Linear thermal buckling analysis of truncated hybrid FGM conical shells
Introduction
Shells are used in various engineering applications such as mechanical and civil engineering structures. Therefore, many investigations are carried out on thermal and mechanical buckling of shells. In recent years, by the development of new types of materials such as the functionally graded materials (FGMs), more researches are carried out on the behavior of shells made of these materials. Shahsiah and Eslami [1], [2] studied the thermal buckling of functionally graded cylindrical shells based on the first order shell theory and the Sanders kinematics relations. Two types of thermal loadings and simply-supported type of boundary conditions are considered. Thermal buckling and postbuckling of FG cylindrical shells with temperature-dependant material properties is presented by Shen [3]. The governing equations are developed based on the classical shell theory taking into account the geometrical nonlinearity using the von-Karman–Donnell type kinematics relations. Applying the singular perturbation technique, buckling temperature and postbuckling equilibrium path are obtained. Wu et al. [4] investigated the thermoelastic stability of FG cylindrical shell based on the Donnells shell theory and presented a closed-form solution for simply supported shell to show the influence of shell geometry and power law index on thermal buckling behavior of cylindrical shell. To show the importance of initial imperfection, Mirzavand and Eslami [5] presented an analysis on thermal buckling of imperfect FG cylindrical shells based on the Wan-Donnell imperfection model. Using the Galerkin method a closed-form solution is provided and the effect of shell geometry and power index of FGM on the instability of the imperfect cylindrical shell is discussed. Kadoli and Ganesan [6] investigated the buckling and free vibration behavior of temperature-dependant clamped–clamped FG cylindrical shells subjected to thermal loads. First order shear deformation theory is utilized to model the FG shell. Fourier series expansion of the displacement variables in circumferential direction is considered. In a recent study, Bagherizadeh et al. [7] presented the thermal buckling analysis of FG cylindrical shell on a Pasternak-type elastic foundation. In this study, the stability equations of the shell are decoupled to establish an equation in terms of only the out-of-plane displacement component.
Free vibration and buckling behavior of FG truncated conical shells subjected to thermal loads is investigated by Bhangale et al. [8] based on the first order shear deformation theory. The effects of initial stresses on the frequencies of the FG shells are studied. Thermoelastic stability analysis of FG truncated conical shells is presented by Sofiyev [9]. Modified Donnell type stability and compatibility equations are derived and the Galerkin method is applied to obtain the closed-form solution. Instability of FG truncated conical shells subjected to thermal and mechanical loads is analyzed based on the classical shell theory and the Sanders nonlinear kinematics relations by Naj et al. [10]. In a series of works Sofiyev studied the buckling of FG truncated conical shells under various mechanical loads, including hydrostatic pressure, external pressure, and combined axial tension and hydrostatic pressure (Sofiyev et al. [11] and Sofiyev [12], [13]). Applying the Galerkin method to stability and compatibility equations, the critical buckling load of the shell is obtained. Recently, Sofiyev [14], [15] presented the thermal and mechanical buckling analysis of FG circular shells resting on a two-parameters elastic foundation by solving the eigen-value problem. The critical buckling loads with and without elastic foundation are obtained using the Galerkin method. Bich et al. [16] studied the instability of FG conical panels based on the classical thin shell theory under axial compression, external pressure, and combination of them. The effects of the initial imperfections on the buckling behavior of FG truncated conical shells are illustrated by Sofiyev [17]. Superposition and Galerkin methods are applied to the modified nonlinear Donnell type stability and compatibility equations and the upper and lower critical axial loads are obtained.
The development of a new class of smart materials and adaptive structures with sensing and actuating capabilities has further improved the performance and reliability of mechanical systems. Among them, piezoelectric materials with a vast range of engineering applications have attracted many researchers in the recent years. Mirzavand and Eslami [18] analyzed the thermal buckling of FG cylindrical shell with piezoelectric layers subjected to combined thermal load and constant applied voltage. Applying the Galerkin method to the governing equations which are based on higher order shear deformation shell theory, results in a closed form solution for the critical buckling temperature difference. In another study, Mirzavand and Eslami [19] investigated the effects of temperature-dependency of the material properties on the stability behavior. An analysis of dynamic thermal buckling of FG cylindrical shells integrated with piezoelectric layers is presented Mirzavand and Eslami [20]. A finite difference based method along with the Runge–Kutta method are employed to evaluate the postbuckling equilibrium path and dynamic buckling temperature difference is concluded based on the Budiansky’s stability criterion. Postbuckling analysis of FG cylindrical shells and panels with piezoelectric layers subjected to combination of mechanical, thermal, and electrical loads is presented in two studies performed by Shen and Liew [21] and Shen [22]. In these studies nonlinear prebuckling deformations and initial imperfections are considered. Singh and Babu [23], [24] investigated the thermal buckling behavior of laminated conical shell and panel incorporated with or without piezoelectric layer subjected to uniform temperature rise. The fundamental relations are developed based on a higher-order shear deformation theory using finite element methods to obtain the critical buckling temperature. It was shown that presence of the piezoelectric layer improves the buckling coefficient depending on the fraction of the piezoelectric material in the laminate.
