Numerical Investigation on Thermal Post-buckling Of Annular Sector Plates Made of FGM Via 3D Finite Element Method

1. Shahsavari^a, K. Asemi ^a*, M. Babaei^b, F. Kiarasi^b,

^aMechanical Engineering Department, Islamic Azad University, Tehran North Branch, Tehran, Iran

^bMechanical Engineering Department, University of Eyvanakey, Semnan, Iran

1.     Introduction

In recent years, functionally graded materials (FGMs) as a kind of thermal barrier materials have been used for structural components subjected to exceedingly high-temperature environments including nuclear reactors and high-speed spacecraft industries. In such conditions, high-temperature 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 post-buckling 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 [1-6] and post-buckling [7-17] analysis of rectangular FG plates and beams, among the plate-type structures, there has been less research attention on thermal buckling [18-25] and post-buckling analysis [26-32] of FGM circular and annular plate. There are a few studies related to buckling and post-buckling of FGM annular sector plates under mechanical load. For instance, based on classical plate theory, Hosseini-Hashemi 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 [34-36] 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 co-authors investigated thermal buckling analysis of symmetric and anti-symmetric laminated composite plates with a cut-out [41], and thermo-mechanical 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 B-spline 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 post-buckling 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 non-linear behavior of the plates.

In this work, a numerical method is carried out by using the nonlinear Green strain tensor and, FEM based on 3D-elasticity theory to study the thermal post-buckling of FGM annular sector plates subjected to uniform temperature rise. The plate is consisting of a composed ceramic-metal 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 non-linear equilibrium equations. The impact of material gradient index, sector angle, aspect ratio, and thickness ratio on the thermal post-buckling behavior of FGM annular sector plates have been examined. The novel point of this paper is the investigation of thermal post-buckling behaviour of FGM annular sector plate with considering various boundary conditions for the first time.

2.     Materials Properties and Geometry

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:

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.

3.     3D Elasticity Theory

Hooke’s law for this structure is considered as:

where:

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:

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:

The cylindrical and natural coordinates are defined by

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 strain-displacement relations (6) can be considered in matrix form in terms of cylindrical coordinate system as:

where the linear part of strain-displacement relations based on natural coordinates can be expressed as:

Consider the three-dimensional 8-node linear brick element (Fig. 2-b). 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:

where:

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

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

where:

Moreover, by substituting (11) into (9), the linear part of strain-displacement relations in terms of natural coordinate system can be expressed as:

where    are:

The non-linear part of strain-displacement relations in terms of cylindrical coordinate system can be considered as:

where:

in which  are as following sub-matrices.

Also, the non-linear part of strain-displacement relations in terms of natural coordinates is as:

where:

where  is given in Ref [48].

The displacement field, as well as the non-homogeneity 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:

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:

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

where:

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

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

where:

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

where:

and

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:

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

where:

where  can be computed as:

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

Stages of the numerical solution of the problem may be summarized as follows;

1- The non-linear 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,

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:

1. a) For annular sector plates with immovable simply supported edges:

1. b) For annular sector plates with immovable simply supported radial edges and free circumferential edges:

1. c) For annular sector plates with immovable simply supported circumferential edges and free radial edges:

For simply supported edges condition, only the value of displacements at the mid-surface of the plate are equal to zero.

4.     Results and Discussions

4.1. Validation

In this part, a comparison study is carried out to show the effectiveness and accuracy of the present method. Thermal post-buckling of annular sector plates is not studied so far. Hence, post-buckling of an FGM square plate subjected to uniform temperature rise is re-considered [12] for verification purposes. Na and Kim [12] studied thermal post-buckling 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 length-to-thickness ratio of a/h=100. Furthermore, the material properties are assumed  as   , ,  and [12] .The post-buckling 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 non-dimensional central displacement w/h. It can be seen that the present results are in an acceptable agreement with those reported in the literature.

4.2. Thermal Post-buckling Of FGM Annular Sector Plate

In this section, the post-buckling 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 Al2O3-Ni with the following geometric and material properties characteristics is assumed:

a= 0.25, 0.5m, b=1m, h=0.025, 0.0125 m, .

, , .

