Thermal stress analysis of functionally graded annular fin

Thermal stress distributions in an annular fin with rectangular profile made of functionally graded material (FGM) are considered. The material properties of annular fin are assumed to be graded along the fin radius as a power-law function while the Poisson’s ratio is taken to be constant. The governing equations are solved analytically for specific value of inhomogeneity parameter of thermal conductivity and all numerical values of inhomogeneity parameters of modulus of elasticity and linear thermal expansion coefficient. The effect of the inhomogeneity parameters on temperature distribution and thermal stresses are presented in graphical form. The formulation is validated with benchmark results in the literature. It is also shown that functionally graded annular fin is subject to lower stresses, although it has higher tip temperature than the homogeneous one.     

Heat transfer enhancement in annular fins using functionally graded material

This study presents a new approach on the heat transfer enhancement of annular fins with constant thickness using functionally graded materials. The thermal conductivity of the annular fin is assumed to be graded along the fin radius as a power‐law function. The resulting fin equation is solved by an approximate analytical method using the mean value theorem. The variable coefficients of second and third terms in the second‐order differential equation of the fin are replaced with their mean values along the fin radius. Several different graphs regarding the computed temperature profile, fin tip temperature, and fin efficiency are plotted with respect to the radii ratio thermo‐geometric parameter, and inhomogeneity parameter. It is demonstrated that the inhomogeneity parameter plays an important role on the heat transfer enhancement of the annular fin. However, for large radii ratios the effect of the inhomogeneity parameter decreases. Finally, it is stated that application of the functionally graded material in the annular fins, enhances the heat transfer rate between the fin and surrounding fluid resulting from the higher fin efficiency in comparison to the homogeneous annular fin. It is hoped that the results obtained from this study arouse interest among thermal designers and heat exchanger industries. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(7): 603–617, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21053     

Application of homotopy perturbation method and inverse prediction of thermal parameters for an annular fin subjected to thermal load

In this work, application of the homotopy perturbation method (HPM) and an inverse solution for estimating unknown thermal parameters such as the variable thermal conductivity parameter (β), the thermogeometric parameter (K), and the nondimensional coefficient of thermal expansion (χ) in an annular fin subjected to thermal stresses is presented. Initially, to obtain the nondimensional temperature distribution from the heat equation, the forward method is employed using an approximate analytical solution based on HPM. Thereafter, a closed form solution for the temperature-dependent thermal stresses is obtained using the classical theory of thermoelasticity coupled with HPM solution containing the temperature distribution. Next, for satisfying a particular stress criterion which makes relevance in selecting appropriate configurations for selecting the finned system, unknown thermal parameters are obtained using an inverse approach based on the Nelder–Mead simplex search minimization technique. The objective function is taken as the sum of square of the residuals between the measured stress field and an initially guessed value which is updated iteratively. It is found that more than one type of temperature distribution may yield a given stress distribution, thereby giving rise to different fin efficiencies. The agreement between the actual and the predicted results was found to be satisfactory.     

Transient analysis of functionally graded material fin under the effect of Lorentz force using the integral transform method for improved electronic packaging

In the present work, transient analysis of longitudinal fin of functionally graded material under the influence of Lorentz force is considered. The governing equations of the linear, quadratic, and exponential functions are solved via Bessel, Legendre, and modified Bessel functions, respectively. The study investigates the effects of Lorentz force on the performance of the fin. The study shows that the transient response of the fin is lower in the linear law function compared with the exponential law function. The power law function shows significant energy‐saving capability than the linear class functionally graded material heatsink. Moreover, the thermal characteristic of the fin improves with an increase in the convective, radiative, magnetic field parameter, and nonhomogeneous index. The temperature prediction of the present study using the integral transform method is verified with the fourth‐order Runge‐Kutta method with an excellent agreement established.     

Thermal buckling analysis of functionally graded perforated annular sector plates using 3D elasticity theory

Thermal buckling of annular sector plates with circular cutouts made of functionally graded material is analyzed in this article. Graded material is considered with two parts of ceramic structure made of zirconia and metal structure which is aluminum. Unlike most conducted research, the direction of property change is observed in three main directions. Thermal loading is assumed as a uniform increase in temperature influencing the whole sector. 3D-finite element method based on elasticity theory is used in this analysis, which resets first and second variations of potential energy of the sector to zero to find equilibrium and stability equations, respectively. Green’s nonlinear strain–displacement relations are used to obtain geometrical stiffness matrix. In the finite element method, unlike most studies, a 3D eight-nodded element is used, which has nodes in the direction of thickness. The circular cutouts of the sector have added to the complexity of the analysis. The finite element formulation is coded in MATLAB. Finally, the effect of different parameters such as dimension and number of cutouts, power law index, and orientation of graded material on the critical thermal buckling temperature is studied.     

Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams

In this work, the thermal effect on the buckling response of the axially functionally graded (AFG) nanobeams is studied based on the nonlocal thermoelasticity theory. Size effects of elastic deformation and heat conduction are considered simultaneously. Non-uniform distribution of temperature along the longitudinal direction of the AFG nanobeams is taken into account and determined by the nonlocal heat conductive law. Equations of motion and the corresponding boundary conditions are derived with the aid of the variational principle within the sinusoidal shear deformation theory and the nonlocal thermoelasticity theory. Ritz method is used to obtain the solutions for the thermal buckling response of the AFG nanobeams with various boundary conditions. Numerical results addressing the significance of the AFG index, the nonlocal parameters of elasticity and heat conduction, and the transverse shear deformation on the buckling behavior are displayed. It is found that, in addition to the nonlocal effect of elasticity, the nonlocal heat conduction plays an important role in analyzing the thermal–mechanical behaviors of the FG nanostructures.     

Thermoelastic analysis of a cylindrical vessel of functionally graded materials

This paper presents a novel method for analyzing steady thermal stresses in a functionally graded hollow cylinder. The thermal and thermoelastic parameters are assumed to arbitrarily vary along the radial direction of the hollow cylinder. The boundary value problem associated with a thermoelastic problem is converted to a Fredholm integral equation. By numerically solving the resulting equation, the distribution of the thermal stresses and radial displacement is obtained. The numerical results obtained are presented graphically and the influence of the gradient variation of the material properties on thermal stresses is investigated. It is found that appropriate gradient can make the distribution of thermal stresses more gentle in the whole structure.     

Thermal fracture model for a functionally graded material with general thermomechanical properties and collinear cracks

In this article, a fracture mechanics model for functionally graded materials (FGMs) with general thermomechanical properties and collinear cracks under thermal loading is proposed. Assuming the thermomechanical properties of FGM strip to be general continuous functions of the coordinate in the thickness direction, the FGM strip is divided into a multilayered medium with the thermomechanical properties varying exponentially in each layer. Using the superposition method, the problem is reduced to a perturbation problem in which the crack surface tractions are the only external forces. Finally, the crack problem is reduced to integral equations with generalized Cauchy kernel and solved numerically. Some typical examples are discussed and the thermal stress intensity factors (TSIFs) for the collinear cracks are presented. The influences of the geometry parameters and the interaction between both collinear cracks on the TSIFs are discussed. Some important conclusions are drawn.     

Multi-objective optimization of annular fin array with B-spline curve based fin profiles

The design of annular fin array with variable thickness fin profiles defined by B-spline curves is studied as a multi-objective optimization problem for simultaneously maximizing heat transfer rate and minimizing thermal stress. Maximization of surface e?ciency and augmentation factor as well as minimization of fin volume are considered as additional objective functions for further assessment of fin array performance. Evaluating the objective values through hybrid spline difference method, different cases are investigated by solving the optimization model by non-dominated sorting genetic algorithm II. The proposed scheme should aid designers in selecting compromise optimal solutions for practical problems.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

In this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.     

Investigation of functionally graded metal foam thermal management system for solar cell

Concentrated photovoltaic cell (CPV) is a solar energy harvesting device that converts solar energy into electrical energy. However, the performance and efficiency of the CPV are heavily dependent on the temperature. Besides, nonuniformity of temperature distribution on the CPV will lead to thermal aging and affects the cycle life. Hence, an effective cooling system is required to remove excess heat generated to ensure that the CPV operates at optimum operating temperature with minimum variation of temperature. Metal foam is a new class of material that possesses huge potential for thermal management. In this study, a functionally graded metal foam is proposed for the CPV thermal management system. Computational thermal fluid dynamic analysis is conducted to investigate the effect of porosity and pore density on the flow field and thermal performance of the aluminum foam heat sink. The investigation results revealed that 10 PPI functionally graded aluminum foam heat sink with two stages of porosity gradient 0.794 and 0.682 produced the lowest pressure drop and highest thermal performance. Temperature difference of 3.9°C was achieved for a solar cell with total heat generation of 900 W under water mass flow rate of 20 gs⁻¹.     

