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

Composite Structures 109 (2014) 130–138 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear stability analysis of imperfect three-phase polymer composite plates in thermal environments Nguyen Dinh Duc a,⇑, Pham Van Thu b a b Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam Institute of Shipbuilding, Nha Trang University, 44 Hon Ro, Nha Trang, Khanh Hoa, Viet Nam a r t i c l e i n f o Article history: Available online November 2013 Keywords: Nonlinear stability Laminated three-phase composite plate Thermal environments Imperfection a b s t r a c t This paper presents an analytical investigation on the nonlinear response of the thin imperfect laminated three-phase polymer composite plate in thermal environments The formulations are based on the classical plate theory taking into account the interaction between the matrix and the particles, geometrical nonlinearity, initial geometrical imperfection By applying Galerkin method, explicit relations of load– deflection curves are determined Obtained results show effects of the fibers and the particles, material, geometrical properties and temperature on the buckling and post-buckling loading capacity of the three phase composite plate, therefore we can proactively design materials and structural composite meet the technical requirements as desired when adjustment components Ó 2013 Elsevier Ltd All rights reserved Introduction Three phase composite is a material consisted of matrix of the reinforced fibers and particles which have been investigated by Vanin and Duc since 1996 They have determined the elastic modulus for three phases composite 3Dm [1] and 4Dm [2] Their findings have shown that the fibers are able to improve the elastic modulus, the particles can resist to the penetration and the heat, reduce the creep deformations and the defects in materials Despite of a large number of applications, our understanding on the structure of three phase composite materials (plate and shell) is not much The general view of three-phase composite can be found in [3] Recently, there are several claims on the deflection and the creep for the three phase composite plate in the bending state [4] These findings have shown that optimal three-phase composite can be obtained by controlling the volume ratios of fiber and particles Plate, shell and panel are basic structures used in engineering and industry These structures play an important role as main supporting component in all kind of structure in machinery, civil engineering, ship building, flight vehicle manufacturing, etc The stability of composite plate and shell is the first and most important problem in optimal design In fact, many researchers are interested in this problem including the studies of the composite plates [4–12] However, researches on the stability of three-phases composite plates and shells are very few Whereas, the choice of a suitable ratio of components materials in three-phases composite is very important in designing new composite materials and predict ⇑ Corresponding author Tel.: +84 37547978; fax: +84 37547724 E-mail address: ducnd@vnu.edu.vn (N.D Duc) 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2013.10.050 mechanical and physical properties of advanced designed materials Therefore, from scientific and practical point of view, it is, therefore, very important and meaningful to carry an investigation on the three phase composite plate and shell Actively choosing material components and ratio of its mixing allows us to decide the advance materials and forecasting its physic-mechanical characteristics Several fundamental references on composite plates and shells are Brush and Almroth [13], Reddy [14] (for laminated composite plate and shell) and Shen [15] (for composite FGM) Some research on the stability