144 



Consider a small element of the stressed body in the shape of a 

 cube oriented so that the faces are perpendicular to the principal stresses. 

 Consider the plane whose normal makes equal angles with the directions of the 

 principal stresses. The normal stresses on this plane do no work because of 

 the constancy of volume. The shearing stress and the corresponding shearing 

 strain are called the octahedral shearing stress and octahedral shearing 

 strain. They are designated hereafter by the symbols r and y respectively 

 and are related to the principal stresses and strains as follows: 



T = J /(cr, - a^)^ + ((T2 - a^)^ + (a^ - <7,)' [k] 



V = |/(£, - e,f + (e, - 63)2 + (63 - £,)=' [5] 



If Equations [2] and [5] are combined to eliminate Cg then 



y = ^VWVe,' + e,e, + e ,^ [6] 



There is some experimental evidence (3) (*+) that in ductile materi- 

 als the quantity t, thought of as a single variable, can be used to define 

 the state of stress in the material. It is considered also that the quantity 

 y can be used to define the state of strain. 



The fundamental assumption (5) is now made that 



T = T(y) [7] 



for the case of static loading. Thus r is completely specified for a given 

 value of y. It is assumed that the stress-strain relation, regardless of the 

 particular type of loading that produced it, will yield points that lie on 

 this curve, which describes a physical property of the material. This means 

 in particular that simple tensile tests and complicated tests producing bi- 

 axial and triaxlal stress conditions can be described in terms of a single 

 generalized stress-strain curve. 



Some evidence for this assumption is found in unpublished work done 

 by Dr. A. Nadal at the Westinghouse Research Laboratories in Pittsburgh. Bi- 

 axial tests were performed on very carefully machined, hollow, tubular test 

 specimens of annealed medium steel in which axial loading and internal pres- 

 sure were used. The ratio of tangential to longitudinal stress was maintained 

 as a constant for a given specimen but a different value of this ratio was 

 taken for each of the ten specimens tested. The measured stresses and strains 

 were used to compute the octahedral shearing stresses and strains. These val- 

 ues were then all plotted with t as the ordinate and y as the abscissa, as 

 shown in Figure 1 . All the points fall remarkably well on a single curve. 

 For a given value of y the maximum deviation of r is about 7 per cent. Gen- 

 erally the deviation is considerably less. 



