DSpace at VNU: A nonlinear stability analysis of imperfect three-phase polymer composite plates

Mechanics of Composite Materials, Vol 49, No 4, September, 2013 (Russian Original Vol 49, No 4, July-August, 2013) A Nonlinear stability analysis of imperfect three-phase polymer composite plates Nguyen Dinh Duc,1* Tran Quoc Quan,1 and Do Nam2 In memory of G A Vanin Keywords: nonlinear stability analysis, laminated 3-phase composite plate, imperfection An analytical investigation into the nonlinear response of a thin imperfect laminated three-phase polymer composite plate consisting of a matrix and reinforcing fibers and particles and subjected to mechanical loads is presented All formulations are based on the classical theory of plates with account of interaction between the matrix and reinforcement, the geometrical nonlinearity, and an initial geometrical imperfection By using the Galerkin method, explicit relations for the load–deflection relationships are determined The effects of reinforcing fibers and particles, material and geometrical properties, and imperfections on the buckling and postbuckling load-carrying capacities of a 3-phase composite plate are analyzed and discussed Introduction Three-phase composites are materials consisting of a matrix and reinforcing fibers and particles They have been investigated by Vanin G A and Duc N D since 1996 [1, 2], who determined the elastic modulus for various 3-phase composites [3, 4] Their findings have shown that fibers are able to improve the elastic modulus, but particles can retard the penetration of heat [5], reduce the creep [6], and hinder the formation of defects in materials [7] Despite the large number of applications, our understanding of their structure is inadequate A general overview of 3-phase composites can be found in [8] Recently, the deflection [9] and the creep of three-phase composite plates in bending have been studied These investigations have shown that an optimal 3-phase composite can be obtained by controlling the volume ratios of fibers and particles Vietnam National University,Hanoi, 144 Xuan Thuy-Cau Giay- Hanoi-Viet Nam University of Engineering and Technology, Vietnam National University, Hanoi * Corresponding author; tel: 84-4-37547978; fax: 84-4-37547724; e-mail: ducnd@vnu.edu.vn Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 49, No 4, pp 519-536 , July-August, 2013 Original article submitted October 31, 2012 0191-5665/13/4904-0345 © 2013 Springer Science+Business Media New York 345 Plates, shells, and panels are the basic structures used in engineering and industry These structures play an important role as the main supporting components in all kinds of structures in machinery, civil engineering, shipbuilding, flight vehicle manufacturing, etc The stability of composite plates and shells is the first and most important problem in an optimal design In fact, many researchers are interested in this problem [8-24] Therefore the investigation on 3-phase composite plates and shells is important both from the scientifical and practical points of view Among the most fundamental references on the composite materials, we can mention monographs [12] (on laminated orthotropic composite plates and shells) and [20] (on composite FGM) Some information on the stability of laminated composite and FGM plates can be obtained in [11-24] In the present paper, we study the stability of a 3-phase polymer composite with an imperfection on an elastic foundation The paper focuses on deriving of an algorithm for calculating