Postbuckling behavior of functionally graded multilayer graphene nanocomposite plate under mechanical and thermal loads on elastic foundations

VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 Original Article Postbuckling Behavior of Functionally Graded Multilayer Graphene Nanocomposite Plate under Mechanical and Thermal Loads on Elastic Foundations Pham Hong Cong1, Nguyen Dinh Duc2, Centre for Informatics and Computing (CIC), Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Advanced Materials and Structures Laboratory, VNU University of Engineering and Technology (UET), 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam Received 08 November 2019 Revised 03 December 2019; Accepted 03 December 2019 Abstract: This paper presents an analytical approach to postbuckling behaviors of functionally graded multilayer nanocomposite plates reinforced by a low content of graphene platelets (GPLs) using the first order shear deformation theory, stress function and von Karman-type nonlinear kinematics and include the effect of an initial geometric imperfection The weight fraction of GPL nano fillers is assumed to be constant in each individual GPL-reinforced composite (GPLRC) The modified Halpin-Tsai micromechanics model that takes into account the GPL geometry effect is adopted to estimate the effective Young’s modulus of GPLRC layers The plate is assumed to resting on Pasternak foundation model and subjected to mechanical and thermal loads The results show the influences of the GPL distribution pattern, weight fraction, geometry, elastic foundations, mechanical and temperature loads on the postbuckling behaviors of FG multilayer GPLRC plates Keywords: Postbuckling; Graphene nanocomposite plate; First order shear deformation plate theory Recently, a new class of promising material known as graphene has drawn considerable attention of the science and engineering communities Graphene is a two-dimensional monolayer of sp2-bonded carbon atoms [1,2] and possesses extraordinarily material properties such as super-high mechanical strength and Introduction Advanced materials have been considered promising reinforcement materials To meet the demand, some smart materials are studied and created such as FGM, piezoelectric material, nanocomposite, magneto-electro material and auxetic material (negative Poisson’s ratio)  Corresponding author Email address: ducnd@vnu.edu.vn https://doi.org/10.25073/2588-1140/vnunst.4972 110 P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 remarkable electrical and thermal conductivities [3-5] It was reported by researchers that the addition of a small percentage of graphene fillers in a composite could improve the composite’s mechanical, electrical and thermal properties substantially [6-8] The research on buckling and postbuckling of the functionally graded multilayer graphene nanocomposite plate and shell has been attracting considerable attention from both research and engineering Song et al [9, 10] studied buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplatelet-reinforced polymer composite plates (excluding thermal load and elastic foundation) Wu et al [11] investigated thermal buckling and postbuckling of functionally graded graphene nanocomposite plates Yang et al [12] analyzed the buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams Shen et al [13] studied the postbuckling of functionally graded graphene-reinforced composite laminated cylindrical