EXACT SOLUTION FOR TEMPERATURE-DEPENDENT BUCKLING ANALYSIS OF FG-CNT-REINFORCED MINDLIN PLATES

This research deals with the buckling analysis of nanocomposite polymeric temperature-dependent plates reinforced by single-walled carbon nanotubes (SWCNTs). For the carbon-nanotube reinforced composite (CNTRC) plate, uniform distribution (UD) and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are assumed. The material properties of FG-CNTRC plate are graded in the thickness direction and estimated based on the rule of mixture. The CNTRC is located in a elastic medium which is simulated with temperature-dependent Pasternak medium. Based on orthotropic Mindlin plate theory, the governing equations are derived using Hamilton’s principle and solved by Navier method. The influences of the volume fractions of carbon nanotubes, elastic medium, temperature and distribution type of CNTs are considered on the buckling of the plate. Results indicate that CNT distribution close to top and bottom are more efficient than those distributed nearby the mid-plane for increasing the stiffness of plates.


INTRODUCTION
Recently, due to the advantage mechanical, physical and electronic properties of CNTs [Salvetat-Delmotte and Rubio, 2002], these advanced materials are considered to be excellent candidates for the reinforcement of polymer composites [Esawi and Farag, 2007;Fiedler et al. 2006].In actual structural applications, CNTRC, as a type of advanced material, have a wide variety of applications in microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS).Hence, knowledge of the bending characteristics of these structures is important.
The problem of bending of thick plates has attracted considerable attention in recent years.The effect of transverse shear deformation on the bending of elastic plates was studied by Reissner [1945].Zenkour [2003] proposed an exact mixed-classical solution for the bending analysis of shear deformable rectangular plates.Buczkowski and Torbacki [2001] used finite element for modelling of thick plates on two-parameter elastic foundation.Based on the boundary element method, the analysis of plates on two-parameter elastic foundations with nonlinear boundary conditions was studied by Chucheepsakul and Chinnaboon [2003].Sladek et al. [2002] investigated meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation.Akhavan et al. [2009aAkhavan et al. [ , 2009b] ] introduced exact solutions for buckling analysis of rectangular Mindlin plates subjected to uniformly and linearly distributed in-plane loading on two opposite edges, simply supported resting on elastic foundation.Postbuckling, nonlinear bending and nonlinear vibration analyses for SWCNTs resting on a two-parameter elastomeric foundation in thermal environments were presented by Shen and Zhang [2011].Heydari et al. [2014] studied an analytical approach for transverse bending analysis of an embedded symmetric laminated rectangular plate using Mindlin plate theory and the surrounding elastic medium simulated using Pasternak foundation.They indicated that the maximum deflection of the laminated plate decreases when considering an elastic medium.
None of the above researchers have considered nanocomposite structures.Reddy et al. [1984] studied the effect of transverse shear deformation on deflection and stresses of laminated composite plates subjected to uniformly distributed load using finite element analyses.The analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method was presented by Ferreira et al. [2003].Swaminathan and Ragounadin [2004] applied an analytical solution for static analyzing of antisymmetric angle-ply composite and sandwich plates.An investigation on the nonlinear bending of simply supported, functionally graded nanocomposite plates reinforced by SWCNTs subjected to a transverse uniform or sinusoidal load in thermal environments was investigated by Shen [2009].Baltacıoğlu et al. [2011] presented the nonlinear static analysis of a rectangular, laminated composite thick plate resting on nonlinear two-parameter elastic foundation with cubic nonlinearity.They used the first-order shear deformation theory for plate formulation and investigated the effects of foundation and geometric parameters of plates on nonlinear deflections.Bending and free vibration analyses of thin-to-moderately thick composite plates reinforced by SWCNTs using the finite element method based on the first order shear deformation plate theory were presented by Zhu et al. [2011].
In the present study, the orthotropic Mindlin plate theory is used for buckling behavior of polymeric temperature-dependent plates reinforced by SWCNTs resting on temperaturedependent elastic medium.For CNTRC plate, both cases of uniform and FG distribution patterns of SWCNT reinforcements are considered.The rule of mixture is used in order to obtain the equivalent material properties of FG-CNTRC plate.The nonlinear governing equations are obtained based on Hamilton's principal along with von Kármán geometric nonlinearity.Exact solution is applied for buckling load of the FG-CNTRC polymeric plate.The effects of the volume fractions of carbon nanotubes, elastic medium, temperature, and distribution type of CNTs on the buckling load of the FG-CNTRC polymeric plate are disused in detail.