of laminated composite and FGM plates can be obtained in [8–12] Recently, in [7] we have studied nonlinear stability of the threephase polymer composite plate under mechanical loads In the present paper, we have studied the nonlinear stability of threephase polymer composite with imperfections in thermal environments The paper focuses on deriving the algorithm for calculating the stability of three-phase composite by analyzing the load– deflection relationship base on the basic equations of laminated composite, while also studies the effect of component material properties, temperature, geometrical properties and imperfection on the stability of three-phase polymer composite plate A special point of the results is to show the algorithms explicit determine the coefficients of thermal expansion of the three-phase composite material on the elastic modulus, coefficients of thermal expansion and the ratio of component material (the elastic modules of three-phase composite material was determined by theoretical and experimental methods are published in [7,16]) Moreover, the first time the article showed performances Hooke’s law relationship of stress–strain three-phase composite plate include the effects of temperature Thereby, we can calculate nonlin- 131 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 ear behavior of three-phase composite plates under temperature loads and determine the effect of the component elements and structure of materials on thermal stability of the plate Determine the elastic modules and the effective thermal expansion coefficients of three-phase composite in which v ¼ À 4m; va ¼ À 4ma : ð6Þ 2.2 Determine the effective thermal expansion coefficient of composite 2.1 Determine the elastic modules of composite The elastic modules of three-phase composites are estimated using two theoretical models of the two-phase composite consecutively: nDm = Om + nD [1,2,7] This paper considers three-phase composite reinforced with particles and unidirectional fibers, so the problem’s model will be: 1Dm = Om + 1D Firstly, the modules of the effective matrix Om which called ‘‘effective modules’’ are calculated In this step, the effective matrix consists of the original matrix and particles, it is considered to be homogeneous, isotropic and have two elastic modules The next step is estimating the elastic modules for a composite material consists of the effective matrix and unidirectional reinforced fibers Assume that all the component phases (matrix, fiber and particle) are homogeneous and isotropic, we will use Em, Ea,Ec; mm, ma, mc; wm, wa; wc to denote Young modulus and Poisson ratio and volume ratio for matrix, fiber and particle, respectively According to Vanin and Duc in [1,2], we can obtain the modules for the effective composite as G ¼ Gm K ¼ Km À wc 5mm ịH ; ỵ wc 10mm ịH ỵ 4wc Gm L3K m ị1 À 4wc Gm Lð3K m ÞÀ1 ð1Þ ð2Þ ; here Lẳ Kc Km ; K c ỵ 4G3m Hẳ Gm =Gc : 10mm ỵ ð7 À 5mm Þ GGmc ð3Þ  can be calculate from G; Kị as E; m Eẳ 9KG 3K ỵ G m ¼ ; 3K À 2G 6K À 2G ð4Þ : We should note that formulas (1) and (2) take into account the nonlinear effects and the interaction between the particles and the base These are different from the other well-known formulas The elastic modules for 3-phase composite reinforced with unidirectional fiber are chosen to be calculated using Vanin’s formulas [17] with six independent elastic modulus E1 ¼ wa Ea ỵ wa ịE ỵ ( E2 ẳ m221 E1 ỵ " ị 8Gwa wa ịma m wa ỵ xwa ỵ À wa Þðxa À 1Þ GGa ;  À 1Þ þ ðva À 1Þðv  À þ 2wa Þ GG 21 wa ịv a 8G  wa ỵ ð1 À wa Þðva À 1Þ GG À wa þ v a #)À1 G   vð1 À wa ị ỵ ỵ wa vị Ga ỵ2 ; v ỵ wa ỵ wa ị GGa G12 ẳ G ỵ wa ỵ wa ị GGa ; G23 ẳ G v ỵ wa ỵ wa ị GGa ; 5ị  ỵ ỵ v  wa Þ GG ð1 À wa Þv a wa ỵ ỵ wa ị GGa " wa ịx ỵ ỵ wa xị GGa m23 m2 ẳ 21 ỵ E22 E11 8G x ỵ wa ỵ wa ị GG a 21 wa ịx 1ị ỵ xa 1ịx ỵ 2wa ị GGa wa þ xwa þ ð1 À wa Þðxa