﻿to Illustrate Elastic Hysteresis. 



519 



increased algebraically, we shall again go through a series of 

 operations in which first 



— — _ \j -f- 2\ 2 , 

 ax 



(20) 



after which (17) will hold. 



By performing such reversals of stress, between limits 

 + F X , sufficient to cause slipping to occur in both directions, 

 we can ultimately reach a cyclical condition ; the stress- 

 strain loop, i. e. the F — x loop, being as shown in fig, 4. 



L . (x„F,) 



The diagram is such that the limits of F are ±F l5 the 

 corresponding limits for x being + x x . 



The typical feature of the cyclic condition is, of course, 

 that the F— x loop is symmetrical with respect to the 

 origin. In our case, the diagram is a parallelogram 

 JKLM, with d¥/dx=\ 1 for the lines KL and MJ, whilst 

 dF/dx = (\x + 2X,) for the lines JK and LM. 



To get the F— m coordinates of K, the intersection of the 

 lines JK and KL, we have for the line JK 



F-(-F,) = (Xi + 2\ 2 ) [>-(-*!)], • • (21) 

 whilst for the line KL we have 



¥ 1 -E = X^i-*) (22) 



These two equations give for the point K : — 



x = 



F = 



Fi— (VfXs)*?! 



(\ 1 +X 2 )F 1 -X 1 (X 1 + 2\ 2 ).r 1 



Xo 



(23) 

 (24) 



