The contact surface in plastic deformation is considered as a field, whose field-line is given by the "generalized rule of gradient" based upon the condition of least resistance applied both to effective and ineffective direction of "relative" slip on solid and liquid surfaces, as shown by equation (1),which is the formulation of the sta tement that the direction of field-lines is the direction of (most rapidly) decreasing frictional resistance [equation (2) ]. The main results are:(1 )The rule of field lines is given by equation (l)and(2). Accordingly, for dry friction, the"resistant" field is divergent; for plastic lubrication, the "lubricating" field is convergent (Fig. 1). There may be combination of both types, as shown in Fig.l(c) and Fig. 4 for the case of plate-rolling. (2)The two field functions, the unit friction (τ) and the unit pressure (p) are governed by equations (3), (4), (8), (9), (11), given in this work, by equations (5), (6), (7), (15) of classical theory and by equations (6), (12), (14), (16)and Fig. 6, all based upon classical concept, but first used in our work. Fig. 6 deals with range of variation of what is termed the "point of friction", that is, (τ, p) point in the stress plane. Equation (12) reads: The point of friction (Q) must be on the segments of the curve τ=Fp, merged in the closed field limited by the three stress circles where F is the "friction-function" defined by equation (6). In these equations, S is are-length of field lines. The other designations are self-evident. (3) A region of sticking in plate-rolling is determined by equation (15) as shown in Fig.6. It is pointed out that the value of the coefficient of friction can be measured by two separate experiments on pressure distribution and on friction-lines.The friction-field of least resistance provides a new fundation for plasticity under pressure.
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