panels under axial compression in thermal environments Stability analysis of multifunctional advanced sandwich plates with graphene nanocomposite and porous layers was considered in [14] Buckling and post-buckling analyses of functionally graded graphene reinforced by piezoelectric plate subjected to electric potential and axial forces were investigated in [15] Some researches using analytical method, stress function method to study graphene structures can be mentioned [16, 17, 18] In [16], the author considered nonlinear dynamic response and vibration of functionally graded multilayer graphene nanocomposite plate on viscoelastic Pasternak medium in thermal environment 2D penta-graphene model was used in [17, 18] From overview, it is obvious that the postbuckling of graphene plates have also attracted researchers’ interests and were studied [9, 10, 11] However, in [9, 10] the authors neither considered thermal load nor elastic 111 foundation In [11], the authors used differential quadrature (DQ) method) but did not mention thermal load, elastic foundation and imperfect elements In addition, in [9, 10, 11] the stress function method was not used to the study Therefore, we consider postbuckling behavior of functionally graded multilayer graphene nanocomposite plate under mechanical and thermal loads and using the analytical method (stress function method, Galerkin method) Nomenclature EGPL , Em aGPL , bGPL , tGPL vGPL , vm GPL , m The Young’s moduli of the GPL and matrix, respectively The length, width and thickness of GPL nanofillers, respectively The Poisson’s ratios with the subscripts “GPL” and “m” refering to the GPL and matrix, respectively The thermal expansion coefficients with the subscripts “GPL” and “m” referring to the GPL and matrix, respectively Functionally graded multilayer GPLRC plate model A rectangular laminated composite plate of length a , width b and total thickness h that is composed of a total of N L on Pasternak foundation model, as shown in Figure Z b a 0.5h Y 0.5h X Winkler layer (KW) Pasternak layer (KG) Figure A FG multilayer GPLRC plate on Pasternak foundation model 112 P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 The three distribution patterns of GPL nanofillers across the plate thickness are shown in Figure In the case of X-GPLRC, the surface layers are GPL rich while this is inversed in OU-GPLRC GPLRC where the middle layers are GPL rich As a special case, the GPL content is the same in each layer in a U-GPLRC plate X-GPLRC O-GPLRC Figure Different GPL distribution patterns in a FG multilayer GPLRC plate Functionally graded multilayer GPLRC plates with an even number of layers are considered in this paper The volume fractions VGPL of the k layer for the three distribution patterns shown in figure are governed by Case 1: UGPLRC Case 2: XGPLRC (k ) GPL V V * GPL (k ) * VGPL  2VGPL 2k  N L  NL V  WGPL WGPL   GPL / m 1  WGPL  in which WGPL is GPL weight fraction; T   bGPL / tGPL  (2) v  vmVm  vGPLVGPL (4) GPL and  m are the mass densities of GPLs and the polymer matrix, respectively The modified Halpin-Tsai micromechanics model [9] that takes into account the effects of nanofillers’ geometry and dimension is used to estimate the effective Young’s modulus of GPLRCs   L LVGPL  TT VGPL E  Em   Em (5)  LVGPL  T VGPL Where  L   aGPL / tGPL  , (6) (1) where k  1,2,3 , N L and N L is the total number of layers of the plate The total volume * fraction of GPLs, VGPL , is determined by * GPL  EGPL / Em    E / E  1 , T  GPL m  EGPL / Em    L  EGPL / Em   T According to the rule of mixture, the Poisson’s ratio v and thermal expansion coefficient  of GPLRCs are 2k  N L   (k ) *  VGPL  2VGPL 1   (3) NL   Case 3: OGPLRC L     mVm   GPLVGPL (7) where Vm   VGPL is the matrix volume fraction Theoretical formulations 3.1 Governing equations Suppose that the FG multilayer GPLRC plate is subjected to mechanical and thermal loads In the present study, the first order shear deformation theory (FSDT) is used to obtain the equilibrium, compatibility equations According to the FSDT, the displacements of an arbitrary point in the plate are given by [19] U  X ,Y , Z   U  X ,Y   Z  X  X ,Y  V  X , Y , Z   V  X , Y   Z Y  X , Y  (8) W  X ,Y , Z   W  X ,Y  By using von Kármán nonlinearity, the nonlinear strains associated with the displacements are obtained as [19] P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122   U , X  W, X      X ,X   XX    X   X       0         YY     Y   z  Y    V,Y  W,Y    Z  Y ,Y              Y ,X   XY   XY   XY  U  V  W W   X ,Y ,Y ,X ,X ,Y     W    XZ   , X X     W   Y    YZ   ,Y where  X and  Y are normal strains and  XY is the shear strain in the middle surface of the plate and  XZ ,  YZ are the transverse shear strains components in the plans XZ and YZ respectively U, V, W are displacement components 0 k  k   B11 B  12     k  B11  B22 B12 B22 0 0 (9) corresponding to the coordinates (X, Y, Z),  X and Y are the rotation angles of normal vector with Y and X axis The stress components of the k layer can be obtained from the linear elastic stress-strain constitutive relationship as k   k  k         XX         YY          B44    T     YZ   0 0 B55   XZ         0    0    XY   where T is the variability of temperature in the environment containing the plate and  XX    YY       YZ     XZ    XY   113      B66  Ek  vE  k   k  E k  k  k  k   , B12  , B44  B55  B66   v2  v2 1  v  (10) (11) According to FSDT, the equations of motion are [19]: N X , X  N XY ,Y  0, (12) N XY , X  NY ,Y  0, (13) QX , X  QY ,Y  N X W, XX  N XYW, XY  NYW,YY  KWW  KG W, XX  W,YY   0, (14) M X , X  M XY ,Y  QX  0, (15) M XY , X  M Y ,Y  QY  0, (16) The axial forces  N X , NY , N XY  , bending moments  M X , M Y , M XY  and shear forces  QX , QY  are related to strain components by T   X0   NX    X  N   0      T  NY    J    Y    C   Y    N  N         XY   XY     XY  (17) 114 P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 T   X0   MX    X  M   0      T  M Y    C    Y    L   Y    M  M         XY   XY    XY   (18) QX   XZ     K  P    QY    YZ  (19) where shear correction factor K  / The stiffness elements of the plate are defined as J N L Z k 1 ij , Cij , Lij     k 1 Z k N L Z k 1 Pij   Q k 1 Z k k  ij Bij  1, Z , Z  dZ ,  i, j  1, 2,3 k dZ ,  i, j  1, ,N , M T T N L Z k 1   Q k 1 Z k (20)  T 1, Z  dZ k  11 (k ) For using later, the reverse relations are obtained from Eq (17) J12 J J C J C C J  C22 J12 J J NY  22 N X  22 11 12 12  X , X  12 22 Y ,Y  12 22 N T      J J J C  C J C J  C J J  J  Y0  N X 12  NY 11  11 22 12 12 Y ,Y  12 11 11 12  X , X  12 11 N T      C C N  XY  XY  33  X ,Y  33 Y , X J 33 J 33 J 33  X0  (21) where   J12  J 22 J11 The stress function F  X , Y  - the solution of both equations (12) and (13) is introduced as N X  F,YY , NY  F, XX , N XY   F, XY (22) By substituting Eqs (21), (18) and (19) into Eqs (14)-(16) Eqs (14)-(16) can be rewritten * KP44 