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- In this paper, an analytical formulation of FGM axisymmetric thick-walled cylinders, based on the plane elasticity theory is presented. The stress and displacements in thick cylindrical shell are calculated using the real, double and complex roots of characteristic equation. Solutions are obtained under generalized plane stress, plane strain and closed-ends cylinder assumptions. It is assumed that the material is isotropic and heterogeneous with constant Poissn's ratio and radially varying elastic modulu. The results have been compared with findings of the researcher (2001) [hoop stress is incorrect], and we have present corrected version as well as supplementary findings. Keywords: Thick-Walled Cylinder, FGM, Plane Elasticity

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Abstract- In this paper, an analytical formulation of FGM axisymmetric thick-walled cylinders, based on the first shear deformation theory (FSDT) is presented. The displacements and maximum stress in thick cylindrical shells are calculated. Solutions are obtained under generalized plane strain assumptions. It is assumed that the material is isotropic and heterogeneous with constant Poissn's ratio and radially varying elastic modulu. The results have been compared with findings of the plane elasticity theory (PET).

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In this paper, free vibration analysis of moderately thick smart FG rectangular plate is presented on the basis of Mindlin plate theory. This structure is composed of a host FG plate and two bonded piezoelectric layers. The plate has two opposite edges simply supported (i.e., Levy-type rectangular plates). The closed circuit piezoelectric layers can be used as an actuator. According to a power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate. Using Hamilton's principle and Maxwell electrostatic equation, six complex coupled equations are introduced. These equations are exactly solved introducing the new potential and auxiliary functions as well as using separation of variables method. The accuracy of the frequencies is verified by the available literature and the finite element method. The present exact solution can accurately predict not only the out of plane, but also the in-plane modes of FG plate. Finally, the effects of various parameters such as boundary conditions, gradient index and thickness of piezoelectric layers on the natural frequency are investigated.

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In this paper, the Meshless Local Petrov-Galerkin (MLPG) method is used to analyze the fracture of an isotropic FGM plate. The stress intensity factor of Mode I and Mode II are determined under the influence of various non-homogeneity ratios, crack length and material gradation angle. Both the moving least square (MLS) and the direct method have been applied to estimate the shape function and to impose the essential boundary conditions. The enriched weight function method is used to simulate the displacement and stress field around the crack tip. Normalized stress intensity factors (NDSIF) are calculated using the path independent integral, J*, which is formulated for the non-homogeneous material. The Edge-Cracked FGM plate is considered here and analyzed under the uniform load and uniform fixed grip conditions. To validate results, at first, homogeneous and FGM plate with material gradation along crack length was analyzed and compared with exact solution. Results showed good agreement between MLPG and exact solution.

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Abstract - In this article , thermal buckling analysis of functionally graded annular sector plate is studied. The mechanical and thermal properties of the functionally graded sector plate are assumed to be graded in the thickness direction . The equilibrium and stability equations are derived based on the first order shear deformation plate theory (FSDT) in conjunction with nonlinear von-karman assumptions. Differential quadrature method is used to discretize the equilibrium and stability equations. In this method a non-uniform mesh point distribution (Chebyshev-Gauss-Lobatto) is used for provide accuracy of solutions and convergence rate . By using this method, there is no restriction on implementation of boundary conditions and various boundary conditions can be implemented along any edges . Finally, The results compared with other researches and the effects of plate thickness, sector angle, annularity, power law index and various boundary conditions on the critical buckling temperature are discussed in details .

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In this paper the meshless local Petrov-Galerkin (MLPG) method is implemented to study the vibration of a Functionally Graded Material (FGM) cylindrical shell. Displacement field equations, based on Donnell and first order shear deformation theory, are taken into consideration. Material properties are assumed to be temperature-dependent and graded in the thickness direction according to different volume fraction functions. A FGM cylindrical shell made up of a mixture of ceramic and metal is considered herein. The set of governing equations of motion are numerically solved by the Meshless method in which a new variational trial-functional is constructed to derive the stiffness and mass matrices so the natural frequencies are obtained in various boundary conditions by using discretization procedure and solving the general eigenvalue problem. The influences of some commonly used boundary conditions, variations of volume fractions and effects of shell geometrical parameters are studied. The results show the convergence characteristics and accuracy of the mentioned method.

