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 a functionally graded piezoelectric rotating hollow cylindrical shell subjected to dynamic loads

In this study, dynamo-thermo-elastic analysis of a rotating piezoelectric hollow cylinder made of functionally material is presented. Coupled differential quadrature and finite difference methods are used to solve boundary/initial value equations of the problem. Material properties are assumed to be graded in radial direction and temperature independent. Numerical results obtained and convergence are studied, and then verified with reported results in literature. Effect of variations of the grading parameter, angular velocity, thermal gradient and ratio of the outer to inner radii on the stresses, radial displacement and electrical potential are presented.     

Transient thermoelasticity analysis of functionally graded thick hollow cylinder based on Green–Lindsay model

The transient thermoelastic response of a thick hollow cylinder made of functionally graded material under thermal loading is studied. The generalized theory of thermoelasticity based on Green?CLindsay model is used in this paper. The thermal and mechanical properties of the functionally graded material are assumed to be varied in the radial direction according to a power law variation as a function of the volume fractions of the constituents. The heat conduction equation and the equation of motion are solved by using Galerkin finite element method. All the finite element calculations were done by using commercial finite element program FlexPDE. The transient temperature, radial displacement, and thermal stresses distribution through the radial direction of the cylinder are plotted. The material composition effect on temperature, radial displacement and thermal stresses is shown.     

Analysis of two-dimensional functionally graded rotating thick disks with variable thickness

Elasticity solutions of two-dimensional functionally graded rotating annular and solid disks with variable thickness are presented. Material properties vary through both the radial and axial directions continuously. Axisymmetric conditions are assumed for the two-dimensional functionally graded disk. The graded finite element method (GFEM) has been applied to solve the equations. The distributions of displacements and stresses in radial and axial directions for four different thickness profiles (constant, linear, concave and convex) and various power law exponents have been investigated. The achieved results show that by the use of functionally graded materials and variable thicknesses, the stresses are reduced, so a higher capability of angular velocity can be obtained. Also, using two-dimensional functionally graded materials leads to a more flexible design in comparison with conventional one-dimensional functionally graded materials. The GFEM solution of a functionally graded thin rotating annular disk has been compared with the published literature and it shows good agreement.     

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.     

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.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere

This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow sphere are expressed as power functions of the radial coordinate. The one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained herein. Some numerical results for the temperature change, displacement, stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated.     

Reliability of stress field in Al–Al2O3 functionally graded thick hollow cylinder subjected to sudden unloading,considering uncertain mechanical properties

In this paper, the reliability analysis and safety evaluation of dynamic stresses are presented for Al–Al₂O₃ functionally graded (FG) thick hollow cylinder subjected to sudden unloading as a mechanical shock loading. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of Al and Al₂O₃ are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. The radial and hoop stresses are obtained by solving Navier equation in displacement form and stress–displacement equations. The FG cylinder is divided to many linear functionally graded elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical problem. The reliability of radial and hoop stresses are calculated in various points across thickness for different grading patterns in functionally graded material (FGM) and several yield stresses. The variability of the dynamic stress reliability of the FG cylinder to the values of coefficients of variation (COVs) is examined and discussed in details.     

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.     

Static deformations of functionally graded polar-orthotropic cylinders with elliptical inner and circular outer surfaces

Functionally graded materials (FGMs) enable one to tailor the spatial variation of material properties so as to fully use the material everywhere. For example, in a hollow circular cylinder one can vary, in the radial direction, the material moduli to make the hoop stress constant. Whereas the problem for a hollow cylinder with the inner and the outer surfaces circular has been studied, that of a cylinder with a circular outer surface and a non-circular inner surface or vice versa has not been investigated. We study here such a plane-strain problem when the cylinder material is polar-orthotropic, material properties vary exponentially in the radial direction, and deformations are independent of the axial coordinate. The problem is challenging since the cylinder thickness varies with the angular position of a point, and the cylinder material is inhomogeneous. Equilibrium equations are solved by expanding the radial and the circumferential displacements in Fourier series in the angular coordinate. The method of Frobenius series is used to solve ordinary differential equations for coefficients of the Fourier series, and boundary conditions are satisfied in the sense of Fourier series. A parametric study has been conducted that delineates effects on stresses of the eccentricity of the ellipse, the material property gradation index and loads applied on boundaries of the cylinder. The analytical solutions presented here will serve as benchmarks for comparing solutions derived by numerical methods.     

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.     

Multiphysical time-dependent creep response of FGMEE hollow cylinder in thermal and humid environment

In this paper, we investigate the history of radial displacement, stresses, electric potential, and magnetic potential of a functionally graded magneto-electro-elastic (FGMEE) hollow cylinder subjected to an axisymmetric hygro-thermo-magneto-electro-mechanical loading for the plane strain condition. The material properties are taken as a power-law function of radius. Using stress-displacement relations, equations of equilibrium, electrostatic and magnetostatic equations, we find a differential equation including creep strains. Initially, eliminating creep strains, we obtain an analytical solution for the primitive stresses and electric and magnetic potential. In the next step, considering creep strains, we find the creep stress rates by applying the Norton law and Prandtl–Reuss equations for steady-state hygrothermal boundary condition. Finally, using an iterative method, we find the time-dependent creep stresses, radial displacement, and magnetic and potential field redistributions at any time. In numerical section, are comprehensively investigate the effects of grading index, hygrothermal environmental conditions, rotating speed, and temperature- and moisture-dependency of elastic constant of FGMEE.

