Abstract

This paper focuses on the three-dimensional (3D) asymmetric problem of functionally graded (FG) truncated conical shell subjected to thermal field and inertia force due to the rotating part. The FG properties are assumed to be varied along the thickness according to power law distribution, whereas Poisson’s ratio is assumed to be constant. On the basis of 3D Green-Lagrange theory in general curvilinear coordinate, the fundamental equations are formulated and then two versions of differential quadrature method (DQM) including polynomial based differential quadrature (PDQ) and Fourier expansion-based differential quadrature (FDQ) are applied to discretize the resulting differential equations. The reliability of the present approach is validated by comparing with known literature where good agreement is reached using considerably few grid points. The effects of different mechanical boundary conditions, temperature fields, rotating angular speed, and shell thickness on the distributions of stress components and displacement in thickness direction for both axisymmetric and asymmetric cases are graphically depicted.

1. Introduction

Sharp discontinuity and delamination problems in monolithic laminated composite materials impelled scientists to come up with the idea of new class of advanced composite material, so-called functionally graded materials (FGMs). FGMs are characterized by smooth variations in material properties from one interface to other based on specific function. Mixture of ceramic phase with metal phase by gradual transition between two phases enables the materials to have the higher wear and thermal and corrosion resistances and avoids the resultant stress concentrations in traditional composite materials [1]. In many industrial applications, FG components are commonly exposed to thermal and mechanical load fields. Hence, determination of the optimum material gradient of FGMs has major rule on reduction of the produced stresses in structure. The papers corresponding to thermoelastic analysis of FGM because of the close concept to present work are brought here. Afsar and Go [2] developed finite element modelled for an axisymmetric rotating FG circular disk under thermoelastic field. Bayat et al. [3] analyzed the FG rotating disk with variable thickness under thermal, bending, and body loads. Peng and Li [4] presented thermoelastic problem of an FG hollow cylinder using Fredholm integral equations. Kiani [5] applied the surface elasticity theory of Gurtin and Murdoch to carry out the thermoelastic analysis of rotating FG nanoplate. Kar and Panda [6] focused on fully nonlinear and linear bending analysis of spherical FG panel under thermomechanical loadings. Eslami et al. [7] studied one-dimensional (1D) problem of a thick-walled FG hollow sphere subjected to thermomechanical loadings. Hosseini and Dini [8] presented analytical solution of thick-walled FG rotating cylinder under magneto-thermomechanical loadings. Natarajan et al. [9] employed the cell-based smoothed finite element method with discrete shear gap technique to investigate the static and dynamic behavior of FG plates. Leu and Chien [10] studied thermoelastic behavior of FG rotating disk with variable thickness involving nonuniform heat source. Mehditabar et al. [11] investigated the magneto-thermoelastic analysis of FG truncated conical shell using DQM approach. Punera et al. [12] presented analytical solutions for thermoelastic analysis of laminated and FG sandwich open cylindrical shell. Xin et al. [13] discussed the thermoelastic responses of the thick-walled FG tube under thermal and mechanical load fields employing Voigt method. Prakash and Ganapathi [14] studied the asymmetric free vibration and thermoelastic stability of FG circular plates by employing finite element method. Based on Gren-Naghdi thermoelasticity theories and using Laplace and Fourier transforms the magneto-thermoelastic analysis of rotating medium was carried out by Kumra et al. [15]. Kordkheili and livani [16] proposed a semianalytical-numerical method to investigate the thermoelastic responses of the FG rotating disks with variable thickness and also in another work the thermoelastic analysis coupled with creep behavior [16] of the same problem was carried out by them. The thermoelastic behavior of rotating FG cylindrical shell with piezoelectric layers applying DQM technique was studied by Saadatfar and Aghaei-Khafri [17]. In their research, the material properties of FGM and FGMP (functionally graded piezoelectric) were assumed to be distributed according to exponential rule across radial direction. Eraslan and Akis [18] performed the elastic-plastic analysis of FG pressurized tube subjected to thermal load. Jabbari et al. [19] investigated the thermoelastic problem of the rotating thick FG cylindrical shell with material gradient in axial direction utilizing multilayer method (MLM). Chikh et al. [20] carried out nonlinear postbuckling analysis of symmetric FG plate under thermomechanical loads. Behravan Rad [21] investigated thermoelastic behavior of FG annular plate resting on a hybrid elastic foundation undergoing asymmetric mechanical load and axial temperature gradient. Isavand et al. [22] studied dynamic analysis of FG steel (FGS) cylindrical panel employing hybrid Fourier series expansion-Galerkin finite element method, Newmark. Mehditabar [23] modelled 3D FG truncated conical shell immersed in magneto-thermomechanical fields.

In this paper 3D asymmetric thermoelastic analysis of rotating FG truncated conical shell under temperature gradient through thickness and centrifugal load duo to rotation of the shell is investigated. The variations of material properties are assumed to be varied along the thickness according to power law rule, whereas Poisson’s ratio is supposed to be constant. Due to intrinsic complexity of existing equations, two versions of DQM technique including PDQ and FDQ in conjunction with Newton-Raphson iteration technique are employed to obtain the numerical results. The validity of the present approach is proved by comparing with that published literature that the good consistency using considerably few grid points is achieved. In this study, the sensitivities of displacement and stress components to different values of thermal loading, thickness of shell, rotating angular speed, and boundary conditions under both axisymmetric and asymmetric loading conditions are graphically plotted and discussed.

2. Governing Equations

The FG truncated conical shell with the origin of the coordinate at vertex of whole cone, as illustrated in Figure 1, is considered. According to Figure 1, the set of orthogonal curvilinear coordinates , , and lie on meridian, circumferential, and thickness directions, respectively. The thickness of the shell is defined as , , and indicate the inner and outer radii of the cone at its small and large edge, respectively, denotes distance from the vertex to its small edge, L is length of the cone along the generator, determines the semivertex angle, and is rotating angular speed. Moreover, , , and indicate the total centrifugal force and its components along and directions, respectively.

