Disk Of Variable Thickness Research Articles

Theoretical Stress Analysis of Rotating Hyperbolic Disk without Singularities Subjected to Thermal Load

A theoretical method for the evaluation of elastic stresses and strains in rotating hyperbolic disks without singularity, also subjected to thermal load, is introduced. The differential equation governing the radial displacement field of the hyperbolic disk without singularity has been deduced from the governing equation of non-linearly variable thickness disks given by the power of a linear function of the radius. A general temperature distribution along the radius expressed by a polynomial relation is considered. To overcome convergence problems of the solution, with hypergeometric functions, analytic continuations of these series have been introduced that allow the extension the theoretical thermoelastic analysis to all cases of technical interest. Then, for the first time the authors are aware, closed-form solutions of the governing equation of hyperbolic disks without singularity are presented.

Effect of Thermal Gradient on Steady State Creep in a Rotating Disc of Variable Thickness

Abstract The steady state creep in a rotating FGM disc having linearly varying thickness has been analyzed in the presence of linear thermal gradient. The disc is made of composite containing silicon carbide particles (SiCp) in a matrix of pure aluminium. The content of SiCp in the disc decreases non-linearly from the inner to the outer radius. The distributions of stresses and strain rates, resulting due to disc rotation, have been estimated by imposing various forms of linear thermal gradient in the FGM disc. The results are also estimated from a similar FGM disc but operating at a constant temperature, estimated by averaging the thermal gradient imposed in the disc. The results indicates that the tangential stress increases near the inner radius but decreases towards the outer radius and the radial stress increases throughout when the FGM disc is assumed to operate under a linear thermal gradient. However, the strain rates in the FGM disc decreases significantly in the presence of thermal gradient as compared to those observed in the FGM disc operating at a constant average temperature.

An Efficient Unified Method for Thermoelastic Analysis of Functionally Graded Rotating Disks of Variable Thickness

Thermoelastic analysis of functionally-graded nonuniform rotating disks subjected to nonuniform change in temperature essentially requires solutions of two-point-boundary-value problems. Spatial variations of thermomechanical properties, thickness, and temperature change constitute governing differential equations with variable coefficients. Closed-form solutions to such equations can be obtained only for specific forms of grading functions. For general variations of the properties, numerical methods must be resorted to, and the complementary functions method (CFM) stands as a viable option. Infusion of CFM into the thermoelastic analysis of functionally-graded rotating disks of variable thickness is a unified novel approach; it can efficiently be used for both heat conduction problems and thermal stress analysis. Complementary functions method reduces the boundary-value problem to an initial-value problem, which can be solved accurately by one of many efficient methods, such as Runge-Kutta method. Material models available in the literature for ceramic-metal disks are used in the analysis. First, the proposed solution scheme is verified using simple closed-form solutions for a disk of uniform properties and then displacement and stresses are calculated for nonuniform heterogeneous disks. Use of CFM has given virtually exact results for few collocation points along the radius. Fifth-order Runge-Kutta method (RK5) has been employed as the numerical method.

Thermal Stresses of Rotating Hyperbolic Disks as Particular Case of Non-Linearly Variable Thickness Disks

A theoretical thermoelastic analysis of the Stodola's hyperbolic disk, axisymmetric and symmetric with respect to the mid-plane, and subjected to thermal load is carried out. A general temperature distribution along the radius expressed by a polynomial relation is introduced. The disk is analyzed as a particular case of the more general disk profile having non-linearly variable thickness given by the power of a linear function of the radius. The differential equation governing the radial displacement field of the Stodola's disk subjected to thermal load is deduced from the equivalent more general equation of this non-linearly variable thickness disk. The governing equation so obtained is analytically integrated and a closed-form solution is presented. Theoretical results are validated by comparing them with those obtained by means of FEA, after performing some significant calculation examples.

