Analysis of origin of shear bands in a reinforcing elastic-plastic body

Abstract

The area of the line of localization of plastic deformation in the section of reinforcement is presented in the form of a band of finite length for the case of plane strain. The banks of the band can slide freely, but in this case they must stay in contact. The theory of microdeformation, which leads to the singular surface of fluidity is used for describing mechanical behavior, which is very important in the problems of localization. We built a closed analytical solution of the problem of building the fields of rates of displacements and changes in stresses in the vicinity of apex of the assigned line of discontinuity of the speeds of displacements. It was established that three different forms of the solution, depending on the results of the solution to the problem of localization, are possible. The point of localization is defined as the point of intersection of the curve defining the dependency of parameters of material on the parameters of load according to the theory of microdeformations, with the curve that defines the boundary between elliptic and hyperbolic regime of the solutions. The field of rates of change in stresses has a root peculiarity. The criterion for ductile fracture (advance of line of discontinuity) was formulated based on the approaches, accepted in the Novozhilov criterion of brittle fracture. In this case it was taken into account that with the ductile fracture we deal with the occurrence of a localized flow, in which the development of line of discontinuity will be determined by average rates of change in stresses in the vicinity of a singular point. The orientation of the line of discontinuity of the rates of displacements and the fields of rates of change in stresses and displacements were defined. The dependence of the length of the line of localization on subcritical stresses is obtained from the limitation of the angle of fracture of the trajectory of load. It was established that the line of localization in the initial state can have dimension comparable with size of the grain. It was shown that localization in the form of the slip line of finite length precedes the localization at the point.