W, XX  W, *XX   KP44  X , X  KP55 W,YY  W,YY   KP55 Y ,Y  F,YY W, XX  W, *XX   F, XY W, XY  W, *XY  (23) *  F, XX W,YY  W,YY   KWW  KG W, XX  W,YY   0, S21F, XXX  S22 F, XYY  S23  X , XX  S24 Y , XY  S25  X ,YY  KP44 W, X  W, *X   KP44  X  0, (24) S31F, XXY  S32 F,YYY  S33  X , XY  S34 Y , XX  S35 Y ,YY  KP55 W,Y  W,Y*   KP55 Y  0, where (25) P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 S21  S23  S24  C J12C11 C12 J11 J C C J  , S22  22 11  66  12 12    J 33   J 22C11  J12C12  C11   C12 J11  C11 J12  C12  L    C12 J 22  C22 J12  C11   J11C22  C12 J12  C12  L  S31   115  66  L12  11 C33C66 C C , S 25  L66  33 66 J 33 J 33 C66 J12C12 J11C22 J C J C   , S32  12 22  22 12 J 33     S33  L12  S34  L66   J C  J C  C C J  C J  C C33C66  L66  22 11 12 12 12  12 11 11 12 22 J 33    C J  C22 J12  C12   J11C22  C12 J12  C22  L C33C66 , S35  12 22 22 J 33   The strains are related in the compatibility equation * * *  X0 ,YY  Y0, XX   XY , XY  W, XY   W, XX W,YY  2W, XY W, XY  W, XX W,YY  W,YY W, XX (26) Set Eqs (21) and (22) into the deformation compatibility equation (26), we obtain  F, XXXX  J C J C C  J11  J12  J 22   F,YYYY   22 11 12 12  33   X , XYY  F, XXYY     J 33    J 33    J C C J C  C12 J 22  C22 J12 C J C J Y ,YYY  12 11 11 12  X , XXX   11 22 12 12  33  Y , XXY    J 33   (27) *  W, XY   W, XX W,YY  2W, XYW, *XY  W, XX W,YY  W,YYW, *XX The system of fours Eqs (23) - (25) and (27) combined with boundary conditions and initial conditions can be used for posbuckling of the FG multilayer GPLRC plate 3.2 Solution procedure Depending on the in-plane behavior at the edges is not able to move or be moved, two boundary conditions, labeled Case and Case will be considered [19]: Case Four edges of the plate are simply supported and freely movable (FM) The associated boundary conditions are W  0, N XY  0, Y  0, M X  0, N X  N X at X  0, a, W  0, N XY = 0,  X  0, M Y  0, NY  NY at Y  0, b (28) Case Four edges of the plate are simply supported and immovable (IM) In this case, boundary conditions are W  U  0, Y  0, M X  0, N X  N X at X  0, a, W  V  0,  X  0, M Y  0, NY  NY at Y  0, b (29) 116 P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 where N X , NY are the forces acting on the edges of the plate that can be moved (FM), and these forces are the jets when the edges are immovable in the plane of the plate (IM) The approximate solutions of the system of Eqs (23)-(25) and (27) satisfying the boundary conditions (28), (29) can be written as W  X,Y   W0 sin  X sin  Y ,  X  X,Y    X cos X sin  Y , (30) Y  X,Y   Y sin  Xcos Y , 1 (31) F  X,Y   A1 cos 2 X  A2 cos 2 Y  N X 0Y  NY X , 2 where   m ,   n , m, n  1,2, are the natural numbers of half waves in the corresponding a b direction X , Y , and W ,  X , Y - the amplitudes which are functions dependent on time The coefficients Ai  i    are determined by substitution of Eqs (30, 31) into Eq (27) as A1  f1W02 , A2  f W02 , A3  f3 x  f 4 y (32) where f1    2  2 W W   h , f   W0 W0   h  ,   0 32  J11 32  J 22 f3   J11 J 33C12   J12 J 33C11    J12 J 33C12    J 22 J 33C11    2C33  J11 J 33  2  J12 J 33  J 22  J 33    f4    J11 J 33C22    J12 J 33C12   J12 J 33C22   J 22 J 33C12    C33  J11 J 33  2  J12 J 33  J 22  J 33    Substituting expressions (30)-(32) into Eqs (23)-(25), and then applying Galerkin method we obtain l11W0  3 mn  b2 m2 N x  a n N y  W0   h   l12 x  l13 y l14 W0   h   x  l15 W0   h   y  l16W0 W0   h W0  2 h   (33) l21 x  l22 y  