Mathematical modeling of rotating disk states

We consider the problem of a rapidly rotating disk in the elastic-plastic state. The piecewise linear plasticity condition in general form is chosen. It is believed that all plastic curves have the common point of intersection which corresponds to uniaxial tension. For external parameters, we obtain the conditions that determine the probability of inception of plastic zones. It is shown that plastic zones could incept in the center of the disk and/or on the boundary of it. The problem in the plastic zone is statically determinate. The case when the plastic zone occupies some central part (core) of the disk, where one regime of plastic condition is fulfilled, is considered. In order to estimate the stress state inside the elastic zone of the disk, equivalent stress which is equal to the chosen plasticity function is defined. In order to define the relationship between plastic deformations and stresses, the piecewise linear plastic potential being equal to the plasticity function is chosen. The plastic incompressible body is considered. The associated flow rule can be integrated so that the problem of getting displacements turns into quasistatic one. The problem of determining displacements in the plastic region leads to a first-order differential equation with respect to the radial component of the displacement vector. Therefore the continuity condition for displacements at the elastic-plastic boundary and the assumption that the displacements in the center of the disk are equal to zero leads to an overdetermined problem. So, only the continuity condition for displacements at the elasto-plastic boundary is accepted. It is assumed that plastic deformations at the elastic-plastic boundary are equal to zero. It is shown that displacements at the center of the disk are equal to zero automatically for all piecewise linear conditions of plasticity apart from the Tresca yield criterion. For the Schmidt–Ishlinskii yield criterion, all deformations at the center of the disk attain finite values. Meanwhile, for other piecewise linear conditions, plastic deformations at the center of the disk attain infinitely large values. This explains the discontinuity of displacements at the center of the disk for the Tresca yield criterion. The calculation results are presented as graphs of stresses, displacements, and deformations.