     

Analysis of elastic wave propagation in a functionally graded thick hollow cylinder using a hybrid mesh-free method

In this paper, a hybrid mesh-free method based on generalized finite difference (GFD) and Newmark finite difference (NFD) methods is presented to calculate the velocity of elastic wave propagation in functionally graded materials (FGMs). The physical domain to be considered is a thick hollow cylinder made of functionally graded material in which mechanical properties are graded in the radial direction only. A power-law variation of the volume fractions of the two constituents is assumed for mechanical property variation. The cylinder is excited by shock loading to obtain the time history of the radial displacement. The velocity of elastic wave propagation in functionally graded cylinder is calculated from periodic behavior of the radial displacement in time domain. The effects of various grading patterns and various constitutive mechanical properties on the velocity of elastic wave propagation in functionally graded cylinders are studied in detail. Numerical results demonstrate the efficiency of the proposed method in simulating the wave propagation in FGMs.     

Green's function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material

Summary A Green's function approach based on the laminate theory is adopted for solving the two-dimensional unsteady temperature field (r, z) and the associated thermal stresses in an infinite hollow circular cylinder made of a functionally graded material (FGM) with radial-directionally dependent properties. The unsteady heat conduction equation is formulated as an eigenvalue problem by making use of the eigenfunction expansion theory and the laminate theory. The eigenvalues and the corresponding eigenfunctions obtained by solving an eigenvalue problem for each layer constitute the Green's function solution for analyzing the unsteady temperature. The associated thermoelastic field is analyzed by making use of the thermoclastic displacement potential function and Michell's function. Numerical results are carried out and shown in figures.     

Piezoelectric Behavior of an Inhomogeneous Hollow Cylinder with Thermal Gradient

An analytical solution to the axisymmetric problem of a radially polarized, radially orthotropic piezoelectric hollow cylinder with a thermal gradient and subjected to various boundary conditions is developed. The elastic coefficients, piezoelectric coefficients, stress-temperature moduli, dielectric coefficient, pyroelectric coefficients, thermal conductivity coefficient, and thermal expansion coefficients of the hollow cylinder are assumed to be graded in the radial direction according to a simple power-law distribution. The governing second-order differential equations are derived from the equilibrium equation, the charge equation of electrostatics, and steady state heat transfer equation through the radial direction of the inhomogeneous hollow cylinder. The displacement, stresses, and potential field distributions in the cylinder are examined. The influence of the inhomogeneity parameter on the numerical results is investigated.     

Transient solution of temperature field in functionally graded hollow cylinder with finite length using multi layered approach

In this paper by using a multi-layered approach based on the theory of laminated composites, the solution of temperature in functionally graded circular hollow cylinders subjected to transient thermal boundary conditions are presented. The material properties are assumed to be temperature-independent and radially dependent. The cylinder has finite length and is subjected to axisymmetric thermal loads. It is assumed that the functionally graded circular hollow cylinder is composed of N fictitious layers and the properties of each layer are assumed to be homogeneous and isotropic. Employing Laplace transform techniques and series solving method for two-dimensional heat conduction equation in the cylinder, solutions for the variation of temperature with time as well as temperature distribution through the cylinder are obtained.     

General analytical solution for elastic radial wave propagation and dynamic analysis of functionally graded thick hollow cylinders subjected to impact loading

An analytical method is proposed for the dynamic response analysis of functionally graded thick hollow cylinders under impact loading. The wave motion equation is solved using an analytical method that is based on the composition of Bessel functions. The mechanical properties are considered as power functions of the radius across the thickness of FG cylinder. The FG cylinder is excited by an impact loading at the inner surface of the cylinder, and the plane strain and axisymmetry conditions are assumed for the problem. The time histories of radial displacement and radial and hoop stresses are presented. Also the dynamic response of the FG cylinder is obtained and discussed for various kinds of power function exponents.     

Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory

This paper is concerned with the investigation of thermoelastic displacements and stresses in a functionally graded spherically isotropic hollow sphere due to prescribed temperature in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). Both the surfaces of the body are free from radial stresses, and the inner surface is subjected to a time-dependent thermal shock whereas the outer one is maintained at constant temperature. The basic equations have been written in the form of a vector–matrix differential equation in the Laplace transform domain which is then solved by an eigenvalue approach. The numerical inversion of the transforms is carried out using a method of Bellman et al. The displacements and stresses are computed and presented graphically. It is found that the variation of the thermophysical properties of a material as well as the thickness of the body strongly influence the response to loading. A comparative study with the corresponding homogeneous material has also been made. The solution of the problem of a spherically isotropic infinite medium containing a spherical cavity has been derived theoretically by tending the outer radius to infinity, as a particular case.     

Thermo-elastic/plastic semi-analytical solution of incompressible functionally graded spherical pressure vessel under thermo-mechanical loading

In this paper, numerical solutions to assess partially plastic and fully plastic deformation behavior of a functionally graded spherical pressure vessel are presented. The modulus of elasticity of the material is assumed to vary nonlinearly in the radial direction and axisymmetric displacements and stresses in the functionally graded spherical vessel subjected to thermal loading and uniform internal pressure are determined using plasticity theory. Tresca??s yield criterion and its associated flow rule are used to formulate different plastic regions for an ideal FG material. In this way, the material property varies by Young??s modulus that may be an arbitrary function of the radial coordinate. Therefore, the material is assumed to be functionally graded in the radial direction. Hence, the general analytical solutions of such equations are not available, the numerical method (semi-analytical) is applied and a new collection of equilibrium equations with small deflections is presented. Accordingly, the radial domain is divided into some virtual sub-domains in which the power-law distribution is used for the thermomechanical properties of the elemental components. By considering the necessary continuity conditions between adjacent sub-domains, jointly with the global boundary conditions, a set of linear differential equations is obtained. Solution of the linear differential equations yields the thermoelastic responses for each sub-domain as exponential functions of the radial coordinate. Subsequently, attributed to centrifugal force, results for the stress, strain, and displacement components along the radius in elastic and plastic area are presented.