﻿to Illustrate Elastic Hysteresis. 531 



Starting with the model in the condition corresponding 

 to the neutral state, and applying a gradually increasing 

 force F, we have at first 



F = (X 1 -f2X 2 >, (9) 



exactly as in § 3. Slip movement will take place when 



\ 2 o; = ^N = T 2 . (40) 



After this, we shall have for further increased values of F : 



T 2 = \ 2 U-y) = (N -/3 i / 2 )(^-2ay). . . (41) 

 The relation between y and x is therefore the cubic equation 



I£ we consider such displacements that (N — /3?/ 2 ) is always 

 positive, it is physically obvious that there must be a real 

 positive root of y for any given value of x (which value must 

 of course be such that \ 2 % > ^oN • The real positive value 

 of y can also be shown to exist by putting successively y = 

 and y = co in the left-hand side of (42), when there appears 

 a change of sign. 



As y goes on increasing, a time will come when T 2 will 

 vanish. This, from (41) , will clearly be the case when 



y = fx/2a, and F = \ lf i/2a. . . . (4.3) 



Further increase of y will result in T 2 becoming negative, 

 i. e. becomino- a compression. If after this has occurred, 

 F is caused to diminish, ultimately becoming a compression, 

 it is easily seen that T 2 will always remain negative, as 

 friction and slope 6 will now be assisting each other. No 

 matter what compression may correspond to T 2 , y cannot be 

 caused to diminish. Thus the value of y given in (43) really 

 corresponds to the beginning of permanent set. For values 

 of F beyond that given by (43), complete recovery cannot 

 be afterwards attained. We may therefore call this value of 

 F the elastic limit. It is to be distinguished from the value 

 of F obtained from (9) and (40), viz. 



F = (X 1± 2X 2 VN (M) 



\ 2 



This latter limit corresponds to the limit of Hookers law. 

 For values of F less than this, Hooke's law will be ac- 

 curately followed, and there will not be any hysteresis 

 effects. 



If F be increased continually beyond the value given by 

 2M2 