Figure 1 

Physical model and system coordinates of the truncated conical shell.

In this study, the elastic modulus, heat conductivity and coefficients of thermal expansion are assumed to be varied only along the thickness direction according to the selected power law function as follows [11]:

, , and are the elastic modulus, coefficient of thermal expansion, and thermal conductivity at the inner surface of shell, respectively, and also represents the FG in-homogeneity constant. Consider , , and as corresponding displacement components in s, , and , respectively; the relationships between strain and displacement based on 3D elasticity formulations in the conical coordinate are expressed as [25]where .

The stress-strain relation according to constitutive equations for isotropic material in matrix form is defined as follows:where and indicate the normal and shear stress components, respectively, and denotes the strain components, represents the temperature distribution through thickness obtained from the steady-state heat conduction equation, and also is the coefficient of thermal expansion. The elastic coefficients for isotropic materials in terms of elastic stiffness and Poisson’s ratio are expressed as follows:

Assuming quasistatic state, the acceleration terms can be eliminated from the equilibrium equations [26]. With this assumption, the equilibrium equations in , , and directions in the presence of body force are given in (5a)–(5c), respectively, asCentrifugal body force components in designated conical coordinate knowing that the cone radius at any point along its generator is are calculated by the following formulations:

Using (2), (3), and (6) and substituting them into (5a)–(5c), the following equilibrium equations in terms of displacement components and temperature in the three directions of , , and are derived in (7a)–(7c), respectively, as follows:The steady-state heat conduction equation without heat generation in the conical coordinate can be written asThe mathematical expressions of mechanical and thermal boundary conditions for the two considered cases including clamped and simply supported at top and bottom of truncated conical shell are described as follows:For both axisymmetric and asymmetric cases, the inner surface of conical shell is kept at zero and the outer surface is varied at constant temperature . Thus, the following mechanical and thermal boundary conditions must be satisfied for the inner and outer surfaces:

3. The Method of Solution

The DQ method as powerful discretization tool is applied to transform a system of resulting differential equations to a system of algebraic equations. The general rule of DQM postulates that derivative of any continuous function with respect to space variable at any discrete point in the domain can be expressed as a linear weighted linear summation of all the functional values in the whole domain as follows [27]:where , , and denote the weighting coefficients for the mth-order derivatives of the function in thickness direction at mesh point , along the generator at mesh point and circumferential direction at mesh point , respectively. Furthermore, , , and are the numbers of sampling points along the s, θ, and , respectively. In this study, polynomial differential quadrature (PDQ) is employed to determine the weighting coefficients along and directions and the Fourier expansion-based differential quadrature (FDQ) is used to compute the weighting coefficient along the circumferential direction. The weighting coefficients are calculated according to following explicit formulations [27]:Using (11) and applying relations in (12a) and (12b), the DQ analogs of the governing, related boundary conditions, and heat transfer equations are achieved. The discretized forms of equations using DQ technique are given in detail in Appendix. The system of algebraic equations discretized by DQM is effectively solved based on the semismooth Newton-Raphson iteration procedure as follows. Consider the vector of unknown variables and the resulting governing and boundary equations : The system of equation with unknown variable vector can be solved using the following Newton-Raphson iteration scheme:where the applied notation kth in the definition of Newton technique is iteration number. It should be noted that the solution starts from . In (14), is successively replaced by previous iteration and this procedure continues until difference between two successive iterations equals designated tolerance or .

4. Numerical Results and Discussion

The coordinates of the grid points are chosen based on the Chebyshev-Gauss-Lobatto relation as follows [27]:

In the s-direction,In the -direction,In the –direction,The geometrical properties of truncated conical shell are considered to be , , , and . The truncated conical shell is assumed to be made up of aluminum with the following material properties [28]:Here, the presented results are based on the nondimensional forms of thickness displacement, temperature, and stresses as defined in the following:In order to illustrate the convergence and accuracy of presented method, the thermoelastic problem of thick-walled FG spherical shell carried out by Eslami et al. [7] is chosen to validate the results obtained by the present approach. The problem considered was the thermoelastic analysis of 1D thick-walled FG spherical shell. Reference [7] assumed the material properties except Poisson’s ratio to be varied along the thickness according to following power law rule:where , , and are the modulus of elasticity, the coefficient of thermal expansion, and thermal conduction coefficient at inner radius, respectively, and is the in-homogeneity constant. The inner and outer radii were assumed to be   and  , respectively. The used material properties were  GPa and °C and also the thermal and mechanical boundary conditions were supposed to be °C, °C,MPa, and . Furthermore, for comparison, the values of circumferential stress and radial displacement for are selected. Figure 2(a) shows the variations of nondimensional values of circumferential stress along radial direction computed by present method and [7]. It is observed from Figure 2(a) that the present numerical technique yields convergence at and also the converged results are in an excellent agreement with those obtained by [7]. For more validation of the present approach, the variations of dimensionless radial displacement along thickness calculated by DQM and with those achieved by [7] are demonstrated in Figure 2(b). According to Figure 2(b), for some values of , the dimensionless radial displacement predicted by present approach is little higher than the values obtained by [7] and with other ranges of is smaller. It can be concluded from Figure 2(b) that the good consistency (maximum discrepancy of 2.5%) between present numerical technique and [7] is depicted that confirms the efficiency and reliability of the present numerical method. For more assurance of the accuracy of present method, another pressure vessel component under thermomechanical loading modelled by [24] is performed in Figures 3(a) and 3(b).