Nonlinear stability of shear deformable eccentrically stiffened functionally graded plates on elastic foundations with temperature-dependent properties

1 Introduction

Since its introduction in 1984 by a group of material scientists in Japan, functionally graded materials (FGMs) have attracted considerable attention in many engineering applications, such as extremely high temperature resistant materials. FGMs are considered for general use as add-on layers for mechanical and structural components for aircraft, spaceships, nuclear plants, and other engineering applications where ultra-high temperatures are expected. FGMs are, essentially, composite materials with the mechanical properties varying smoothly and continuously across the thickness. FGMs could be either fabricated by gradually varying the volume fraction of the constituent materials or combining different materials of varying composition factions. With the advantage of not having material mismatch or physical interface like conventional composite materials, FGMs are less prone to failures while subjected to extreme thermo-mechanical loadings. The inclusion of reinforcing ridges on an FGM plate provides further advantages for thermal diffusion (increasing surface area) and structural integrity, as can be seen in the heat sink application (Figure 1). With the increased usage of these FGM structures, it is important to understand the nonlinear post-buckling behaviors of eccentrically stiffened FGM panels with temperature-dependent properties.

Figure 1:

(A) Geometry and coordinate system of an eccentrically stiffened moderately thick FGM plate on an elastic foundation, (B) different heat sinks made of stiffened ceramic FGM materials.

To date, there have been a number of studies on the stability of eccentrically stiffened FGM plates and shells. However, these studies have only concentrated on thin structures using classical plate and shell theories. Not much consideration has been given to eccentrically stiffened moderately thick FGM plates and shells with shear deformation behaviors, especially when material properties depend on temperature. An overview on studies that apply shear deformation theory to FGM plates and shells is provided as follows.

Shen [1] has investigated the thermal post-buckling analysis of imperfect laminated plates based on a higher-order shear deformation theory. In another research, Ma and Wang [2] have discussed thermo-elastic buckling behavior of functionally graded circular/annular plates based on first-order shear deformation plate theory. Lanhe [3] has studied the thermal buckling and post-buckling behavior of simply supported FGM rectangular plates based on the first-order shear deformation plate theory. Jahanghiry et al. [4] have applied the stability analysis of a FGM microgripper subjected to nonlinear electrostatic and temperature variation loadings. Asemi et al. [5] have investigated the three-dimensional natural frequency analysis of anisotropic functionally graded annular sector plates resting on elastic foundations. Akavci et al. [6] have applied the first-order shear deformation theory for symmetrically laminated composite plates on elastic foundations. Duc and Tung [7] have investigated the stability of simply supported rectangular functionally graded plates under in-plane compressive, thermal, and combined loads. The equilibrium and compatibility equations for FGM plates, which use simple power-law distribution (P-FGM), are derived by using the first-order shear deformation theory [7] and the higher-order shear deformation of the plates [8]. In recent work, Duc and Cong [9] investigated the nonlinear post-buckling of symmetric plates made of Sigmoid-type functionally graded material (S-FGM) resting on elastic foundations based on higher-order shear deformation plate theory in thermal environments. In these studies [7], [8], [9], Duc et al. used shear deformation theory to study the nonlinear static stability of unstiffened FGM moderately thick plates. Malekzadeh et al. [10] studied the buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates on elastic foundations; the formulation was based on the first-order shear deformation theory.

Designed and fabricated with high temperature resistance capability, FGM plates can be used as structural components operating in ultra-high temperature environments subjected to extremely high thermal gradients. In such environments, material properties of constituents forming FGM are often affected negatively. It is, therefore, important to take into account the temperature-dependent effects of material properties in order to ensure that analyses are accurate and reliable. The discussion following provides a review of several studies on the stability of FGM plates and shells with temperature-dependent material properties. Reddy and Chin [11] have investigated the thermo-mechanical analysis of functionally graded cylinders and plates. Yang et al. [12], conversely, have analyzed the thermo-mechanical post-buckling of cylindrical panels that are made of FGMs with temperature-dependent properties. Shen [13] has investigated the thermal post-buckling behaviour of shear deformable FGM plates with temperature-dependent properties (without stiffeners). Sepahi et al. [14] discussed the thermal buckling and post-buckling analysis of functionally graded annular plates with temperature-dependent material properties. Nonlinear thermal buckling and post-buckling analyses of imperfect thickness variation of a temperature-dependent bidirectional functionally graded cylindrical shell has been investigated by Shariyat and Asgari [15]. In Ghiasian et al. [16] investigated the thermal buckling of shear deformable temperature-dependent circular/annular FGM plates. Malekzadeh [17] studied the three-dimensional thermal buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates using differential quadrature method.

