Creep modeling in functionally graded rotating disc of variable thickness

Creep behavior of rotating discs made of functionally graded materials with linearly varying thickness has been investigated. The discs under investigation are made of composite containing silicon carbide particles in a matrix of pure aluminum. The creep behavior of the composite has been described by threshold stress based creep law by assuming a stress exponent of 5. The effect of imposing linear particle gradient on the distribution of stresses and strain rates in the composite disc has been investigated. The study indicates that with increase in particle gradient in the disc, the radial stress increases throughout the disc, whereas the tangential and effective stresses increase near the inner radius but decrease near the outer radius. The steady state strain rates in the composite disc, having gradient in the distribution of reinforcement, are significantly lower than that observed in a disc having uniform distribution of reinforcement.     

Effects of gradient on stress distribution in rotating functionally graded solid disks

This paper presents the elastic analysis of a rotating sandwich solid disk composed of three-layered perfectly-bonded composites. The center and outer regions of the solid disk are two homogeneous isotropic media, and the middle region is a transition zone made of functionally graded materials (FGM). Two cases are considered. One is on FGM with power-law gradient. For this case, explicit expressions for full elastic fields are derived. The other is on FGM with any radial nonhomogeneity, for which an analytic approach is proposed to reduce the problem into a Fredholm integral equation. The validity of the latter method is ensured by comparing numerical results of the elastic fields with the exact ones for the case of power-law gradient. For a general case, the current study presents the influences of the gradient, width variation of the FGM, and rotational velocity on the distribution of the radial and hoop stresses for a rotating composite disk made of aluminum and zirconia. The method presented in this paper may help engineering designers in choosing appropriate gradients and materials to acquire the optimal state of a rotating functionally graded disk.     

Quasi-three-dimensional elastic stresses in rotating disks

A method is presented to obtain stresses and displacements in rotating disks by taking into account the effect of out-of-plane restraint conditions at the hub. The stresses and displacements are obtained in a non-dimensional form, presented in the form of graphs and compared with the generalized plane stress solution.     

Elastic stresses in a rotating anisotropic annular disk of variable thickness and variable density

Closed-form solutions for stresses and displacement in an anisotropic rotating circular disk of variable thickness and variable density are obtained. It is found that the stresses and displacement induced are lower when the mass density of the material of the disk increases radially. Numerical results showing the degree of anisotropy and density variation on stresses and displacement are presented graphically.     

Three-dimensional solution for transient thermal stresses of functionally graded rectangular plate due to nonuniform heat supply

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a functionally graded rectangular plate due to nonuniform heat supply. The thermal and thermoelastic constants of the rectangular plate are assumed to vary exponentially in the thickness direction. The transient three-dimensional temperature is analyzed by the methods of Laplace and finite cosine transformations. We obtain the three-dimensional solution for the simple supported rectangular plate. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the nonhomogeneity of the material is investigated.     

On the rotating elastic–plastic solid disks of variable thickness having concave profiles

In this paper, an analytical solution for the elastic–plastic stress distribution in rotating variable thickness solid disks is presented. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. It is shown that depending on the shape of the disk profile, the radial stress in the central region may exceed the circumferential stress. The plastic zone which develops away from the axis of the disk consists of three annular regions governed by different mathematical forms of the yield criterion. The propagation of these plastic regions with increasing angular velocity is obtained together with the distributions of stresses and deformations in nondimensional forms.     

Elastic stresses and failure of rotating cross-ply laminate disks

Centrifugal force acting on a rotating disk induces in-plane loads in the radial and circumferential directions. An application of fiber-reinforced composite materials to a rotating disk can satisfy the demand for increasing its rotating speed. However, previous researches have been confined to single lamina disks. In this paper, a stress analysis of rotating disks made of multidirectional laminates is performed. The Tsai-Wu failure criterion is applied to the strength analysis of the rotating laminate disks, from which the maximum allowable speeds are obtained. The cross-ply ratios providing the maximum allowable speeds are obtained for the glass/epoxy and carbon/epoxy cross-ply laminate disks with CD geometry. This paper was recommended for publication in revised form by Associate Editor Heoung Jae Chun Kyo-Nam Koo received his B.S. and M.S. degrees in Aerospace Engineering from Seoul National University, Korea, in 1987 and 1989, respectively. He then received his Ph.D. degree from Korea Advanced Institute of Science and Technology in 1994. Dr. Koo had worked with Korea Aerospace Research Institute for seven years after his education. He is currently a Professor at the Department of Aerospace Engineering at University of Ulsan in Korea since 2001. His research interests include composite materials, structural dynamics, finite element analysis, and aeroelasticity.     

