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–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.     

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.     

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.     

Specific features of stress and strain states of disks and disk flywheels with rigid reinforcing coatings

An approach to calculation of rotating disks with reinforcing coatings taking into account emerging boundary layers is formulated within linear spatial elasticity theory. Analysis of the calculated stress fields in a disk’s base material and in coatings enables one to explain two different phenomena: a significant increase in the load capacity of disks and the energy capacity of flywheels whose face areas are covered with high-modular coatings whose thickness is 0.1–0.6 mm and a reduction in the load capacity (in some cases) if thicker coatings are used. Applicability limits of the standard theory used to calculate disks, in which the stress state is assumed to be a 2D one and boundary layers are absent, are studied.     

Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading

This investigation aims to study the random stresses in a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading using a hybrid numerical method. The mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The stresses are obtained by solving Navier equation and using Galerkin finite element and Newmark finite difference methods. The Monte Carlo simulation is used to generate the random mechanical properties for the problem. The failure probabilities and time history analysis of stresses are determined for various coefficient of variation considering various grading patterns of mechanical properties. The presented hybrid numerical method is effective, with high capability for stochastic analysis of dynamic and transient analysis of FG structures with various boundary conditions.     

The anisotropic rotating disks

This paper analyzes the stresses generated, in closed form, in rotating disks or cylinders made of general orthotropic material. The radial and circumferential stresses in a rotating circular disk are functions of radial co-ordinates only and shear stress is equal to zero. However, the deformed shape does not remain circular. The stresses and displacements found in a rotary elliptical disk are nonaxisymmetric. Since the success of this method is hinged on the fact that the boundary condition integral is integrable, the same technique can also be used to solve some other type of boundary value problem as long as the boundary conditions can be integrated.     

功能梯度压电材料壳体热残余应力

功能梯度压电材料结构成型冷却后会出现热残余现象,影响结构强度.借鉴复合材料层合结构的研究方法,将功能梯度压电材料球壳和圆柱壳沿厚度分为若干层,各层视为均匀材料,根据层间连续条件导出递推关系,得到显式的力—电—热多场耦合热残余解.统一了多层功能梯度压电材料壳体和连续功能梯度压电材料壳体热残余解.对于前者,其解为精确解;对于后者,其解为渐近解,随层数增加而收敛于精确解.其解也适用于功能梯度压电材料涂层.该方法对材料性能的变化方式(函数)没有要求,适应性强.并讨论影响热残余应力和界面强度的因素,球壳因双曲率的影响,热残余应力显著大于柱壳.     

Approximate solution of limit angular speed for externally loaded rotating solid disk

The present work proposes a numerical method to investigate the stress and deformation states of rotating disks using variational method taking the radial displacement field as unknown variable. Assuming a series approximation following Galerkin's principle, the solution of the governing partial differential equation is obtained. The analysis is carried out for various disk geometries and the effect of geometry and loading parameters on the stress and deformation states is investigated. Limit angular speed of the disks is also calculated for various system parameters and reported in dimensionless form. The results are validated with those of other researchers for appropriate values of the system parameters and very good agreement is observed. It is observed that the present methodology is quite robust, stable and realistic and the documented tables and curves may be used by the practicing engineers as design monograms.     

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.     

空心旋转圆盘的安定分析

本文用解析法与计算机分析相结合对理想弹塑性材料等厚度空心旋转圆盘作了安定分析,得到各种不同内外半径比的空心旋转圆盘的安定极限转速,为有关工程设计提供了必要的理论依据。     

An experimental study of the burst strength of rotating disks

Details are given of a spinning rig for burst tests on metal disks in vacuo at speeds in excess of 100,000 rpm. The permanent strain distributions and the instability and fracture conditions are observed in the spinning of disks of vacuum remelted alloy steel. It is shown for hollow disks of uniform thickness that at instability two “necks” form at either side of the bore and the bore becomes oval with the direction of the minimum diameter passing through the two “necked” regions.Good correlation between theoretical and experimental strain distributions at instability are obtained provided the ratio of outside to inside radius of the disk is greater than 10. The theory is based on a rigid-plastic material and it is thought that better correlation for radius ratios less than 10 would be achieved if elastic strains were taken into account in the theory.     