FORMULATION
As shown in Figure 1, a CNTRC plate with length L, width b and thickness h is considered.The CNTRC plate is surrounded by an orthotropic elastomeric temperature-dependent medium which is simulated by k, ξ G and η G correspond Winkler foundation parameter, shear foundation parameters in ξ and η directions, respectively.Four types of CNTRC plates namely as uniform distribution (UD) along with three types of FG distributions (FGA, FGO, FGX) of CNTs along the thickness direction of a CNTRC plate is considered.In order to obtain the equivalent material properties two-phase Nanocomposites (i.e.polymer as matrix and CNT as reinforcer), the rule of mixture [Esawi and Farag, 2007] is applied.According to mixture rule, the effective Young and shear moduli of CNTRC plate can be written as: 4 The effects of the volume fractions of carbon nan distribution type of CNTs on the buckling load disused in detail.

Formulation
As shown in Fig. 1, a CNTRC plate with length L , CNTRC plate is surrounded by an orthotropic el which is simulated by k,  G and  G correspo foundation parameters in ξ and η directions, res namely as uniform distribution (UD) along with th FGX) of CNTs along the thickness direction of a obtain the equivalent material properties two-phas and CNT as reinforcer), the rule of mixture (Esaw mixture rule, the effective Young and shear moduli , ) 1 ( distribution type of CNTs on the buckling load disused in detail.

Formulation
As shown in Fig. 1, a CNTRC plate with length L , CNTRC plate is surrounded by an orthotropic el which is simulated by k,  G and  G correspo foundation parameters in ξ and η directions, res namely as uniform distribution (UD) along with th FGX) of CNTs along the thickness direction of a obtain the equivalent material properties two-phas and CNT as reinforcer), the rule of mixture (Esaw mixture rule, the effective Young and shear moduli , ) 1 ( where: E r11 , E r22 and G r11 -indicate the Young's moduli and shear modulus of SWCNTs, E m , G m -represent the corresponding properties of the isotropic matrix. The scale-dependent material properties, η j (j = 1, 2, 3), can be calculated by matching the effective properties of CNTRC obtained from the MD simulations with those from the rule of mixture.V CTN and V m are the volume fractions of the CNTs and matrix, respectively, whose sum equals one.The uniform and three types of FG distributions of the CNTs along the thickness direction of the CNTRC plates take the following forms: Similarly, the thermal expansion coefficients in the longitudinal and transverse directions respectively (α 11 and α 22 ), Poisson's ratio (v 11 ) and the density (ρ) of the CNTRC plates can be determined as: , : , 2 1 ) ( : where where CNT w , m  and CNT  are the mass fraction of the CNT, the densities of the matrix and CNT, respectively.Similarly, the thermal expansion coefficients in the longitudinal and transverse directions respectively ( 11 and 22  ), Poisson's ratio ( 12  ) and the density (  ) of , 2 1 ) ( : where where CNT w , m  and CNT  are the mass fraction of the CNT, the densities of the matrix and CNT, respectively.Similarly, the thermal expansion coefficients in the longitudinal and transverse directions respectively ( 11 and 22  ), Poisson's ratio ( 12  ) and the density (  ) of the CNTRC plates can be determined as where  are the thermal expansion coefficients of the CNT and matrix, respectively.It (11) , : , 2 1 ) ( : where where CNT w , m  and CNT  are the mass fraction of the CNT, the densities of the matrix and CNT, respectively.Similarly, the thermal expansion coefficients in the longitudinal and transverse directions respectively ( 11 and 22  ), Poisson's ratio ( 12  ) and the density (  ) of the CNTRC plates can be determined as where  is assumed as constant over the thickness of the FG-CNTRC plates.
(12) where: v r12 and v m are Poisson's ratios of the CNT and matrix, respectively.In addition, α r11 , α r22 and α m are the thermal expansion coefficients of the CNT and matrix, respectively.
It should be noted that v 12 is assumed as constant over the thickness of the FG-CNTRC plates.