À 1Þ GGa  ỵ 1ịm  ma ịwa v m21 ẳ m ;  wa ỵ wa ịva 1ị GG wa ỵ v a # ; Similar to the elastic modulus, the thermal expansion coefficient of the three-phase composite materials were also identified in two steps: First, to determine the coefficient of thermal expansion of the effective matrix The current paper uses the Duc’s result from [18] for calculating the thermal expansion coefficient of effective matrix a ẳ am ỵ ac am ị K c 3K m ỵ 4Gm ịwc ; K m 3K c ỵ 4Gm ị ỵ 4K c K m ÞGm xc ð7Þ in which a⁄ is the effective thermal expansion coefficient of effective matrix; am, ac are the thermal expansion coefficients of original matrix and particle, respectively Then, determining two coefficients of thermal expansion of the three-phase composite using formulas from [17] of Vanin " à à a1 ¼ a À ða À a À1 a Þwa E1 Ea þ 8Ga ðma À mÞð1 À wa Þð1 þ ma ị # ; wa ỵ xwa ỵ À wa Þðxa À 1Þ GGa m À m21 a2 ẳ a ỵ a a1 ịm21 a aa ị1 ỵ ma ị : m ma 8ị To numerical calculating, we chosen three phase composite polymer made of polyester AKAVINA (made in Vietnam), fibers (made in Korea) and titanium oxide (made in Australia) with the properties as in Table [16] The results of elastic modules of composite materials for different volume ratios of component materials written in (5) are given in Table [7], in which 14 variant cases of different volume ratios of component three-phase composite materials respectively are given in Table [7] Fig illustrates the SEM images of the structure of the twophase composite polymer with the based phase composed of the glassy polyester fiber (without the particles) in 1D (all the composite fibers are reinforced in one direction) Fig illustrates the structure of three-phase 1Dm in the presence of the TiO2 particles From the above illustrations, it is obviously that the more the particles are doped, the finer the material is, in other words, the less the holes are These results also mean that we can increase the elastic modulus as well as strengthens the penetration resistance of the materials by doping the particles Figs and show the SEM pictures of our proposed fabricated composite structures These pictures are taken by ourself using the SEM instrumentation at the Laboratory for Micro-Nano Technology, University of Engineering and Technology, Vietnam National University, Hanoi Also, we made these composite material samples in the Institute of Ship building, Nha Trang University Governing equations Consider a three phase composite plate with midplane-symmetric The plate is referred to a Cartesian coordinate system x, y, z, where xy is the mid-plane of the plate and z is the thickness coordinator, Àh/2 z h/2 The length, width, and total thickness of the plate are a, b and h, respectively In this study, the classical theory is used to establish governing equations and determine the nonlinear response of composite plates [13–15] 132 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 Table Properties of the component phases for three-phase considered composite Component phase Young modulus, E (GPa) Poisson ratio, m Matrix polyester AKAVINA (Vietnam) Glass fiber (Korea) Titanium oxide TiO2 (Australia) 1.43 22 5.58 0.345 0.24 0.20 Table Elastic modules for three-phase composite materials E1 (GPa) E2 (GPa) m12 G12 (GPa) G23 (GPa) 0.8043 0.8722 0.9457 1.0256 1.1132 1.2097 1.3167 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 1.0513 1.0116 0.9457 0.8496 0.7190 0.5486 0.3327 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 wa = const wc – Particle’s ratio increase Case Case Case Case Case Case Case 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 wc = const wa – Fiber’s ratio increase Case Case Case Case Case Case Case 10 11 12 13 14 24.2929 20.9035 17.5209 14.1451 10.7762 7.4144 4.0598 7.9971 7.3880 6.8385 6.3402 5.8860 5.4702 5.0880 Fig Black–White SEM image of 1Dm composite