Among the abovementioned researches, the FGMs do not have stiffeners. Recently, eccentrically stiffened FGM plates and shells studied and used popularly in the industry have drawn researchers’ attentions. Bich et al. [18] studied the nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells based on classical shell theory. In other work [19], Bich et al. discussed a semi-analytical approach to investigate the nonlinear dynamic behavior of imperfect eccentrically stiffened FGM shallow shells, which take damping into account when subjected to mechanical loads. In other work, Bich et al. [20] derived the governing equations of motion for eccentrically stiffened FGM cylindrical panels with geometric imperfections. The characteristics of free vibration and nonlinear responses are investigated. The nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elastic foundations has been further studied by Dung and Thiem [21].

Recently, Duc and Quan [22] investigated the nonlinear post-buckling for imperfect eccentrically stiffened thin FGM double curved thin shallow shells on elastic foundations using a simple power-law distribution in thermal environments. In [23], Duc and Cong studied the nonlinear post-buckling of imperfect eccentrically stiffened thin FGM plates with temperature-dependent material properties under temperature while resting on elastic foundations using a simple power-law distribution (P-FGM) and the classical plate theory.

From the aforementioned review, to the best of our knowledge, it has been shown that there are no publications on the nonlinear stability of a moderately thick FGM plate with stiffeners in a thermal environment using the first-order shear deformation plate theory. Moreover, in elevated temperature conditions, both the FGM plates and the stiffeners are deformed, which results in further difficulty in calculating the thermo-mechanical behaviors of FGM plates and stiffeners.

This article focuses on studying the buckling and post-buckling of an eccentrically stiffened functionally graded moderately thick plate on elastic foundations under thermal loads, with both FGM plates and stiffeners having temperature-dependent properties. The governing equations are derived within the framework of the first-order shear deformation plate theory, accounting for both the von Karman nonlinearity and initial imperfections. The article also analyses and discusses the effects of material and geometrical properties, temperature, elastic foundations, and eccentric stiffeners on the buckling and post-buckling loading capacity of the functionally graded moderately thick plate in thermal environments.

2 Functionally graded plates on elastic foundations

Consider a ceramic–metal eccentrically stiffened moderately thick FGM plate of length a, width b, and thickness h resting on an elastic foundation. A coordinate system (x, y, z) is established, in which (x, y) plane is on the middle surface of the plate and z is the thickness direction (-h/2≤z≤h/2), as shown in Figure 1.

Functionally graded material in this paper is assumed to be made from a mixture of ceramic and metal with the volume-fractions given by a power law, as

where the subscripts m and c refer to the metal and ceramic constituents, respectively. According to the mentioned law, the Young’s modulus and thermal expansion coefficient can be expressed in the form

where Poisson’s ratio (v) is assumed to be a constant and E_cm =E_c -E_m , α_cm =α_c -α_m .

A material property (Pr), such as the elastic modulus (E), and the thermal expansion coefficient (α) can be expressed as a nonlinear function of temperature [7], [23], as

in which T=T₀+ΔT(z) and T₀=300 K (room temperature); P_-1, P₀, P₁, P₂, P₃ are coefficients characterizing the constituent materials, and ΔT is the temperature rise from stress free initial state. In short, we will use T-D (temperature-dependent) for the cases in which the material properties depend on temperature. Otherwise, we use T-ID for the temperature-independent cases. The material properties for the latter scenario have been determined by Equation (3) at room temperature, i.e. T₀=300 K.