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In this paper, the free vibration of a two-dimensional functionally graded circular cylindrical shell is analyzed.To describe the material properties of the two-phased FGM material Mori–Tanaka micromechanical model is used. The spatial derivatives of the equations of motion and boundary conditions are discretized using the methods of generalized differential – Integral quadrature (GDIQ). To validate the results, comparisons are made with the solutions for FG cylindrical shells available in the literature. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies. The results of this study show that the values of natural frequency of 2D FGMs are higher than those of 1D FGMs in parallel conditions. Furthermore, application of a confining elastic foundation increases the value of natural frequencies.

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Functionally graded materials have been taken into consideration by many researchers in the last two decades. Gradual changes of mechanical properties in FGMs decrease stress concentration, crack initiation and propagation and delamination. Many of the present and potential applications of FGM contain contact loading.This kind of loading causes surface crack initiation which is followed by subcritical crack propagation.Thus, propagation of surface cracks is one of the most important failure mechanisms in FG structures. In this article two dimensional sliding contact of a rigid flat punch on a homogeneous substrate with an FGM coating is studied. Plane strain condition is considered in this problem. The Properties of the substrate and the FGM layer are assumed to be elastic and the Poisson’s ratio is assumed to be constant. The modulus of elasticity in the graded layer is calculated based on TTO model approximation. This model defines a parameter q which considers the microstructural interactions. The governing equations are solved by Finite Difference method by means of MATLAB software. The influence of different parameters such nonhomogeneity,q, the dimensions of the punch, the thickness of the graded layer and the coefficient of friction on the mode I and II stress intensify factors are investigated.

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In this paper, the thermoelastic behavior of cylindrical sandwich shell with functionally graded (FGM) core under thermal shock is presented. Thermo mechanical properties of FGM layer are assumed to be independent of temperature and also, very continuously and smoothly functions in the radial direction as a nonlinear power function. The analytical solutions of governing partial differential equations for each layer of cylinder are solved by using Laplace transform and power series method. Mechanical boundary conditions and continuity equations for interfaces are used to obtain unknown parameters that get in recurrence equations for each layer of cylinder. The results in Laplace domain transferred to time domain by employing the fast inverse Laplace transform method (FLIT).The effects of FGM’s power on the dynamic characteristics of the FG thick sandwich cylindrical shell are studied in various points across the thickness of cylinder. The analytical presented method provides an appropriate field for analysis of transient radial and hoop stresses in a cylinder on various thermo mechanical load. Accuracy of gained equations is evaluated by similar articles. The results have a good agreement with published data in pervious researches.

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This paper presents an elastic parametric analysis for the purpose of investigating the limit angular speed, displacement and stresses in rotating disks made of functionally graded materials (FGMs) based on Tresca yield criterion. The material properties obey the power law in radial direction. The Poisson’s ratio due to slight variations in engineering materials is assumed constant. For different values of inhomogeneity constant, limit angular speed, displacement and stresses in radial direction are plotted and for the commencement of the plastic flow, different states are investigated. state1: onset of plastic flow at the inner radius, state2: onset of plastic flow at the outer radius, state3: onset of plastic flow as the simultaneously at both radii and state4: onset of plastic flow between the inner and outer radii. To the best of the researchers’ knowledge, so far, in the papers which have been dealing with the investigation of onset yield analysis, the density and yield stress has been assumed constant; however, in this paper by assuming varying density and yield stress in rotating disks made of functionally graded materials and comparing results obtained by fixing these parameters, it has been observed that taking the density as a constant value is wrong and varying it has significant effects on the stresses.

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To investigate, understanding and predicting dynamic fracture behavior of a cracked body, dynamic stress intensity factors (DSIFs) are important parameters. In the present work interaction integral method is presented to compute static and dynamic stress intensity factors for three-dimensional cracks contained in the functionally graded materials (FGMs), and is implemented in conjunction with the finite element method (FEM). By a suitable definition of the auxiliary fields, the interaction integral method which is not related to derivatives of material properties can be obtained. For the sake of comparison, center, edge and elliptical cracks in homogeneous and functionally graded materials under static and dynamic loading are considered. Then material gradation is introduced in an exponential form in the two directions in and normal to the crack plane. Then the influence of the graded modulus of elasticity on static and dynamic stress intensity factors is investigated. It has been shown that, material gradation has considerable reduce influence on DSIFs of functionally graded material in comparison with homogenous material. While, static stress intensity factors can decrease or increase, depend on the direction of gradation material property.

