Thermal stability analysis of eccentrically stiffened Sigmoid-FGM plate with metal–ceramic–metal layers based on FSDT

This paper researches the thermal stability of eccentrically stiffened plates made of functionally graded materials (FGM) with metal – ceramic – metal layers subjected to thermal load. The equilibrium and compatibility equations for the plates are derived by using the first-order shear deformation theory of plates, taking into account both the geometrical nonlinearity in the von Karman sense and initial geometrical imperfections with Pasternak type elastic foundations. By applying Galerkin method and using stress function, effects of material and geometrical properties, elastic foundations, temperature-dependent material properties, and stiffeners on the thermal stability of the eccentrically stiffened S-FGM plates in thermal environment are analyzed and discussed. Subjects: Materials Science; Mechanical Engineering; Structural Mechanical Engineering


ABOUT THE AUTHOR
In Vietnam, Nguyen Dinh Duc is one of the wellknown scientists in mechanical science. He is a full professor of Vietnam National University, Hanoi.
Professor Nguyen Dinh Duc is the Head of Advanced Materials and Structures Laboratory of University of Engineering and Technology -Vietnam National University, Hanoi. He had graduated from Hanoi State University since 1984 and completed his Ph.D degree in 1991 at Moscow State University, Russia. Since 1997, he had become Doctor of Science (Dr. Habilitation) promoted by Russian Academy of Sciences. Webpage: http://uet.vnu.edu.vn/~ducnd/, http:// irgamme.uet.vnu.edu.vn/gs-tskh-nguyen-dinhduc/

PUBLIC INTEREST STATEMENT
In recent years, there has been significant interest in the development of functionally graded materials (FGMs) for engineering applications. FGM materials have been used in aerospace, nuclear, and microelectronics engineering applications, where the materials are required to work in extreme temperature environments. It is also important for these materials to maintain their structural integrity, with minimum failures due to material mismatch. The focus of this manuscript is on a theoretical analysis on the thermal stability of eccentrically stiffened plates made of FGMs with metal -ceramic -metal layers (S-FGM) subjected to thermal load. Both the FGM plate and the outside stiffeners are deformed under temperature and having temperature-dependent properties. The influences of the material and geometrical properties, elastic foundations, temperaturedependent material properties, and outside reinforced stiffeners on the thermal stability of the eccentrically stiffened S-FGM plates in thermal environment are analyzed and discussed. The outcomes from this work are important to composite engineers and designers.

Introduction
Since its first introduction in 1984 by a group of material scientists in Japan (Koizumi, 1997), functionally graded materials (FGMs) have attracted considerable attention in many engineering applications such as extremely high-temperature-resistant materials. To date, there have been a number of studies on the stability of eccentrically stiffened FGM plates. However, these studies have only been concentrated on thin structures using classical plate theory. Not much consideration has been given to eccentrically stiffened thick Sigmoid-FGM (S-FGM) plates with shear deformation behaviors, especially when material properties depend on temperature. An overview on studies that apply shear deformation theory to FGM plates is provided in the following part. Wu (2004) has studied the thermal buckling and post-buckling behavior of simply supported FGM rectangular plates based on the first-order shear deformation plate theory. Ma and Wang (2004) studied the thermoelastic buckling behavior of functionally graded circular/annular plates based on first-order shear deformation plate theory. Duc and Tung studied the mechanical and thermal postbuckling of shear deformable FGM plates with temperature-dependent properties (Duc & Tung, 2010). Shen (1997) studied the thermal post-buckling analysis of imperfect laminated plates using a higher order shear deformation theory. Duc and Tung (2011) studied mechanical and thermal post-buckling of higher order shear deformable functionally graded plates on elastic foundations. Seren Akavci et al. used the first-order shear deformation theory for symmetrically laminated composite plates on elastic foundation (Seren Akavci, Yerli, & Dogan, 2007). Ghiasian et al. studied the thermal buckling of shear deformable temperature-dependent circular/annular FGM plates (Ghiasian, Kiani, Sadighi, & Eslami, 2014). Duc and Cong (2015) studied the nonlinear vibration of thick FGM plates on elastic foundation subjected to thermal and mechanical loads using the first-order shear deformation plate theory. In these studies (Seren Akavci et al., 2007;Duc & Cong, 2015;Duc & Tung, 2010Ghiasian et al., 2014;Ma & Wang, 2004;Shen, 1997), the authors used shear deformation theory to study the nonlinear static stability of unstiffened FGM thick plates. Dung and Nga (2013) studied the nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first-order shear deformation theory but without temperature. Shen (2007) studied the thermal post-buckling behavior of shear deformable FGM plates with temperature-dependent properties. In Bich, Nam, and Phuong (2011), studied the nonlinear post-buckling of eccentrically stiffened functionally graded plates and shallow shells based on classical shell theory. The nonlinear stability of eccentrically stiffened functionally graded imperfect plates resting on elastic foundations has been further studied by Dung and Thiem (2012). In (Duc, 2014;Duc & Cong, 2014), 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. Swaminathan, Naveenkumar, Zenkour, and Carrera (2015) studied the stress, vibration, and buckling analyses of FGM plates-A state-of-the art review. Thai and Kim (2015) studied a review of theories for the modeling and analysis of functionally graded plates and shells. Reddy and Chin (1998) studied the thermo-mechanical analysis of functionally graded cylinders and plates.
From the above review, to the best of our knowledge, it has been shown that there are no publications on the nonlinear stability of a thick S-FGM plate (with metal -ceramic -metal layers) reinforced by stiffeners in a thermal environment using the first-order shear deformation plate theory. This paper will focus on studying the buckling and post-buckling of an eccentrically stiffened functionally graded thick plate on elastic foundations under thermal loads with both S-FGM plates and stiffeners having temperature-dependent properties and thermal deformations. The paper also analyzes 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 plate in thermal environments.