The reaction-deflection relationship of the Pasternak foundations is given by

in which w is the deflection of the plate, k₁ is the Winkler foundation modulus and k₂ is the shear layer foundation stiffness of the Pasternak model.

3 Theoretical formulation

The present study uses first-order shear deformation plate theory to establish the governing equations and determine the buckling loads and post-buckling paths of the eccentrically stiffened moderately thick FGM plates.

The strains across the plate thickness at a distance z from the middle surface are given by

where

in which u, v are the displacement components along the x, y directions; and ϕ_x , ϕ_y are the rotations in the (x, z) and (y, z) planes, respectively.

Hooke’s law for an FGM plate under thermal conditions is defined as

For stiffeners in thermal environments with temperature-dependent properties, we have proposed its form be adopted from [22], [23] as follows:

From Eqs. (6), the geometrical compatibility equation can be written as

In order to investigate the moderately thick FGM plates with stiffeners in the thermal environment, we have assumed that all elastic moduli of the FGM plate and stiffeners are temperature-dependent, and both the FGM plate and stiffeners are deformed in the presence of thermal loading. Hence, the geometric parameters, the plate’s shape and stiffeners, are varied through the deformation process due to temperature change. We have assumed that the thermal stress in the stiffeners is subtle and distributes uniformly throughout the whole panel; therefore, it can be ignored. The Lekhnitsky smeared stiffeners technique can be adopted for eccentrically stiffened moderately thick FGM plates under temperatures as follows [22], [23]:

with

and

and

After the thermal deformation process, the geometric shapes of the stiffeners can be determined as follows [22], [23]:

For later use, the reverse relations are obtained from Equation (10) as follows:

Substituting Equation (15) into Equation (10) yields

where the coefficients Aij∗, Bij∗, Cij∗ and Dij∗ are given in Appendix A.

The nonlinear equilibrium equations of an eccentrically stiffened moderately thick FGM plate on elastic foundations, based on the first-order shear deformation plate theory, are [24]

Considering Equations (17a) and (17b), a stress function f(x, y) may be defined as

Substituting the expressions of M_x , M_y , M_xy in Equation (16), and Q_x , Q_y in Equation (10) into Equation (17c–17e), we obtain

in which w^*(x, y) is a known function representing the initial small imperfections of the plate [25].

The system of Equation (19) include four unknown functions (w, ϕ_x , ϕ_y , and f), so it is necessary to find the fourth equation relating to these functions by using the compatibility equation, Equation (9). For this purpose, substituting the expressions of εx0, εy0, γxy0 from Equations (15) into Equation (9), we get

Equations (19) and (20) are nonlinear equations in terms of the four dependent unknown functions (w, ϕ_x , ϕ_y , and f) used to investigate the buckling and post-buckling of an eccentrically stiffened functionally graded moderately thick plate on elastic foundations subjected to compression, thermal, and combined loads.

Three cases of boundary conditions (BCs), labeled Cases 1, 2, and 3, will be considered:

Case 1. Plate edges are simply supported and freely movable (FM). The associated boundary conditions are

Case 2. The edges are simply supported and immovable (IM). The associated boundary conditions are

Case 3. The edges are simply supported, and uniaxial edge loads operate in the x direction. The edges x=0, a and x=0, b are considered freely movable, but the other two are load-free and immovable. For this case, the boundary conditions are

Here, N_x0, N_y0 are the pre-buckling force resultants in directions x and y, respectively, for Case 1 and the first boundary condition (BC) of Case 3; N_x0, N_y0 represent fictitious compressive edge loads, which set immovable edges for Case 2 and the second BC of Case 3.

To solve Equations (19) and (20) for unknowns w, ϕ_x , ϕ_y and f, and with consideration of the boundary conditions of Equations (21)–(23), we assume the following approximate solutions:

where α=mπa, β=nπb,m, n=1, 2, … are the number of half waves in the x, y directions, respectively; and W is the amplitude of deflection. Also, λ_i (i=1–4) and F_i (i=1–3) are coefficients to be determined.