Transient thermal stress analysis of a functionally graded thick hollow cylinder with temperature-dependent material properties

This article deals with the study of temperature distribution and thermal stresses of a functionally graded thick hollow cylinder with temperature dependent material properties. All the material properties except Poisson’s ratio are assumed to be dependent on temperature and spatial coordinate z. The two-dimensional transient heat conduction equation is solved under convective heat transfer condition with varying point heat source. The influence of inhomogeneity parameters on the thermal and mechanical behavior is examined. Numerical computations are performed for ceramic-metal-based functionally graded material, in which alumina is selected as ceramic and nickel as metal.     

Thermoelastic vibration analysis of functionally graded skew plate using nonlinear finite element method

Numerical investigation of nonlinear free vibration of functionally graded skew (FGS) plate in the thermal environment is presented. The mathematical model is proposed for the first time based on higher order shear deformation theory in conjunction with Green–Lagrange-type geometric nonlinearity for the FGS plate subjected to a thermal load. The material properties are considered to be temperature dependent and are graded along the thickness direction as per simple power law of distribution in terms of volume fraction of the constituent phase. The governing algebraic equations are derived using Hamilton’s principle, and the solutions are obtained using the direct iterative method. The proposed finite element model has discretized into an eight-noded quadratic serendipity elements. To validate the model, the obtained results are compared with the available literature. The influence of volume fraction index, skew angle, temperature change, aspect ratio, side–thickness ratio, and boundary conditions on the linear and nonlinear frequency of skew functionally graded material plate is examined and discussed in detail.     

Thermoelastic analysis of laminated and functionally graded sandwich cylindrical shells with two refined higher order models

Analytical solutions for laminated and functionally graded sandwich open cylindrical shells under mechanical and thermal loads are presented using a refined higher order shear and normal deformation theory. Temperature variation through thickness is assumed as thickness coordinate polynomial. Present study also extends the classical thickness criteria with more reliable extension to moderately thick shells. Navier solution method is used to solve system of equations derived using principle of minimum potential energy for all edges diaphragm supported. Two kinds of sandwich panels with core or face sheets made of thickness graded material are studied. Several examples are numerically evaluated to establish the accuracy of present models.     

Surface elasticity-based modeling of rotating functionally graded nanoshafts in thermal environments

Thermoelastic field analysis of a rotating functionally graded nanoshaft (RFGNS) in thermal environments is of interest. The governing equations of the rotating nanoshaft with varying material properties along the radial direction are obtained. Two nonclassical boundary conditions, namely, fixed-free and free–free, are established accounting for the surface energy effect. Using finite element method and Hamilton's principle, the thermoelastic field within RFGNS is evaluated. The effects of power-law index, aspect ratio, temperature, angular velocity of the RFGNS, and surface energy on the displacements and stresses are displayed in detail.     

Buckling of functionally graded plates under thermal,axial, and shear in-plane loading using complex finite strip formulation

A complex finite strip method was used to study the buckling of functionally graded plates (FGPs) under thermal and mechanical (longitudinal, transverse, and shear in-plane) loading. The mechanical characteristics of FGPs were assumed to vary through the thickness, according to power law distribution. The nonlinear temperature distribution in the direction of the plate thickness was assumed according to thermal conduction steady state conditions. In complex finite strip method, the polynomial Hermitian functions were assumed in the transverse direction and the complex exponential functions were used in the longitudinal direction to evaluate the standard and geometric stiffness matrices that have the ability of calculating the critical shear stress in contrast to trigonometric shape functions. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. In addition, numerical results for FGPs with different boundary conditions were presented and compared with those available in the literature and the interaction curves of mechanical and thermal buckling capacity of FGPs were obtained.     

Thermal behavior of functionally graded solid sphere with nonuniform heat generation

This article deals with the thermoelastic analysis of the functionally graded solid sphere due to nonuniform heat source inside the body under the constant surface temperature. The sphere material is considered to be graded along the radial direction where an exponentially varying distribution is assumed. Also, the material assumed with constant Poisson’s ratio. The implicit finite difference scheme is used to determine the transient temperature, radial displacement, and stress field within the sphere. The results are illustrated numerically and graphically for functionally graded solid sphere consists of metal and ceramic.     

Thermal analysis of a longitudinal fin with variable thermal properties by recursive formulation

The present study supplies a new approach to calculate thermal performance of a singular fin with variable thermal properties. With discrete model, the singular fin can be divided into many sections. Then, each section can be combined together to obtain the whole solution of the fin by recursive numerical formulation. The recursive formulas for both conditions with and without heat transfer on fin tip are derived in the present study. Finally, several examples including composite and boiling mode of a fin have been successfully simulated to demonstrate the validity of the present approach.