388 REPORTS ON THE STATE OF SCIENCE, ETC. 
Though the strains are known at any epoch of constant strain-range, the real 
stresses are not known, even at the skin. Consider the epoch at which the semi- 
range of skin-strain is ON,, and suppose the stress calculated from the torque by the 
usual formula on the assumption of perfect elasticity (i.e. on assumption of linear 
stress distribution) is represented by N,D,. If the strain and stress each vary as the 
distance from axis, then the strain and stress must be proportional to each other, 
which is obviously wrong, except for regions of the bar near the axis. This point 
may be illustrated (fig. 16) by joining the points O and D,; the straight line OD, 
represents proportionality between stress and strain from axis to skin. It is clear 
that both stress and strain cannot have a linear distribution. If the stress did vary 
as distance from axis, then the distribution of strain, in the case of torsion of a solid 
round bar, would be represented by some line, not straight, such as A, BO in fig. 14. 
As mentioned previously, such a distribution of strain is scarcely conceivable. It 
Fic. 16 
seems clear, then, that the stress cannot have a linear distribution, except near the 
axis, where the elasticity will not be impaired. Part of the real stress-strain graph 
through the body of the bar will evidently be the line OA for all epochs such as 
mentioned above. In order to ascertain, if possible, the parts of the true stress-strain 
curve, other than OA, for any epoch of the test (an epoch immediately subsequent to 
a number of cycles sufficient to establish constancy of ranges of stress and of strain 
throughout the specimen is considered only), it is necessary to examine the true 
stress-strain curve for a thin-walled tube under alternating torsion. 
Hollow Tubular Specimens under Alternating Torsion. 
If the thickness of wall is less than about 20 per cent. of the radius of the tube, the 
actual stress at the middle of the wall can be calculated with very little error. What- 
ever may be the variation of stress through the wall, it is obvious that the stress at 