three-phase material (25% of fibers and 10% of particles) B @ 1 e0x ex kx C B B C ey A ẳ @ e0y C A ỵ z@ ky A; cxy kxy c0xy ð9Þ where Table Variant cases of different volume ratios of matrix, fibers and particles B @ Case wm wa wc 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 10 11 12 13 14 0.5 0.3 0.2 0.55 0.25 0.2 0.6 0.2 0.2 0.65 0.15 0.2 0.7 0.1 0.2 0.75 0.05 0.2 0.8 0.0 0.2 wm wa wc 1 1 e0x u;x ỵ w2;x =2 w;xx kx B B C B C C ey A ¼ @ v ;y ỵ w2;y =2 C w ; k ẳ A @ yA @ ;yy A; 2w;xy kxy c0xy u;y ỵ v ;x ỵ w;x w;y 10ị in which u, v are the displacement components along the x, y directions, respectively Hooke law for a composite plate is defined as [19] 10 ex À a1 DT rx Q 011 Q 012 Q 016 B C B C 0 C B @ ry A ¼ @ Q 12 Q 22 Q 26 A @ ey À a2 DT A ; 0 cxy rxy k Q 16 Q 26 Q 66 k k ð11Þ in which Q 011 ¼ Q 11 cos4 h þ Q 22 sin h þ 2ðQ 12 þ 2Q 66 ịsin hcos2 h; Q 012 ẳ Q 12 cos4 h ỵ sin hị ỵ Q 11 ỵ Q 22 À 4Q 66 Þsin hcos2 h; Q 016 ẳ Q 12 Q 22 ỵ 2Q 66 ịsin h cosh ỵ Q 11 Q 12 2Q 66 ịsinhcos3 h; Q 022 ẳ Q 11 sin h ỵ Q 22 cos4 h ỵ 2Q 12 ỵ 2Q 66 ịsin hcos2 h; Q 026 ¼ ðQ 11 À Q 12 À 2Q 66 Þsin h cosh ỵ Q 12 Q 22 ỵ 2Q 66 ịsinhcos3 h; Q 066 ẳ Q 66 sin h ỵ cos4 hị ỵ ẵQ 11 ỵ Q 22 2Q 12 ỵ Q 66 ịsin h cos2 h; 12ị and Q 11 ẳ Q 22 ẳ Q 12 ¼ Fig Black–White SEM image of 1D composite two-phase material (25% of fibers without particles) E1 À EE21 m212 ¼ E1 ; À m12 m21 E2 E2 ¼ Q 11 ; À EE21 m212 E1 E1 À EE21 m212 Q 66 ¼ G12 ; ¼ m12 Q 22 ; ð13Þ N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 where h is an angle between the fiber and the coordinate system The force and moment resultants of the composite plates are determined by [13,14] Ni ¼ Mi ¼ n Z X hk kẳ1 hk1 n Z hk X kẳ1 hk1 ẵri k dz; Nx;x ỵ Nxy;y ẳ 0; Nxy;x ỵ Ny;y ẳ 0; P1 f;xxxx ỵ P2 f;yyyy ỵ P3 w;xxyy ỵ P4 w;xxxy ỵ P5 w;xyyy ỵ P6 w;xxxx i ẳ x; y; xy; ỵ 2Nxy w;xy ỵ Ny w;yy ẳ 0; where i ¼ x; y; xy: P1 ¼ BÃ21 ; Substitution of Eqs (9) and (11) into Eq (14) and the result into Eq (14) give the constitutive relations as x y xy ðNx ; Ny ; Nxy ị ẳ A11 ; A12 ; A16 ịe þ ðA12 ; A22 ; A26 Þe þ ðA16 ; A26 ; A66 ịc ỵ B11 ; B12 ; B16 ịkx ỵ B12 ; B22 ; B26 ịky ỵ B16 ; B26 ; B66 ịkxy DTẵa1 A11 ; A12 ; A16 ị ỵ a2 A12 ; A22 ; A26 ފ; ðM x ; My ; M xy Þ ¼ B11 ; B12 ; B16 ịe0x ỵ B12 ; B22 ; B26 ịe0y ỵ B16 ; B26 ; B66 ịc0xy P2 ẳ B12 ; P3 ẳ B11 ỵ B22 2BÃ66 ; P4 ¼ 2BÃ26 À BÃ61 ; 2BÃ16 P5 ẳ B62 ; P6 ẳ B11 B11 ỵ B12 B21 ỵ B16 B61 ; P ẳ B12 B12 ỵ B22 B22 ỵ B26 B62 ; P8 ẳ ỵ ỵ B16 B62 ỵ B12 B11 ỵ B22 B21 ỵ B26 B61 ỵ 4B16 B16 ỵ 4B26 B26 ỵ 4B66 B66 ; P9 ẳ B11 B16 ỵ B12 B26 ỵ B16 B66 ỵ B16 B11 ỵ B26 B21 ỵ B66 B61 ; P10 ẳ B12 B16 ỵ B22 B26 ỵ B26 B66 ỵ B16 B12 ỵ B26 B22 ỵ B66 B62 ; B11 B12 B12 B22 21ị ỵ D11 ; D12 ; D16 ịkx ỵ D12 ; D22 ; D26 ịky ỵ D16 ; D26 ; D66 ịkxy DTẵa1 B11 ; B12 ; B16 ị ỵ a2 B12 ; B22 ; B26 ފ; f(x, y) is stress function defined by 15ị where n   X Aij ẳ Q 0ij hk hk1 ị; kẳ1 n  X k 20ị þ P7 w;yyyy þ P8 w;xxyy þ P9 w;xxxy þ P10 w;xyyy ỵ Nx w;xx 14ị zẵri k dz; 133 Nx ¼ f;yy ; N y ¼ f;xx ; Nxy ¼ Àf;xy : ð22Þ For an imperfect composite plate, Eq (20) are modified into form as [7,8] i; j ¼ 1; 2; 6;   2 Q 0ij hk À hkÀ1 ; Bij ¼ k¼1 Dij ¼ n    1X 3 Q 0ij hk hk1 ; k kẳ1 k P1 f;xxxx ỵ P2 f;yyyy ỵ P3 w;xxyy ỵ P4 w;xxxy ỵ P w;xyyy ỵ P w;xxxx i; j ẳ 1; 2; 6; 16ị ỵ P7 w;yyyy ỵ P8 w;xxyy ỵ P9 w;xxxy ỵ P10 w;xyyy       ỵ f;yy w;xx ỵ w;xx 2f ;xy w;xy þ wÃ;xy þ f;xx w;yy þ wÃ;yy ¼ 0; i; j ẳ 1; 2; 6: 23ị The nonlinear equilibrium equations of a composite plate based on the classical theory are [13–15] in which w⁄(x, y) is a known function representing initial small imperfection of the plate The geometrical compatibility equation for an imperfect composite plate is written as Nx;x ỵ Nxy;y ẳ 0; 17aị Nxy;x ỵ Ny;y ẳ 0; 17bị e0x;yy ỵ e0y;xx c0xy;xy ẳ w2;xy w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy M x;xx þ 2M xy;xy ỵ M y;yy ỵ Nx w;xx ỵ 2N xy w;xy ỵ Ny w;yy ẳ 0: 17cị 