Document Type : Research Paper
Authors
^{1} Mechanical Engineering Department, Islamic Azad University, Tehran North Branch, Tehran, Iran
^{2} Mechanical Engineering Department, University of Eyvanakey, Semnan, Iran
Abstract
Keywords
Numerical Investigation on Thermal Postbuckling Of Annular Sector Plates Made of FGM Via 3D Finite Element Method
^{a}Mechanical Engineering Department, Islamic Azad University, Tehran North Branch, Tehran, Iran
^{b}Mechanical Engineering Department, University of Eyvanakey, Semnan, Iran
KEYWORDS 

ABSTRACT 
Thermal postbuckling analysis 3D elasticity FGM annular sector plate Finite element method 
This paper presents the thermal postbuckling analysis of functionally graded annular sector plates subjected to uniform temperature rise for the first time. The plate is consisting of a composed ceramicmetal material which the volume fraction of the component materials is assumed to change continuously through the thickness via a simple power law distribution.3D elasticity theory and nonlinear Green strain tensor are used to derive the governing equations which are extended based on the principle of virtual work and solved via the graded finite element method. The nonlinear equilibrium equations are solved by applying the Newton–Raphson procedure. The influences of material gradient exponent, various sector angles, thickness ratio, aspect ratio on the thermal postbuckling response of FGM annular sector plates subjected to uniform temperature rise are presented. Results indicate that the thermal postbuckling response of FGM annular sector plates can be considered as a bifurcation point following a stable postbuckling path. 
In recent years, functionally graded materials (FGMs) as a kind of thermal barrier materials have been used for structural components subjected to exceedingly hightemperature environments including nuclear reactors and highspeed spacecraft industries. In such conditions, hightemperature which is induced compressive stresses will be developed in the constrained plates; therefore, it will lead to buckling. Annular sector plates are often used as structural components in many engineering applications and are subjected to different thermal loading conditions. Hence, the thermal buckling and postbuckling response of FGM annular sector plates are of incited interest for engineering design. Gas turbine rotating blades made of FGMs can be considered as one of the applications of FGM annular sector plates subjected to high thermal loadings.
Although many researches have been conducted on the thermal buckling [16] and postbuckling [717] analysis of rectangular FG plates and beams, among the platetype structures, there has been less research attention on thermal buckling [1825] and postbuckling analysis [2632] of FGM circular and annular plate. There are a few studies related to buckling and postbuckling of FGM annular sector plates under mechanical load. For instance, based on classical plate theory, HosseiniHashemi et al. [33] performed DQ analysis for buckling behavior of radially FG circular and annular sector plates witch located on the Pasternak elastic foundation. Based on FSDT theory, Naderi and Saidi [3436] employed an analytical solution to investigate the buckling behavior of relatively thick FG annular and sector plates located on the Winkler elastic foundation. Asemi et al. applied 3D elasticity and finite element method to study biaxial [37] and shear [38] buckling analysis of FGM annular sector plates resting on Winkler elastic foundation with fully or partially supported boundary conditions. Also, limited investigation focused on the thermal buckling of FGM annular sector plate, for example, Saidi and Baferani [39] investigated thermal buckling analysis of relatively thick FG annular sector plates. The equilibrium and stability equations are obtained based on FSDT. Finally, an analytical method was applied to solve it. In another study, Jabbarrzadeh et al [40] utilized FSDT and Von Karman’s assumptions to study the thermal buckling of FG annular sector plate. Differential quadrature method was applied to solve the equilibrium and stability equations. Based on FSDT, Shaterzadeh and his coauthors investigated thermal buckling analysis of symmetric and antisymmetric laminated composite plates with a cutout [41], and thermomechanical buckling analysis of FGM plates with an elliptic cutout [42]. In addition, Shaterzadeh et al [43] applied FSDT and finite element method to study the thermal buckling analysis of perforated FGM plates. 3D thermomechanical buckling analysis of perforated annular sector plates with multiaxial material heterogeneities based on curved Bspline elements was presented by Shariyat et al [44]. Based on 3D elasticity and Green’s nonlinear strain assumptions, Behzad et al [45] carried out the investigation about the thermal buckling analysis of FG perforated annular sector plates. Shaterzadeh et al [46] studied the stability analysis of FG annular sector plate under thermomechanical loading via 3D elasticity theory and using finite element method.
From the above literature review, it can be seen that the nonlinear Von Karman’s assumptions and plate theories have been mostly applied to investigate postbuckling analysis of functionally graded material plates. In this paper, 3D elasticity theory and green’s assumptions are employed. 3D elasticity leads to more realistic and accurate results than the other plate theories. Furthermore, the complete Green strain tensor is more acceptable for analyzing the geometrically nonlinear behavior of the plates.
In this work, a numerical method is carried out by using the nonlinear Green strain tensor and, FEM based on 3Delasticity theory to study the thermal postbuckling of FGM annular sector plates subjected to uniform temperature rise. The plate is consisting of a composed ceramicmetal material which the volume fraction of the component materials is assumed to change continuously through the thickness via a simple power law distribution. Three various boundary conditions have been considered such as: 1 Immovable simply supported edges. 2 Immovable simply supported radial edges and free circumferential edges. 3 Immovable simply supported circumferential edges and free radial edges. The principle of virtual work is utilized to derive the governing equations and the iterative Newton–Raphson procedure is applied to solve the nonlinear equilibrium equations. The impact of material gradient index, sector angle, aspect ratio, and thickness ratio on the thermal postbuckling behavior of FGM annular sector plates have been examined. The novel point of this paper is the investigation of thermal postbuckling behaviour of FGM annular sector plate with considering various boundary conditions for the first time.
A cermet FGM annular sector plate is considered. h, a, b and β are thickness, inner and outer radius of the sector, respectively. The geometry of the problem are shown in Fig. 1.
The material properties of the FGM annular sector plate across the thickness are defined as follows:

(1) 
in which Q indicates material property including modulus of elasticity and coefficient of thermal expansion , and subscripts m and c related to the metal and ceramic component, respectively. Poisson’s ratio is considered to be constant along with the thickness.
Hooke’s law for this structure is considered as:

(2) 
where:

(3) 
where is the thermal strain components and and are the stress and strain tensor components in cylindrical coordinates, respectively, and the elasticity matrix D is as:
(4) 
Fig. 1. Geometry of annular sector plate
It is assumed that the elasticity modulus E varies along the thickness direction whereas Poisson’s ratio ν is assumed to be constant. Therefore Matrix ϕ is constant.
The Green strain displacement relations in cylindrical coordinates are assumed as follows:

(5) 
(6) 
The cylindrical and natural coordinates are defined by

(7) 
where are through r, θ and z coordinate axes, respectively. are the inner radius, outer radius, sector angle, and thickness of each annular sector element.
Hence, the linear part of straindisplacement relations (6) can be considered in matrix form in terms of cylindrical coordinate system as:

(8) 
where the linear part of straindisplacement relations based on natural coordinates can be expressed as:

(9) 

(10) 
Consider the threedimensional 8node linear brick element (Fig. 2b). In comparison to the conventional brick elements, material properties are among the nodal degrees of freedom. The displacement field vector of an optional point of the element may be related to the nodal displacement vectors of the element through the shape function matrix N, as:

(11) 
where:

(12) 

(13) 
Matrix N based on the natural coordinates can be expressed as [47]:

(14) 

(a) 

(b) 

Fig. 2. Schematic of an annular sector element. 

Replacing (11) into (8), the linear part of straindisplacement can be expressed as:

(15) 
where:
(16) 
Moreover, by substituting (11) into (9), the linear part of straindisplacement relations in terms of natural coordinate system can be expressed as:

(17) 
where are:
(18) 
(19) 
The nonlinear part of straindisplacement relations in terms of cylindrical coordinate system can be considered as:

(20) 
where:

(21) 
in which are as following submatrices.
(21) 

(22) 
Also, the nonlinear part of straindisplacement relations in terms of natural coordinates is as:

(23) 
where:

(24) 
where is given in Ref [48].
The displacement field, as well as the nonhomogeneity of the mechanical properties of the FGM plate, can be determined based on their nodal values. Therefore, the graded finite element method can be applied to achieve to effectively trace smooth variations of the material properties at the element level. Thus, the shape functions analogous to those of the displacement field can be expressed:

(25) 
in which is the modulus of elasticity for node i. and are vectors of shape functions and modulus of elasticity of each element, respectively.
Hence, Equation (4) can be rewritten as:

(26) 
The principle of virtual work could be used to achieve governing equations, which is defined as:

(27) 
where:

(28) 
To extend the variation of strain energy, the definition of Green strain tensor and its variation are expressed as:

(29) 
where B and in term of cylindrical and natural coordinates are as follow, respectively:

(30) 

(31) 
where:

(32) 
where and based on natural coordinates are noted in reference [48]. Therefore, Equation (27) can be written in a compact form as:

(33) 
where:
(34) 


(35) 

and

(36) 
The first term of the expanded form of the stiffness matrix is linear and is related to , the second and third ones are nonlinear which are linearly dependent on the unknown variables and their summation is defined by . The fourth one is nonlinear which is the quadratic function of the unknown variables and is defined by .
By assembling the element matrices, the governing finite element equations of the FGM annular sector plate are:

(37) 
The postbuckling behavior could be obtained with the gradually rising in used thermal load. There are the nonlinear terms in the stiffness matrix, so an iterative technique must be applied. Newton–Raphson iterative procedure is employed to solve nonlinear equilibrium equations [47]. The process at the n+1th iteration is as follows:

(38) 
where:

(39) 
where can be computed as:

(40) 
The sum of stiffness and geometric stiffness matrix ( is tangent stiffness matrix and can be expressed as follows:

(41) 
Stages of the numerical solution of the problem may be summarized as follows;
1 The nonlinear stiffness matrices (Eq. (34)) of the elements are determined based on the nodal displacements achieved at the previous loading increment. At the beginning of the solution procedure, these components are zero.
2 The overall stiffness and force matrices of the plate are constructed through assemblage of the stiffness and force matrices of the individual elements.
3 The boundary conditions are imposed.
4 The tangential stiffness matrix is established using Eq (41).
5 Eq. (39) is solved for determination of the required increments in the nodal displacements to reach the solution.
6 If the following convergence criterion is not met, the iterative solution has to be continued from step 1,

(42) 
where is a sufficiently small number, e.g., 0.0001.
7 In case of convergence occurrence, the results have to be saved (e.g., the load values and the resulting lateral deflections), the load is incremented and the solution process is continued from step 1.
8 Steps 1 to 7 are repeated till the assigned final values of the loads are reached.
In the present investigation, three various types of displacement boundary conditions are assumed as follow:

(43) 

(44) 

