Elastic deformation of the rotating functionally graded annular disk with rigid casing

An accurate solution for a rotating functionally graded annular disk is presented. Material properties of the present annular disk are assumed to be graded in the radial direction according to a simple exponential-law distribution. The inner surface of the disk is pure metal whereas the outer surface of the disk is pure ceramic. The boundary condition of rigid casing is considered herein, that is the vanishing of the radial displacement at the outer surface. The boundary condition at the inner surface of the disk is taken to be vanishing either radial displacement or radial stress. Analytical solutions for the elastic deformation of the rotating functionally graded annular disks subjected to these boundary conditions are obtained. Numerical results for radial displacement, circumferential and radial stresses are presented. Comparisons between the different rotating homogeneous and functionally graded annular disks are made at the same angular velocity. The results show that distributions of stresses and displacement through the radial direction of the rotating annular disk vary with different parameters.     

Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness

A thermo elastic analysis is presented for axisymmetric rotating disks made of functionally graded material (FGM) with variable thickness. Material properties are assumed to be temperature-dependent and graded in the radial direction according to a grading index power law distribution. The temperature field considered is assumed to be uniformly distributed over the disk surface and varied in the radial direction. Semi-analytical solutions for the displacement field are given for solid disk and annular disk under free-free and fixed-free boundary conditions. The effects of the thermal field, the material grading index and the geometry of the disk on the displacement and stress fields are investigated. Results of this study emphasize on the crucial role of the temperature-dependent properties in a high temperature environment. A comparison of these results with the reported ones in the literature that is temperature-dependent versus temperature-independent suggests that a functionally graded rotating disk with concave thickness profile can work more efficiently than the one with uniform thickness irrespective of whether the material properties are assumed to be temperature-dependent or temperature-independent.     

Thermal stress in rotating functionally graded hollow circular disks

The analysis of thermoelastic problem of a rotating functionally graded hollow circular disk is made. The hollow disk is assumed to have varying material properties along the radial direction. An analytical method is presented to investigate steady thermal stresses in a functionally graded circular annulus rotating with constant angular velocity about its central axis. The associated boundary value problem is reduced to a Fredholm integral equation. The thermal stresses and radial displacement are obtained by numerically solving the resulting equation. A comparison of the numerical results with the exact ones for material properties of special power-law profile confirms the effectiveness of the method. For generally varying material parameters, numerical results are presented graphically to show the effects of gradient parameter, temperature change, angular velocity and thickness of the disk on the distribution of thermal stresses and radial displacement.     

Material tailoring for orthotropic elastic rotating disks

The tailoring of elastic moduli in the radial direction is studied to design a fiber-reinforced orthotropic linear elastic rotating disk with constant radial or hoop stress or constant in-plane shear stress. For fibers arranged in concentric circles the axes of material symmetry coincide with the radial and the circumferential directions. However, when fibers are aligned along helices, the orientation of material principal axes varies with the radial coordinate of a point. For a solid disk made of an orthotropic material with Young’s moduli proportional to each other, we give explicit expressions for the required variations of the elastic moduli with the radius to attain a given state of stress. For a rotating annular disk composed of a fiber-reinforced composite with fibers placed along concentric circles, the required radial variation of the volume fraction of fibers is calculated numerically and exhibited graphically. For fibers of known volume fraction laid along helices, the radial variation of the fiber orientation angle is determined. We have also analyzed the material tailoring problem for a disk of variable thickness. Results presented herein should help structural engineers and material scientists optimally design rotating disks composed of radially inhomogeneous materials.     

Three dimensional elasticity solution for static and dynamic analysis of multi-directional functionally graded thick sector plates with general boundary conditions

In this study, the three dimensional static and dynamic behavior of a thick sector plate made of two-directional functionally graded materials (2D-FGMs) is investigated. Material properties are assumed to be graded in both radial and thickness directions according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the 3D theory of elasticity. Employing 3D graded finite element method (GFEM) based on the Hamilton’s principle and Rayleigh–Ritz energy method, the equations are solved in space and time domains. In the case of static analysis, the sector plate is subjected to a uniform pressure load and for dynamic analysis is subjected to an impact loading. The effects of material gradient index, boundary condition and thickness to radius ratio of the sector plate on the static and dynamic responses are presented and discussed.     