Effect of acceleration stresses on the yielding of rotating disks

The influence of shearing stresses arising due to angular acceleration in a rotating disk on yielding has been examined using von Mises yield criterion. Yield loci are plotted for both uniform and non-uniform thickness disks. The values of angular velocity and angular acceleration are reported and compared with the values given by Reid,⁴ which is based on Tresca's yield criterion.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

An exact solution for functionally graded piezoelectric laminates in cylindrical bending

In the present paper, the exact solutions of a simply supported functionally graded piezoelectric plate/laminate under cylindrical bending are derived. With similar derivation procedure as that used for Stroh formalism, the eigenrelation and general solutions for the problems can be expressed in very concise forms, which are convenient for further treatments of both analytic and numerical studies. The exact solutions can be served as benchmarks to verify and improve various approximate theories and numerical methods. To show the influence of material gradients, numerical examples based on the exact solutions are given, and some properties of the mechanical and electric responses of the plates under mechanical and electrical forces are discussed.     

Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions

Analytical solutions for the elastic–plastic stress distribution in rotating variable thickness annular disks are obtained under plane stress assumption. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. The thickness of the disk is assumed to vary in parabolic form in radial direction which leads to hypergeometric differential equations for the solution. It is shown that, depending on the boundary conditions used, the plastic core may contain one, two or three different plastic regions governed by different mathematical forms of the yield criterion. The expansion of these plastic regions with increasing angular velocity is obtained together with the distributions of stress, displacement and plastic strain. It is also shown mathematically that in the limiting case the variable thickness disk solution reduces to the solution of rotating uniform thickness disk.     

Dynamic creep rupture of a rotating disk of variable thickness

An analysis is provided of the finite strains of rotating disks followed by damage and creep rupture front motion as described by Kachanov's hypothesis. The creep is represented by the generalized Norton-Odqvist law for the true stresses and logarithmic strains. An iterative method of solution is proposed. It allows for the inclusion of inertia forces in the formulation of equilibrium. Two numerical examples illustrate the deformation process, ductile and ductile-brittle creep rupture.     

Effect of material inhomogeneity on the rotating functionally of a graded orthotropic hollow cylinder

The effect of the material inhomogeneity on the stress field of a rotating orthotropic infinite hollow cylinder made of functionally graded materials is investigated. The original functionally graded hollow cylinder is approximated by the laminate model, of which the solution will gradually approach the exact one with the increase in number of fictitious layers. The analysis is performed by means of the state space method. The validity of the solution is verified by utilizing the exact solution of a special non-homogeneous rotating hollow cylinder and earlier results. The distributions of the radial and tangential stresses of the internally pressurized rotating functionally graded hollow cylinder for various material inhomogeneity parameters are depicted graphically. The degree of the orthotropy on the stress fields is also discussed.     

Temperature and thickness effects on thermal and mechanical stresses of rotating FG-disks

In the present paper, radial and hoop thermal and mechanical stress analysis of a rotating disk made of functionally graded material (FGM) with variable thickness is carried out by using finite element method (FEM). To model the disk by FEM, one-dimensional two-degree elements with three nodes are used. It is assumed that the material properties, such as elastic modulus, Poisson’s ratio and thermal expansion coefficient, are considered to vary using a power law function in the radial direction. The geometrical and boundary conditions are in the shape of two models including thermal stress (model-A) and mechanical stress (model-B). In model-A there exists no pressure in both external and internal layers, and there is a temperature distribution considered as a second order function in the radial direction of the rotating disk. In this case, the temperature dependency of the material properties is considered and a hyperbolic type is assumed for the geometry of the disk. In model-B, there is a constant pressure only on the internal layer and a pressure on the internal layer of the disk without temperature distribution but with different types of surface profiles. Furthermore, the displacements and stresses for various power law indices (N) and angular velocities are calculated and compared to other results in the literature. The effect of varying thicknesses and dependency of material properties on temperature distribution is investigated.     

Flexure of orthotropic rotating disks

The influence of off-set in mounting the blades on the disk, termed “eccentricity”, and the centrifugal stiffening on the stresses and deflection induced in laterally loaded orthotropic disks of variable thickness is analysed. The analysis is based on a series solution for the differential equation governing the deflection of the disk. Numerical results showing the effects of anisotropy, eccentricity and rotation on the stresses and deflection of the disk are presented graphically. It is shown that the stress due to radial bending moment reduces significantly with the increase in the degree of anisotropy.     