Functionally graded plates on elastic foundations
Consider a eccentrically stiffened thick S-FGM plate (metal -ceramic -metal) 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.
By applying a Sigmoid power-law distribution, Young's modulus and thermal expansion coefficient can be expressed in the form (Duc & Tung, 2010): where N is volume-fraction index, the subscripts m and c refer to the metal and ceramic constituents, respectively, Poisson's ratio (v) is assumed to be a constant and 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 (Touloukian, 1967), as: where T = T 0 + ΔT( z ) and T 0 = 300 K (room temperature); P −1 , P 0 , P 1 , P 2 , P 3 are coefficients characterizing the constituent materials; and ΔT is the temperature rise from stress-free initial state. In short, T-D (temperature-dependent) will be used for the cases in which the material properties depend on temperature. Otherwise, T-ID will be used for the temperature-independent cases. The material properties for the latter scenario have been determined by Equation (2) at room temperature, i.e. T 0 = 300 K.

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 S-FGM plates.
The strains across the plate thickness at a distance z from the middle surface are given by: where where 0 x , 0 y are normal strains, 0 xy is shear strain on the mid-plane of the plate, 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.
(1) Hooke's law for an S-FGM plate under thermal conditions is defined as: For stiffeners in thermal environments with temperature-dependent properties, its form proposed adapted from Duc and Cong (2014), is as follows: where E 0 , v 0 , α 0 are Young's modulus, Poisson's ratio, and thermal expansion coefficient of stiffeners, respectively.
From Equation (4), the geometrical compatibility equation can be written as: In order to investigate the FGM plates with stiffeners in the thermal environment, the materials' moduli with temperature-dependent properties have been taken into account. In addition, it can be assumed that all elastic moduli of FGM plates and stiffener are temperature-dependent and they are deformed in the presence of temperature. Hence, the geometric parameters, the plate's shape and stiffeners are varied through the deforming process due to the temperature change. Assuming that the thermal stress of stiffeners is subtle which distributes uniformly through the whole plate structure, it can be ignored. Lekhnitsky smeared stiffeners technique can be adapted from Bich et al. (2011) for eccentrically stiffened FGM plate under temperatures as follows: where A 1 , A 2 are cross-section areas of stiffeners; d 1 , d 2 are spacing of the longitudinal and transversal stiffeners; I 1 , I 2 are second moments of cross-section areas; z 1 , z 2 are eccentricities of stiffeners with respect to the middle surface of plate; b 1 , b 2 are width of longitudinal and transversal stiffeners; h 1 , h 2 are thickness of longitudinal and transversal stiffeners; and specific expressions of coefficients A ij , B ij , D ij are given in Appendix A and After the thermal deformation process, the geometric shapes of stiffeners can be determined as follows (Duc & Cong, 2014).
For later use, the reverse relations are obtained from Equation (8), as follows Substituting Equation (11) into Equation (8), yields: The nonlinear equilibrium equations of an eccentrically stiffened S-FGM plate on elastic foundations, based on the first-order shear deformation plate theory (Reddy, 2004), are: Considering Equation (13a), a stress function f(x, y) may be defined as: The three equations (13b) and (13c) become: Substituting the expressions of M x , M y , M xy in Equation (12), and Q x, Q y in Equation (9) into Equation (15), we obtain: Q x,x + Q y,y + N x w ,xx + 2N xy w ,xy + N y w ,yy − k 1 w + k 2 w ,xx + w ,yy = 0, N x = f ,yy , N y = f ,xx , N xy = −f ,xy .
The system of Equations (16) include four unknown functions (w, ϕ x , ϕ y, and f); so it is necessary to find the fourth equation relating to these functions using the compatibility equation (Equation 7). For this purpose, substituting the expressions of 0 x , 0 y , 0 xy from Equation (11) into Equation (7), we get: Equations (16) and (17) 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 plate on elastic foundations subjected to compression, thermal and combined loads. The boundary condition (BC), will be considered: The edges are simply supported and immovable (IM). The associated BCs are here, N x0 , N y0 are the pre-buckling force resultants in directions x and y, respectively.
