The Problem of Propagation of a Plastic State

Summary The author considers the problem on propagation of a plastic state in an infinite plane with a circular hole, subject to the action of symmetrical forces causing assigned displacements on the boundary. Author’s hypothesis are reduced to the following. 1. Matter of the plane can be only in two states: elastic or plastic; moreover, the plastic zone is expressed in Lagrangian coordinates (ϱ 0; ϑ 0) by the inequalities $$ r_0 R_0 (t)$$ The author restricts himself to the consideration of only the motions when R 0(t) is an increasing function. 2. The elastic state of the matter is described by the usual equations of linear elasticity within small quantities of the highest orders. 3. The plastic state is described by the known Saint-Venant equations. 4. The displacement vector, stress tensor, and velocity vector of a particle remain continuous while crossing the boundary between the elastic and plastic zones. The author indicates the method of computation of all quantities characterizing the motion, i.e., the radius R0(t), components of the displacement at any point of time in both zones, components of the stress tensor also in both zones and flow lines in the plastic zone. The research shows that if the angle of rotation given on the internal contour increases too rapidly relative to the radius of expansion of the inner hole, then the characteristics of the plastic zone become tangent to the inner contour, and the problem losses its meaning. From a physical standpoint, this is related to the fact that the break of the matter occurs along the inner boundary. The rate of growth of the maximal angle of rotation for large values of the radius of expansion of the inner hole is approximately equal to 3 ln r.