﻿536 Mr. S. Lees on a Simple Model 



carve M 2 PjJ 2 , which is symmetrical with K 2 N]L 2 about the 

 origin, is similarly obtained from the curve PJ (compare 

 with fig. 7b)*. 



Fig. 15. 



(c) With | F x | > F N , we get hysteresis loops whatever be 

 the value of | F \. The mean-stress-strain-curve, i. e. the 

 locus of the point given by the mean of the extreme stresses 

 and the mean of the extreme strains for an instantaneous 

 loop with the given | F x | corresponding to any values of F , 

 is therefore a curve like UOV, of limited length (see fig. 12). 

 Provided permanent set does not take place, the length of 

 UOY will be determined by both of the values of | F 1 and 

 i'FjI, and not merely by | F |, as in § 9 (c). A limitation is 

 set up to the value of | F 1 + | F : | by the restriction that no 

 permanent set shall take place. The algebraic details are 

 not given here. 



§ 14. Application of Model to Shear Stresses. 



The model of § 3 and its modification of § 10 can also 

 be used to illustrate elastic hysteresis for shear stresses. 

 Referring, e. g. to fig. 3, we have only to imagine the 

 force F now applied to B 2 instead of A l5 and the equal and 

 opposite force applied to C 2 instead of A 2 , to get a model 

 showing shearing action. We are here neglecting the con- 

 sequent tilting of the portions of the model ; this tilting may 

 be avoided by supposing the component parts to be con- 

 strained to move parallel to the forces F by suitable friction- 

 less guides. Whilst it is not intended that any comparison 



* Notice, however, that our theory would make the dotted Hue N'L' 

 of fig. 7 b always coiucide with a curve of the mean-stress-strain 

 diagram, which apparently is not the case. 



