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.     

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.     

Numerical analysis of elastic–plastic rotating disks with arbitrary variable thickness and density

A unified numerical method is developed in this article for the analysis of deformations and stresses in elastic–plastic rotating disks with arbitrary cross-sections of continuously variable thickness and arbitrarily variable density made of nonlinear strain-hardening materials. The method is based on a polynomial stress–plastic strain relation, deformation theory in plasticity and Von Mises’ yield condition. The governing equation is derived from the basic equations of the rotating disks and solved using the Runge–Kutta algorithm. The proposed method is applied to calculate the deformations and stresses in various rotating disks. These disks include solid disks with constant thickness and constant density, annular disks with constant thickness and constant density, nonlinearly variable thickness and nonlinearly variable density, linearly tapered thickness and linearly variable density, and a combined section of continuously variable thickness and constant density. The computed results are compared to those obtained from the finite element method and the existing approaches. A very good agreement is found between this research and the finite element analysis. Due to the simplicity, effectiveness and efficiency of the proposed method, it is especially suitable for the analysis of various rotating disks.     

Elastoplastic deformations of rotating parabolic solid disks using Tresca''s yield criterion

Analytical solutions for the stress distribution in rotating parabolic solid disks are obtained. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening. It is shown that, the deformation behavior of the convex parabolic disk is similar to that of the uniform thickness disk, but in the case of concave parabolic solid disk, it is different. In the latter, the plastic core consists of three different plastic regions with different mathematical forms of the yield criteria. Accordingly, three different stages of elastic–plastic deformation occur. All these stages of elastic–plastic deformation are studied in detail. It is also shown mathematically that in the limiting case the parabolic disk solution reduces to the solution of rotating uniform thickness solid disk.     

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.     

Elastoplastic analysis of nonlinearly hardening variable thickness annular disks under external pressure

Based on von Mises’ yield criterion, deformation theory of plasticity and Swift’s hardening law, elasto-plastic deformation of variable thickness annular disks subjected to external pressure is studied. A nonlinear shooting method using Newton’s iterations with numerically approximated tangent is designed for the solution of the problem. Considering a thickness profile in the form of a general parabolic function, the condition of occurrence of plastic deformation at the inner and outer edges of the annular disk is investigated. A critical disk profile is determined and the corresponding elastic–plastic stresses as well as the residual stress distribution upon removal of the applied pressure are computed and discussed.     

加速旋转变厚度锥壳和圆盘中的应力

通过求解球坐系中的弹性力学平衡方程，得到线性变厚度锥壳和圆盘在角加速旋转中位移和应力的封闭形式解，从而发现锥壳式圆盘中的剪应力只与几何尺寸、材料密度和角加速度有关，而与材料的弹性常数无关，文末的数字结果表明，变厚度圆盘的变形和应力与等厚度圆盘相比有较大的不同。     

Elastic–plastic stresses in linearly hardening rotating solid disks of variable thickness

The distribution of stress, displacement and plastic strain in a rotating elastic–plastic solid disk of variable thickness in a power function form is investigated. The analysis is based on Tresca's yield condition, its associated flow rule and linear strain hardening material behavior. An analytical solution is obtained and numerical results are presented for different values of the geometric parameters. The validity of the solution is demonstrated by comparing the results with those for a uniform thickness disk available in the literature.     

Stress and strain fields in rotating elastic/plastic annular disks of pressure-dependent material

The Drucker–Prager yield criterion is used in conjunction with its associated flow rule to find the elastic/plastic stress and strain distributions within the rotating annular disks under plane stress conditions. The main distinguished feature of the model, as compared to typical models used for analysis of disks, is that the material is plastically compressible. Using an approach proposed elsewhere, the solution for strain rates is reduced to one nonlinear ordinary differential equation and two linear ordinary differential equations. These equations can be solved one by one, which significantly simplifies the numerical treatment and increases the accuracy of solution.     

Determination of elastic and plastic characteristics of TiC–TiNi alloys by the ultrasonic resonance method

Elastic characteristics and propagation velocities of ultrasonic waves in a TiC–TiNi composite material are determined by the ultrasonic resonance method. The values of the elastic moduli of the solid composite obtained are used to estimate its plastic properties. The effect of various additives on the elastic and plastic properties of the composite is studied.     

Rotor oscillation and stability in complex motion

Precession vibration of a rigid disk with unequal axial moments of inertia is considered when the axis of rotation turns; the disk is located asymmetrically on a flexible axle. Periodic solutions of the equations of motion and the amplitude-frequency relations are obtained for various values of the angular velocity of the axis of rotation. The critical rotational velocities of disks with various moments of inertia are defined in terms of the gyroscopic forces. The stability of motion is analyzed for various angular velocities of the rotating axis. State Technical University of Building and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 104–107, July, 1999.     