the stability of the composites by analyzing the load–deflection relationship obtained from the basic equations for laminated composites The effects of the material properties of components, geometrical properties, imperfections, and of the elastic foundation on the stability of 3-phase polymer composite plates are also examined Determination of the Elastic Moduli of a 3-Phase Composite The elastic moduli of 3-phase composites are estimated by consecutively using two theoretical models of a 2-phase composite: nDm = Om + nD [1-10] This paper considers a 3-phase composite reinforced with particles and unidirectional fibers, so the model of the problem will be 1Dm= Om + 1D First, the moduli of the effective matrix Om, called the “effective moduli,” are calculated At this step, the effective matrix consists of the original matrix and particles and is considered to be homogeneous and isotropic The next step is estimating the elastic moduli for a composite material consisting of the effective matrix and unidirectional reinforcing fibers Assuming that all the components (matrix, fibers, and particles) are homogeneous and isotropic, we denote by Em , Ea , Ec , n m ,n a ,n c , y m ,y a , and y c Young’s moduli, the Poisson ratios, and the volume fractions of matrix, fibers, and particles, respectively As in [1, 2], we obtain the moduli for the effective composite in the form G = Gm K = Km where L= −ψ c ( − 5ν m ) H + ψ c ( − 10ν m ) H , + 4ψ c Gm L ( 3K m ) −1 − 4ψ c Gm L ( 3K m ) −1 (1) , (2) Kc − K m Gm / Gc − , H= 4Gm G Kc + − 10ν m + ( − 5ν m ) m Gc E and n can be calculate from G and K : E= KG 3K + 2G , ν = 3K + G K − 2G We should note that formulas (1) and (2) take into account the nonlinear effects and the interaction between particles and the base and differ from the well-known formulas The elastic moduli for the 3-phase composite reinforced with unidirectional fibers are calculated as in [24]: 346 E11 = ψ a Ea + (1 −ψ a ) E + 8Gψ a (1 −ψ a ) (ν a −ν ) −ψ a + xψ a + (1 −ψ a ) ( xa − 1) G Ga , TABLE Elastic Moduli of 3-Phase Composite Materials E1, GPa Case E2, GPa ν12 G12, GPa G23, GPa 1.8616 1.6747 1.5093 1.3618 1.2296 1.1103 1.0021 2.7174 2.4451 2.2044 1.9900 1.7978 1.6244 1.4672 1.4840 1.4974 1.5093 1.5199 1.5295 1.5382 1.5461 2.6771 2.4247 2.2044 2.0103 1.8382 1.6843 1.5461 ψ a = const , y c increases 18.2019 17.8415 17.5209 17.2338 16.9751 16.7404 16.5273 8.0967 7.4411 6.8385 6.2829 5.7687 5.2916 4.8474 0.8043 0.8722 0.9457 1.0256 1.1132 1.2097 1.3167 ψ c = const , y a increases 10 11 12 13 14 24.2929 20.9035 17.5209 14.1451 10.7762 7.4144 4.0598 E22 where 1.0513 1.0116 0.9457 0.8496 0.7190 0.5486 0.3327   G G χ (1 −ψ a ) + + ψ a χ   (1 −ψ a ) ( χ − 1) + ( χ a − 1)( χ − + 2ψ a ) G G ν  a a  + =  21 +  E G G G  11 −ψ a + χψ a + (1 −ψ a )( χ a − 1) χ + ψ a + (1 −ψ a )   Ga Ga   ( ) −1     ,    G G χ + ψ a + (1 −ψ a ) Ga Ga G12 = G , G23 = G , G G −ψ a + (1 + ψ a ) (1 −ψ a ) χ + + χψ a Ga Ga + ψ a + (1 −ψ a ) 7.9971 7.3880 6.8385 6.3402 5.8860 5.4702 5.0880 ( ν 23 E22 )  G G (1 −ψ a ) ( x − 1) + ( xa − 1) ( x − + 2ψ a )  (1 −ψ a ) x + (1 + ψ a x ) Ga  Ga =− + −  G E11 8G G x + ψ a + (1 −ψ a ) −ψ a + xψ a + (1 −ψ a ) ( xa − 1)  G G a  a ν 21 ν 21 = ν − ( χ + 1) (ν −ν a )ψ a G −ψ a + χψ a + (1 −ψ a ) ( χ a − 1) Ga (3)   ,    , x = − 4ν For numerical calculations, we chose a 3-phase polymer composite made of a polyester AKAVINA matrix (made in Vietnam), glass fibers (made in Korea), and titanium oxide (made in Australia) with the following properties: AKAVINA — E = 1.43 GPa and n = 