472 Prof. Osborne Keynolds on the Dilatancy of Media 



the strains and stresses in any other condition if the particles 

 were frictionless and rigid. 



If the particles were rigid the medium would be absolutely 

 without resilience, and hence the only energy of which it would 

 be susceptible would be kinetic energy ; so that, supposing 

 the motion slow, the work done upon any group in distorting 

 it would be zero. Thus, supposing a contraction in one direc- 

 tion and expansion at right angles, then if p x be the stress in 

 the direction of contraction, and p 9 , p z the stress at right 

 angles, a being the contraction, b and c expansions, 



p a a+Pyb+p z c = ; 



or, supposing b = c, Py—pz, 



p x a +p y (a + c) = 0. 



With friction the relation will be different ; the friction always 

 opposes strain, i. e. tends to give stability. 



It is a very difficult question to say exactly what part fric- 

 tion plays ; for although we may perhaps still assume without 

 error, 



p y _ 1 — sin cj> 



p x ~~ 1 -t- sin (j> 



where <£ is the angle of repose, we cannot assume that tan <f> 

 has any relation to the actual friction between the molecules. 



The extreme value of <f> is a matter of arrangement ; as in 

 the case of shot, which would pile equally well although 

 without friction. 



Supposing the grains rigid, the relations between distortion 

 and dilation are independent of friction ; that is to say, the 

 same distortion of any bounding surfaces must mean the same 

 internal distortion whatever the friction may be. 



The only possible effect of friction would be to render the 

 grains stable under circumstances under which they would 

 not otherwise be stable ; and hence we might with friction be 

 able to bring about an alteration of the boundaries other than 

 the alteration possible without friction ; and thus we might 

 possibly obtain a dilation due to friction. How far this is the 

 case can be best ascertained by experiment. 



In the case of a granular medium, friction may always be 

 relaxed by relieving the mass of stress, and any stability due 

 to this cause would be shown by shaking the mass when in a 

 condition of no stress. 



But before applying this test, it is necessary to make per- 

 fectly sure that during the shaking the boundary spheres do 

 Dot change position. 



Another test of the effect of friction is by comparing the 