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Careful and numerical analysis eccentrically stiffened shells in the industry is a major step forward in the design of these shells. In this paper, a careful analysis of post-buckling behavior of eccentrically stiffened FGM thin circular cylindrical shells is surrounded by an elastic foundation and external pressure is presented. The two parameter elastic foundation based on Winkler and Pasternak elastic model is assumed. Stringer and ring stiffeners are internal. Shell properties and eccentrically stiffened are FGM. Fundamental relations and equilibrium equations are derived based on the smeared stiffeners technique and the classical theory of shells and according to von- Karman nonlinear equations. The three-term approximation for the deflection shape, including the pre-buckling, linear buckling shape and nonlinear buckling shape was chosen that using the Galerkin method, the critical load and post-buckling pressure-deflection curves is calculated. The effects of different dimensional parameters, buckling modes, volume fraction index and number of stiffeners are investigated. Numerical results show that stiffeners and elastic foundation enhance the stability of the shells. Increasing the shell thickness, reducing the volume fraction index, raising the number of Stringer and ring stiffeners and applying foundation elastic, causes the critical buckling load is increased, too.

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In this paper, the effect of nonhomogeneous parameter in orthotropic Functionally Graded Material(FGM) in a cracked layer is investigated. It is assumed that the mechanical and thermal properties of material are dependent on x-coordinate (collinear with crack surfaces) in exponential form. The problem is solved for internal and edge crack in two way, integral equations and generalized differential quadrature method. Thermal loading is in a way that temperature distribution in the layer is uniform. Because of variation in mechanical and thermal properties, stress distribution due to this loading is not uniform. In the solution of problem with integral equations method, first, thermo-elasticity problem with no cracks and then isothermal crack problem are separately solved. Afterward with these solutions, the main problem will be solved. In order to solve isothermal crack problem, after conversion and simplifying the equations in orthotropic material, Navier's equations will be solved with the Fourier. Numerical solution of the problem is the generalized differential quadrature element method that is being presented for verification of the results of the integral equations for a specific state in the diagram format. Also the effect of temperature on intensity factor with various values of nonhomogeneous parameter is investigated.

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In this paper, FGMs are used as non-uniform materials in high temperature environments specially in various industries like aircraft, aerospace vehicles, nuclear plants and engineering structures. Different industries use them in thin and thick walled spherical pressure vessels. Based on the governing equations, differential equation of stresses is obtained in plastic state that can be widely used in the study of vessel and pipe behavior in elasto- plastic state. It is discussed on the temperature distribution and stress - strain relationships in the tube under internal pressure and temperature difference. Properties of these materials are considered as variable parameters function of radius. In this work, effects of these parameters have been investigated on yielding and the yield temperatures and stress changes under different loading in during thickness of the tubes. Furthermore it is shown that tubes structure can be optimized by choosing appropriate parameters. In fact, by study the four parameters, internal and external temperature, internal pressure and the yield stress, together, can be better analyzed on a functionally graded cylindrical tube that this work has been shown at the end of the article.

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In this paper, a new inverse method has been presented for identifying the distribution of material properties and volume fraction index of rectangular functionally graded (FG) material plates. This method benefits from vibration analysis of FG plates accompanied by a novel and efficient meta-heuristic optimization algorithm called Drops Contact Optimization (DCO) algorithm, being proposed for the first time in this article. The presented algorithm relies on the initial population and mimics the behavior of water drops in different level of contacting successively with a fluid surface. Through using the second shear deformation theory and applying the Hamilton principle, the motion equations are derived and, subsequently, the natural frequencies of the considered FG plates are obtained. The outcomes relevant to considered different material phases and various length to thickness ratios are achieved and compared with those available in the literature. Making a comparative study of the obtained results with five well-known optimization algorithms confirms that the proposed DCO algorithm produces better performance in convergent speed and accurate characterization of FG materials.

