﻿to Illustrate Elastic Hysteresis. 513 



§ 2. Discussion of the Problem. 



In connexion with elastic hysteresis, it seems natural to 

 invoke viscous effects analogous to those found in viscous 

 fluids. A very obvious idea is to introduce for a material 

 undergoing cyclical variations of stress, a stress term always 

 depending upon the rate of change of the corresponding- 

 strain. Thus, e.g., if / denote stress, s denote the corre- 

 sponding strain, we may try 



fy=Ks + \s, (1) 



where K and X are constants. For the type of stress con- 

 sidered, K would be the ordinary modulus of elasticity. If 

 we now make s go through a cycle given by (£ = time) 



s = s cos pt y (2) 



we get different values of / which can be plotted against s, 

 giving rise to a stress-strain loop *. This loop is clearly an 



ellipse, as in fig. 1. For on eliminating t between 



/= s (Kcosp£— Xpsmpt) .... (3) 



and (2), an ellipse arises. It is not difficult to verify that 

 the area of the ellipse is proportional to both p and s 2 . 

 Thus with such an assumption, the area of the loop, for a 

 given 5 , will diminish indefinitely as the speed of fluctuation 



* In plotting stress-strain loops, we can always make the straight line 

 /=Ks have any slope we please by suitably choosing the scale for 

 /ands; but in general this will give us a loop of minute proportions, 

 for breadth. Having chosen a suitable slope for the above line, the loop 

 can be magnified by representing the divergence (/— Ks), of a point of 

 the loop from the straight line corresponding to no hysteresis, on a scale 

 any number of times that of/. This will be assumed to have been done 

 for the stress-strain diagrams shown in this paper. 



Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 L 