Formation of large-scale dust lanes in spiral arms of galaxies

We consider the interaction of interstellar dust grains with a galactic shock in the gaseous component. Typical parameters of dust grains and spiral density waves imply that the formation of large-scale dust lanes at the front of a galactic shock is possible only in models taking into account a self-focusing phenomenon. In the case of an isothermal flow of interstellar gas through a spiral arm in a model with a gaseous disk of variable thickness, dust lanes can be projected onto the region of increased gas density, although this is not associated with a galactic shock. The dust density peak derived from the classical model of a galactic shock (isothermal flow and a constant thickness of the gaseous disk) is appreciably shifted downstream of the gas flow, so that it does not outline the gas density maximum.

Large deflection thermoelastic analysis of shear deformable functionally graded variable thickness rotating disk

In this paper, large deflection thermoelastic analysis of functionally graded (FG) solid and hollow rotating axisymmetric disk with uniform and variable thickness subjected to thermo-mechanical loading is studied. To achieve this goal, in addition to the uniform transverse loading, the rotating FG disk is subjected to a thermal gradient along the radial direction. Three different profiles (convex, linear and concave) are considered for variable thickness disk. The mechanical and thermal properties of FG disk are assumed to vary continuously along the radial direction by the Mori–Tanaka distribution. The nonlinear formulations are based on first order shear deformation theory (FSDT) and large deflection von Karman equations. The dynamic relaxation (DR) method combined with the finite difference discretization technique is employed to solve the equilibrium equations. Effects of grading index, angular velocity, boundary condition, thermal loading, ratios of thickness to radius and thickness profile of the disk are studied in detail. Also some linear and nonlinear analysis with different thickness-to-radius ratios is carried out based on classical plate theory (CPT) and FSDT to consider the effect of shear deformation and nonlinearity on the results.

Homogenization of a quasilinear parabolic problem with different alternating nonlinear fourier boundary conditions in a two-level thick junction of the type 3:2:2

We investigate the asymptotic behavior of a solution of a quasilinear parabolic boundary-value problem in a two-level thick junction of the type 3:2:2. This junction consists of a cylinder on which thin disks of variable thickness are e-periodically threaded. The thin disks are divided into two levels, depending on their geometric structure and the conditions imposed on their boundaries. In this problem, we consider different alternating, inhomogeneous, nonlinear Fourier conditions; moreover, the Fourier conditions depend on additional perturbation parameters. We prove theorems on the convergence of a solution of this problem as e → 0 for different values of these parameters.

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.

Dynamic Analysis of Rotating Annular Disk of Variable Thickness under Uniform Axial Pressure

The large-amplitude free vibration analysis of a rotating annular disk of exponentially varying thickness, subjected to uniform axial pressure, is presented. The entire formulation is energy-based and uses variational principles to derive the governing equations. The study is carried out up to the onset of yielding, which occurs due to the application of two types of external loadings, namely body force due to rotation and uniform axial pressure. The results are compared for three exponential disks having the same mass. The effects of the individual loadings and their combination on the free vibration dynamic behavior are presented.

Mathematical modelling of steady state creep in a functionally graded rotating disc of variable thickness

A mathematical model has been developed to investigate steady state creep in a rotating disc having linearly varying thickness. The disc is assumed to be composed of functionally graded (FG) composite containing linearly varying content of silicon carbide particles (SiCp) reinforced in a matrix of pure aluminium (Al). The disc material undergoes steady state creep as described by a creep law based on threshold stress with a stress exponent of 5. The mathematical model developed has been employed to investigate the effect of imposing various linear gradients of SiCp on the creep performance of the FG disc. The study indicates that with the increase in SiCp gradient, the radial stress increases throughout the disc. However, the increase in SiCp gradient results in increase of tangential and effective stresses near the inner radius of the disc but a decrease near the outer radius. The tangential as well as radial strain rates in the FG disc reduce to a significant extent with the increase in SiCp gradient.

Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods

In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.

Stability Loss of Rotating Elastoplastic Discs of the Specific Form

A method of calculating a possible stability loss by a rotating circular annular disc of variable thickness is suggested within the theory of perfect plasticity with the help of small parameter method. A characteristic equation for a critical radius of a plastic zone is obtained as a first approximation. The formula for the critical angular velocity, determining the stability loss of the disc according to the self-balanced form, is derived. The method using which we can take into account the disc’s geometry and loading parameters is also specified. The efficiency of the proposed method is shown in Section 5 while considering an illustrative example. The values of critical angular velocity of rotating are found numerically for different parameters of the disc.