l23W0 W0  2 h   l24 W0  h   (34) l31 x  l32 y  l33W0 W0  2 h   l34 W0  h   (35) where l11  3 mn  b m2 KG  a n KG  a n P55 K  b m2 P44 K  3a 2b n mCT  3a 2b K w n m l12  3b2 3m2 P44 Kan, l13  3a 2n2 P55 Kbm, l14  32 f3m2n2 l15  32 4m2n2 f , l16  6n3 6m3  f1  f  P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 117 l22  3b 2 3m3 S21 f n  3S 22 m 3n3 f a  3S 24 m 2n 2a 2b l21  3b 2 3m3 S21 f 3n  3S 22 m 3n3 f 3a  3a 3 2n 3S 25 l23  256 S21 f1m 2 b , l24  3b 2 mP44 Ka n l31  3n m2 S33 b a  3n 3m3 S31 f 3b  3a n3 S32 f 3m l32  3b3 m3 S34  3a n 2 S35 mb  3n 3m3 S31 f 4b  3a n3 3S32 f 4m  3a 2b P55 Km, l33  256a n 2 S32 f , l34  3a n P55 Kmb 3.3 Mechanical postbuckling analysis Consider the FG multilayer GPLRC plate hinges on four edges which are simply supported and freely movable (corresponding to case 1, all edges FM) Assume that the FG multilayer GPLRC plate is loaded under uniform compressive forces FX and FY (Pascal) on the edges X=0, a, and Y= 0, b, in which N X   FX h, NY   FY h (36) Substituting Eq (36) into Eqs (33)-(35) leads to the system of differential equations for studying the postbuckling of the plate   l22l34  l24l32  l12   l21l34  l24l31  l13  W0 l11    l21l32  l22l31   l22l31  l21l32   W0   h    l l  l l  l  l l  l l  l  W02    22 33 23 32 12  21 33 23 31 13   l22l31  l21l32   W0   h    l21l32  l22l31  (37)   l22l33  l23l32  l14  l21l33  l23l31  l15    l22l34  l24l32  l14  l21l34  l24l31  l15     W   W  l22l31  l21l32     l21l32  l22l31   l22l31  l21l32     l21l32  l22l31  l16W0 W0  2 h   3 mnh  b m FX  a n FY  3.4 Thermal postbuckling analysis Consider the FG multilayer GPLRC plate with all edges which are simply supported and immovable (corresponding to case 2, all edges IM) under thermal load The condition expressing the immovability on the edges, U = (on X = 0, a) and V = (on Y = 0, b), is satisfied in an average sense as b a a b   U , X dXdY  0,  V 0 0 ,Y dXdY  (38) From Eqs (9) and (21) of which mentioned relations (22) we obtain the following expressions J12 J J C J C C J  C22 J12 NY  22 N X  22 11 12 12  X , X  12 22 Y ,Y     J  J 22 T  12 N  W, X   J J J C  C12 J12 C J C J V,Y  N X 12  NY 11  11 22 Y ,Y  12 11 11 12  X , X     J J  12 11 N T  W,Y   U,X  (39) 118 P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 Substituting Eqs (30)-(32) into Eqs (39), and substituting the expression obtained into Eqs (38) we have N x  n11  X  n12Y  n13W02  NT (40) N y  n21  X  n22Y  n23W0  NT   J11 J 22  a n f3  J122  a n f3  n11   2    a b mn  J11 J 22  J122    J112 C11 ab2 m  J122 C11 ab m  n12   n13     J11 J 22  a n f  J122  a n f     a 2b2 mn  J11 J 22  J122    J11 J 22 C12 a 2bn  J122 C12 a 2bn    J12 a n3 m  J11 3b m3n  8 a 2b mn  J11 J 22  J122  n21    b J122 m2 f3 b J122 C12 am  4  2   a b mn  J11 J 22  J12    J 22 J11 b2 m2 f3  J 22 J11 C12 ab 2m  n22   nJ122 C22 a 2b   b J122 m2 f  4  2   a b mn  J11 J 22  J12    J 22 J11 b m2 f  J 22 J11 C22 a 2bn  m  n23   b nJ12  J 22 a n3 3m   8 a 2b2 mn  J11 J 22  J122  Substituting (40) into Eqs (33)-(35) leads to the basic equations used to investigate the postbuckling of the plates in the case all IM edges p1 W0  p W  p W  p W W  2 h  W0   h  0  p5 where W   b m 3 mn  a n 3 mn  NT W0   h   l22l34  l24l32  l12   l21l34  l24l31  l13 ,  l21l32  l22l31   l22l31  l21l32   l22l34  l24l32   n21 a n 3 mn  n11b m 3 4mn  l14  p2   l21l32  l22l31   l21l34  l24l31   n22 a n 3 mn  n12b m 3 mn  l15   ,  l22l31  l21l32   l22l33  l23l32   n21 a