A mathematical model for the formation and development of defects in metals

Abstract

Summary To simulate metal forming processes, the formation and development of defects in metals, one has to solve relevant boundary value problems. The progress in the theory of plasticity is obvious (for example, the slip-line method, the finite element method, etc.,), yet it retains too many unsolved problems to be applied to attain these ends. A mathematical model for the formation and development of continuity defects in metals under deformation cannot be constructed within the theory of plasticity alone (or any other section of continuum mechanics) because of the fundamental axiom of continuity. The proposed mathematical model of continuity defect formation deals with a new boundary value problem, in which the classical problem of plasticity is supplemented with a kinetic ordinary differential equation for a scalar functional depending on the stress-strain state and temperature histories. This kinetic ordinary differential equation is written for each material particle. The functional is called “metal damage”, , caused by microdiscontinuities. Here we present s new technique for solving rather general boundary value problems, which can be characterized by the following: microdamage and macrofragmentation; the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; nonisothermal flow; rapid flow with inertial forces; nonstationary state; movable boundaries; changeable and nonclassic boundary conditions etc.