THE DETERMINATION OF THE WORKABILITY DIAGRAM

One of the major aims of the optimization of technological parameters of the metal forming processes is the possibility of producing pieces without defects. There are a number of various theoretical approaches to the fracture analysis in metal forming processes. The classical variational principle of the theory of ideally plastic solids (the upper bound method) was used in [1-4] to predict the impending fracture in extrusion and rolling. An obvious drawback of this approach is the lack of a fracture criterion in the model of the process. The modification of the variational principle suggested in [5] allows one to introduce the stress fracture criterion (criterion of the maximum stress). The fracture analysis in the rolling of a three-layered sandwich sheet, based on this modified principle, was carried out in [6]. Although the upper bound method is widely applied to the analysis of metal forming processes including the numerical simulation [7,8], this method encounters essential difficulties in the study of some steady-state processes (if either the material is not ideally plastic or the shape of the deformation zone is not known in advance) and in using a number of friction laws. Another approach to the fracture analysis of metal forming processes is based on the introduction of a damage parameter into the model of the process. In some models, this parameter is calculated using the stress-strain state found in advance [9], while in other models, both the stress-strain state and the damage parameter are determined from a coupled system of equations [10, II]. Such models correspond to the realistic character of fracture in ductile materials. However, the solutions provided by these models and, especially, the models of [10, 11] are too complicated for engineering applications. Moreover, the tests aimed at finding the parameters of these models require much efforts. A possible set of tests for the identification of the parameters of one of the models was suggested in [12]. In the engineering practice, a workability diagram is often used to predict the impending fracture in metal stamping [ 13] and forging processes [14, 15]. A review of publications devoted to workability diagrams is presented in [16]. Since the workability diagram does not take into account the influence of the deformation history but reflects only the dependence of the limit strain on the type of the stress state, additional hypotheses are required in the cases of complex loading paths. In many cases, these hypotheses also introduce the concept of the material damage measure [16]. Another criterion is suggested in [17]. It permits one to apply directly the original workability diagram to the estimation of the fracture probability. In the present paper, we study some features of the solutions which are necessary for the application of this approach. The original workability diagram is plotted in the coordinates "degree of stress state triaxiality"-"equivalent plastic strain", which allows one to take into account two basic influences determining the fracture of ductile metals. The upsetting by dies of various shapes is the most frequently used type of tests for the construction of the workability diagram [18, 19]. One of the basic drawbacks of this type of tests is the effect of friction which complicates the material flow character. For this reason, the calculations are often based on rough assumptions such as the method of thin sections [20-22]. Refined calculations can be carried out by using other approximate methods, for example, the method presented in [23]. Nevertheless, the friction law in the metal forming process leads to substantial difficulties in the modeling [24-26] and can influence significantly the results of calculations. However, in many cases, the fracture occurs on a free surface of the specimen depending on the process parameters and material properties. Such a fracture was observed in the upsetting of cylindrical specimens [19], the tension of notched bars (see, for example, [27, 28]), and the upsetting of plates by dies of complex shape [17]. This observation is used here to demonstrate that in such cases, the parameter suggested in [17] can be determined by measurements of free surface strains. These measurements can be performed easily by the embedded grid method [29-31], the method of microstructure variations [32], and other well known experimental methods. The material is assumed to satisfy the von Mises yield condition and associated yield law. The approach suggested is used for the determination of a point of the workability diagram [15] based on the results of the upsetting of a disk by spherical dies.