ORTHOTROPIC STRESS-STRAIN RELATIONS
The constitutive equation for stresses σ and strains ε matrix in thermal environment may be written as follows: The constitutive equation for stresses σ and strains ε matrix in thermal environment may be written as follows where ij C denotes temperature-dependent elastic coefficients which can be expressed as . , ,  (13) where: C ij denotes temperature-dependent elastic coefficients which can be expressed as: where ij C denotes temperature-dependent elastic coefficients which can be expressed as   may be obtained using rule of mixture (i.e.Eqs.(1-7)).

Nonlinear Mindlin plate theory
Based on Mindlin plate theory, the displacement field can be expressed as (Reddy, 1984) )  The von Kármán strains associated with the above displacement field can be expressed in the following form (14) Noted that C ij and α 11 , α 22 may be obtained using the rule of mixture (i.e.Eqs.(1-7)).

NONLINEAR MINDLIN PLATE THEORY
Based on Mindlin plate theory, the displacement field can be expressed as (Reddy, 1984): where ij C denotes temperature-dependent elastic coefficients which can be expressed as  The von Kármán strains associated with the above displacement field can be expressed in the following form (15) where: (u x , u y , u z ) denote the displacement components at an arbitrary point (x, y, z) in the plate, and (u, v, w) are the displacement of a material point at (x, y) on the mid-plane (i.e.z = 0) of the plate along the x-, y-, and z-directions, respectively; ) , ( y x  The von Kármán strains associated with the above displacement field can be expressed in the following form: ), (

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the ), (

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the work done by the elasomeric medium, W .The strain energy can be written as ), (

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the work done by the elasomeric medium, W .The strain energy can be written as (18) x z x ), (

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the work done by the elasomeric medium, W .The strain energy can be written as ), (

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the work done by the elasomeric medium, W .The strain energy can be written as where: (ε xx , ε yy ) are the normal strain components and (γ yz , γ xz , γxy ) are the shear strain components.

ENERGY METHOD
The total potential energy V of the CNTRC plate is the sum of strain energy U and the work done by the elastomeric medium W. The strain energy can be written as: 7 components.

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energ work done by the elasomeric medium, W .The strain energy can be written as where the stress resultant-displacement relations can be written as , ) ,

Energy method
The total potential energy,V , of the CNTRC plate is the sum of strain energy, U and the work done by the elasomeric medium, W .The strain energy can be written as Combining of Eqs. ( 8)-( 15) yields where the stress resultant-displacement relations can be written as , ) , where the stress resultant-displacement relations can be written as: The total potential energy,V , of the CNTRC plate is the work done by the elasomeric medium, W .The strain energ where the stress resultant-displacement relations can be wr , ) , In which K is shear correction coefficient.The extern dependent elastomeric medium and a uniform load on be written as where q is related to elastic medium.Elastic f where K and G are spring and shear constants.

Governing equations
The governing equations can be derived by Hamilton' In which K is shear correction coefficient.The external work due to orthotropic temperature-dependent elastomeric medium and a uniform load on upper surface of the CNTRC plate can be written as: In which K is shear correction coefficient.The ex dependent elastomeric medium and a uniform loa be written as where q is related to elastic medium.Elasti where K and G are spring and shear constants.

Governing equations
The governing equations can be derived by Hamil  23) and (24) into Eq.( 28), the where: q is related to elastic medium.
Elastic foundation force can be expressed as In which K is shear correction coefficient.The exte dependent elastomeric medium and a uniform load be written as where q is related to elastic medium.Elastic where K and G are spring and shear constants.

