In the recent years, the theory related to friction-lines has been independently developed both in Europe and here, according the information reached the author in 1956. These investigations are based on the condition of least frictional resistance at a point.In 1954, a variational equation (2a) on classical basis was given on a meeting for the related research. This equation was based on the frictional work at point. In as much as the condition of least force at a point has became acceptable, there seems no reason to object the condition of least work at a point, that is, the frictional work along the path of an element of area. Thus, the above equation are further investigated in this paper.For the case of short range slip occuring in processes such as plane forging under small reduction, the last term of this equation is zero, the rule of gradient follows. Therefore, the rule of gradient holds only for short range slip or instantaneous friction-lines.For long range slip, this equation leads to the "rule of isoclinic-gradient", (equation 9), which states that the gradient line of pressure (p) is the isoclinic curve for frictional force τ(Fig. 2). The angle of inclination (φ) changes along the pressure contour according equation (11), and along the friction-line according to equaiion (14). The function (τ) has the general nature of equation (16). Examples for long range slip is given in Fig. 3 and 4. Continuous divergent long range slip can only be generated by point or line-sourse in extrusion. The singularities in the case of short range slip are not real sourses.In this analysis, the frictional force is regarded as a shear stress on pure mechanical basis, without assuming its physical nature.For complete details of the paper, see[10].
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