Discussion – Supplement To Technical Publications No. 1782 - Symposium On Cohesive Strength – Class C, Iron Steel Division, No. 372; Class E, Institute Of Metals Division, No.449 - Bridgman, P. W.

P. W. BRIDGMAN.-Owing to a misunderstanding, I did not see Dr. McAdam's and Dr. MacGregor's remarks on my paper on Flow and Fracture (Metals Technology, December 1944, Pp. 32-38), until after their publication. Both Dr. McAdam and Dr. MacGregor speak of my analysis for the stress distribution at the neck of a tension specimen as containing several assumptions; in particular, that the strain is uniform across the section of the neck, that there is no hoop tension at the periphery, and that the plasticity conditions of von Mises hold. As far as the mathematics goes, the situation is no different in this problem of plastic flow than in a conventional problem of elasticity. That solution is unique which satisfies the boundary conditions on the stresses, the stress equations of equilibrium, and in addition the constitutive equations connecting stress and strain in the elastic case, or a corresponding set of equations, of which the von Mises relations are a special form, in the plastic case. In the particular case of the necked tension specimen I have found that there is a solution satisfying the boundary conditions, the stress equations of equilibrium, and the von Mises plasticity conditions in which the strain is uniform across the section, and there is no hoop tension at the periphery. This is therefore the solution, and the uniformity of strain and vanishing of hoop tension at the periphery are to be described as discoveries, over which the operator has no control, rather than as assumptions. The only part of the solution that can properly be described as an assumption is the von Mises plasticity condition. With regard to this it is to be noted that the condition is needed only in a much weakened form, for the analysis uses the condition only at the neck, where the strain is approximately constant. Thus, in the extreme case referred to in a footnote of my paper (p. 37, Met. Tech. Dec. 1944) the actually measured strain across the section varies from 2.4 to 2.6. Obviously it is a much weaker assumption to postulate that the von Mises condition holds in the strain range 2.4 to 2.6 than to assume it to hold for the entire range of strain from o up to 2.6, which would be necessary if the solution applied to the entire tension specimen rather than to the immediate region of the neck only. Furthermore, it is to be commented that under the special conditions of this problem the von Mises condition becomes identical with the maximum shearing-stress criterion of plastic flow; this amounts also to a weakening of the restrictions imposed by the von Mises condition. Dr. MacGregor refers to actual experiments of his own in which the strain was found not to be uniform across the neck. It is to be emphasized, however, that these were experiments on bars of rectangular section, and that the nonuniformity established was a nonuniformity around the perimeter, not a nonuniformity from the surface toward the axis. My method of solution applies only to a circular cross section. It can be said that the solution that satisfies the boundary conditions, etc., would definitely not be expected to demand a uniform strain for noncircular sections. In discussing my results, Dr. McAdam uses a definition of "flow stress" different from mine, so that the immediate applica-