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Previous investigations have shown that Lüders' 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 Lüders' 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 Lüders' 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
Lüders' 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.