Algorithms of the Method of Statically Admissible Discontinuous Stress Fields (SADSF) – Part II: The Algorithms for Solving Limit Stress Fields Around Isolated Nodes of Stress Discontinuity Lines

SUMMARY OF THE WHOLE PAPER: By now, the SADSF method is practically the only tool of shape design of complex machine elements that provides an effective solution even to the problems of 3D distribution of the material, and at the same time it is still enough user-friendly to be useful for engineers. This unique property of the method is due to the existence of its simple application version. When using it, a design engineer does not need to solve by oneself any statically admissible field – which could be very difficult – but obtains such a solution by assembling various ready-made particular solutions. The latter are in general obtained by means of individual and complex analyses and provided to a designer in a form of libraries. The algorithms presented in this paper break up with the individual approach to a particular field. The algorithms are the first ones of general character, as they apply to the fundamental problems of the method. The algorithms enable solving practically any boundary problem that one encounters in constructing 2D statically admissible, discontinuous stress fields, first of all the limit fields. In the presented approach, one deals first with the fields arising around isolated nodes of stress discontinuity lines (Parts II and III), then integrates these fields into 2D complex fields (Part IV). The software, created on the basis of the algorithms, among other things, allows one to find all the existing solutions of the discontinuity line systems and present them in a graphical form. It gives the possibility of analysing, updating and correcting these systems. In this way, it overcomes the greatest difficulty of the SADSF method following form the fact that the systems of discontinuity lines are not known a priori, and appropriate relationships are not known either, so that they could be found only in an arduous way by postulating the line systems, and verifying them. Application version of the SADSF method is not described in this paper; however, a reference is given to inform the reader where it can be found. SUMMARY OF PART II: In the paper, the author introduces the sets of conditions that create the algorithms of the functions on which one defines the boundary problems met in the search for discontinuous limit fields existing around isolated nodes. Among those, there are functions describing states of stress in the component homogeneous regions, the parameters of lines that separate these regions, and, first of all, the formulae for determining the domains based on the general conditions of existence. These formulae play a key role in numerical implementations of the method. The fields satisfying the Huber-Mises yield condition are of primary choice however, the derived relationships have a general meaning. To emphasise this fact one presents not only the areas of existence valid for the Huber-Mises condition, but also the areas obtained for several other yield conditions applicable to plastically homogeneous materials. The knowledge of the areas opens the possibility of developing the method of search for the fields that obey these conditions, and for algorithmizing this method. This could be applied even for the fields that are characterised by arbitrary, admissible states of stress. One also presents, basing on a mathematically complete set of conditions, typical formulations of problems concerning the fields around the nodes. One discusses the balance between the set of conditions and the unknowns, as well as the transformations into global systems connected with complex fields. One consequently applies parametrisation of the yield conditions, which not only reduces the number of unknowns and leads to simple, recursive forms of the formulae, but, first of all, makes it possible to find the formulae for generation of domains, without which numerical solution of the fields and algorithmization of the method would not be possible at all.