Ductile Materials under Combined Stress. 123 



54. Conclusions prooable for general type of Stress. — Con- 

 sidering- the three-dimensional view (fig. 20, p. 93), if the 

 shearing-stress law were true the observations plotted in space 

 between Ox, Oy, Oz, or two tensions and a third stress, would 

 fall upon the two planes, EAB, EDB. We have in our 

 experiments investigated such combinations of stresses as 

 give rise to half the line AE, i. e. to FA, to AB, and to BD' 

 which is half BD. 



Now a similar figure may be drawn for the strains, and 

 since the traces on the planes ??2 — and ?73 = in such a 

 figure are straight lines, and since the strains are so small 

 that no quadratic or higher power can have appreciable 

 effect, except in the improbable case of their coefficients 

 being extremely large, the function representing this relation 

 is linear. Since by experiment the value of the intermediate 

 strain has no effect on the yield-point relation, the coefficient 

 of t] 2 in the linear relation is zero. The linear relation there- 

 fore reduces to the equation of the line AE (or the plane 

 AEB). This line, as determined by experiment, being 

 equally inclined to the axes, the corresponding relation is 

 that the shearing stress or strain is constant. 



55. Effect of a Uniform Volumetric Stress. — In order to 

 ascertain the relationship of the stresses at the yield-point 

 more accurately, sequences of tests in which torsion and 

 tension combined in various amounts should be compared, 

 and the inclination of the line AE more accurately deter- 

 mined. From this point of view a direct compression-test 

 would be of much assistance. Now the linear relation may 

 be written 



OT i — ^s + M^i + 'cr 2 ) =C, 



since this contains two independent constants. Putting 

 vr 2 = we get the case of tension only and the yield-point 



stress from the equation is " . Putting «r 2 = — vt x we get 



the case of pure torsion, and the yield -point stress is then ±C. 

 The mean of the experimental results gives for the ratio of 

 these yield-point stresses the value 0*52, for which the value 

 of X is 0"04 ; and the relation becomes 



OTj — -5T 2 + 0*04 (utj + nr 3 ) = q , 



where q is the value of the yield-point shearing-stress in 

 pure torsion. 



56. Convenient view of general type of Stress. — Since the 

 stress ellipsoid 'ar 1 a; 2 + 'uy 2 y Q + ^ 3 z 2 =:l 7 where -33- l3 ot 2 , -b7 3 are in 



