Ductile Materials under Combined Stress. 105 



In the tables of results the next column contains the 

 maximum shearing-stress ; this takes place in a plane per- 

 pendicular to the axis of mean principal stress, and is that of 

 the plane through the mean axis bisecting the angles between 

 the other principal stresses: its value is half the difference be- 

 tween the greatest and least principal stresses. It will be noticed 

 in the case of those experiments in which the stresses in the 

 tangential plane are of opposite signs, that this tangential 

 plane is the plane of the greatest shear, but that in the 

 other cases the plane passes through the axis of the 

 tube. The value of the maximum shearing-stress in cases 

 involving torsion is, from the value of the principal stresses, 



vW+* 



43. Calculation of the Strains. — Owing to the elastic- 

 limit effect the measured strains vary more than the stresses, 

 especially as the elastic-limit effect is practically absent in a 

 test when the immediately preceding test is of the same type. 



In the simple torsion tests the value of the shearing-strain 

 or slide <£ is obtained directly from the reading by the use of 

 the factor. If i be the angle of twist of the specimen in a 

 length I, r the radius of the specimen, d the distance to the 

 scale, measured in the same units as x the scale-reading, we 

 have 



. r . r x 

 ^=l l= -ld' 



The principal component strains which produce the slide cf> 

 are r) 1 = ^<\>; v- 2 =—if'> and %=0. 



The effect of an axial stress is not only to produce an axial 

 strain, but to produce a uniform strain of the opposite sign 

 in all directions at right angles to it, and bearing to it, in 

 amount, the value of Poisson's ratio (er). The value of the 

 axial strain was determined from the corresponding scale- 

 reading by the use of a factor; the value of the resulting 

 contractile strain was obtained by multiplying this by the 

 value of a- deduced from the values of E and C for the speci- 

 men, in preference to using the results of the diametral 

 extensometer. 



In the combined tension and torsion tests, the value of the 

 principal strains was obtained from the observed value of the 

 axial strain e x and of the slide <f>, and by taking the value of 

 the circumferential strain as — aei. Since (f> is the only slide 

 the strain ellipsoid becomes 



e x x 2 + e 2 y 2 + e 5 z 2 + $xy= 1, 

 w here e 2 and e 3 are the strains perpendicular to e^ The 



