GENERAL STRESS SYSTEM. 43 



LINES OF UNALTERED DIRECTION. 



It was shown above that, in general, three diameters of the strain ellip- 

 soid have the same direction after strain as before strain.* It is usual 

 to assume that these same lines retain their direction during tbe process 

 of strain,f but this appears to be true only under certain limitations. 



If the displacements a and b are connected by the equation a = mb; 

 formula (7), which assigns the position of the lines of unchanged direc- 

 tion in the x y plane, becomes : 



f—e 



nK(W 



and the position of the axes of the principal ellipse at the inception of 

 strain is given by — 



&•* 



tan 1 v — ! — f— . 



e—j 



1 fence one may write- 



in -flf , n ,'4 hi . ,., ) 



tan , _ t_ | cot 1 .„ ± yj^^-^, + cot- 1 ». | . 



In this formula v Q depends solely upon the direction of the external 

 force relatively to the resistance and not upon its intensity. Conse- 

 quently, if the tan * is to preserve its initial values throughout the strain- 

 ing process, m must be constant. Now, the displacements may lie such 

 that a or b is zero throughout deformation, and m is then constantly zero 

 or infinity. It may also happen that a = b, so that m = 1 . and this case 

 also involves no hypothesis as to a relation between stress and strain in 

 homogeneous matter; but if m is a finite quantity differing from unity, 

 the assumption that m is constant is equivalent to the hypothesis that 

 the ratio of the displacements bears a constant relation to the ratio of the 

 stress components which produce them. This hypothesis is only justifi- 

 able when the strain is very small. 



When there is no rotation, or when a = b, the elastic cube acts as if it 

 rested upon. an inflexible support and were affected by stresses axially 

 disposed. When one of the displacements a or b disappears, the strain 

 involves only axial deformations and scission. This again implies the 

 presence of an inflexible support or an equivalent rotating system of 

 forces. Hence the lines which have the same direction after strain as 



* Two of these diameters may < <>i n<- 1< l<- and both of these may become imaginary. 



f Thomson and Tait speak of these lines as unaltered indirect! luring the change of straii 



they may have meant by rather than during. Nat. Phil., section 181. 