24ị From the constitutive relations (18) in conjunction with Eq (22) one can write Calculated from Eq (15) e0x ẳ A11 Nx ỵ A12 Ny ỵ A16 Nxy B11 kx B12 ky B16 kxy ỵ DT a1 D11 þ a2 DÃ12 ; e0y ¼ Ẫ12 Nx þ Ẫ22 Ny ỵ A26 Nxy B21 kx B22 ky B26 kxy ỵ DTa1 D21 ỵ a2 D22 ị; exy ẳ A16 Nx ỵ A26 Ny ỵ A66 Nxy À BÃ16 kx À BÃ26 ky À BÃ66 kxy ỵ DT a1 D16 ỵ a2 D26 ; ð18Þ where A22 A66 À A226 A16 A26 À A12 A66 A12 A26 À A22 A16 ¼ ; Ẫ12 ¼ ; Ẫ16 ¼ ; D D D A11 A66 À A216 A12 A16 À A11 A26 A11 A22 À A212 AÃ22 ¼ ; AÃ26 ¼ : AÃ66 ¼ ; D D D 2 D ¼ A11 A22 A66 A11 A26 ỵ 2A12 A16 A26 A12 A66 A16 A22 ; 19ị A11 B11 ẳ A11 B11 þ Ẫ12 B12 þ Ẫ16 B16 ; BÃ12 ¼ Ẫ11 B12 ỵ A12 B22 ỵ A16 B26 ; ẳ A11 B16 ỵ A12 B26 ỵ A16 B66 ; ẳ A12 B12 ỵ A22 B22 ỵ A26 B26 ; ẳ A16 B11 ỵ A26 B12 ỵ A66 B16 ; ẳ A16 B16 ỵ A26 B26 ỵ A66 B66 ; ẳ A11 A12 ỵ A12 A22 ỵ A16 A26 ; ẳ A12 A12 ỵ A22 A22 ỵ A26 A26 ; ẳ A16 A12 ỵ A26 A22 ỵ A66 A26 : B21 ẳ A12 B11 ỵ A22 B12 ỵ A26 B16 ; B16 BÃ22 BÃ61 BÃ66 DÃ12 DÃ22 DÃ26 À w;yy wÃ;xx : B26 ẳ A12 B16 ỵ A22 B26 ỵ A26 B66 ; B62 ẳ A16 B12 ỵ A26 B22 ỵ A66 B26 ; D11 ẳ A11 A11 ỵ A12 A12 ỵ AÃ16 A16 ; DÃ21 DÃ16 ¼ ¼ AÃ12 A11 AÃ16 A11 ỵ ỵ A22 A12 A26 A12 ỵ ỵ A26 A16 ; AÃ66 A16 ; Substituting once again Eq (18) into the expression of Mij in (15), then Mij into Eq (17c) leads to e0x ẳ A11 f;yy ỵ A12 f;xx À AÃ16 f;xy À BÃ11 kx À BÃ12 ky B16 kxy ỵ DT a1 D11 ỵ a2 D12 ; e0y ẳ A12 f;yy ỵ A22 f;xx À AÃ26 f;xy À BÃ21 kx À BÃ22 ky À B26 kxy ỵ DT a1 D21 ỵ a2 D22 ; e0xy ẳ A16 f;yy ỵ A26 f;xx AÃ66 f;xy À BÃ16 kx À BÃ26 ky À BÃ66 kxy ỵ DT a1 D16 ỵ a2 D26 : ð25Þ Setting Eq (25) into Eq (24) gives the compatibility equation of an imperfect composite plate as [7,8] AÃ22 f;xxxx ỵ E1 f;xxyy ỵ A11 f;yyyy 2A26 f;xxxy 2A16 f;xyyy ỵ B21 w;xxxx ỵ B12 w;yyyy ỵ E2 w;xxyy ỵ E3 w;xxxy ỵ E4 w;xyyy   w2;xy w;xx w;yy ỵ 2w;xy w;xy w;xx w;yy w;yy w;xx ẳ 0; 26ị where E1 ẳ 2A12 ỵ A66 ; E2 ẳ B11 ỵ B22 2B66 ; E3 ẳ 2B26 B61 ; 27ị E4 ẳ 2B16 ỵ B62 : Eqs (23) and (26) are nonlinear equations in terms of variables w and f and used to investigate the stability of thin composite plates subjected to thermal loads 134 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 In the present study, the edges of composite plates are assumed to be simply supported and four edges of the plate are simply supported and immovable (IM) In this case, boundary conditions are The in-plane condition on immovability at all edges, i.e u = at x = 0, a and v = at y = 0, b, is fulfilled in an average sense as [8,11,12] w ¼ u ¼ Mx ¼ 0; Nx ¼ Nx0 at x ¼ 0; a; Z w ¼ v ¼ M y ¼ 0; Ny ¼ Ny0 at y ẳ 0; b; 28ị b Z where Nx0, Ny0 are fictitious compressive edge loads at immovable edges The approximate solutions of w and f satisfying boundary conditions (28) are assumed to be [7,8] w; w ị ẳ W; lhị sin km x sin dn y; 29aị f ẳ A1 cos 2km x ỵ A2 cos 2dn y ỵ A3 sin km x sin dn y ỵ A4 1 cos km x cos dn y ỵ Nx0 y2 ỵ Ny0 x2 ; 2 29bị a Z @u dxdy ¼ 0; @x a Z 0 b @v dydx ẳ 0: @y 32ị From Eqs (10) and (18) one can obtain the following expressions in which Eq (22) and imperfection have been included @u ¼ Ẫ11 f;yy ỵ A12 f;xx A16 f;xy ỵ B11 w;xx þ BÃ12 w;yy þ 2BÃ16 w;xy @x À Á þ DT DÃ11 a1 þ DÃ12 a2 À w2;x À w;x wÃ;x ; @v à à à ¼ A12 f;yy ỵ A22 f;xx A26 f;xy ỵ B21 w;xx þ BÃ22 w;yy þ 2BÃ26 w;xy @y À Á þ DT DÃ21 a1 þ DÃ22 a2 À w2;y À w;y wÃ;y : ð33Þ km = mp/a, dn = np/b W is amplitude of the deflection and l is imperfection parameter The coefficients Ai(i = 1Ä4) are determined by substitution of Eqs (29a) and (29b) into Eq (26) as Substitution of Eqs (29a) and (29b) into Eq (33) and then the result into Eq (32) give fictitious edge compressive loads as d2n A1 ẳ WW ỵ 2lhị; 32A22 k2m Ny0 ẳ J W ỵ J WW ỵ 2lhị ỵ J DT; A3 ẳ A4 ẳ Q 22 À Q 21 ðQ Q À Q Q Þ Q 22 À Q 21 ð30Þ Q 22 À Q 21 ðQ Q À Q Q Þ k2m d2n À P4 ðQ Q À Q Q ị Q 22 Q 21 # ỵ P6 k4m þ P7 d4n þ P8 k2m d2n W Q 22 Q 21   1 ỵ km dn P1 ỵ P2 WW ỵ 2lhị A22 A11   ab 4 d þ k WðW þ lhÞðW þ 2lhÞ À 64 Ẫ22 n A11 m W ỵl ; ineWW ỵ 2lị W þl þ b4 WðW ð35Þ Á ab À Nx0 k2m ỵ Ny0 d2n W ỵ lhị ẳ 0; b1 ¼ 32hmnp2 ðQ Q À Q Q Þ ; Q 22 À Q 21 3a2 b X 2    ðQ Q ÀQ Q Þ m4 p n4 p 1Q3Þ P ðQ QQ 42 Q ỵ P ỵ P ỵ P 2 a Q ÀQ b ÀQ   16 n2 p4 ðQ Q ÀQ Q Þ ðQ Q ÀQ Q ị m3 np4 m b2 ẳ þ P þ P À P 7; 2 2 3b a Q ÀQ Q ÀQ a b X6 ðQ Q ÀQ Q Þ mn3 p4 ÀP5 Q ÀQ ab ! 