﻿514 Mr. S. Lees on a Simple Model 



is caused to diminish indefinitely. But this is quite contrary 

 to the facts obtained by experiment. 



The results obtained by using a formula * of the type 



/= Ks + \s+fjLS+VS+ , ... (4) 



are quite analogous. On putting s = s cospt as before, we 

 shall get 



f=s {cospt (K—/j,p 2 + )— p sin pt (X— vp 2 + )}, (5) 



and on plotting / against s, we shall again get an ellipse. 

 Further, whilst the area of the ellipse will not follow quite 

 so simple a law as before, it will easily be seen that as p is 

 made to approach zero as limit, the area (for a given s Q ) will 

 do likewise. This, again, is contrary to experience. 



It may be said, in passing, that elastic hysteresis loops are 

 not found in exact experiment to be ellipses at all. A nearer 

 approximation to actual shapes can be found by assuming 

 the loops to be lenticular. Such a shape can be got, e.g. by 

 taking 



f=K,±/3s*, (6) 



where the sign of the s 2 term is always so to be taken as to 

 make the frictional stress term oppose the change in strain. 



Pig. 2. 



-F=Ks 



Taking s = s Q cospt as before, it will be found that we get a 

 diagram which is the result of eliminating t between this 

 relation and 



/= ~Ks Q cospt ±fis<?p 2 sin 2 pt (7) 



If the loop be considered to be described in the clockwise sense, 

 the plus sign will be taken from t=—7r/2p to t = ir/2p. The 

 loop is then seen to be lenticular, as in fig. 2. Here, again, 



* See Maxwell, ' Collected Papers,' vol. ii. p. 623. 



