STRESS SYSTEMS IX THE SOEDEia;ESS WlfAl'I'ED CONNECTION 1097 



^here o-q is the initial stress, a the final stress, k Boltzmann's constant, 

 T the absolute temperature, ^o a time determined from the equation 



kT , /^Ato 



<7o = --T-Jog( -^ ) (3) 



and A and P are constants in the fundamental equation 



^- = -Ae-'"-'"'"' (4) 



dt 



where H is the activation energy in the absence of any stress. 



The result of (2) is that for times large compared to ^o , the stress dif- 

 ference (To — 0-, plotted on logarithmic paper will be a straight line 

 relation. Hence, in Fig. 2, for times in the order of forty years, the in- 

 dicated stress level is 60 per cent of 8,000 pounds per square inch or 

 4,800 pounds per square inch. 



Another method of extrapolation which essentially makes use of (4) 

 is to measure the rates of relaxation at different temperatures and de- 

 termine an activation energy for each stress level by comparing the 

 logarithms of the times for the same stress levels as a function of tem- 

 perature. The constant A in (4) is usually considered to have a tempera- 

 ture factor T in it and is usually written as A'T. With this relation, (4) 

 can be written 



da _-(H-&a)lkT 



da = dt (5) 



^'7'g-(fl-^<r)/A:r A'T 



Integrating this equation between the limits ao and a, we have 



Hence, if the quantity 



is small compared to unity, we have 



-0{co-a)lkT 



(H - fia) = k loge (/1//2) / 



7\ Y, 



(7) 



As shown by Fig. 3, the vakie of jS is about () calorics per pound per 

 square inch and hence \i aa — a is at least 800 pounds/square inch (i.e., 

 for all values of relaxation of 0.9 or less) this factor will be less than 1 

 per cent for all temperatures used and hence we can use (7) to evaluate 

 values of the activation energy H-^a. 



