Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Stress intensity factors for two offset parallel circular cracks: Part II — Semi-infinite solid

With the aid of the formulation in [1] (R. Muki, Progress in Solid Mechanics (North-Holland, 1961)) for general three-dimensional asymmetric problems and the superposition principle, Part II of this work makes use of the method in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [2]) to examine the interaction of arbitrarily located penny-shaped cracks in an infinite elastic solid to the case of a semi-infinite solid. As in Part I for the infinite body, the problem of a semi-infinite solid containing two penny-shaped cracks is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved for some special cases when cracks are far apart and far away from the boundary. Some asymptotic solutions are presented and comparisons are made with the results for the special case where there is only one crack under axisymmetric loading.     

基于无网格法的非均匀弹性地基上变厚度加筋板弯曲与固有频率分析

采用基于移动最小二乘近似的无网格方法并结合一阶剪切变形理论,分析了非均匀弹性地基上变厚度加筋板的弯曲和固有频率问题.首先,用节点对变厚度板和筋条分别进行离散,导出变厚度板和筋条的势能;其次,利用筋条与变厚度板之间的位移协调条件将筋条的节点参数转换为板的节点参数,再将两者的势能进行叠加得到变厚度加筋板的总势能,并根据能量...     

Coupled shear-torsional motion of a rubber support system

The finite amplitude, coupled shear-torsional motion of a circular disk supported between identical rubber spring cylinders is studied. The material of the springs is assumed to be an incompressible elastic material. The oscillatory motion oscillatory of the disk is studied for two different cases. In the first case, the material of the spring is assumed to be an incompressible elastic material whose response functions are constants. Typical examples include the Mooney-Rivlin model. The motion of the disk in this case is governed by two independent equations whose closed form solutions are noted. For the second case, the material of the spring is assumed to be an incompressible quadratic material. The motion oscillatory of the disk in this case is governed by two coupled nonlinear differential equations. The stability properties of small shearing oscillation superimposed on finite torsion and small torsional oscillation superimposed on finite shearing are studied.     

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material(FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.     

Transverse Vibrations of a Centrally Clamped Rotating Circular Disk

The transverse vibrations of a circular disk of uniform thickness rotatingabout its axis with constant angular velocity are analyzed. The resultsspecialized to the linear case of disks clamped at the center and free atthe periphery are in good agreement with those reported in the literature.The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitraryspatially fixed transverse loads. Using the method of multiple scales, wedetermine a set of four nonlinear ordinary-differential equations governingthe modulation of the amplitudes and phases of two interacting modes. Thesymmetry of the system and the loading conditions are reflected in thesymmetry of the modulation equations. They are reduced to an equivalentset of two first-order equations whose equilibrium solutions aredetermined analytically. The stability characteristics of thesesolutions is studied; the qualitative behavior of the response isindependent of the mode being considered.     

Limit angular velocities of variable thickness rotating disks

Elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form. Analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion. An efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks. Von Mises yield criterion and its flow rule is used to estimate plastic limit angular velocities. Both linear and nonlinear hardening material behaviors are treated numerically. The results are verified by comparing with those of uniform thickness rotating solid disks available in the literature. Elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile.     

MHD viscous flow between torsionally oscillating disks

The unsteady motion of an incompressible, viscous, stratified fluid between two parallel infinite disks maintained at different temperatures is studied under the influence of a uniform transverse magnetic field. The whole system is under rigid rotation in the initial state and perturbations are created by the small amplitude torsional oscillations of the disks. The time required for the transient velocity and temperature to decay is found for various ranges of the values of the forcing frequency of the disks. The steady state velocity and temperature distributions represent boundary layers on the disks and an interior flow. The interplay between the Hartmann number and the Ekman number in determining the boundary layers on the disks is discussed.     

