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to Illustrate Elastic Hysteresis. 521 



the stages KL and MJ) through a slip on each occasion of 

 amount 'iy x . 



The area of the hysteresis loop clearly vanishes when 

 y 1= = 0, i.e. when 



F 1 =(X 1 + 2\ 2 >-i (28) 



Using (26), we get for the corresponding limiting values of 

 X\ and Fj : — 



v =^— 



A-2 



Fj _ / aN(X 1 + 2XQ _> 



A-2 



The limiting loop for which the area just vanishes is shown 

 in fig. 4 as the straight line PON. It is easy to verify 

 from (23) and (24) that N is the middle point of KL, and 

 similarly that P is the middle point of JM *. 



It is to be understood that for values of x\ and ¥i less than 

 those given by (29), the F— x relationship is always ex- 

 pressed by a portion of the straight line NP, whose slope 

 is the same as that of both JK and LM, and is that cor- 

 responding to the neutral state. This statement arises 

 from the fact that for such values of x and F, there will be 

 no slip. 



From (26) and (27) we can get two other expressions for 

 the area of the hysteresis loop (when it exists) in the forms : 



8 M y(^ 1 -g) = ^[F 1 - ^ +> -> ]. . ^30) 



From these we see that the area has a value proportional to 

 the excess of x y over the limiting value just referred to ; 

 alternatively, it is proportional to the excess of Fj over its 

 corresponding limiting value. 



§ 6. Comparison of Results obtained with Experiment. 



A large amount of data is available concerning elastic 

 hysteresis, but, for the moment, reference will only be made 

 to the paper by Smith & Wedgwood, loc. cit. figs. 2-5. The 

 static stress-strain loops obtained by these authors for tension 

 and compression of a material in the cyclic state agree in 

 several particulars with the theory just outlined. Thus 

 elastic hysteresis does not occur till the limits of stress 



* To get the model to the neutral state from the cyclical state, 

 we arrange for it to be put into either of the conditions corresponding 

 to the points N or P, and then take off the load F. 