Considering the boundary conditions of Equations (21)–(23), the imperfections of the plate are assumed as

where the coefficient μ, varying between 0 and 1, represents the size of the imperfections [25].

After substituting Equations (24) and (25) into Equations (19b), (19c) and (20), the coefficients λ_i (i=1–4) and F_i (i=1–3) are found as

and specific expressions of coefficients f_i (i=1–3) and L_j (j=1–4) are given in Appendix B.

Introduction of Equations (24) and (25) into Equation (19a), and applying the Galerkin method for the resulting equation yields

Equation (27), derived for odd values of m, n, is used to determine the nonlinear buckling and post-buckling response of eccentrically stiffened moderately thick FGM plates in thermal environments. An interesting characteristic of Equation (27) is the temperature-dependent components, which appear in the B11∗, B21∗, B22∗, B66∗, D11∗, D12∗, D21∗, … coefficients, as shown in Appendix A.

4 Numerical results and discussion

In this section, the components of the material are silicon nitride Si₃N₄ (ceramic) and SUS304 stainless steel (metal). The material properties (P_r) in (2) are shown in Table 1, and Poisson’s ratio is chosen to be v=0.3.

4.1 Comparison of results

In verification of the present formulation for buckling and post-buckling behaviors of the plates, thermal post-buckling of a simply supported square moderately thick FGM plate is analyzed. The plate is exposed to a uniform temperature field with all immovable edges and without foundation interaction for unstiffened moderately thick FGM plates. Figure 2 gives thermal post-buckling load-deflection curves for isotropic plates (v=0.3) according to the present approach, in comparison with Shen’s results [13] using an asymptotic perturbation technique. As can be seen, a good agreement is obtained in this comparison.

Figure 2:

Comparisons of thermal post-buckling load-deflection curves for isotropic plates.

In particular, for the case of an FGM plate without stiffeners with the conditions A1T=A2T=0,I1T=I2T=0 and where the combination of plate materials consists of aluminum E_m =70 GPa and alumina E_c =380 GPa and Poisson’s ratio v=0.3, we have compared the numerical results of unstiffened moderately thick FGM plates with Duc and Tung’s results reported in [8]. These have been presented in Figure 3 and show that there is good agreement between our findings for the post-buckling of moderately thick FGM plates and the findings of other researches.

Figure 3:

Comparison of post-buckling curves for unstiffened moderately thick FGM plates under compression.

Table 2 shows the effects of the volume fraction index N and the aspect ratio a/b on the difference of buckling temperature of moderately thick Sigmoid-FGM plates in comparison with Shen’s results [13] and Duc and Tung’s results [7] with T-D material properties. These comparisons show that the results obtained in this paper are in good agreement with the existing results, thus verifying the reliability and accuracy of the present method.

Table 2

Comparisons of buckling temperatures ΔT_b (K) for moderately thick Sigmoid-FGM plates under a uniform temperature rise (T-D properties, b/h=20 and v=0.29).

a/b		N=0	N=1	N=2	N=5
T-ID
1.0	Duc and Tung [7]	205.6	313.7	351.1	378.0
	Shen [13]	206.8	315.1	352.6	388.8
	Present	203.9	313.5	350.8	386.7
2.0	Duc and Tung [7]	129.1	197.0	220.5	243.0
	Shen [13]	129.9	198.1	221.7	244.4
	Present	128.9	196.8	220.3	242.8
T-D
1.0	Duc and Tung [7]	182.4	265.3	292.0	316.1
	Shen [13]	184.6	268.2	295.1	319.5
	Present	182.4	266.3	293.3	317.6
2.0	Duc and Tung [7]	118.9	174.9	193.3	210.4
	Shen [13]	120.4	176.9	195.6	212.9
	Present	119.7	176.1	194.7	211.9

4.2 Eccentrically stiffened moderately thick FGM plates on elastic foundations

In order to provide continuity between the plate and the stiffeners, we assume that the stiffeners are made of full metal (E₀=E_m ) when putting them on the metal-rich side of the plate; and conversely, we assume full ceramic stiffeners (E₀=E_c ) on the ceramic-rich side of the plate. The parameters for the stiffeners are

h1=0.08m, h2=0.08m, b1=0.008m, b2=0.008m,d1=0.15m, d2=0.15m, E0=Em.

