154 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 11, NO. 7 



standards of performance for the comparative study of different 

 bodies, a^ process may be called ideally irreversible if the observed 

 effect is in exact agreement with the amount calculated from the prin- 

 ciple of superposition. The departure of an actual process from this 

 arbitrary standard might be taken as a measure of the irreversible 

 peculiarities of the body, just as the deviation of a real fluid from the 

 equation of state of an ideal gas may be taken as a measure of its in- 

 trinsic properties. 



Simple harmonic motion and the cyclic state. As an example of com- 

 putation by the superposition method, consider a body subjected to a 

 load 



X = Rsin cor (16) 



If this body follows the drift function (6) with constants independent 

 of load, integrating (11) gives 



y = 



^-'"' + sin {o:t - 4>) 



(17) 



where tan <^ = co/m. From (1) the total displacement x can be found. 

 After sufficient time that the exponential term of (17) has dwindled 

 to an insignificant amount, the cyclic state may be considered es- 

 tablished; the displacement then is given approximately by 



" sin (coi - <^) (18) 



X 



=ii.. 



hi '4:)]* 



where tan$ = AE{m/oo). The hysteresis loop will be found by elimi- 

 nating the time {T = t) between the load equation (16) and the dis- 

 placement equation (18). Since (16) and (18) represent two simple 

 harmonic motions at right angles, of the same period but differing in 

 amplitude and phase, the hysteresis loop will be an ellipse. The area 

 of this ellipse will represent the energy dissipated into heat per cycle, 

 after the cyclic state has been reached. 



Calculation of recovery curve. Suppose a load R suddenly applied, 

 kept on for a time interval to , then suddenly released. What will be 

 the after-effect z (residual displacement) at an interval t' after the 

 instant of release? Since x^ is now zero, the after-effect will equal 

 the yield y at time t = to + t' due to a load increment aX = R 3.t 

 time T = followed by a second increment A A' = —Rat time t = to. 

 Hence by (10) 



z = R[F{to +t') - Fit')] (19) 