Governing equations
The governing equations can be derived by Hamilto  23) and (24) into Eq.( 28), the be obtained as follow (26) where: K and G are spring and shear constants.

GOVERNING EQUATIONS
The governing equations can be derived by Hamilton's principal as follows: In which K is shear correction coefficient.The external work du dependent elastomeric medium and a uniform load on upper sur be written as where q is related to elastic medium.Elastic foundation where K and G are spring and shear constants.

Governing equations
The governing equations can be derived by Hamilton's principal  23) and (24) into Eq.( 28), the stress resulta be obtained as follow (27) Substituting Eqs. ( 22) and (25) into Eq.( 27) yields the following governing equations: In which K is shear correction coefficient.The external work due to orthotropi dependent elastomeric medium and a uniform load on upper surface of the CN be written as where q is related to elastic medium.Elastic foundation force can be where K and G are spring and shear constants.

Governing equations
The governing equations can be derived by Hamilton's principal as follows Substituting Eqs. ( 23) and (24) into Eq.( 28), the stress resultant-displacemen be obtained as follow (28) .

Numerical Results and discussion
A computer program is prepared for the numerical solution of resting on an elastic temperature-dependent foundation.Here reinforcements with the material properties listed in Table 1.

NUMERICAL RESULTS AND DISCUSSION
A computer program is prepared for the numerical solution of buckling of CNTRC plates resting on an elastic temperature-dependent foundation.Here, Poly methyl methacrylate (PMMA) is selected for the matrixes which have constant Poisson's ratios of v m = 0.34, temperature-dependent thermal coefficient of v m = (1 + 0.0005ΔT)×10 -6 /K, and temperature-dependent Young moduli of E m = (3.52-0.0034T)GPa in which T = T 0 +ΔT and T 0 = 300 K (room temperature).In addition, (10,10) SWCNTs are selected as reinforcements with material properties listed in Table 1.The elastomeric medium is made of Poly dimethylsiloxane (PDMS) whose temperature-dependent material properties of which are assumed to be v m = 0.48 and E s = (3.22-0.0034T)GPa in which T = T 0 +ΔT and T 0 = 300 K (room temperature) [Shen, 2009].
It should be noted that the mass fraction (w CNT ) of the UD and FG distribution of CNTs in the polymer are considered equal for the purpose of comparisons.As can be seen, the buckling load of FGA-and FGO-CNTRC plates are lower than buckling load of UD-CNTRC plates while the FGX-CNTRC plate have higher buckling load with respect to three other cases.It is due to the fact that the stiffness of CNTRC plates changes with the form of CNT distribution in the matrix.However, it can be concluded that CNT distribution close to top and bottom are more efficient than those distributed nearby the midplane for increasing the stiffness of plates.In addition, increasing the thickness can decrease the buckling load of the CNTRC plate.This is because with increasing the thickness, the stability of plate decreases.
The effect of the CNT volume fraction on the buckling load of the CNTRC plate with respect to thickness is shown in Figure 3.It can be found that thickener plate can decrease the buckling load of the CNTRC plate.It is also observed that Advances in Science and Technology Research Journal Vol. 10 (29), 2016 158 increasing the CNT volume fraction increases the buckling load of the CNTRC plate.This is due to the fact that the increase of CNT volume fraction leads to a harder structure.Meanwhile, the effect of CNT volume fraction becomes more considerable at lower thickness.
The effect of the elastic temperature-dependent medium on the buckling load of the CNTRC plate with respect to thickness is illustrated in Figure 4.In this figure three cases are considered which are without elastic medium, Winkler medium and Pasternak medium.As can be seen, considering elastic medium increases buckling load of the CNTRC plate.It is due to the fact that considering elastic medium leads to stiffer structure.Furthermore, the effect of the Pasternak-type is higher than the Winkler-type on the buckling load of the CNTRC plate.It is perhaps due to the fact that the Winkler-type is capable to describe just normal load of the elastic medium while the Pasternak-type describes both transverse shear and normal loads of the elastomeric medium.
The effect of temperature on the buckling load of the CNTRC plate, with respect to the thickness, is demonstrated in Figure 5.The same as other figures, increasing the thickness decreases

