﻿534 Mr. S. Lees on a Simple Model • 



process of slip. Along KL the loss by friction is therefore 



2 P 1 T 2 dy = 2 P 1 (N -/fy 2 )(>-2a2/)<fy, . (51) 



J-Vx J-Vx 



whilst along the path MJ, the loss is 



2 (~ yi T 2 dy=-2 f" fl (N - i 8y»)( A4 + 2ay)4 (52) 

 Jyi Jy-i 



By changing y into (say) — u in (52), we see that (51) and 

 (52) give the same result, as might be expected. Hence the 

 total loss by friction in describing the loop is 



4 j** (N -/3/)(^-2ay)^ = 8 m (n, - f^ 2 ) . (53) 



This is the required area of the loop. It clearly vanishes 

 when the slip y x vanishes. 



When the slip y Y is small, a first approximation for y 1 is 

 from (45), 



Vi= (\ 2 ^i-^No)/(\ 3 -2aN ). . . . (54) 



A nearer approximation is therefore 



yi ~ X 2 -2aN + / ^ (A 2 -2aN f ' ' (55) 



Hence from (53) and (55), as x x and y Y increase, the area of 

 the loop increases at a greater rate than (A2^i— A&N ). This 

 result may be compared with the results of § 5, and the 

 comments in § 6. 



§ 12. Asymmetric Cyclic condition for Modified Model. 



We can work out the case of a hysteresis loop produced 

 with unsymmetrical stress limits, following the procedure 

 of § 7. A typical resulting loop is shown in fig. 12, which 

 shows a case with the mean stress F positive. The point L 

 will still lie on the dotted curve NL of fig. 11, whilst the 

 point J will also lie on the dotted curve JP of the same 

 figure. The new curves KL and MJ will not, however, be 

 symmetrical about the origin 0. As F is increased, the 

 point J will ultimately coincide with P, and further increase 

 of F will result in J lying on the curve PT, which is the 

 continuation to the right of P, of the curve JP. A limiting 

 case of elastic hysteresis will arise when F is increased to 

 such an extent that JK just touches the curve JPT. This 