Figures 4 and 5 illustrate the effects of eccentric stiffeners on the nonlinear response of moderately thick FGM plates under compression (Figure 4) and thermal loads (Figure 5). Table 3 shows the different values of buckling compression and thermal load of eccentrically stiffened moderately thick FGM plates and unstiffened FGM plates. In the case of unstiffened FGM plates, F_xb =2296.4 MPa, ΔT_b =235.7 K, and in the case of moderately thick FGM plates with stiffeners, F_xb =2935.2 MPa, ΔT_b =348.7 K with T-D material properties. It is clear that the stiffeners can enhance the mechanical and thermal loading capacity for the imperfect and perfect moderately thick FGM plates. A similar conclusion has been reported for nonlinear static and dynamic analysis of eccentrically stiffened thin FGM plates and shells [22], [23].

Figure 4:

Effect of eccentric stiffeners on the post-buckling of moderately thick FGM plates under compression loads.

Figure 5:

Effect of eccentric stiffeners on the post-buckling of moderately thick FGM plates under thermal loads.

Table 3

Buckling compression and thermal loads of moderately thick FGM plates with T-ID and T-D material properties B_a =1, B_h =20, N=1, μ=0, K₁=0, K₂=0.

	Fxb (MPa)		ΔTb (K)
	Unstiffened	Stiffened	Unstiffened	Stiffened
T-ID	2296.4	2935.2	272.8	348.7
T-D	2296.4	2935.2	235.7	293.4

Figure 6 shows the effects of temperature-dependent material properties on the nonlinear stability of eccentrically stiffened moderately thick FGM plates under thermal load. There is a comparison between the thermal post-buckling curves of both perfect and imperfect FGM plates with T-D and T-ID material properties. It is apparent that T-D material properties make the FGM plate considerably weaker under thermal load. It is understandable that the temperature has a negative effect on material properties, like the decline of elastic modulus (E) and the increase in the coefficient of expansion, especially when W/h is large enough.

Figure 6:

Thermal post-buckling behavior of eccentrically stiffened moderately thick FGM plates with T-ID and T-D material properties.

Figures 7 and 8 present the effects of volume fraction index on the post-buckling of eccentrically stiffened moderately thick FGM plates under compression and thermal loads. These post-buckling curves show that the loading ability of FGM plates became worse with the increase of N.

Figure 7:

Effects of volume fraction index on the post-buckling of eccentrically stiffened moderately thick FGM plates under compression load.

Figure 8:

Effects of volume fraction index on the post-buckling of eccentrically stiffened moderately thick FGM plates under thermal load.

Figures 9 and 10 show the effects of the elastic foundations on the nonlinear response of eccentrically stiffened moderately thick FGM plates with temperature-dependent material properties. Elastic foundations are recognized to have strong impact, as demonstrated by curve 1 and 2, which show that the ability of sustaining compression and thermal load will increase if the effects of elastic foundations enhance from (K₁=0, K₂=0) to (K₁=100, K₂=0). Furthermore, Pasternak’s elastic foundation (K₂) is more powerful than Winkler’s foundation (K₁), which is proven by curve 3 with K₁=100, K₂=10 and curve 4 with K₁=50, K₂=20.

Figure 9:

Effects of elastic foundations on the post-buckling of eccentrically stiffened moderately thick FGM plates under compression load.

Figure 10:

Effects of elastic foundations on the post-buckling of eccentrically stiffened moderately thick FGM plates under thermal load.

Figures 11 and 12 show the effects of imperfections on post-buckling response of the eccentrically stiffened FGM plates under compressive and thermal loads. In the post-buckling period, the imperfections have actively affected the load bearing ability of the plate. In other words, the loading ability increases together with μ.