To solve Equations (16) and (17) for unknowns w, ϕ x , ϕ y, and f, and with consideration of the BC (18), the following approximate solutions (Dung & Nga, 2013) are assumed: where α = mπ/a, β = nπ/b, 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 BC (18), the imperfections of the plate are assumed as: where the coefficient μ, varying between 0 and 1, represents the size of the imperfections. (19) and (20) into Equations (16b), (16c), and (17), the coefficients 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 A. (19) and (20) into Equation (16a), and applying the Galerkin method for the resulting equation yields:
w = W sin x sin y, x = 1 cos x sin y + 2 sin 2 x, y = 3 sin x cos y + 4 sin 2 y, f = F 1 cos 2 x + F 2 cos 2 y + F 3 sin x sin y + N x0 y 2 ∕2 + N y0 x 2 ∕2, (20) w * = h sin x sin y, (22), derived for odd values of m, n, is used to determine the nonlinear buckling and postbuckling response of eccentrically stiffened thick FGM plates in thermal environments. An interesting characteristic of Equation (22)  A simply supported FGM plate with immovable edges under thermal loads is considered. The condition expressing the immovability on the edges, u = 0 (on x = 0, a) and v = 0 (on y = 0, b), is generally fulfilled as (Shen, 2007): From Equations (4) and (11), one can obtain the following expressions in which Equation (14) and imperfection have been included: Substitute Equations (19) and (20)    In the case of T-D, both sides of Equation (27) are temperature-dependent, which makes it very difficult to solve. Fortunately, a numerical technique using an iterative algorithm has been applied to determine the buckling loads, as well as to determine the deflection-load relationship in the postbuckling period of the eccentrically stiffened FGM plate. Given more details, including the material parameter N, the geometrical parameter (b/a, b/h), and the value of W/h, we can determine ΔT in Equation (27), as follows. First, the initial step for ΔT 1 on the right-hand side in Equation (27) is chosen with ΔT = 0 (since T = T 0 = 300 K, the initial room temperature). In the next iterative step, the known value of ΔT 1 found in the previous step is replaced to determine the right-hand side of Equation (27), ΔT 2 . This iterative procedure will stop at the kth step if ΔT k satisfies the condition Here, ΔT is a desired solution for the temperature and ξ is a tolerance used in the iterative steps. This is also an interesting point to be solved in this article.

Numerical results and discussion
In this section, the components of the material are silicon nitride Si 3 N 4 (ceramic) and SUS304 stainless steel (metal). The material properties (Pr) in Equation (2) are shown in Table 1 and Poisson's ratio is chosen to be v = 0.3.
In particular, for the case of an S-FGM plate without stiffeners with the conditions: A T 1 = A T 2 = 0, I T 1 = I T 2 = 0 and ceramic -metal -ceramic layers which are compared the numerical results of unstiffened thick S-FGM plates with Duc and Tung (2010). As can be seen, a good agreement is obtained in this comparison (Figure 2).    (1) and (2), which show that the ability of sustaining compression and thermal load will increase if the effects of elastic foundations enhance from (K 1 = 0, K 2 = 0) to (K 1 = 100, K 2 = 0). Furthermore, Pasternak's elastic foundation (K 2 ) is more powerful than Winkler's foundation (K 1 ), which is proven by curve (3) with K 1 = 100, K 2 = 10, and curve (4) with K 1 = 50, K 2 = 20.

Property
Material affected the load bearing ability of the plate. In other words, the loading ability increases together with μ. Figure 7 shows the effects of temperature-dependent material properties on the nonlinear stability of eccentrically stiffened thick S-FGM plates under thermal load. There is a comparison between

Conclusions
This paper presents an analysis approach, in conjunction with an iterative procedure, to investigating the buckling and post-buckling behavior of the eccentrically stiffened S-FGM plates under thermal load. The formulation is based on the first-order shear deformation theory, accounting for both the von Karman nonlinearity and initial imperfections. The paper also analyzes 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 eccentrically stiffened S-FGM plate in thermal environments.