Influences of initial stresses on guided waves in functionally graded hollow cylinders

Limitations of the manufacturing technology result in the existence of initial stresses in functionally graded material (FGM) structures. In the context of the theory of “Mechanics of Incremental Deformations,” the guided wave characteristics in FGM hollow cylinders under initial stresses in the radial and axial directions are investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the method is discussed through numerical examples. It is found that the influences of initial stresses on the longitudinal waves and on the torsional waves are quite distinct, and that the influences of initial stresses in the axial direction are very different from those in the radial direction, both on the dispersion curves and on the displacement and stress distributions.     

Using the third-order shear deformation theory to examine magnetic-mechanical buckling of a two-dimensional functionally graded cylindrical panel with longitudinal and circumferential stiffeners

In this study, the magnetic-mechanical buckling of a cylindrical panel made of two-dimensional functionally graded materials (2D-FGMs) has been investigated. The panel contains longitudinal and circumferential stiffeners and has been subjected to a uniform magnetic field as well as axial load. Material properties of the cylindrical panel are assumed to vary continuously in radial and thickness directions as a function of the volume fraction of the components. The magnetic field has been exerted radially. Equilibrium and stability equations have been derived using both Hamilton's principle and principle of minimum potential energy based on the third-order shear deformation theory (TSDT). The generalized differential quadrature method (GDQ) has been employed to solve the coupled differential equations. Moreover, the effect of geometry, load, magnitude of the magnetic field, number of stiffeners, and volume fraction coefficient on the critical buckling load has been determined. The results are in good agreement with the previous related works.     

Finite element analysis of thermoelastic contact problem in functionally graded axisymmetric brake disks

An analysis of thermoelastic contact problem of functionally graded (FG) rotating brake disk with heat source due to contact friction is presented. Finite element method (FEM) is used. The material properties of disk are assumed to be represented by power-law distributions in the radial direction. The inner and outer surfaces considered are metal and ceramic, respectively. Pure material is considered for the brake pad. Coulomb contact friction is assumed as the heat source. It is divided into two equal parts between pad and brake disk which leads to thermal stresses. Mechanical response of FG disks are compared and verified with the known results from the literatures. The results show that the maximum value of radial displacement in mounted FG brake disk is not at outer surface. It is found that the all areas between pad and brake disk is in full-contact status when the ratio of pad thickness to brake disk thickness is 0.66. It is observed that the total strain due to thermomechanical load is negative for some parts of the disks, whereas, the thermal strains are always positive. It can be concluded that gradation index of the metal-ceramic has significant effect in the thermomechanical response of FG disks.     

Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells

An exact elasticity solution is presented for the free and forced vibration of functionally graded cylindrical shells. The functionally graded shells have simply supported edges and arbitrary material gradation in the radial direction. The three-dimensional linear elastodynamics equations, simplified to the case of generalized plane strain deformation in the axial direction, are solved using suitable displacement functions that identically satisfy the boundary conditions. The resulting system of coupled ordinary differential equations with variable coefficients are solved analytically using the power series method. The analytical solution is applicable to shallow as well as deep shells of arbitrary thickness. The formulation assumes that the shell is made of a cylindrically orthotropic material but it is equally applicable to the special case of isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced composite materials. The homogenized elastic stiffnesses of isotropic materials are estimated using the self-consistent scheme. In the case of fiber-reinforced materials, the effective properties are obtained using either the Mori–Tanaka or asymptotic expansion homogenization (AEH) methods. The fiber-reinforced composite material studied in the present work consists of silicon-carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The natural frequencies, mode shapes, displacements and stresses are presented for different material gradations and shell geometries.     