Figure 11:

Effect of imperfection on post-buckling of eccentrically stiffened moderately thick FGM plates under compression.

Figure 12:

Effect of imperfection on post-buckling of eccentrically stiffened moderately thick FGM plates under thermal load.

Figure 13 illustrates the effects of varying temperature on the response of the plates under compressive load. It seems that temperature makes the loading ability of both perfect and imperfect plates worse. In addition, when temperature changes (ΔT≠0), imperfect plates are more deflective despite no compression, which is performed by the intersection between dashed lines and the horizontal axis.

Figure 13:

Effects of the temperature field on the post-buckling curves for eccentrically stiffened moderately thick FGM plates under compression load.

There is the same response when plates are concurrently under increasing temperature and different values of mechanical load, which is shown in Figure 14. The compression on the borders makes the thermal loading ability of perfect plates reduce considerably. Overall, the ability of sustaining thermal and mechanical load is limited to FGM plates by temperature and the compression field.

Figure 14:

Effects of a compressive load (F_x ) on the thermal post-buckling behavior of eccentrically stiffened moderately thick FGM plates with T-ID and T-D material properties.

Figure 15 evaluates how the ratio of a/b affects the nonlinear stable response of immovable FGM plates under thermal load. In this figure, two values of the ratio a/b are considered, and the results are compared with each other between the perfect and imperfect plates when considering the dependence and independence of temperature on material properties. The result shows that T-D material properties and increasing the ratio of a/b are two factors that reduce the FGM plate’s ability to withstand thermal load and makes the thermal post-buckling curves lower.

Figure 15:

Effect ratio a/b on the thermal post-buckling curves of eccentrically stiffened moderately thick FGM plates.

Figure 16 shows the effects of boundary conditions (two edges FM x=0, a and two edges IM y=0, b) on the post-buckling of eccentrically stiffened moderately thick FGM plates under compressive loads. The curves in the case of FM have been plotted using Equation (30), whereas the curves in the case of IM have been plotted using Equation (36) with ΔT=0. This has illustrated that the boundary conditions have a significant effect on the buckling and post-buckling of moderately thick FGM plates. Although the perfect FGM plates have only been buckled in the case of large loads, the loading ability of the imperfect FGM plate in the post-buckling period, as well as under the boundary condition of the edge y=0, b, is better than those of the perfect one.

Figure 16:

Effects of boundary conditions (FM and IM) on the post-buckling of eccentrically stiffened moderately thick FGM plates under compressive loads.

5 Conclusions

This article uses first-order shear deformation plate theory to study the stability of an eccentrically stiffened functionally graded moderately thick plate with temperature-dependent properties on elastic foundations. We summarize the main findings as follows:

• We established the basic equations based on the first-order shear deformation plate theory for the static nonlinear behavior of eccentrically stiffened moderately thick FGM plates under compression, thermal and thermo-mechanical loads.

• In particular, this paper used loop algorithms to successfully determine the buckling and post-buckling load-deflection curves for material properties that depend on temperature.

• The stiffeners have an obvious influence on the loading ability of thick FGM plates. Specifically, the eccentrically stiffened moderately thick FGM plates are better at loading than unstiffened moderately thick FGM ones.

• The elastic foundation parameters, K₁ and K₂, strongly affect the buckling and post-buckling behavior of an eccentrically stiffened moderately thick FGM plate. The effect of the Pasternak foundation (K₂) on buckling and post-buckling of an eccentrically stiffened moderately thick FGM is larger than the Winkler foundation (K₁).

• The material properties depending on temperature have an obvious impact on the static nonlinear behavior of eccentrically stiffened moderately thick FGM plates under mechanical and thermal loads.

• The influences of the volume fraction index N, geometrical parameters (ratio a/b, imperfection) and boundary conditions on the loading ability of eccentrically stiffened moderately thick FGM plates were analyzed and discussed.

• The appropriate comparison between the obtained results in the specific cases with other publications has proven the confidence and accuracy of our calculation and methods.