Development of instability in a rotating elastoplastic annular disk

Abstract

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 Keywords: annular flat disk, axisymmetric elastoplastic problem, instability, rotating shaft, shape perturbation method, critical rotation rateIntroduction. The theory of elastoplastic [1,5,10,[13][14][15]21] and other compound media allows for plastic strains. Plastic zones [16,17] cause stress redistribution. The stress is maximum at the elastic-plastic boundary. In this connection, it is of special interest to locate the elastic-plastic boundary in elastoplastic problems.Plastic-equilibrium problems arise even when dealing with simplest stress-strain states in which the stresses and strains depend on one coordinate only. Among them are many problems of practical importance such as the equilibrium of cylindrical pipes and spherical vessels or the uniform rotation of cylinders and disks under high loads or at high angular rates. Since the yield point can be exceeded in some zones of very rapidly rotating cylinders or disks, it has long been recognized that it is necessity to use highly ductile materials for heavy shafts of steam turbines or massive cylindrical rotors of big turbogenerators, which are subjected mainly to centrifugal stresses [11].At high rotation rates, a plastic zone appears at the center of a solid disk or along the internal edge of a disk with a hole. As the rotation rate is increased, the plastic zone extends over the entire disk. This increase is approximately 12% for the solid disk [11]. However, the disk can no longer be used as before, even prior to the loss of load-carrying capacity [7,19,20] occurring when the plastic zone reaches the outer edge of the disk. This is because of the instability of the disk in elastoplastic state-it takes a flat equilibrium shape different from the initial circular shape. In [6-8, etc.], the corresponding critical rotation rate w * for disks made of an incompressible material was determined and the loss of load-carrying capacity of rotating solid disks was studied.Here we outline a method, based on the perturbation method [2][3][4]8], to determine the critical rotation rate for an elastoplastic homogeneous isotropic flat annular disk tightly set on a shaft. The radius of the shaft is equal to the radius of the hole in the disk. The inside surface of the disk is subject to the pressure exerted by the shaft during rotation.1. Problem Formulation. Small deviations from the circular shape of a disk and misaligned fit on the shaft are known to have an insignificant effect (to 3%) on the loss of its load-car...

Analysis of Dynamics of Boundary Shape Perturbation of a Rotating Elastoplastic Radially Inhomogeneous Plane Circular Disk: Analytical Approach

“…Now, in contrast to Section 3, the elastic domain is homogeneous and represents zone D2e (Figure 1(b)), therefore [14,15] …”

“…In the analysis of plane stress strain state this method was employed to obtain approximate critical values of the plastic zone dimensions and angular velocity of continuous homogeneous circular discs [13,14], ring-shaped discs [15] including those loaded along the contour by additional radial forces [16], stepped discs and some arbitrary profile discs [17] as well as simplest inhomogeneous discs. This proves efficiency of the analytical method of boundary shape perturbation (with the use of simplest numerical procedures at certain stage) which reduces essentially the amount of calculations and at the same time facilitates fruitful application of various numerical techniques [18][19][20] for stability and strength calculation of turbine and other massive discs.…”