COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 387 
A,B and BO must be identical, and any distortion producing an angle such as A, BO 
must be common to every section, except perhaps near the ends of the twisted 
specimen. A displacement of A to A, at each section would be due to a sliding or 
distortion common to each portion along the length of the specimen, and a distortion 
of this kind would not be produced by equal and opposite torsional couples except 
perhaps near the gripped ends. Thus any deviation of these radial lines AO from 
straightness is difficult to imagine for torsional strain of the amount met with in fatigue 
tests. 
Consider a uniform bar under a bending moment uniform over its length—the 
conditions the author is herein concerned with. It is clear that the strained shapes 
of equal slices A and B (fig. 15) must be exactly alike, and that such slices must fit 
HriGs 15 
together to make the strained whole. It seems impossible that these conditions can 
be fulfilled unless the sectional surfaces of the slices remain plane. It appears to 
the writer—possibly, however, by defect of imagination—that any other than linear 
distribution of strain is not conceivable. 
In the following work linearity of strain distribution will be assumed. In the 
tests to be cited, the strains at the skins of the specimen were measured ; with 
linearity of strain distribution from the axis, the strains at any radius are, of course 
ascertainable. 
Stress-Strain Curves for Alternating Stress Tests. 
Tf a test-specimen is subjected to increasing stress of equal + and — maxima 
a series of hysteresis loops is obtained such as shown in fig. 16. At any one constant 
range of stress the width of the loop increases, and if the range of stress is not too 
great, this width increases to a maximum value and remains constant at that value. 
The increase of width to a maximum is illustrated by curves in a former paper by 
the writer.1 We will suppose that the loops of fig. 16 represent a series of these 
constant maximum loops for a corresponding series of increments of stress. We may 
suppose that the loops are obtained for increments of either torsion or bending. The 
loops then represent the cyclic stress-strain condition at the skin of the specimen 
when constancy of cyclic strain has been reached. ‘The writer has found that points 
such as B, C, D, lie on a straight line when the ordinates parallel to OY are the semi- 
ranges of the torque (or bending moment) ; or when(what amounts to the same thing) 
the ordinates are semi-ranges of stress at the skin calculated from the torque 
or bending moment by the formula founded on the assumption of perfect elasticity. 
In the writer’s experiments the ranges of strain at the skin—not the whole loop—were 
observed ; and the stress-strain curves (an example of which is the curve ORDF of 
fig. 20) from which he begins his argument are of values of these ranges (or rather 
semi-ranges), when constancy of range has been reached, plotted to the semi-ranges 
of stress calculated on the assumption of perfect elasticity. 
_ Considering an abscissa ON of any point C, on B, C, D, (fig. 16) with linearity of 
strain from the axis outwards, the strain at any radius 7 in the body of a specimen 
of outside radius ry will be ON x= at the epoch when the constant value of the 
0 
skin-strain ON has been reached at a range of ‘ calculated’ stress NC,. That is, 
distances from O along the z-axis represent, to the scale on which ON represents 
the semi-range of strain at the skin, the semi-ranges of strain at corresponding radii; 
in other words, at any one such epoch, abscisse represent both radii and semi- 
‘ranges of strain at those radii. 
1’ Proc. Inst. Mech. Eng., Feb. 1917. See fig. 8, p. 129; fig. 12, p. 134. 
