Analytic calculation of the energy characteristics of slip line fields in plane plastic deformation problems

Abstract

An analytic implementation of the slip line method (the method of characteristics) is proposed for statically definable problems of the plane plastic deformation of an ideal rigid plastic medium. The solution of the Riemann problem (the initial characteristic problem) with boundary conditions defined by power series is represented in terms of generalized hypergeometric functions. Other boundary-value problems (the Cauchy problem and the mixed problem) reduce to the equivalent Riemann problems. A mixed problem with Prandtl friction in a curvilinear contact surface and a Cauchy problem with arbitrary smooth initial data are treated. An equation governing the form of the free surface is obtained. The analytic representations of the radii of curvature of the characteristics in. the physical plane and in the plane of the velocity hodograph are used to calculate the dissipation power in the plastic domain. Problems of reduction in terms of long and short wedge-shaped matrices using the proposed energy approach are considered as an application.