(45) 
For simply supported edges condition, only the value of displacements at the midsurface of the plate are equal to zero.
In this part, a comparison study is carried out to show the effectiveness and accuracy of the present method. Thermal postbuckling of annular sector plates is not studied so far. Hence, postbuckling of an FGM square plate subjected to uniform temperature rise is reconsidered [12] for verification purposes. Na and Kim [12] studied thermal postbuckling of all edges immovable simply supported FG plates subjected to uniform temperature rise using the three dimensional 18 node element. To convert the annular sector plate of the present study to a square plate, the sector angle is assumed as a small value rad and inner and outer radiuses of the plate are chosen as large values: a=200 m, b=200.2 m, and h=0.02. These geometric dimensions lead to nearly a square plate with lengthtothickness ratio of a/h=100. Furthermore, the material properties are assumed as , , and [12] .The postbuckling path of the center point of the plate for n=1 is achieved and compared with Ref [12]. Fig. 3. indicates the variations of temperature versus nondimensional central displacement w/h. It can be seen that the present results are in an acceptable agreement with those reported in the literature.
In this section, the postbuckling analysis of all edges immovable simply supported FGM annular sector plates for different values of volume fraction index, sector angles, aspect ratio, and thickness ratio is presented. For this purpose, an FGM annular sector plate made of Al2O3Ni with the following geometric and material properties characteristics is assumed:
a= 0.25, 0.5m, b=1m, h=0.025, 0.0125 m, .
, , .
Postbuckling paths are presented according to the nondimensional transverse displacement .. It should be noted that all the figures and contours are plotted by Matlab software. First, an immovable simply supported FGM annular sector plate is considered (a=0.5m, b=1m, h=0.025 m). The influence of power law exponents on the postbuckling paths of the plate subjected to uniform temperature rise is shown in Fig. 4. This result corresponds to an annular sector plate with 0 and transverse displacement of the center point of the plate. Fig. 4. indicates that there is a sudden change in the loaddeflection curve and the response of the plate can be introduced as a primarysecondary equilibrium path. Fig. 4. shows that for the homogeneous plate, the response of the plate is the bifurcationtype buckling, and the plate remains flat in the prebuckling regions. Although, a nonlinear stable equilibrium path is in the postbuckling area. FGM plates start to deflect immediately after the thermal load is applied. However, there is a sudden change in the loaddeflection curve. The asymmetric distribution of the material properties in the thickness direction causes this behavior. The coupling between thermal forces and bending moments is created due to the asymmetric distribution of the material properties. In this case, the remained force passes through the midplane, not neutral surface of the plate. Hence, an extra bending moment influences on the plate, and the simply supported edges cannot tolerate the additional moments, and the plate starts to transverse displacement by initiation of applying the thermal load. For the cases that the movable simply supported FGM plates were subjected to inplane normal loads [48], there is not any sudden change in the loaddeflection curve. Also, the response of the plate could not be considered as a primarysecondary equilibrium path, and FGM plates show nonlinear bending behavior with unique and stable equilibrium paths.
A comparison between [48] and present results denotes that in the present study when all edges of the plate are immovable (u=v=w=0), the behavior of the FGM plate subjected to thermal loads is considered as a primarysecondary equilibrium path. Also, results show that by increasing the power law exponent, the volume fraction of ceramic constituent causes decreases, and hence, the plate becomes softer and experiences more transverse displacement. Moreover, by increasing the power law exponent, the strength of the plate decreases at the postbuckling region. Figs. 5 ac show the deflection of the plate for different temperatures and n=1, z=h/2.
Fig. 3. A comparison between postbuckling equilibrium curves obtained by the present 3D elasticity procedure and Ref [12] for immovable simply supported FGM square plate subjected to uniform temperature rise ( z=h/2).
Fig. 4. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges (h=0.025 m and 0 )
Fig. 5. The postbuckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (h=0.025 m and 0 ) a) T=520 ,
The influence of thickness ratio on postbuckling paths of alledgeimmovable simply supported FGM annular sector plate subjected to uniform temperature rise is also studied. Therefore, the thickness of the plate is considered to be h=0.0125 m. Fig. 6 indicates postbuckling paths of the center point of the plate with 0 for various power law exponent. A comparison between Figs. 4. and 6. demonstrates that by decreasing the thickness of the plate, the strength of the plate at the buckling and postbuckling region is decreased and the plate experiences more lateral deflection. Figs. 7 ac shows the deflection of the plate for different temperatures and for n=1, z=h/2.
