A New Method for Finding Initial Extension of Plastic Region for Elastoplastic Torsion of Variable Yield Sections without Strain Hardening

摘要

In this paper membrane analogy was developed to predict initial development of plastic deformation for prismatic bars with variable yield stress. In pervious works this method was used for constant yield stress. In this article we used mathematical methods to develop it for variable yield stress. It was supposed that the yield stress is a function of, minimum distance between every point in the cross section of the bar to the boundary of the section.This supposition is not haphazardly because in heat treatment of shafts the yield stress in interior points depends on the gradient of temperature therefore is approximately dependent on the minimum distance between every point in the bar section to the boundary of the section. First a mathematical method was introduced to extend membrane analogy to this case. Then using this proof, a new software package (TORSION BAR) was developed for calculating the torsion function for full plastic deformation of shafts. In the next step by use of the developed software, the elastic function of torsion was found. In addition by integration from the torsion function the diagram of torque-angel of rotation was found. For extending the method to find an approximate analysis a piecewise smooth torsion function was supposed. This assumption redounds to decoupling of the differential equations of torsion. This method is very fast and exact to find the initial development of the plastic region.