0.45; glass fibers — E = 22 GPa and n = 0.24; TiO2 — E = 5.58 GPa and n = 0.20 [25] 347 TABLE Cases of Different Volume Fractions of Fibers and Particles Case ym ya yc Case ym ya yc 0.5 0.2 0.3 0.55 0.2 0.25 0.6 0.2 0.2 0.65 0.2 0.15 0.7 0.2 0.1 0.75 0.2 0.05 0.8 0.2 0.0 0.5 0.3 0.2 0.55 0.25 0.2 10 0.6 0.2 0.2 11 0.65 0.15 0.2 12 0.7 0.1 0.2 13 0.75 0.05 0.2 14 0.8 0.0 0.2 a b Fig Black-and-white SEM images of 1D (a) and 2D (b) 2-phase composite materials (25% fibers, without particles) The elastic moduli of the composite materials calculated by formulas (3) for different volume fractions of components are given in Table The 14 cases considered are detailed in Table Figure illustrates SEM images of the structure of 2-phase polymer composites with glassy polyester fibers (without particles) In the 1D composite, all fibers are oriented in one direction, but in the 2D one — in two perpendicular directions Figure illustrates the structure of 3-phase 1Dm and 2Dm composites, respectively, containing TiO2 particles From these images, which were taken using SEM in the Laboratory of Micro-Nano Technology of the University of Engineering and Technology, Vietnam National University, Hanoi, it is seen that the greater amount of particles is introduced, the finer is the resulting material Governing Equations Consider a 3-phase midplane-symmetric composite plate The plate is referred to a Cartesian coordinate system ( x, y,z ), where xy is the midplane of the plate and z is the thickness coordinate, − h £ z £ h The length, width, and total thickness of the plate are a , b and h , respectively In this study, the classical theory of shells is used to establish the governing equations and to determine the nonlinear response of composite plates [12, 15] According to this theory, 348 b a Fig Black-and-white SEM images of 1Dm (a) and 2Dm (b) 3-phase composite materials (25% fibers, without particles)  ε   ε x0   k x      x    ε y  =  ε y0  + z  k y  ,        γ xy   γ xy   k xy  (4) where  ε   u + w2  ,x ,x  x       εy = v, y + w,2y  ,      u + v + w w   γ xy ,x , y     , y ,x  k   −w  , xx  x     k y  =  − w, yy  ,      k xy   −2 w, xy    with u and v the displacement components along the x and y directions, respectively Hooke’s law for the composite plate is expressed as σ  ′  Q11  x   ′  σ y  =  Q12    Q′  σ xy  k  16 where ′ Q12 ′ Q22 ′ Q26 ′  Q16 ′  Q26 ′  Q66 ε   x  ε y  ,   k  γ xy  k ′ = Q11 cos θ + Q22 sin θ + 2(Q12 + 2Q66 ) sin θ cos θ , Q11 ′ = Q12 (cos θ + sin θ ) + (Q11 + Q22 − 4Q66 ) sin θ cos θ , Q12 ′ = (Q12 − Q22 + 2Q66 ) sin θ cos θ + (Q11 − Q12 − 2Q66 ) sin θ cos3 θ , Q16 ′ = Q11 sin θ + Q22 cos θ + 2(Q12 + 2Q66 ) sin θ cos θ , Q22 ′ = (Q11 − Q12 − 2Q66 ) sin θ cos θ + (Q12 − Q22 + 2Q66 ) sin θ cos3 θ , Q26 with (5) ′ = Q66 (sin θ + cos 4θ ) + Q11 + Q22 − 2(Q12 + Q66 ) sin θ cos θ Q66 Q11 = E1 E1 , = E2 −ν 12ν 21 − ν 12 E1 349 Q22 = Q12 = E2 E = Q11 , E E1 − ν 12 E1 ν E1 = 12 , Q66 = G12 E2 Q22 − ν 12 E1 Here, θ is the angle between fibers and the coordinate system The force and moment resultants of the composite plates are determined by [13] n Ni = ∑ hk ∫ k =1 hk −1 n Mi = ∑ hk ∫ k =1 hk −1 σ i k dz , i = x, y, xy, (6) z σ i k dz , i = x, y, xy Insertion of Eqs (4) and (5) into Eq (6) gives the constitutive relations ( N x , N y , N xy ) = ( A11 , A12 , A16 )ε x0 + ( A12 , A22 , A26 )ε y0 + ( A16 , A26 , A66 )γ xy0 +( B11 , B12 , B16 )k x + ( B12 , B22 , B26 )k y + ( B16 , B26 , B66 )k xy ( M x , M y , M xy ) = ( B11 , B12 , B16 )ε x0 + ( B12 , B22 , B26 )ε y0 + ( B16 , B26 , B66 )γ xy0 +( D11 , D12 , D16 )k x + ( D12 , D22 , D26 )k y + ( D16 , D26 , D66 )k xy , (7) where n Aij = ∑ (Qij′ ) k (hk − hk −1 ), k =1 Bij = Dij = n ∑ (Qij′ )k (hk2 − hk2−1 ), k =1 n ∑ (Qij′ )k (hk3 − hk3−1 ), i, j = 1, 2, k =1 The nonlinear equilibrium equations of the composite plate are [13]: N x, x + N xy , y = N xy , x + N y , y = , M x, xx + M xy , xy + M y , yy + N x w, xx + N xy w, xy + N y w, yy = (8) From relations (7), it follows that * * * ε x0 = A11 N x + A12 N y + A16 N xy − F11k x − F12 k y − F16 k xy , * * * ε y0 = A12 N x + A22 N y + A26 N xy − F21k x − F22 k y − F26 k xy , * * * ε xy = A16 N x + A26 N y + A66 N xy − F61k x − F62 k y − F66 k xy , where 350 * A11 = A22 A66 − A26 A A − A12 A66 A A − A22 A16 * * , A12 = 16 26 , A16 = 12 26 , ∆ ∆ ∆ (9) * A22 = 2 A11 A66 − A16 A A − A11 A26 A A − A12 * * , A26 = 12 16 , A66 = 11 22 , ∆ ∆ ∆ 2 ∆ = A11 A22 A66 − A11 A26 + A12 A16 A26 − A12 A66 − A16 A22 , * * * * * * F11 = A11 B11 + A12 B12 + A16 B16 , F12 = A11 B12 + A12 B22 + A16 B26 , * * * * * * F16 = A11 B16 + A12 B26 + A16 B66 , F21 = A12 B11 + A22 B12 + A26 B16 , * * * * * * F22 = A12 B12 + A22 B22 + A26 B26 , F26 = A12 B16 + A22 B26 + A26 B66 , * * * * * * F61 = A16 B11 + A26 B12 + A66 B16 , F62 = A16 B12 + A26 B22 + A66 B26 , * * * F66 = A16 B16 + A26 B26 + A66 B66 Introducing Eq (9) into the expression of M ij in relations (7) and then the results into Eq (8), we have N x, x + N xy , y = , N xy , x + N y , y = , P1 f, xxxx + P2 f, yyyy + P3 w, xxyy + P4 w, xxxy + P5 w, xyyy + P6 w, xxxx + P7 w, yyyy + P8 w, xxyy + P9 w, xxxy + P10 w, xyyy + N x w, xx + N xy w, xy + N y w, yy = with (10) P1 = E12 , P2 = E21 , P3 = E11 + E22 − E33 , P4 = E32 − E13 , P5 = E31 − E23 , P6 = E14 , P7 = E25 , P8 = E15 + E24 + E36 , P9 = E16 − E34 , P10 = E26 − E35 , E11 = F11 , E12 = F21 , E13 = F61 , E21 = F12 , E22 = F22 , E23 = F62 , E31 = F16 , E32 = F26 , E33 = F66 , E14 = B11 F11 + B12 F21 + B16 F61 − D11 , E15 = B11 F12 + B12 F22 + B16 F62 − D12 , E16 = B11 F16 + B12 F26 + B16 F66 − D16 , E24 = B12 F11 + B22 F21 + B26 F61 − D12 , E25 = B12 F12 + B22 F22 + B26 F62 − D22 , E26 = B12 F16 + B22 F26 + B26 F66 − D26 , E34 = B16 F11 + B26 F21 + B66 F61 − D16 , E35 = B16 F12 + B26 F22 + B66 F62 − D26 , E36 = B16 F16 + B26 F26 + B66 F66 − D66 , where f(x,y) is the stress function defined by N x = f, yy , N y = f, xx , N xy = − f, xy (11) For an imperfect composite plate, Eqs (10) are modified to the form [22, 23] P1 f, xxxx + P2 f, yyyy + P3 w, xxyy + P4 w, xxxy + P5 w, xyyy + P6 w, xxxx + P7 w, yyyy + P8 w, xxyy ( ) ( ) ( ) + P9 w, xxxy + P10 w, xyyy + f, yy w, xx + w,*xx − f, xy w, xy + w,*xy + f, xx w, yy + w,*yy = 0, (12) where w* ( x, y ) is a known function representing an initial small imperfection of the plate The geometrical compatibility equation for an imperfect composite plate is written as * * * ε x0, yy + ε y0, xx − γ xy , xy = w, xy − w, xx w, yy + w, xy w, xy − w, xx w, yy − w, yy w, xx (13) From constitutive relations (9), in conjunction with Eq (11), one can obtain that * * * ε x0 = A11 f, yy + A12 f, xx − A16 f, xy − F11k x − F12 k y − F16 k xy , * * * ε y0 = A12 f, yy + A22 f, xx − A26 f, xy − F21k x − F22 k y − F26 k xy , * * * ε xy = A16 f, yy + A26 f, xx − A66 f, xy − F61k x − F62 k y − F66 k xy (14) 351 Inserting Eqs (14) into Eq (13) gives the compatibility equation for an imperfect composite plate [16, 21-23] * * A11 f, xxxx + E1 f, xxyy + A22 f, yyyy + F21 w , xxxx + F12 w , yyyy + E2 w , xxyy − E3 w , xxxy + E4 w , xyyy − w,2xy − w, xx w, yy + w, xy w,*xy − w, xx w,*yy − w, yy w,*xx = , where ( ) (15) * * E1 = A12 − A66 , E2 = F11 + F22 + F66 , E3 = F26 + F61 , E4 = F16 + F62 Equations (12) and (15) are nonlinear in the variables w and f and are used to investigate the stability of thin composite plates subjected to mechanical loads In the present study, the edges of composite plates are assumed to be simply supported Depending on in-plane restraints at the edges, three cases of boundary conditions, labeled as Cases 1, 2, and 3, will be considered [13, 6,21] Case All four edges of the plate are simply supported and freely movable (FM) The associated boundary conditions are = w N= M x = , N x = N x at x = 0, a , xy = w N= M y = , N y = N y at y = 0, b xy (16) Case All four edges of the plate are simply supported and immovable (IM) In this case, the boundary conditions are w= u= M x = , N x = N x at x = 0, a , w= v= M y = , N y = N y at y = 0, b (17) Case All edges are simply supported The edges x = 0, a are freely movable, whereas the edges y = 0, b are immovable In this case, the boundary conditions are defined as w N= M x = , N x = N x at x = 0, a , = xy w= v= M y = , N y = N y at y = 0, b , (18) where N x0 and N y0 are in-plane compressive loads at the movable edges (i.e., Case and the first of Case 3) or fictitious compressive edge loads at the immovable edges (i.e., Case and the second of Case 3) The approximate solutions w and f satisfying boundary conditions (16)-(18) are assumed to be [22, 23] ( ) w, w* = (W , µ h ) sin λm x sin δ n y , f = A1 cos 2λm x + A2 cos 2δ n y + A3 sin λm x sin δ n y + A4 cos λm x cos δ n y + 1 N x0 y + N y x2 , 2 (19) where λm = mπ / a , δ n = nπ / b , W is the deflection amplitude, and m is the imperfection parameter The coefficients Ai (i = 1−3) are determined by inserting Eqs (19) into Eq (15): 352 A1 = ⋅ δ n2 * λm2 32 A22 W (W + µ h) , A2 = A3 = ⋅ λm2 * δ n2 32 A11 −( F21λm4 + F12δ n4 + E2 λm2 δ n2 ) A4 = * * A22 λm + A11 δ n + E1λm2 δ n2 λmδ n ( E3λm2 + E4δ n2 ) * * A22 λm + A11 δ n + E1λm2 δ n2 W (W + µ h) , W , W Introducing Eqs (19) into Eq (12) and applying the Galerkin procedure to the resulting equation yield − F λm4 + F12δ n4 + E2 λm2 δ n2 F21λm4 + F12δ n4 + E2 λm2 δ n2 ab [ P1 21 + P λ δn * m * * * 4 A22 λm + A11 δ n + E1λm2 δ n2 A22 λm + A11 δ n + E1λm2 δ n2 * * A22 λm + A11 δ n + E1λm2 δ n2 + P5   * A22 + λmδ n  P1 F21λm4 + F12δ n4 + E2 λm2 δ n2 + P3 − λm2 δ n2 + P4 λm2 δ n4 ( E3λm2 + E4δ n2 ) * * A22 λm + A11 δ n + E1λm2 δ n2 + P2 λm4 δ n2 ( E3λm2 + E4δ n2 ) * * A22 λm + A11 δ n + E1λm2 δ n2 − P6 λm4 − P7δ n4 − P8 λm2 δ n2 ]W  4 ab   W (W + µ h ) −  * δ n + * λm  W (W + µ h ) (W + µ h ) *  A11  64  A22 A11  8λmδ n ( F21λm4 + F12δ n4 + E2 λm2 δ n2 ) * * 3( A22 λm + A11 δ n + E1λm2 δ n2 ) W (W + µ h ) − (20) ab N x λm2 + N y 0δ n2 (W + µ h ) = , ( ) where m and n are odd numbers This is the basic governing equation for the nonlinear response of 3-phase polymer composite plates under mechanical loading conditions Nonlinear Stability Analysis Consider a simply supported polymer composite plates whose all edges are movable Let us analyze two cases of mechanical loads 4.1 Polymer composite plates under axial compressive loads Let a polymer composite plate be subjected to an axial compressive load Fx uniformly distributed along its two curved = q 0= , N y 0, and N x = − Fx h , and Eq (20) leads to edges x = In this case, Fx = b11W + b21 where b11 = b21 = − P1 − P3 W W (W + µ ) + b31 + b4W W + µ , W +µ W +µ ( −32mBa2 ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * m Ba4 + A11 n + E1m n Ba2 ) 3nBh ( A22 