The Lüders' Lines on Mild Steel

TLDR: In this paper, it was shown that the Luders' lines on the outer surface of mild steel tubes have the same or approximately the same inclination to an axis of simple pull or simple push.

Abstract: Previous investigations have shown that Luders' lines on specimens of mild steel and wrought iron, strained in tension, are inclined at about 50 deg. to the axis of pull. For tests in compression the information available is not precise, and though the angle of the lines with the direction of the compression is commonly understood to be about 40 deg., some doubt has been thrown on this point. The author had found previously that the lines are well developed on the surface of mild steel tubes. Since it was easy to obtain a compressive stress of practically uniform distribution in tubes under end pressure, while at the same time a hoop tensile stress could be induced by internal fluid pressure, the author confined his attention to the lines on tubular specimens. These were of mild steel, either hot or cold drawn, and most of them were annealed. They varied in bore from 2½ in. to 3 in., and in thickness of wall from about 0.08 in. to 0.125 in. Four sets of tubes were tested under end loading, simultaneously (for the most part) with internal water pressure, these loadings being so arranged as to give at the yield point ratios of the longitudinal compressive stress to hoop tensile stress ranging from 0.25 to infinity. The Luders' lines on the outer surface appeared at the yield point indicated by the extensometer - i.e., their appearance coincided with the commencement of the large "yield" strain. In all cases where there were lines on the inner and outer surfaces of a tube, an inner and outer line, and also the ends of these lines, were found to be radially opposite; showing that the lines, were traces of surfaces or canals of disturbance which passed through the tube wall, and indicating, moreover, that the disturbance spread spirally onwards, and not outwardly from a line initially formed on the more severely stressed inner surface. The inclination a of the lines to the axis of the tube was found to vary in the following way for tubes under end load and internal pressure: For longitudinal stress/hoop stress = 0.25, a = 42 deg. (about). For longitudinal stress/hoop stress = 1, a = 45 deg. (about). For longitudinal stress only, a = 50 deg. (about). For values between 0.25 and 1, and between 1 and infinity, a became progressively larger. Some tests of tubes under external water pressure and longitudinal tension gave results similar to those above mentioned. The directions of the lines with respect to the axes of like stress were practically the same, though the inclinations to the axes of the tubes were now complementary to the values of a quoted above. The conclusion is drawn that the Luders' surfaces have the same or approximately the same inclination to an axis of simple pull or simple push. With stresses of opposite sign at right angles to each other the lines and surfaces are more inclined to the stress of greater intensity, and with equal intensities the surfaces are at about 45 deg. The author suggests that if a shear stress of given intensity be the only condition for that spreading of plastic strain which commences at the yield point, then there is no reason why there should be any variation in the angle of the Luders' lines and surfaces. The variation suggests that not only the maximum shear stress - i.e., the half difference of the greatest and least principal stresses - but also the magnitudes of the principal stresses are factors in producing the yielding condition in mild steel.