In order to study the influence of sector angle on thermal postbuckling behavior of plate, an immovable simply supported FGM annular sector plate with , is considered. Influence of power law exponents on center point deflection of the plate with , is shown in Figs. 8. and 9., respectively. A comparison between results of , and 0 indicate that by increasing the sector angle, the strength of plate at buckling and postbuckling areas is decreased and the plate experiences more lateral deflection. Figs. 10 ac and Figs. 11 ac show the deflection of the plate for different temperatures with , , respectively (n=1, z=h/2). These results indicate that by increasing the temperature in the deep postbuckling regime, higher mode shapes are appeared. Also, these figures denote that, by increasing the sector angle, the number of buckling waves is increased.
The influence of aspect ratio on postbuckling paths of alledgeimmovable simply supported FGM annular sector plate subjected to uniform temperature rise is also studied. Therefore, the inner radius of the plate is supposed to be a=0.25 m (b=1 m, h=0.025 m, )). Fig. 12. indicates postbuckling paths of the center point of the plate with different power law exponent. A comparison between Figs. 4 and 12 demonstrates that by decreasing the inner radius of the plate, the strength of the plate at buckling and postbuckling areas is decreased. Figs. 13. ac shows the deflection of the plate for different temperatures through the postbuckling path and for n=1, z=h/2.
Now, the influences of various boundary conditions on thermal postbuckling of FGM annular sector plate subjected to uniform temperature rise have been investigated (a=0.5 m, b=1 m, h=0.025 m, ). The influence of power law exponents on the center point deflection of FG annular sector plate with immovable simply supported radial edges and free circumferential edges is shown in Fig. 14. It can be seen that the postbuckling response of the FGM plate can be considered as a primarysecondary equilibrium path with a stable postbuckling regime. By applying the free edges condition, the strength of the plate at postbuckling is noticeably decreased. Figs. 15. ac shows deflection of the plate for different temperatures through the equilibrium path for n=1, z=h/2.
Fig. 6. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges (h=0.0125 m and 0 )
Fig. 7. The postbuckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (h=0.0125 m and 0 ), a) T=120 , b) T=180 , c) T=1000
Fig. 8. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges (h=0.025 m and )
Fig. 9. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges (h=0.025 m and )
Fig. 10. The post buckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (h=0.025 m and )), a) T=480 , b) T=580 , c) T=1000
Fig. 11. The post buckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (h=0.025 m and )), a) T=510 , b) T=610 , c) T=1000
Postbuckling paths of FGM plate with immovable circumferential edges and free radial edges with various power law exponent are shown in Fig. 16. The influence of power law exponent on postbuckling paths is similar to the other boundary conditions. But by comparing the results of various boundary conditions, it is shown that in this case, the buckling and postbuckling of the plate are increased. Figs. 17. ac shows deflection of the plate for different temperatures through the equilibrium path for n=1, z=h/2. Finally, for giving a clear sense of the benchmark results, the critical temperature values of the presented curves extracted from postbuckling paths are reported in Table 1.
Fig. 12. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges (a=0.25, h=0.025 m and )
Fig. 13. The post buckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (a=0.25, h=0.025 m and ), a) T=360 , b) T=410 , c) T=1000
Fig. 14. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable simply supported edges at for different power law exponent (a=0.5 m, h=0.025 m and )
Fig. 15. The postbuckling deformation type of annular sector plate made of FGM with immovable simply supported edges at for (n=1, z=h/2), (h=0.025 m and ),
Fig. 16. Influence of different power law exponent on postbuckling curves of annular sector plate made of FGM with immovable circumferential edges and free radial edges (a=0.5 m, h=0.025 m and )
Fig. 17. The post buckling deformation type of annular sector plate made of FGM with immovable simply supported edges at r=a, b for (n=1, z=h/2), (h=0.025 m and ), a) T=560 , b) T=700 , c) T=1000
Table 1. The Influence of different power law exponent on critical temperature of immovable simply supported FGM annular sector plates
N 
a=0.5m, b=1m h=0.025 m 
a=0.5m, b=1m h=0.0125 m 
a=0.5m, b=1m h=0.025 m 
a=0.25, b=1m h=0.025 m 
n=0 
714.54 
186.703 
689.923 
503.226 
n=1 
538.412 
136.084 
507.203 
369.194 
n=3 
535.814 
133.0929 
505.518 
366.82 
n=10 
533.221 
130.0932 
501.139 
362.977 
In this study, the 3D elasticity approach is used to investigate the thermal postbuckling analysis of functionally graded annular sector plates. The GreenLagrange nonlinear straindisplacement relation is assumed for large deflections. The governing equations are extended via the principle of virtual work and solved based on a GFEM. Finally, Newton–Raphson procedure is applied to solve the nonlinear equilibrium equations. The influences of material gradient exponent, various sector angles, thickness ratio, aspect ratio, and three various boundary conditions on the thermal postbuckling response of FGM annular sector plates have been carried out.
Some of the innovations in the present study are:
Furthermore, the following conclusions are obtained from the results:
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