m 4π Ba4 ( F21m Ba4 + F12 n + E2 m n Ba2 ) * n Bh2 ( A22 m Ba4 + A1*1n + E1m n Ba2 ) m 2π Ba2 ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * Bh2 ( A22 m Ba4 + A11 n + E1m n Ba2 ) − P5 ) m n 2π Ba2 ( E3 m Ba2 + E4 n ) * * Bh6 ( A22 m Ba4 + A11 n + E1m n Ba2 ) − P2 − P4 + P6 (21) , n 2π ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * Bh2 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m 4π Ba4 ( E3 m Ba2 + E4 n ) * * Bh6 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m 4π Ba4 n Bh2 + P7 n 2π Bh2 + P8 m 2π Ba2 Bh2 , 353 b31 = and P1 = P1 h 4mnBa2  P1 P2  4   , b41 = −π + ( n + m Ba ) , 2 * *  * * n B 3Bh2  A22 16 A A A h 22 11 11   , P2 = P2 h , P3 = P3 h , P4 = P4 h , P5 = * * * * A11 = hA11 , A22 = hA22 , E1 = hE1 , E2 = E4 = E4 h ,W= P5 h , P6 = P6 h , P7 = P7 h , P8 = P8 h3 , E E2 F F , F21 = 21 , F12 = 12 , E3 = 33 , h h h h W b b , Bh = , Ba = h h a For a perfect composite plate ( µ = 0) subjected only to an axial compressive load Fx , Eq (21) leads to Fx = (b11 + b31 ) W + b21 + b41W , from which the upper buckling compressive load may be obtained at W → : Fx upper = b21 = − P1 − P3 m 4π Ba4 ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * n Bh2 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m 2π Ba2 ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * Bh2 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m n 2π Ba2 ( E3 m Ba2 + E4 n ) − P5 * * Bh6 ( A22 m Ba4 + A11 n + E1m n Ba2 ) − P4 + P6 − P2 n 2π ( F21m Ba4 + F12 n + E2 m n Ba2 ) * * Bh2 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m 4π Ba4 ( E3 m Ba2 + E4 n ) * * Bh6 ( A22 m Ba4 + A11 n + E1m n Ba2 ) m 4π Ba4 n Bh2 + P7 n 2π Bh2 + P8 m 2π Ba2 Bh2 We are also interested in finding the lower buckling load To this end, we consider Fx = Fx (W ) at odd numbers m dFx and n, and from equation = , we get that dW 4mn3 Ba2  P1 P   +  * * *   A* m Ba4 + A11 n + E1m n Ba2 ) 3( A22  22 A11  −π  4  n + m Ba *   A* A 11  22  −32mnBa2 ( F21m Ba4 + F12 n + E2 m n Ba2 ) W th = − Since dFx dW (b11 + b31 ) = − 2b41 + > 0, the lower buckling load is W =W th Fxlower = − (b11 + b31 ) 4b41 + b21 4.2 Numerical results and discussion of 3-phase composite plates The results presented in this section, which are found from Eq (21), correspond to the deformation mode with the half-wave numbers m= n= The numerical calculation were performed for two following cases: 6-layer asymmetric plates with the stacking sequence 0/45/–45/45/–45/90 and 5-layer symmetric plates with the stacking sequence 45/–45/0/45/–45 354 a b F 109, GPa x F 109, GPa x 2 W/h W/h Fig Pressure–deflection curves Fx–W/h of 6-layer 3-phase polymer composite plates with ya = 0.2 at yc = (1), 0.1 (2), and 0.2 (3) (a) and with yc = 0.2 at ya = (1), 0.1 (2), and 0.2 (3); a/b = 1, b/h = 30, and m = 0.1 а 12 b F 109, GPa 2.5 x 10 Fx.1010, GPa 2.0 1.5 1.0 0.5 W/h W/h Fig The same for 5-layer plates b а Fx.109, GPa Fx.109, GPa 3 1 3 1 W/h W/h Fig Pressure–deflection curves Fx–W/h of 6-layer (a) and 5-layer (b) 3-phase polymer composites plates with an initial imperfection m = (1), 0.1 (2), 0.2 (3), and 0.3 (4); m = n = 1, b/a = 1, b/h = 30, ya = 0.2, and yc = 0.15 Figure illustrates the effect of particles and fibers on the buckling of 6-layers 3-phase plates and Fig — the same for 5-layer 3-phase plates 355 b a F 109, GPa x F 109, GPa x 5 2 W/h 3 W/h Fig Pressure–deflection curves Fx–W/h of 6-layer (a) and 5-layer (b) 3-phase polymer composite plates with b/h = 20 (1), 30 (2), 40 (3), and 50 (4); a/b = 1, m = 0.1, ya = 0.2, and yc = 0.15 b а 