Development of instability in a rotating elastoplastic annular disk

The paper proposes a method, based on perfect-plasticity and perturbation theories, for instability analysis of an annular flat disk tightly set on a shaft with no interference fit. The perturbed elastoplastic state of the rotating disk is analyzed by determining the stress–strain state of a fixed elastic annular plate under in-plane loading. A characteristic equation of the first order for the critical radius of the plastic zone in the disk subject to internal pressure is derived. The critical rotation rate is calculated for different parameters of the disk     

Effect of an axial throughflow on buoyancy-induced flow in a rotating cavity

In this paper large-eddy simulation is used to study buoyancy-induced flow in a rotating cavity with an axial throughflow of cooling air. This configuration is relevant in the context of secondary air systems of modern gas turbines, where cooling air is used to extract heat from compressor disks. Although global flow features of these flows are well understood, other aspects such as flow statistics, especially in terms of the disk and shroud boundary layers, have not been studied. Here, previous work for a sealed rotating cavity is extended to investigate the effect of an axial throughflow on flow statistics and heat transfer. Time- and circumferentially-averaged results reveal that the thickness of the boundary layers forming near the upstream and downstream disks is consistent with that of a laminar Ekman layer, although it is shown that the boundary layer thickness distribution along the radial direction presents greater variations than in the sealed cavity case. Instantaneous profiles of the radial and azimuthal velocities near the disks show good qualitative agreement with an Ekman-type analytical solution, especially in terms of the boundary layer thickness. The shroud heat transfer is shown to be governed by the local centrifugal acceleration and by a core temperature, which has a weak dependence on the value of the axial Reynolds number. Spectral analyses of time signals obtained at selected locations indicate that, even though the disk boundary layers behave as unsteady laminar Ekman layers, the flow inside the cavity is turbulent and highly intermittent. In comparison with a sealed cavity, cases with an axial throughflow are characterised by a broader range of frequencies, which arise from the interaction between the laminar jet and the buoyant flow inside the cavity.     

Critical state of imbalanced rotating anisotropic disks with small radial and shear moduli

Critical speeds and the mass center movement of an imbalanced, circumferentially stiff, radially compliant, rotating annular disk on a stiff shaft and bearings are evaluated. It is demonstrated that recently developed hoop-wound composite material disks having elastomeric resin and carbon fibers can enter a critical state before reaching the circumferential strength limit of the material if certain material and geometric relationships are met.     

Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius

An analytical procedure for evaluation of elastic stresses and strains in rotating conical disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius is presented. The procedure is based on two independent integrals of the hypergeometric differential equation describing the displacement field; this procedure is just as general as the one found in technical literature, but less intricate and more reliable. General unpublished relations of stress state and displacement field in conical disk 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.     

Singularities in a Solution to a Rotating Orthotropic Disk with Temperature Gradient

A semi-analytical solution is obtained for a rotating stress-free edge disk of constant thickness and density. In the plastic range, the Hill’s quadratic orthotropic yield criterion is adopted. In the elastic range, the Hooke’s law holds with thermal effects included. The analysis of singularities performed may be used for correct implementation of numerical codes and preliminary engineering design.     

Evaluation of elastic compensation using elastic/plastic rotating circular disk problems

A version of elastic compensation is evaluated in the context of stress and deformation analysis of elastic/plastic rotating circular disks of both constant and variable thicknesses undergoing small deflections. An iterative incremental method is combined with finite difference methodology to generate information about the entire quasistatic loading histories of such disks. The evaluation process involves comparison of representative numerical results with corresponding predictions existing in the literature.     

Formation of a regular sequence of vortex loops around a rotating disk in stratified fluid

A regular system of vortex loops in annular flow behind the edge of a disk of 6–11 cm in diameter rotating in stratified fluid is first visualized using the shadowgraph techniques (classical method with the Foucault knife and the “filament-in-focus” method). Clearly outlined vortex loops are observed in strongly and weakly stratified fluids over a wide range of the angular disk rotation velocities. The dimensions of the vortex flow region depend on the stratification (buoyancy period), the angular velocity of rotation, and the disk diameter. Extended lengthy filaments which form spiral and irregular patterns inside the loops are clearly expressed in the fine flow structure. The filaments connect the neighboring loops. The trajectories of motion of the upper edges of the loops, i.e., the sources of short internal waves, are traced.     

Axisymmetric flows of an incompressible fluid between movable rotating disks

Within the Karman family of exact solutions of the Navier-Stokes equations, some non-selfsimilar solutions are considered to the problem of unsteady incompressible flow between two rotating disks one of which moves along a common rotation axis. Three classes of the flow regimes are studied: (i) a flow between the non-rotating disks, (ii) a flow between the disks rotating with identical angular velocities, and (iii) a flow between the disks rotating with opposite velocities. Examples of exact rotationally symmetric solutions for the inviscid-fluid equations, satisfying the no-slip conditions, are given.     

变速旋转部件的弹塑性和安定分析

本文研究了变速旋转部件(空心长轴、等厚度和变厚度圆盘)的弹塑性和安定分析,得到了在弹性极限、塑性极限和安定情况下角速度和角加速度之间的关系曲线,并讨论了屈服极限和圆盘厚度对这些曲线的影响。