 2014: 38(3): 926-937. [48] Shu C, Richards BE. Application of generalized differential quadrature to solve twodimensional incompressible Navier-Stokes equations. Int J Numer Meth Fluids 1992; 15: 791-798. [49] Setoodeh AR, Karami G. Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-wise FEM, Engineering Structures, 2004, 26: 211–220. [50] Wang CM, Wang CY, Reddy JN. Exact solutions for buckling of structural members. CRC Press LLC, 2005. 30

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[51] Bloom F, Coffin D. Handbook of thin plate buckling and postbuckling. 2nd Edition, Chapman & Hall/CRC, 2001. [52] Shukla KK, Nath Y. Analytical solution for buckling and post-buckling of angle-ply laminated plates under thermomechanical loading. International Journal of Non-Linear Mechanics 2001; 36: 1097-1108. [53] Girish J, Ramachandra LS. Thermomechanical postbuckling analysis of symmetric and antisymmetric composite plates with imperfections. Composite Structures 2005; 67: 453–460. [54] Shariyat M. Thermal buckling analysis of rectangular composite plates with temperature-dependent properties based on a layerwise theory. Thin-Walled Structures 2007; 45: 439–452. [55] Pandey R, Shukla KK, Jain A. Thermoelastic stability analysis of laminated composite plates: An analytical approach. Commun Nonlinear Sci Numer Simulat 2009; 14: 16791699. [56] Zhen W, Cheung YK, Lo SH, Wanji C. Effects of higher-order global–local shear deformations on bending, vibration and buckling of multilayered plates. Composite Structures 2008; 82: 277–289. [57] Shiau L-C, Kuo S-Y, Chen C-Y. Thermal buckling behavior of composite laminated plates. Composite Structures 2010; 92: 508–514. [58] Shen H-S. Hygrothermal effects on the postbuckling of axially loaded shear deformable laminated cylindrical panels. Composite Structures 2002; 56: 73–85. [59] Shen H-S. The effects of hygrothermal conditions on the postbuckling of shear deformable laminated cylindrical shells. International Journal of Solids and Structures 2001; 38; 6357-6380. [60] Shen H-S. Hygrothermal effects on the postbuckling of composite laminated cylindrical shells. Composites Science and Technology 2000; 60: 1227-1240. [61] Chao L‐P, Shyu S‐L. Nonlinear buckling of fiber‐reinforced composite plates under hygrothermal effects, Journal of the Chinese Institute of Engineers 1996, 19:6, 657-667.

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