12 F 109, GPa x 14 F 109, GPa x 12 10 10 8 4 W/h 0.5 1.0 1.5 2.0 2.5 W/h Fig The same at b/a = 0.5 (1), 1.0 (2), 1.5 (3), and 1.75 (4); b/h = 30, m = 0.1, ya = 0.2, and yc = 0.15 From Figs and 4, it is seen that an increase in the volume fraction of particles and fibers increases the load-carrying ability of the plates However, the effect of fibers is stronger, and the symmetric plate has a higher load-carrying ability than the asymmetric one Figure depicts the effect of imperfection on the buckling of 6- and 5-layer plates The results show only subtle changes when the degree of imperfection increases However, the 6-layer imperfect plates have a little higher load-carrying ability than the 5-layer ones Figures and illustrate the effect of the geometrical parameters b/h and b/a on buckling of the plates It is not surprising that the thicker a plate (b/h is large) and the larger its length/width (b/a) ratio, the higher the load-carrying ability of the plate From the results illustrated in the above figures, the following conclusions can be drawn • Increasing the volume fractions of fibers and particles in 3-phase polymer composites improves their load-carrying ability, but the effect of fibers is stronger • The load-carrying ability of symmetric plates is higher than that of asymmetric ones • The geometry of the plates affects their stability significantly 356 Concluding Remarks The paper presents an analytical investigation of the nonlinear response of 3-phase polymer composite plates subjected to mechanical loadings All formulations are based on the classical theory of plates with account of a geometrical nonlinearity and initial imperfections The Galerkin method is used to obtain explicit expressions for the load–deflection relationships The study reveals the effects of fibers and particles and the geometrical parameters and imperfections of the plates on their nonlinear response to mechanical loadings Acknowledgment In loving memory of Professor Vanin G A , it is our greatest hounor to devote our findings to him on the 5th anniversary (2008-2012) of his death and the 82th anniversary of his birth (1930-2012) This work was supported by Project QGDA.12.03 of Foundation for the Science and Technology Development of Vietnam National University, Hanoi The authors are grateful for this financial support References G A Vanin and Nguyen Dinh Duc, “Theory of spherofibrous composites 1: Input relations, hypotheses, and models,” J Mechanics of Composite Materials, 32, No 3, 291-305 (1996) G A.Vanin and Nguyen Dinh Duc, “Theory of spherofibrous composites 2: The fundamental equations,” J Mechanics of Composite Materials, 32, No 3, 306-316 (1996) G A Vanin and Nguyen Dinh Duc, “Determination of rational structure of a composite with a filled matrix 1: Reinforcement scheme 3Dm,” J Mechanics of Composite Materials, 33, No 2, 155-160 (1997) Nguyen Dinh Duc, “The determination of the rational structure of a composite with a filled matrix 2: Reinforcement scheme Dm,” J Mechanics of Composite Materials, 33, No 3, 370-376 (1997) Nguyen Dinh Duc, “The thermoelastic expansion of a spherofibre composite,” J Mechanics of Composite Materials, 33, No 2, 251-257 (1997) Nguyen Dinh Duc, “The creep by the shear of an orthogonally reinforced spherofibre composite,” J Mechanics of Composite Materials, 32, No 6, 770-779 (1996) Nguyen Dinh Duc, “The theory of adhesive strength for spherofibres composite,” J Mechanics of Composite Materials, 33, No 5, 644-65 (1997) Dinh Khac Minh Bending of Three-Phase Composite