﻿526 - Mr. S. Lees on a Simple Model 



value o£ F corresponding to the point N ; call this value F^ 

 (given by equation (11)). Keeping | !F X | constant and slowly 

 allowing F to increase from zero, we ultimately get (on the 

 mean-stress-strain curve) to the point Ni, corresponding to 

 the maximum stress F^. Further increase of the maximum 

 stress can only be obtained with slip, and such maximum 

 stresses must correspond to points lying on the line KL. 

 During the rapid variation of Fj, whenever the F— x point 

 on the diagram leaves the line KL, it will do so to travel 

 along a line (like RS) parallel to PON, and must return to 

 the line KL at the same point that it left it (since during this 

 travel there is no slip). Hence the line of the mean-stress- 

 strain curve corresponding to the description of the line KL 

 will be K 2 L 2 , a line parallel to KL (see again fig. 7). The 

 complete mean-stress-strain diagram for the complete cyclic 

 variation of F is therefore given by J 2 K 2 L 2 M2. It will be 

 noticed that it is of similar type to the static stress-strain 

 loop of" fig. 4, i. e. its sides are parallel to the sides of the 

 parallelogram JKLM of fig. 4. 



If we draw in the static hysteresis loop J 3 K 3 L 3 M3 cor- 

 responding to the range ± (| F | 4- |Fj|), we see at once 

 that the area of the mean-stress-strain loop J 2 K 2 L 2 M 2 is 

 (Fn — [Fi|)/Fm- times that of J 3 K S L 3 M3, and is therefore 

 by (30) given by 



. . . (34) 



(c) We lastly consider the case of |F 1 | greater than F^. 

 In this case, the rapid variations in |F X | will always cause 

 the F — x or state point of the model to move from the line 

 KL to the line MJ (or vice versa). A point on the mean- 

 stress-strain curve is therefore always to be regarded as the 

 geometrical centre (or centre of gravity) of a loop in 

 the form of a parallelogram like J'K'L'M' of fig. 7 a, such 

 loop corresponding to some asymmetric cyclic condition. 

 The locus of such centre of area is clearly a straight line 

 UOV, passing through and lying parallel to KL or JM. 

 In such a case, therefore, the area of the mean-stress-strain 

 loop vanishes. For a given |F | and [F^, it is easily 

 verified that the x coordinate in the diagram (fig. 7 a) of 

 the end point V of this line is given by | F |/X 2 , the corre- 

 sponding value of F being |F |. 



We may compare these results for mean-stress-strain loops 

 with those given by Smith & Wedgwood (loc. cit.). 