4mnp2 P1 P2 ỵ ; b3 ¼ à 3a2 b X A22 A11 ! p4 n4 m4 b4 ẳ ỵ ; 16X A22 b4 A11 a4 36ị where X ẳ J k2m ỵ J d2n ;  ẳ W: W h ð37Þ For a perfect composite plate (l = 0), Eq (37) leads to equations that determining buckling temperature DTb of the plate It can be obtained by taking the limit function DTðWÞ with W ! ðQ Q Q Q ị ỵ km dn WW ỵ lhị Q 22 Q 21 ỵ b3 ỵ 2lị; W; Q Q À Q Q Þ W where " ab ðQ Q À Q Q Þ ðQ Q À Q Q Þ P1 42 km ỵ P2 42 dn Q2 À Q1 Q À Q 21 P DT ẳ b1 W ỵ b2 W; where Qi(i = 1Ä4) = Qi(km, dn) was mentioned in Appendix A Subsequently, substitution of Eqs (29a) and (29b) into Eq (23) and applying the Galerkin procedure for the resulting equation yield ỵP 34ị where Ji(i = 16) = Ji(km, dn) are give in Appendix A Subsequently, setting Eq (34) into Eq (31) give k2m A2 ¼ WðW þ 2lhÞ; 32Ẫ11 d2n ðQ Q À Q Q ị Nx0 ẳ J W ỵ J WW ỵ 2lhị ỵ J DT; 31ị where m, n are odd numbers This is basic equation governing the nonlinear response of three-phase polymer composite plates under thermal loads Nonlinear stability analysis 4.1 Plate three-phase under uniform temperature rise Consider a simply supported polymer composite plates subjected to temperature environments uniformly raised from stress free initial state Ti to final value Tf and temperature difference DT = Tf À Ti is constant DT b ¼ 32hmnp2 ðQ Q À Q Q Þ 3a2 b X Q 22 À Q 21 : ð38Þ 4.2 Numerical results and discussion The asymmetric plate with simply supported boundary condition subjected to in-plane compressive loads as well as a uniform temperature rise, the bifurcation buckling load (temperature) does not exist [20] Hence, the numerical calculation will be calculated for three following cases of symmetric plates [19,20]: five-layers plate with the fiber angle 0/90/0/90/0, five-layers plate with the fiber angle 45/45/0/45/45 and four-layers plate with the twisted angle 0/45/À45/90 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 135 Fig Effects of particle’s ratio wc on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges) Fig Effects of fiber angles góc quâ´n on the nonlinear response of symmetric three-phase polymer composite plates with five-layers 0/90/0/90/0 and 45/45/0/ 45/45 under uniform temperature rise (immovable edges) Fig Effects of fiber’s ratio wa on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges) Fig Effects of mode (m, n) on the nonlinear response of symmetric three-phase polymer composite plate with five layers 0/90/0/90/0 under uniform temperature rise (immovable edges) Fistly, we consider the effects of fiber angle on buckling of five-layers plate with two case of the fiber angle: 0/90/0/90/0 and 45/45/0/45/45 From Fig 3, easily see that the plate with fiber angle 0/90/0/90/0 has a better thermal loading ability than the other one We also calculate thermal buckling loads with different modes (m, n) = (1, 1), (1, 3), (3, 1), (3, 3) Fig shows the effect of parameters (m, n) on nonlinear buckling of five-layers symmetric plate with the fiber angle 0/90/0/90/0 We can see that thermal buckling load is at minimum when (m, n) = (1, 1) In another word, buckling happens at first with (m, n) = (1, 1) Therefore, from now on, all of our figures and calculations are based only on (m, n) = (1, 1) Then, we consider the effects of material parameter and geometry on nonlinear response of the symmetric five-layers three phase plate with the fiber angle 0/90/0/90/0 and the symmetric four-layers three phase plate with the fiber angle 0/45/À45/90 Fig Effects of ratio particle’s ratio wc