In this paper, buckling behavior of FG conical shell integrated with piezoelectric layers under combined thermal and electrical loads is investigated. The surface-bonded piezoelectric layers are considered as actuators with constant applied voltage. In addition, as the piezoelectric layer is assumed to be thin, only the transverse component of the electric field is considered. Material properties of FG shell are assumed to vary continuously through the thickness direction following a power law form. Using the variational approach, equilibrium and stability equations are obtained based on the classical shell theory and the Sanders nonlinear kinematics relations. The shell is simply supported at both ends and it is assumed to be subjected to a uniform temperature distribution through the thickness direction. The prebuckling forces are obtained considering membrane solutions of linear equilibrium equations. Applying the Galerkin method to the stability equations results in an eigen-value problem which provides the critical buckling temperature difference. Finally, numerical results are presented to illustrate the effects of various parameters such as applied actuator voltage, shell geometry, and the power law index of FGM on thermal buckling behavior of the shell.
Section snippets
Governing equations
A truncated conical shell of height H and semi-vertex angle β is considered. It is assumed that shell consists of a host FG layer of thickness h and two piezoelectric layers of thickness ha as actuators that are perfectly bonded to the host layer. The curvilinear coordinate system is defined as (x, θ, z), where x and θ coincides with generator and circumferential directions, respectively, and z is perpendicular to the x − θ plane and its direction is inwards normal of the conical shell, as shown in
Prebuckling analysis
The equilibrium equations and the possible boundary condition are obtained in Eqs. (13), (15). Although the critical buckling temperature may be found based on the stability equations [25], but the equilibrium equations are analyzed to define the prebuckling behavior of the shell and prebuckling force resultants. Since the deflections in prebuckling state are small, the linear form of equilibrium equations may be considered [25]. In addition, the bending of shell walls affects the narrow
Stability equation
The critical buckling temperature difference is obtained based on the stability equations. Therefore, stability equations are derived on the basis of minimum potential energy criterion. While the first variation of the total potential energy provides the equilibrium equations, the second variation results in the stability equations. To obtain the second variation, the displacement components of a neighboring state are defined as:where u0, v0, and w0 correspond to the
Solution procedure
Eqs. (13), (21) present the equilibrium and stability equations of a truncated FG conical shell with piezoelectric layers. According to the stability Eq. (21), the force and moment resultants in stability state are obtained as:in which the linear forms of strains and curvatures are given as:
Results and discussion
In this section, some numerical results are provided and the results are compared with those reported in the literature to prove the efficiency of the present study. The piezoelectric FG conical shell is assumed to be composed of a mixture of Zirconia and Aluminum for the FGM substrate and PZT−5 A for the piezoelectric layer. The Young modulus and thermal expansion coefficient are Ec = 151 GPa and αc = 10−5/°C for Zirconia and Em = 70 GPa and αm = 2.3 × 10−5/°C for Aluminum [18]. The PZT − 5 A properties are Ea
Conclusion
Thermal buckling of piezoelectric FG conical shell under uniform temperature distribution through the thickness and constant applied voltage is analyzed. The material properties vary continuously through the thickness direction following the power law rule. The governing equations are obtained based on the classical shell theory and the Sanders nonlinear kinematics relations. The Galerkin method is utilized to obtain the closed-form solution and extracting the critical buckling temperature
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2020, Composite StructuresCitation Excerpt :In this approach, the transverse stress equilibrium at the layer intersections was enforced via Lagrange multipliers, which inevitably increased the problem size. On the application side, FGM shell modeling has also been investigated in the context of free vibration [29–41] and dynamic response [42–44], instability [45–57], thermal-mechanical behavior [58–62], reinforcement with carbon nanotubes [63–69] and stiffeners [70–76], material elastoplasticity [77–79], variable thickness [80], active/passive control [81–83], nanoscale modeling [84], and extended FEM [85,86]. Despite the wide variety of work reported in literature, the FGM shell formulations were all developed under traditional finite element approaches.