Stresses and displacements in a rotating skew-faced disk

An analytical solution is presented for the stresses and displacements in a thin isotropic, non-axisymmetric disk rotating with a constant angular velocity.The asymmetry is such that the disk faces are skewed and therefore the disk thickness changes in the circumferential and the radial directions.Assuming a state of plane stress, the governing system of partial differential equations is presented in polar coordinates.A harmonic analysis transforms the problem into one having a sole independent variable.A second asymptotic expansion renders it possible to obtain closed form solutions for the disk stresses and for the displacements.Integral convergence tests are carried out for checking the completeness of the solution.     

变厚度回转构件的优化设计

论述了变厚度回转构件优化设计的一种数值方法。为该构件的厚度构造一个连续的函数 ,给出它的体积和转动惯量的计算表达式 ,提出变厚度回转构件应力分析计算的数值方法 ,然后将此设计问题转化为数值优化问题。目标函数定义为转动惯量与体积之比。该优化设计方法新颖通用性强 ,可应用于任何盘形旋转构件     

Elastic-plastic stress distribution in a rotating hyperbolic disk with rigid inclusion

The elastic-plastic stress distribution of a rotating hyperbolic annular disk mounted on a circular rigid shaft is investigated in this work. The exact solution presented is based on the usual assumptions of plane stress, Tresca's yield condition, its associated flow rule and linear strain hardening material behaviour. According to the present analysis the plastic core consists of three parts with different forms of the yield condition. The theory presented is illustrated by numerical results. The results show that the plastic flow and maximum stresses are influenced by the thickness parameter.     

Effects of diffusion on lubricant distribution under flying headon thin-film disks

Lubricants on thin-film disks have large effects on head–disk interface characteristics. They reduce head and disk wear while thick lubricant film increases friction force between them and lubricant transfer onto head surfaces. Therefore, we have to know the lubricant behavior in many cases. Lubricant depletion due to disk rotation has been studied very well. However, the effects of flying heads have not been understood systematically until now. We developed a simulation program to numerically calculate the change in lubricant thickness under a flying head on a thin-film magnetic disk. The program included the effects of centrifugal force, shear stress from the air due to disk rotation with a flying head, and the effect of lubricant diffusion. We first calculated a change in lubricant thickness under a flying head using previously published data without the effect of diffusion. Calculated results showed fairly good agreement with the published experimental data with very high peaks on both sides of the flying head rails. With the introduction of diffusion effects, these peaks became moderate and the calculated result agreed very well with the experimental data. The coefficient of diffusion obtained to best fit to the experimental data was close to that reported in a literature. We analyzed the effects of air shear stress patterns under flying head on the change in lubricant distribution. We found that the side shapes had large effect on the distribution. We also confirmed that our program could calculate lubricant depletion on rotating disks without a flying head.     

Thermal stresses in anisotropic discs

A general solution is given for the elastic thermal stresses in a solid or hollow anisotropic disc of constant thickness when subjected to a temperature rise which varies symmetrically in an axial direction and arbitrarily with the radius. Graphs demonstrate the variations of stress distribution and maximum stresses with the degree of anisotropy and other parameters.     

Decay rate of edge effects in cross-ply-laminated hollow cylinders

The decay rates of stresses and displacements due to self-equilibrated loads acting on the end of cross-ply-laminated hollow cylinders are found by using the theory of three-dimensional elasticity. The study assumes that the periodic displacement and stresses fields used decay away exponentially from the end load region. The use of a recursive and successive approximation method leads to an eigenmatrix whose eigenvalues represent the decay rates of the problem. By this approach, the composite cylinders may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thicknesses. The eigenvalue problem has always a dimension of 3×3, regardless of the number of the layers. The decay rates for either symmetric or antisymmetric cross-ply-laminated cylinders are found and presented. The effects of material properties on decay rates are investigated for a typical eight-layered anti-symmetric cross-ply-laminated cylinder. The displacement and stress distributions across thickness of cylinders for selected problems are also presented.     

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.