Ductile Materials under Combined Stress 

 and substituting in (2), 



20 1 + a 1 +\, <l>i + P, 



1 + #1 — -r — ~k~, 4>i + 010U 



97 



1- 



2 



2 

 «1 



1 + v 



o, 



«1 



= 0. 



From which we get 



fi^fa^ + aj. 



So that to the second order of small quantities, 



#2 = 20! + a — fa 2 , 



tj 2 = (fi 1 + (f> 1 (2d 1 + a l ), 



z 2 =l-20 l 2 -20 1 cc i 



-«l-il 



We now proceed to find x } y, z from equations similar to 

 (1), (2), and (3), namely, 



#h + VV2 4 s£ 2 = #8& + ytfz + *2?2, 



(8) 

 (9) 



x y z = 0, ... 

 & % ?2 



#2 #2 z 2 



and a? 2 + 2/ 2 + ^ 2 =l. 



Equation (8) becomes 



= (20 1 + « 1 - < /> 1 2 )(1 + 2 ) + {0i + 1 2^+^}(1 + 2 )^ 2 

 - +(l-2^-2^ 1 -^-^)(l-0 2 -^) ; . (10) 

 or to the first order, 



x(l + 2 )+y<f> 2 +z(l-0 2 )=20 1 +a 1 +l-0 2 . . (10 a) 



.-. x + z = l. 



Equation (9) becomes 



x 



2 



0, 2 



1 + 02- f-^, ft(l+*,), 1-02-^ 



20 x + <xi% fa 1 



Phil. Mag. S. 5. Yol. 50. No. 302. July 1900. 



= 0; 



H 