on the nonlinear response of three-phase polymer composite four-layers plate under uniform temperature rise (immovable edges) 136 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 Fig Effects of fiber’s ratio wa on the nonlinear response of three-phase polymer composite four-layers plate under uniform temperature rise (immovable edges) Fig 10 Effects of imperfection on the nonlinear response of symmetric three-phase polymer composite four-layers plate under uniform temperature rise (immovable edges) Fig Effects of imperfection on the nonlinear response of symmetric three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges) Figs and represent the effect of the particles and the fibers on buckling of the five-layers three phase plate Similarly, Figs and represent the effects of those on buckling of four-layers three-phase plate In Figs 5–8, we can realize that the increase of the particles and fibers density will increase the thermal loading ability of the plates Especially notice on the curve in Fig (fiber ratio wa = 0.2; particles ratio wc = 0.1) and curve in Fig (fiber ratio wa = 0.1; particles ratio wc = 0.2), easily notice that with the same reinforced component ratio wa + wc = 0.3, the titanium oxide particles make the thermal capacity of the composite plate better and stronger The titanium oxide particles play an important role in thermal resistance of the polymer matrix so we can replace a part of expensive fibers by particles of titanium oxide in order to reduce the price However, the five-layers symmetric plate has a better thermal loading ability than the four-layers symmetric one The same conclusion was also found in study case of the three phase laminated composite plates under mechanical loads [7] Figs and 10 represent the imperfection effects on buckling of five-layers and four-layers symmetric plates The results show the Fig 11 Effects of ratio b/h on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges) subtle change with the increase of the imperfection degree However, with the same geometrical size of the plates, the imperfect five-layers imperfect plate has a better thermal loading ability than the four-layers imperfect plate Figs 11–14 illustrates the geometrical effects b/h and b/a on buckling of plates It is not surprising that the thicker the plate (b/h) is large) and the larger the length/width (b/a) ratio is, the better thermal loading ability on buckling of plates is Concluding remarks The paper presents an analytical investigation on the nonlinear response of three-phase polymer composite plates subjected to thermal loads The formulations are based on the classical theory of plates taking into account geometrical nonlinearity and initial imperfection Galerkin method is used to obtain explicit expressions of load–deflection curves From the obtained results in this paper, we make the following conclusions: 137 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 – With the same thickness and size, the thermal loading ability of the symmetric five-layers plate is better than that of the symmetric four-layers plates – The geometry affects significantly on the stability of the composite plates The advantage of study approach in this paper is the nonlinear response of three-phase polymer composite plates which is presented explicitly on parameters of polymer matrix, fibers and particles and also with the geometrical parameters, imperfection and ratio of components in composite, so when adjusting the phase component, we can design and structural new materials meet the technical requirements Acknowledgment Fig 12 Effects of ratio b/h on the nonlinear response of three-phase polymer composite four-layers plate under uniform temperature rise (immovable edges) This work was supported by Project code 107.02-2013.06 in Mechanics of the National Foundation