CONCLUSIONS
Based on orthotropic temperature-dependent Mindlin polymeric plate theory, buckling analysis of an embedded CNTRC plate was studied in this paper.CNT distributions in polymer were considered as UD, FGA, FGX and FGO.The rule of mixture was used for obtaining the material properties of FG-CNTRC plate.The nanocomposite system was surrounded in a temperature-dependent elastic medium.Using strain-displacement relation, energy method and Hamilton's principle, the governing equations were derived.In order to obtain the buckling load of the FG-CNTRC plate, Navier method was performed.The effects of the volume fractions of carbon nanotubes, elastic medium, temperature, thickness and distribution type of CNTs were considered.The results indicate that considering elastic medium increases, buckling load of the FG-CNTRC plate decreases with increasing temperature.It was also concluded that buckling load gets larger as the CNT volume fraction increases.Furthermore, the lowest and highest buckling load were respectively obtained for FGO-and FGX-CNTRC plates.

G
indicate the Young's respectively, and m E , m G represent the correspond scale-dependent material properties, j  (j= 1, 2 effective properties of CNTRC obtained from the M mixture.CNT V and m V are the volume fractions of the sum of them equals to unity.The uniform and t along the thickness direction of the CNTRC plates (2) 4

G
indicate the Young's respectively, and m E , m G represent the correspond scale-dependent material properties, j  (j= 1, 2 effective properties of CNTRC obtained from the M mixture.CNT V and m V are the volume fractions of the sum of them equals to unity.The uniform and t along the thickness direction of the CNTRC plates : w CTN , ρ m and ρ CTN are the mass fraction of the CNT, the densities of the matrix and CNT, respectively.

Fig. 1 .
Fig. 1.Configurations of the SWCNT distribution in a CNTRC plates: (a) UD CNTRC plate; (b) FG-A CNTRC plate; (c) FG-O CNTRC plate; (d) FG-X CNTRC plate plate along the x-, y-, and z-directions, respectively; the normal to the mid-plane about x-and y-directions, respectively.
the normal to the mid-plane about x-and y-directions, respectively. x of the normal to the mid-plane about x-and y-directions, respectively.

(
PMMA) is selected for the matrix which have constant Po temperature-dependent thermal coefficient of  ).In addition,(10,10) made of Poly dimethylsiloxane (PDMS) which the temperature-d of which are assumed to be 48 ) (Shen, 2009).Depicted in Fig. 2 is the buckling load for the UD and three versus thickness.It should be noted that the mass fraction distribution of CNTs in polymer are considered equal for the pur be seen, the buckling load of FGA-and FGO-CNTRC plates are UD-CNTRC plates while the FGX-CNTRC plate have higher b three other cases.It is due to the fact that the stiffness of CNT form of CNT distribution in matrix.However, it can be conclude to top and bottom are more efficient than those distributed (50)Solving the above equation, yields the buckling load of system.

Fig. 4 .Fig. 5 .
Fig. 4. Effects of elastic medium on the buckling behavior of CNTRC plates 12 r  and m  are Poisson's ratios of the CNT and matrix, respectively.In addition, 11 r  , 22 r  and m  are the thermal expansion coefficients of the CNT and matrix, respectively.It should be noted that 12 Advances in Science and Technology Research Journal Vol. 10 (29), 2016

Table 1 .
Temperature-dependent material properties of (10, 10) SWCNT (L= 9.26 nm, R= 0.68 nm, h= 0.067 nm, the buckling load of the CNTRC plate.It can be also found that the buckling load of the CNTRC plate decreases with increasing temperature which is due to higher stiffness of CNTRC plate with lower temperature.