Hydrodynamic drag versus roughness for rotating disks

Stylus measurements of the microroughness of rotating disks and their significant correlation with hydrodynamic drag measurements were studied. The roughest disks (R_a ≈ 16 μm) were found to have drag coefficients about 30% greater than those of the smoothest disks (R_a ≈ 0.14 μm). For a Reynolds' number of 1.5 × 10⁶, the following empirical relationship between the drag coefficient C, the average roughness R_a and a parameter designated the peak count wavelength λ_pc was obtained:

$\text{C=}\text{bR}{}_{\mathrm{a}}\text{λ}{}_{\mathrm{pc}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ C}{}_{\mathrm{o}}$

where .Other surface parameters and functions were measured in addition to R_a and λ_pc; however, it seems that knowledge of an amplitude-sensitive parameter and a wavelength-sensitive parameter is adequate for characterizing increases in the drag of rotating disks due to surface roughness. R_a is the preferred amplitude-sensitive parameter because it is the most widely used. We propose λ_pc as the wavelength parameter because it is sensitive to the larger profile features rather than to the fine structure.However, there is a need for all workers in the field to standardize on measuring the same parameters so that the results of one group can be easily related to the work of other groups.     

Thermoelastic analysis of non-uniform pressurized functionally graded cylinder with variable thickness using first order shear deformation theory(FSDT) and perturbation method

Recently application of functionally graded materials(FGMs) have attracted a great deal of interest. These materials are composed of various materials with different micro-structures which can vary spatially in FGMs. Such composites with varying thickness and non-uniform pressure can be used in the aerospace engineering. Therefore, analysis of such composite is of high importance in engineering problems. Thermoelastic analysis of functionally graded cylinder with variable thickness under non-uniform pressure is considered. First order shear deformation theory and total potential energy approach is applied to obtain the governing equations of non-homogeneous cylinder. Considering the inner and outer solutions, perturbation series are applied to solve the governing equations. Outer solution for out of boundaries and more sensitive variable in inner solution at the boundaries are considered. Combining of inner and outer solution for near and far points from boundaries leads to high accurate displacement field distribution. The main aim of this paper is to show the capability of matched asymptotic solution for different non-homogeneous cylinders with different shapes and different non-uniform pressures. The results can be used to design the optimum thickness of the cylinder and also some properties such as high temperature residence by applying non-homogeneous material.     

Analytical and numerical methods of solution of three-dimensional problem of elasticity for functionally graded coated half-space

In the present paper, a three-dimensional problem of elasticity for homogeneous half-space with gradient coating is considered. Poisson's ratio of the layer is constant and its Young's modulus is a power function of the distance from the surface of the half-space. The surface of non-homogeneous half-space is under tangential loading applied in circular area. Analytical solution is obtained using Fourier integral transform technique. The analytical solution of the problem in continuous dependence of the Young modulus is compared with the solution of the problem in which the inhomogeneous layer is modeled by the package of homogeneous layers. In sub-layers of a package and in substrate is constructed an analytical solution satisfying the conditions of ideal mechanical contact at interlines. Good agreement between results is obtained.     

A four variable refined plate theory for nonlinear cylindrical bending analysis of functionally graded plates under thermomechanical loadings

In this paper we present a new application for a four variable refined plate theory to analyse the nonlinear cylindrical bending behavior of functionally graded plates subjected to thermomechanical loadings. This recent theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The non-linear strain-displacement relations in the von Karman sense are used to study the effect of geometric non-linearity. The solutions are achieved by minimizing the total potential energy and the results are compared to the classical and the first-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the nonlinear cylindrical bending behavior of functionally graded plates.     

Vibration of functionally graded cylindrical shells

Functionally gradient materials (FGMs) have attracted much attention as advanced structural materials because of their heat-resistance properties. In this paper, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to a volume fraction power-law distribution. The results show that the frequency characteristics are similar to that observed for homogeneous isotropic cylindrical shells and the frequencies are affected by the constituent volume fractions and the configurations of the constituent materials. The analysis is carried out with strains–displacement relations from Love’s shell theory and the eigenvalue governing equation is obtained using Rayleigh–Ritz method. The present analysis is validated by comparing results with those in the literature.