n 3 mn  n11b m 3 mn  l14  p3   l21l32  l22l31  2  l21l33  l23l31   n22 a n 3 mn  n12b2 m2 3 mn  l15   ,  l22l31  l21l32  l l  l l  l l l  l l  l p4   n23a n 3 mn  n13b m 3 mn  l16  , p5  22 33 23 32 12  21 33 23 31 13  l21l32  l22l31   l22l31  l21l32  p1  l11  (41) P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 119 Numerical example and discussion 4.1 Validation of the present formulation The plate (a×b×h = 0.45m×0,45m×0.045m) is reinforced with GPLs with dimentions aGPL  2.5m, bGPL  1.5m,h GPL  15nm The material properties of epoxy and GPL are presented in Table In addition, GPL weight fraction is 0.5% and the total number of layers In table 1, the critical buckling load of FG multilayer GPLRC plate under biaxial compreession (kN) are also compared with those presented in Song et al [9], in which the authors used a two step perturbation technique [20] to solve differential equations N L  10 According to Table 2, the errors of critical buckling load with Ref [9] are very small, indicating that the approach of this study is highly reliable Table Material properties of the epoxy and GPLs [9] Material properties Epoxy GPL Young’s modulus (GPa) 3.0 1010 Density (kg.m-3) 1200 1062.5 Poisson’s ratio 0.34 0.186 Thermal expansion coefficient 60 5.0 106 / K   Table Comparison of critical buckling load of FG multilayer GPLRC plate under biaxial compreession (kN) WGPL U-GPLRC X-GPLRC Present Ref [9] % different Present Ref [9] % different Pure epoxy 2132.3 2132.3 2132.3 2132.3 0.2% 3547.6 3550.9 0.0929 4181.8 4081.3 2.462 0.4% 4962.3 4968.9 0.1328 6224.7 6025.1 3.313 0.6% 6376.4 6386.3 0.155 8265.0 7966.3 3.75 0.8% 7789.8 7803.1 0.1704 10304.0 9905.7 4.021 1% 9202.7 9219.2 0.179 12341.0 11843.6 4.2 4.2 Postbuckling Postbuckling curves of the FG multilayer GPLRC plate with different GPL distribution patterns is shown in figures and It can be seen that the postbucking strength of pattern X is the best, next is pattern U and the least pattern O 600 500 Perfect (=0) Imperfect (=0.1) T (K) 1: U-GPLRC 400 2: X-GPLRC 3: O-GPLRC (2) 300 200 (1) (3) 100 WGPL=0.5%, KG=0, KW=0 0 Figure Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of GPL distribution pattern 0.5 W0/h 1.5 Figure Postbuckling curves of the FG multilayer GPLRC plate under thermal load: Effect of GPL distribution pattern P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 120 600 500 0.6 0.4 200 X-GPLRC: WGPL=0.5%, KG=0, KW=0 0.5 W0/h 1.5 100 Figure Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of imperfection Figures and show effects of imperfection on buckling and postbuckling curves of the FG multilayer X-GPLRC plate under uniaxial compressive and thermal loads In postbuckling period, those suggest us that the imperfect properties have affected actively on the loading ability in the limit of large enough W0/h In other words, the loading ability increases with µ 1: WGPL=0 (Pure epoxy) 2: WGPL=0.3% 3: WGPL=0.5% 4: WGPL=0.7% 0.8 5: WGPL=1% 1.2 X-GPLRC: WGPL=0.5%, KG=0, KW=0 0.5 W0/h 1.5 Figure Postbuckling curves of the FG multilayer GPLRC plate under thermal load: Effect of imperfection Figures and shows the effects of GPL weight fraction WGPL on the postbuckling behavior of the FG multilayer X-GPLRC plate under uniaxial compressive and thermal loads As expected, the postbucking strength of the FG multilayer X-GPLRC plate increased with WGPL, i.e., with the volume content of GPL in the plate 600 Perfect (=0.0) Imperfect (=0.1) Perfect (=0) Imperfect (=0.1) 500 1: WGPL=0 (Pure epoxy) 2: WGPL=0.3 % 3: WGPL=1 % 400 T (K) 1.4 Fx(GPa) 400 300 0.2 0 =0 =0.1 =0.3 =0.5 700 T (K) Fx(GPa) 0.8 800 =0 =0.1 =0.3 =0.5 0.6 300 (1) (3) 200 0.4 (2) 0.2 0 100 X-GPLRC, KG=0, KW=0 X-GPLRC, KG=0, KW=0 0.5 W0/h 1.5 Figure Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of GPL weight fraction 0 0.5 W0/h 1.5 Figure Postbuckling curves of the FG multilayer GPLRC plate under thermal load: Effect of GPL weight fraction P.H Cong, N.D Duc / VNU Journal of Science: Natural Sciences and Technology, Vol 35, No (2019) 110-122 0.7 Perfect (=0.0) Imperfect (=0.1) 0.6 1: bGPL/tGPL=10 2: bGPL/tGPL=102 3: bGPL/tGPL=103 4: bGPL/tGPL=104 Fx(GPa) 0.5 0.4 0.3 (3) (2) (1) 0.1 WGPL=0.5%, KG=0, KW=0 0 0.5 W0/h 1.5 Figure Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of GPL length-tothickness ratio Fx(GPa) 0.5 0.4 Perfect (=0.0) Imperfect (=0.1) 1: aGPL/bGPL=1 2: aGPL/bGPL=10 3: aGPL/bGPL=20 (1) (3) (2) 0.3 0.2 0.1 0 O-GPLRC: WGPL=0.5%, KG=0, KW=0 0.5 W0/h 1.5 Perfect (=0.0) Imperfect (=0.1) 0.8 KW=0.1GPa/m, KG=0.01GPa.m KW=0,KG=0 0.6 0.4 KW=0.1GPa/m,KG=0 0.2 0 O-GPLRC, WGPL=0.5% 0.5 W0/h 1.5 Figure 11 shows the effects of the elastic foundations on the postbuckling behavior of FG multilayer GPLRC plate Elastic foundations are recognized to have strong impact, as demonstrated by curve (KW = 0, KG = 0) and (KW=0.1Gpa/m, KG = 0.01Gpa.m), which show that the ability of sustaining compression load will increase if the effects of elastic foundations enhance from (KW=0, KG = 0) to (KW=0.1Gpa/m, KG = 0.01Gpa.m) Conclusions The postbuckling behavior of FG multilayer GPLRC plate under mechanical and thermal loads is investigated based on the FSDT Some remarkable results are listed following - The postbucking strength of pattern X is the best, next is pattern U and the least pattern O - Elastic foundation models have a positive influence on postbuckling curves, specifically making postbucking strength decrease - Increasing the values of GPL weight fraction makes postbucking strength capacity better - Effect of geometry and dimension of GPL is also discussed and demonstrated through illustrative numerical examples 0.7 0.6 1.2 Figure 11 Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of elastic foundations (4) 0.2 1.4 Fx(GPa) Figures and 10 illutrates the effects of GPL width-to thickness ratio bGPL/tGPL and length-towidth ratio aGPL/bGPL on the postbuckling behavior of the FG multilayer O-GPLRC plates Figure demonstrates the increased uniaxial compressive postbuckling load – carrying capability of FG multilayer O-GPLRC plates when bGPL/tGPL increases Figure 10 presents the decreased uniaxial compressive postbuckling load–carrying capability of FG multilayer OGPLRC plates when aGPL/bGPL increases 121 Figure 10 Postbuckling curves of the FG multilayer GPLRC plate under uniaxial compressive load: Effect of GPL length-to-width ratio Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2018.04 The authors are grateful for 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Therefore, we consider postbuckling behavior of functionally graded multilayer graphene nanocomposite plate under mechanical and thermal loads and using the analytical method (stress function method,... considered nonlinear dynamic response and vibration of functionally graded multilayer graphene nanocomposite plate on viscoelastic Pasternak medium in thermal environment 2D penta -graphene model