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In this paper, an exact analysis of thermal post-buckling behavior of eccentrically stiffened functionally graded (FG) thin circular cylindrical shells subjected to thermal radial loading and surrounded by elastic foundation, is presented. Stringer and ring stiffeners are assumed to be placed on the inner surface of the FG cylinder shell and the material properties of the shell and stiffeners are assumed to be temperature dependent and continuously graded in the thickness direction. The elastic medium around the circular cylindrical shell is modeled by a two parameter elastic foundation based on the Winkler and Pasternak model. Fundamental relations and equilibrium equations are derived based on the smeared stiffeners technique and the classical theory of shells according to the von- Karman nonlinear equations. By using the Galerkin method, the thermal post-buckling response of eccentrically stiffened FG thin circular cylindrical shells are obtained. In order to validate the method, the obtained results are compared with available solutions and in continue, the effects of different parameters such as volume fraction exponent, number of stiffeners and elastic foundation parameters, on the thermal post-buckling response of eccentrically stiffened FG thin circular cylindrical shells are considered. Numerical results show that stiffeners and elastic foundation enhance the stability of the FG shells. Moreover, increasing the shell thickness, reducing the volume fraction index, increasing the number of Stringer and ring stiffeners and applying stiffer elastic foundation lead to increase the thermal post-buckling response of stiffened FG circular cylindrical shells.

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Identification of the thermal conductivity of a functionally graded material (FGM) is considered as an inverse heat conduction problem. ‎In this investigation, the measurements of the temperatures on the portion of the 2D body where heat flux is specified as the boundary ‎condition and/or the heat flux on the portion of the boundary where temperature is specified as the boundary condition are used as ‎additional data needed to identify the thermal conductivity of the FGM domain in an inverse procedure. The thermal conductivity is ‎approximated as a quadratic function of only one direction, and therefore three constant coefficients should be estimated simultaneously.‎‏ ‏The solution of the direct heat conduction problem for‏ ‏FGM domain is obtained using the boundary elements method (BEM).‎‏ ‏The ‎imperialist competitive algorithm (ICA) which is an evolutionary and meta-heuristic global optimization is used to identify the constants ‎in the thermal conductivity function of the quadratic FGM. An inverse computer code is developed which employs the boundary ‎temperature and heat flux measurements data obtained by solving the direct boundary elements code with known thermal conductivity. ‎To show the feasibility and effectiveness of the developed inverse code, a number of example problems are solved and results are ‎verified.‎

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This research presents the study on mechanical buckling of thick-walled cylindrical shell made of functionally graded materials with ring supported under uniform axial and lateral loads. The mechanical properties of shell are variable along the thickness direction. First the governing equations on the buckling of the FGM cylindrical shell supported with ring are established based on third-order shear deformation theory. Then the governing characteristic equations were employed, using energy method and by applying the Ritz technique. In the following with solving characteristic equations, the critical load buckling of the FGM thick-walled cylindrical shell supported with axial and lateral loads are calculated. The boundary conditions represented by end conditions of the FGM shell are the following: clamped-clamped and free-free. To verify the validity of the proposed analytical method the results of this research are compared with the results came from using the finite element software. Finally, the effects of the different parameters such as thickness variations, boundary conditions, loading conditions and geometrical parameters of shell and ring on the buckling behavior of FGM thick-walled cylindrical shell are investigated. The results showed that by increasing the FGM volume fraction power in the shell structure, the critical buckling load increases and the location of the ring support has the significant effect on the critical buckling load. The results presented can be used as an important benchmark for researchers to validate their numerical and analytical methods.

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Exact and numerical solution of eccentrically stiffened panels in the industry is a major step forward in the design of these structures. This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FG cylindrical panels on elastic foundations subjected to hygro-thermo-mechanical loads. The stiffeners are assumed to be spiral-type. The panel has the initial geometrical imperfection. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. The elastic foundation is considered based on Winkler and Pasternak proposed model. Governing equations are derived basing on the Lekhnitsky smeared stiffeners technique and classical shell theory incorporating Von Karman-Donnell geometrical type nonlinearity. Explicit relations of load–deflection curves for FG cylindrical panels are determined by applying stress function and Galerkin method. The effects of angel of stiffener, different dimensional parameters, volume fraction index, initial geometrical imperfection, the stiffness of elastic foundation and moisture concentration on the postbuckling of FG panel are investigated. Also effects of temperature gradient through the thickness and effects of different boundary conditions are investigated for thermo-mechanical loading. The obtained results are validated by comparing with those in the literature.