Thermo-piezo-magneto-mechanical stresses analysis of FGPM hollow rotating thin disk

Thermo-piezo-magnetic behavior of a functionally graded piezo-magnetic (FGPM) rotating disk, under mechanical and thermal loads is investigated. The mechanical, thermal and magnetic properties, except Poisson’s ratio, are assumed to depend on variable r and they are expressed as power functions in radial direction of the disk using mathematical modeling. Temperature distribution is obtained using a steady-state one dimensional heat transfer equation considering the boundary conditions of the symmetrical disk. Stress and displacement correlations, including mechanical, magnetic and thermal terms are defined using elasticity theory. Substituting these relations in the mechanical and magnetic equilibrium equations, lead ultimately to provision of a system of coupled second-order ordinary differential equations in terms of displacement and magnetic potential. Using these differential equations, physical characteristics including displacement, temperature, magnetic potential and distributions of radial and circumferential stresses are investigated graphically for a range of non-homogeneous parameters i.e., elastic stiffness, thermal expansion and thermal conductivity. Hence, the effect of non-homogeneity on the stresses, displacement, temperature and magnetic potential are demonstrated. Results of this investigation could be applied for optimum design of FGPM hollow rotating thin disks.     

Steady‐State Creep Analysis of Functionally Graded Rotating Cylinder

Axi‐symmetric component like cylinder etc. has to operate under severe thermo‐mechanical loads, which cause significant creep. It, thus, reduces its service life. The present study investigates the steady‐state creep in a functionally graded rotating cylinder at constant angular speed. The cylinder is made up of aluminium matrix and reinforced with silicon carbide particles. The thermal gradient in the functionally graded rotating cylinder is estimated by performing finite element analysis on ANSYS software (ANSYS Inc., Canonsburg, USA). The creep behaviour of the cylinder has been described by threshold stress‐based creep law. The creep parameters are obtained by conducting regression analysis. The mathematical models have been developed to describe steady‐state creep in the cylinder. The study reveals that the radial, tangential, axial and effective stresses in the cylinder are significantly affected by the presence of particle gradient alone as well as with the presence of particle & thermal gradient both. It has been found that the creep rates have been reduced significantly by imposition of particle and thermal gradients together and thus increases the service life of cylinder.     

Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads

In this paper, a thick truncated hollow cone with finite length made of two-dimensional functionally graded materials (2D-FGM) subjected to combined loads as internal, external and axial pressure is considered. The volume fraction distribution of materials and geometry are assumed to be axisymmetric but not uniform along the axial direction. The Finite Element Method based on the Rayliegh-Ritz energy formulation is applied to obtain the elastic behavior of the functionally graded thick truncated cone. By using this method, the effects of semi-vertex angle of the cone and the power law exponents on the distribution of different types of displacements and stresses are considered. The results show that using 2D-FGM leads to a more flexible design so that both the maximum stresses and stress distribution can be controlled by the material distribution.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems

Summary Closed form solutions to functionally graded rotating solid shaft and rotating solid disk problems are obtained under generalized plane strain and plane stress assumptions, respectively. The nonhomogeneity in the material arises from the fact that the modulus of elasticity of the material varies radially according to two different continuously nonlinear forms: exponential and parabolic. Both forms contain two material parameters and lead to finite values of the modulus of elasticity at the center. Analytical expressions for the stresses at the center are determined. These limiting expressions indicate that at the center of shaft/disk: (i) the stresses are finite, (ii) the radial and the circumferential stress components are equal, and (iii) the values of the stresses are independent of the variation of the modulus of elasticity. It is also shown mathematically that the nonhomogeneous solutions presented here reduce to those of homogeneous ones by an appropriate choice of the material parameters describing the variation of the modulus of elasticity.     

Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness

We analyze axisymmetric deformations of a rotating disk with its thickness, mass density, thermal expansion coefficient and shear modulus varying in the radial direction. The disk is made of a rubberlike material that is modeled as isotropic, linear thermoelastic and incompressible. We note that the hydrostatic pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field. The problem is analyzed by using an Airy stress function φ. The non-homogeneous ordinary differential equation with variable coefficients for φ is solved either analytically or numerically by the differential quadrature method. We have also analyzed the challenging problem of tailoring the variation of either the shear modulus or the thermal expansion coefficient in the radial direction so that a linear combination of the hoop stress and the radial stress is constant in the disk. For a rotating annular disk we present the explicit expression of the thermal expansion coefficient for the hoop stress to be uniform within the disk. For a rotating solid disk we give the exact expressions for the shear modulus and the thermal expansion coefficient as functions of the radial coordinate so as to achieve constant hoop stress. Numerical results for a few typical problems are presented to illuminate effects of material inhomogeneities on deformations of a hollow and a solid rotating disk.     

Effect of anisotropy on steady state creep in functionally graded cylinder

The steady state creep in transversely isotropic functionally graded cylinder, operating under internal and external pressures, has been investigated. The cylinder is composed of functionally graded material (FGM) containing silicon carbide whiskers in a matrix of 6061Al. The creep behavior of the FGM has been described by a threshold stress based creep law. The effect of anisotropy on creep stresses and creep rates in the FGM cylinder has been analysed and compared with an isotropic FGM cylinder. The anisotropy is represented by a parameter α, defined as the ratio of radial (or axial) and tangential yield strength. The study reveals that in an anisotropic FGM cylinder i.e. when α deviates from unity, radial and tangential stresses are marginally affected whereas axial and effective stresses are significantly affected as compared to those in an isotropic FGM cylinder. The strain rates as well as inhomogeneity in strain rates in the FGM cylinder decrease significantly when α reduces from 1.3 to 0.7. The magnitude of stresses, strain rates and inhomogenity in strain rates in the FGM cylinder, subjected to internal pressure alone, could be significantly reduced by subjecting it to both internal and external pressures though the stress inhomogenity in the FGM cylinder increases.     

Application of Differential Transform Method to Thermoelastic Problem for Annular Disks of Variable Thickness with Temperature-Dependent Parameters

This article analyzes the one-dimensional steady temperature field and related thermal stresses in an annular disk of variable thickness that has a temperature-dependent heat transfer coefficient and is capable of temperature-dependent internal heat generation. The temperature dependencies of the thermal conductivity, Young’s modulus, and the coefficient of linear thermal expansion of the disk are considered, whereas Poisson’s ratio is assumed to be constant. The differential transform method (DTM) is employed to analyze not only the nonlinear heat conduction but also the resulting thermal stresses. Analytical solutions are developed for the temperature and thermal stresses in the form of simple power series. Numerical calculations are performed for an annular cooling/heating fin of variable thickness. Numerical results show that the sufficiently converged analytical solutions are in good agreement with the solutions obtained by the Adomian decomposition method and give the effects of the temperature-dependent parameters on the temperature and thermal stress profiles in the disk. The DTM is useful as a new analytical method for solving thermoelastic problems for a body with temperature-dependent parameters including material properties.     

Mechanical buckling of two-dimensional functionally graded cylindrical shells surrounded by Winkler–Pasternak elastic foundation

In this research, buckling analysis of a two-dimensional, functionally graded, cylindrical shell that has been embedded in an outer elastic medium in the presence of combined axial and transverse loading based on third-order shear deformation shell theory is numerically investigated. Variations of the shell properties are considered to be continuous through length and thickness. Winkler–Pasternak foundation and simply supported boundary conditions have been applied. The problem has been solved using the generalized differential quadrature method. Geometrical, load, and foundation parameters beside functionally graded power indexes effects on the critical buckling load have been studied.     

Static,free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique

In this paper the authors derive a higher-order shear deformation theory for modeling functionally graded plates accounting for extensibility in the thickness direction.The explicit governing equations and boundary conditions are obtained using the principle of virtual displacements under Carrera’s Unified Formulation. The static and eigenproblems are solved by collocation with radial basis functions.The efficiency of the present approach is assessed with numerical results including deflection, stresses, free vibration, and buckling of functionally graded isotropic plates and functionally graded sandwich plates.