Laminated Plates in Shipbuilding Industry PhD thesis in Engineering Maritime University, Hai Phong, Vietnam (2011) Nguyen Dinh Duc and Dinh Khac Minh, “Bending analysis of three-phase polymer composite plates reinforced with glass fibers and titanium oxide particles,” J Computational Mater Sci., 49, No 4, 194-198 (2010) 10 Nguyen Dinh Duc, Dinh Khac Minh, and Pham Van Thu, “Bending analysis of three-phase polymer composite plates reinforced with glass fiber and titanium oxide particles including the creep effect,” Int J of Aeronautical and Space Sci., 11, No 4, 361–366 (2010) 11 Hoang Van Tung, Elastic Stability of FGM Plates and Shells PhD thesis in Mechanics University of Sciences – Vietnam National University, Hanoi, (2011) 12 D O Brush and B O Almroth, Buckling of Bars, Plates, and Shells, McGraw-Hill, New York, (1975) 13 L.C Bank and J Yin, “Buckling of orthotropic plates with free and rotationally restrained edges,” Thin-Walled Structures, 24, 83-96 (1996) 14 Christian Mittelstedt, “Stability behaviour of arbitrarily laminated composite plates with free and elastically restrained unloaded edges,” Int J of Mech Sci., 49, 819-833 (2007) 15 J N Reddy, “Analysis of functionally graded plates,” Int J Numer Meth Eng 47, 663-684 (2000) 357 16 E Feldman and J Aboudi, “Buckling analysis of functionally graded plates subjected to uniaxial loading”, J Compos Struct., 38, 29-36 (1997) 17 R Javaheri and M R Eslami, “Buckling of functionally graded plates under in-plane compressive loading,” ZAMM, 82, No 4, 277-283 (2002) 18 B A Samsam Shariat and M R Eslami, “Buckling of thick functionally graded plates under mechanical and thermal loads,” J Compos Struct., 78, 433-439 (2007) 19 K M Liew, J Yang, and S Kitipornchai, “Postbuckling of piezoelectric FGM plates subjected to thermo-electromechanical loading,” Int J Solids Struct., 40, 3869-3892 (2003) 20 Shen Hui-Shen, Functionally Graded materials, Nonlinear Analysis of Plates and Shells, CRC Press, Taylor & Francis Group, London, Newyork, (2009) 21 Nguyen Dinh Duc and Hoang Van Tung, “Mechanical and thermal postbuckling of shear deformable FGM plates with temperature-dependent properties,” J Mechanics of Composite Materials, 46, No 5, 461-476 (2010) 22 Nguyen Dinh Duc and Hoang Van Tung, “Mechanical and thermal postbuckling of higher-order shear deformable functionally graded plates on elastic foundations,” J Composite Structures, 93, 2874-2881 (2011) 23 D H Bich, V H Nam, and N T Phuong, “Nonlinear postbuckling of eccentrically stiffened functionally graded plates and shallow shells,” Vietnam J of Mech, 33, No 3, 131-147 (2011) 24 G A Vanin, Micromechanics of Composite Materials, Kiev, “Nauka Dumka”, (1985) 25 Nguyen Dinh Duc and Dinh Khac Minh, “Experimental study on the mechanical properties of a three-phase polymer composite reinforced with glass fibers and titanium oxide particles,” Vietnam J of Mechanics, 33, No 2, 105-112 (2011) 358 ... and of the elastic foundation on the stability of 3-phase polymer composite plates are also examined Determination of the Elastic Moduli of a 3-Phase Composite The elastic moduli of 3-phase composites... nonlinear response of 3-phase polymer composite plates subjected to mechanical loadings All formulations are based on the classical theory of plates with account of a geometrical nonlinearity and... * A1 1 = A2 2 A6 6 − A2 6 A A − A1 2 A6 6 A A − A2 2 A1 6 * * , A1 2 = 16 26 , A1 6 = 12 26 , ∆ ∆ ∆ (9) * A2 2 = 2 A1 1 A6 6 − A1 6 A A − A1 1 A2 6 A A − A1 2 * * , A2 6 = 12 16 , A6 6 = 11 22 , ∆ ∆ ∆ 2 ∆ = A1 1