for Science and Technology Development of Vietnam – NAFOSTED The authors are grateful for this financial support Appendix A Q ẳ A22 k4m ỵ A11 d4n ỵ E1 k2m d2n ; Q ẳ 2A26 k3m dn ỵ 2A16 km d3n ; Q ẳ B21 k4m ỵ B12 d4n ỵ E2 k2m d2n ; Q ẳ E3 k3m dn ỵ E4 km d3n ; J1 ¼ J2 ¼ J3 J4 Fig 13 Effects of ratio b/a on the nonlinear response of three-phase polymer composite five-layers plate under uniform temperature rise (immovable edges) J5 J6 AÃ22 L1 À AÃ12 L3 AÃ11 AÃ22 À AÃ2 12 AÃ22 L2 À AÃ12 L4 ; ; AÃ11 AÃ22 À AÃ2 12 Aà H1 À AÃ12 H2 ¼ À 22à à ; A11 A22 À Ẫ2 12 Ẫ L3 À Ẫ12 L1 ¼ 11 ; Ẫ11 AÃ22 À AÃ2 12 Aà L4 À AÃ12 L2 ¼ 11 ; AÃ11 AÃ22 À AÃ2 12 Aà H2 À AÃ12 H1 ¼ À 11à à ; A11 A22 À AÃ2 12 where L1 ¼ L2 ¼ L3 ¼ Fig 14 Effects of ratio b/a on the nonlinear response of three-phase polymer composite four-layers plate under uniform temperature rise (immovable edges) – Increasing the density of fibers and particles in three phase composite polymer improves the thermal loading ability of the composite plates However, the effects of the titanium oxide particles on thermal capacity of the composite plates are stronger than those of the glass fibers L4 ¼ À à A11 dn ỵ A12 km ðQ Q À Q Q Þ Q Q Q Q ị ỵ 4A16 32 ab km dn Q 22 À Q 21 Q À Q 21 À à Á à ỵ B11 km ỵ B12 dn ; k2m ; À à Á à A12 dn ỵ A22 km Q Q Q Q Þ ðQ Q À Q Q ị ỵ 4A26 32 ab km dn Q 22 À Q 21 Q À Q 21 À ỵ B21 km ỵ B22 d2n ; d2n ; H1 ẳ D11 a1 ỵ D12 a2 ; H2 ẳ D12 a1 ỵ D22 a2 : References [1] Vanin GA, Duc ND The determination of rational structure of spherofibre composite 1: Models 3Dm J Mech Compos Mater 1997;33(2):155–60 138 N.D Duc, P Van Thu / Composite Structures 109 (2014) 130–138 [2] Duc ND The determination of rational structure of spherofibre composite 2: Models 4Dm J Mech Compos Mater 1997;33(3):370–6 [3] Minh DK Bending of three-phase composite laminated plates in shipbuilding industry PhD thesis in Engineering Maritime University, Hai Phong, Vietnam; 2011 [4] Duc ND, Minh DK Bending analysis of three-phase polymer composite plates reinforced by glass fibers and titanium oxide particles J Comput Mater Sci 2010;49(4):194–8 [5] Bank LC, Yin J Buckling of orthotropic plates with free and rotationally restrained edges Thin-Wall Struct 1996;24:83–96 [6] Mittelstedt Christian Stability behaviour of arbitrarily laminated composite plates with free and elastically restrained unloaded edges Int J Mech Sci 2007;49:819–33 [7] Duc ND, Quan TQ, Nam D Nonlinear stability analysis of imperfect three phase polymer composite plates J Mech Compos Mater 2013;49(4):345–58 [8] Duc ND, Cong PH Nonlinear postbuckling of symmetric S-FGM plates resting on elastic foundations using higher order shear deformation plate theory in thermal environments J Compos Struct 2013;100:566–74 [9] Feldman E, Aboudi J Buckling analysis of functionally graded plates subjected to uniaxial loading J Compos Struct 1997;38:29–36 [10] Javaheri R, Eslami MR Buckling of functionally graded plates under in-plane compressive loading ZAMM 2002;82(4):277–83 [11] Samsam Shariat BA, Eslami MR Buckling of thick functionally graded plates under mechanical and thermal loads J Compos Struct 2007;78:433–9 [12] Liew KM, Yang J, Kitipornchai S Postbuckling of piezoelectric FGM plates subjected to thermo–electro-mechanical loading Int J Solids Struct 2003;40:3869–92 [13] Brush DO, Almroth BO Buckling of bars, plates and shells New York: McGrawHill; 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A1 2 A2 6 À A2 2 A1 6 ¼ ; A 12 ¼ ; A 16 ¼ ; D D D A1 1 A6 6 À A2 16 A1 2 A1 6 À A1 1 A2 6 A1 1 A2 2 À A2 12 Ẫ22 ¼ ; Ẫ26 ¼ : A 66 ¼ ; D D D 2 D ẳ A1 1 A2 2 A6 6 A1 1 A2 6 ỵ 2A1 2 A1 6 A2 6 À A1 2 A6 6 À A1 6 A2 2 ; ð19Þ... a2 ẳ a ỵ a a1 ịm21 a aa ị1 ỵ ma ị : m À ma ð8Þ To numerical calculating, we chosen three phase composite polymer made of polyester AKAVINA (made in Vietnam), fibers (made in Korea) and titanium