On pressure perpendicular to the shear planes in finite pure shears , and on the lengthening of loaded wires when twisted

Abstract

In the ‘ Philosophical Magazine,' vol. 9, 1905, p. 397, I gave an analysis of the stresses in a pure shear which appeared to show that if ε is the angle of shear and if n is the rigidity, then a pressure nε2 exists perpendicular to the planes of shear. That analysis is, I believe, faulty in that the diagonals of the rhombus into which a square is sheared are not the lines of greatest elongation and contraction, and are not at right angles after the shear, when second order quantities are taken into account, i. e., quantities of the order of ε2; I think the following analysis is more correct, and though it does not give a definite result, it leaves the existence of a longitudinal pressure an open question. The question appears to be answered in the affirmative by some experiments, described in the second part of the paper, in which loaded wires were found to lengthen when twisted by a small amount proportional to the square of the twist. I.— Stresses in a Pure Shear. Let a square ABCD (fig. 1) of side a be sheared into EFCD by motion through AE = d, the volume being constant. The angle of shear is ADE = ε, and tan ε = d/a exactly ; neglecting ε3, we may put ε = d/a.