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.

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.

Steady state creep in a rotating composite disc of variable thickness

Abstract The steady state creep in a rotating disc having variable thickness and made of isotropic aluminium–silicon carbide particulate composite has been investigated. Using threshold stress based creep law, the general expressions for stresses and strain rates in the discs have been obtained. The expressions are used to calculate the distributions of stresses and strain rates in different discs viz (i) disc having constant thickness (ii) disc having linearly varying thickness and (iii) disc having hyperbolically varying thickness. The volume of different discs is kept the same. The study revealed that the stresses and strain rates in the disc could be reduced to a significant extent by varying the disc profile. The linearly varying disc exhibits the lowest values of stresses and strain rates compared to those observed in hyperbolic or uniform thickness disc.

Elastic–plastic analysis of rotating disks having non-linearly variable thickness: residual stresses by overspeeding and service stress state reduction

Two models for evaluation of elastic and elastic–plastic stress and strain in non-linear variable thickness rotating disks, either solid or annular, subjected to such a rotational speed to determine stresses beyond yielding are presented. The first model regards the elastic field and the second concerns the elastic–plastic field, assuming that material hardening is isotropic and follows a most general hardening power-law. In the elastic region, a non-homogeneous hypergeometric differential equation is obtained. Such differential equation, which solves the elastic problem in closed form for non-linear variable thickness disks, with thickness given by the power of a linear function which may define a fourfold infinity of profiles, is integrated in closed form. As concerns the elastic–plastic analysis, first of all the introduction is made of a correlation between equivalent plastic strain and equivalent stress according to Von Mises, which is more general than those known from literature. In the case of isotropic hardening a second-order, non-homogeneous, non-linear differential equation is found, which governs the stress state in the plastic region of the disk. The procedure allows to calculate stress and displacement states in the two regions—plastic and elastic—of the disk subjected to prestressing by overspeeding. Several examples of disks, also prestressed by overspeeding, are considered (annular and solid, convex or concave, and linear tapered disks); the matching of results of the theoretical model and those obtained by means of FEA is very good. Lastly, the residual stress state can be found in a prestressed disk by overspeeding and the stress state in the actual operating condition in the same disk is evaluated.

Homogenization of elliptic problems with alternating boundary conditions in a thick two-level junction of type 3:2:2

The asymptotic behavior of solutions to boundary value problems for the Poisson equation is studied in a thick two-level junction of type 3:2:2 with alternating boundary conditions. The thick junction consists of a cylinder with e-periodically stringed thin disks of variable thickness. The disks are divided into two classes depending on their geometric structure and boundary conditions. We consider problems with alternating Dirichlet and Neumann boundary conditions and also problems with different alternating Fourier (Neumann) conditions. We study the influence of the boundary conditions on the asymptotic behavior of solutions as e → 0. Convergence theorems, in particular, convergence of energy integrals, are proved. Bibliography: 31 titles. Illustrations: 1 figure.

Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness

In this paper, the analytical and numerical solutions for rotating variable-thickness solid disk and numerical solution for rotating variable-thickness annular disk are presented. The outer edge of the solid disk and the inner and outer edges of the annular disk are considered to have clamped boundary conditions. Two different cases for the radially varying thickness of the solid and annular disks are given. The numerical solution as well as the analytical solution is available for the first case of the solid disk while the analytical solution is not available for the second case of the annular disk. Both analytical and numerical results for displacement and stresses will be investigated for the first case of radially varying thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second case of radially varying thickness is investigated. Finally, the distributions of displacement and stresses will be presented and the appropriate comparisons and discussions are made at the same angular velocity.

Vortices in a mesoscopic superconducting disk of variable thickness

In this paper we study the formation of vortices in a mesoscopic superconductingdisk of thickness equal to or of the same order as the coherence lengthξ(T). The roughness of the lower and upper surfaces of the disk is taken into account. We alsocompute the total induced current and the magnetization for the disk in the presence of anexternal applied magnetic field.

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.