Post-buckling paths are presented according to the non-dimensional 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 post-buckling 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 load-deflection curve and the response of the plate can be introduced as a primary-secondary equilibrium path. Fig. 4. shows that for the homogeneous plate, the response of the plate is the bifurcation-type buckling, and the plate remains flat in the pre-buckling regions. Although, a non-linear stable equilibrium path is in the post-buckling area. FGM plates start to deflect immediately after the thermal load is applied. However, there is a sudden change in the load-deflection 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 mid-plane, 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 in-plane normal loads [48], there is not any sudden change in the load-deflection curve. Also, the response of the plate could not be considered as a primary-secondary equilibrium path, and FGM plates show non-linear 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 primary-secondary 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 post-buckling region. Figs. 5 a-c show the deflection of the plate for different temperatures and n=1, z=h/2.

Fig. 3. A comparison between post-buckling 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 post-buckling curves of annular sector plate made of FGM   with immovable simply supported edges (h=0.025 m and 0 )

Fig. 5. 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 0 ) a) T=520 ,

1. b) T=600 , c) T=1000

The influence of thickness ratio on post-buckling paths of all-edge-immovable 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 post-buckling 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 post-buckling region is decreased and the plate experiences more lateral deflection. Figs. 7 a-c 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 post-buckling 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 post-buckling areas is decreased and the plate experiences more lateral deflection. Figs. 10 a-c and Figs. 11 a-c 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 post-buckling 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 post-buckling paths of all-edge-immovable 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 post-buckling 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 post-buckling areas is decreased. Figs. 13. a-c shows the deflection of the plate for different temperatures through the post-buckling path and for n=1, z=h/2.

Now, the influences of various boundary conditions on thermal post-buckling 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 post-buckling response of the FGM plate can be considered as a primary-secondary equilibrium path with a stable post-buckling regime. By applying the free edges condition, the strength of the plate at post-buckling is noticeably decreased. Figs. 15. a-c 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 post-buckling curves of annular sector plate made of FGM with immovable simply supported edges (h=0.0125 m and 0 )

Fig. 7. The post-buckling deformation type of annular sector plate made of FGM with immovable simply supported edges for (n=1, z=h/2), (h=0.0125 m and 0 ), a) T=120 , b) T=180 , c) T=1000

Fig. 8. Influence of different power law exponent on post-buckling 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 post-buckling 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

Post-buckling 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 post-buckling 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 post-buckling of the plate are increased. Figs. 17. a-c 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 post-buckling paths are reported in Table 1.

Fig. 12. Influence of different power law exponent on post-buckling 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 post-buckling 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 post-buckling 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 ),

1. a) T=310 , b) T=410 , c) T=1000

Fig. 16. Influence of different power law exponent on post-buckling 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

5.     Conclusions

In this study, the 3D elasticity approach is used to investigate the thermal post-buckling analysis of functionally graded annular sector plates. The Green-Lagrange nonlinear strain-displacement 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 non-linear equilibrium equations. The influences of material gradient exponent, various sector angles, thickness ratio, aspect ratio, and three various boundary conditions on the thermal post-buckling response of FGM annular sector plates have been carried out.

Some of the innovations in the present study are:

• Analyzing post-buckling of FGM annular sector plates subjected to uniform temperature rise for the first time.
• Using 3D elasticity theory instead of the approximate plate theories that shows the behavior of the plate more real.
• The Green-Lagrange nonlinear strain-displacement relation is taken into account for large deflections.
• The coupling between thermal load and bending moments of FGM is studied.
• The influences of three various boundary conditions on the post-bucking response of plate have been presented.

Furthermore, the following conclusions are obtained from the results:

• The post-buckling behavior of the FGM annular sector plate for different boundary conditions can be noticed as primary-secondary equilibrium path.
• While the power law exponent increases, the post-buckling resistance of the plate is decreased.
• By increasing the sector angle, the post-buckling strength of the plate decreases and higher mode shapes appears in the deep post-buckling regime.
• While the thickness of the plate decreases, the post-buckling strength of the plate is noticeably decreased.
• The post-buckling strength of the plate with immovable circumferential edges is much more than the other boundary conditions.

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