Transient squeeze flow of viscoplastic materials

The transient, axisymmetric squeezing of viscoplastic materials under creeping flow conditions is examined. The flow of the material even outside the disks is followed. Both cases of the disks moving with constant velocity or under constant force are studied. This time-dependent simulation of squeeze flow is performed for such materials in order to determine very accurately the evolution of the force or the velocity, respectively, and the distinct differences between these two experiments, the highly deforming shape and position of all the interfaces, the effect of possible slip on the disk surface, especially when the slip coefficient is not constant, and the effect of gravity. All these are impossible under the quasi-steady state condition used up to now. The exponential constitutive model, suggested by Papanastasiou, is employed. The governing equations are solved numerically by coupling the mixed finite element method with a quasi-elliptic mesh generation scheme in order to follow the large deformations of the free surface of the fluid. As the Bingham number increases, large departures from the corresponding Newtonian solution are found. When the disks are moving with constant velocity, unyielded material arises only around the two centers of the disks verifying previous works in which quasi-steady state conditions were assumed. The size of the unyielded region increases with the Bingham number, but decreases as time passes and the two disks approach each other. Their size also decreases as the slip velocity or the slip length along the disk wall increase. The force that must be applied on the disks in order to maintain their constant velocity increases significantly with the Bingham number and time and provides a first method to calculate the yield stress. On the other hand, when a constant force is applied on the disks, they slow down until they finally stop, because all the material between them becomes unyielded. The final location of the disk and the time when it stops provide another, probably easier, method to deduce the yield stress of the fluid.     

Frictional contact of two rotating elastic disks

We study the problem of constrained uniform rotation of two precompressed elastic disks made of different materials with friction forces in the contact region taken into account. The exact solution of the problem is obtained by the Wiener-Hopf method.An important stage in the study of rolling of elastic bodies is the Hertz theory [1] of contact interaction of elastic bodies with smoothly varying curvature in the contact region under normal compression. Friction in the contact region is assumed to be negligible. If there are tangential forces and the friction in the contact region is taken into account, then the picture of contact interaction of elastic bodies changes significantly. Although the normal contact stress distribution strictly follows the Hertz theory for bodies with identical elastic properties and apparently slightly differs from the Hertz diagram for bodies made of different materials, the presence of tangential stresses results in the splitting of the contact region into the adhesion region and the slip region. This phenomenon was first established by Reynolds [2], who experimentally discovered slip regions near points of material entry in and exit from the contact region under constrained rolling of an aluminum cylinder on a rubber base. The theoretical justification of the partial slip phenomenon in the contact region, discovered by Reynolds [2], can be found in Carter [3] and Fromm [4]. Moreover, Fromm presents a complete solution of the problem of constrained uniform rotation of two identical disks. Apparently, Fromm was the first to consider the so-called “clamped” strain and postulated that slip is absent at the point at which the disk materials enter the contact region.Ishlinskii [5, 6] gave an engineering solution of the problem on slip in the contact region under rolling friction. Considering the problem on a rigid disk rolling on an elastic half-plane, we model this problem by an infinite set of elastic vertical rods using Winkler-Zimmermann type hypotheses. Numerous papers of other authors are surveyed in Johnson’s monograph [7].The exact solution of the problem on the constrained uniform rotation of precompressed rigid and elastic disks under the assumptions of Fromm’s theory is contained in the papers [8, 9]. In the present paper, we generalize the solution obtained in [8, 9] to the case of two elastic disks made of different materials.     

Free axisymmetric vibration of FGM circular plates

Based on the three-dimensional theory, this work presents a direct displacement method to investigate the free axisymmetric vibration of transversely isotropic circular plates, whose material is functionally graded and properties obey the exponential law along the thickness direction of the plate. Under two boundary conditions, the solution satisfies all basic equations and the Corresponding boundary condition at every point. Thus, it is three-dimensional exact. Numerical examples are presented and compared with previous works. The present method can also be extended to the case of arbitrary distribution of the material properties along the thickness direction of the plate.     

Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in non-linear variable thickness rotating disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. Thickness variation of disks is described by means of a power of linear function, which can be used to describe a fourfold infinity of actual disk profiles. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this theoretical procedure is just general and does not present limitations and drawbacks of the approaches as the one found in technical literature. General unpublished relations of stress state and displacement field in non-linear variable thickness disks subjected, under elastic conditions, to thermal gradient, and featuring a density variation along the radius are defined. Particular consideration is given to some industrial example of turbine rotors carrying hub and rim with buckets on periphery or radial blades on lateral surfaces. The analytical results obtained by using the new general relations perfectly match those obtained by FEA and overlap those concerning the special cases of tapered conical disks found in literature.     

On approximate large strain relations for a shell of revolution

A series of geometric and constitutive relations is studied for large axisymmetric strain of elastic shells of revolution. The kinematic assumption employs a modified Kirchhoff hypothesis which accounts for thickness changes but neglects transverse shear deformation. Calculations are presented for cylindrical and spherical shells composed of incompressible materials with two types of strain energy density function: Mooney-Rivlin (rubber) and exponential (biological tissue). Comparison of results for large bending at a clamped edge demonstrates the accuracy and limitations of various approximations for stress and strain. The computations indicate that the stress resultants are quite sensitive to the details of the asymmetric motion of points relative to the reference surface.     

On finite deformation of viscoelastic rotating disks

The governing equations for a rotating disk under large elastic or viscoelastic deformations, are formulated in terms of two coupled first-order ordinary differential-integral equations with explicit derivatives. The constitutive equations developed by Christensen are used for the formulation. A recurrent formula for the hereditary integrals is obtained. This recurrent formula makes it possible for the same governing equations and numerical methods to be applied to both elastic and viscoelastic problems.     

Von mises yield criterion and nonlinearly hardening variable thickness rotating annular disks with rigid inclusion

A computational model is developed to investigate inelastic deformations of variable thickness rotating annular disks mounted on rigid shafts. The von Mises yield condition and its flow rule are combined with Swift’s hardening law to simulate nonlinear hardening material behavior. An efficient numerical solution procedure is designed and used throughout to handle the nonlinearities associated with the von Mises yield condition and the boundary condition at the shaft–annular disk interface. The results of the computations are verified by comparison with an analytical solution employing Tresca’s criterion available in the literature. Inelastic stresses and deformations are calculated for rotating variable thickness disks described by two different commonly used disk profile functions i.e. power and exponential forms. Plastic limit angular velocities for these disks are calculated for different values of the geometric and hardening parameters. These critical angular velocities are found to increase as the edge thickness of the disk reduces. Lower plastic limit angular velocities are obtained for disks made of nonlinearly hardening materials.     

Free axisymmetric vibration of transversely isotropic piezoelectric circular plates

Based on three-dimensional elastic theory of piezoelectric materials, the axisymmetric state space formulation of piezoelectric laminated circular plates is derived. Finite Hankel transforms are used and the boundary variables in free terms are replaced, for two kinds of boundary conditions, to obtain ordinary differential equations with constant coefficients. Regarding the axisymmetric free vibration problem, two exact solutions for two different boundary conditions are found. Discarding piezoelectric effect, the exact solutions for transversely isotropic circular laminates are also obtained through the same procedure. Numerical examples are given and compared with those of Finite Element Method (FEM) .     

任意厚度层合闭口柱壳的轴对称温度应力

基于层合柱壳混合状态方程和边界条件的弱形式，在轴对称情况下，建立了两端固支层合闭口柱壳的温度应力混合方程，给出了任意厚层合柱壳在温度荷载和机械荷载共同作用下的解析解。     

Numerical computation of an impinging jet with uniform wall suction

The paper presents numerical predictions of a turbulent axisymmetric jet impinging onto a porous plate, based on a finite volume method of solving the Navier-Stokes equations for an incompressible air jet with the K–ε turbulence model. The velocity and pressure terms of the momentum equations are solved by the SIMPLE (semi-implicit method for pressure-linked equation) method. In this study, non-uniform staggered grids are used. The parameters of interest include the nozzle-to-wall distance and the suction velocity. The results of the present calculations are compared with available data reported in the literature. It is found that suction effects reduce the boundary layer thickness and increase the velocity gradient near the wall.