7i*6 Dr. Chree on the Stresses in the Earth's Crust 



centre of the stressed area ; consequently the strains and 

 stresses vary inversely as the square of this distance. For 

 instance, at a point at a distance z vertically below the centre 

 of the stressed area, a'/z being small, 



zz = STa' 2 /(2z 2 ) j' # * ' [OJ) 



while in the plane of the loaded area, but outside it, a'/r 

 being small, ^ 



-.rl-=6ct> = (l-2 V )Ta' 2 l2r\) 



r;J. }■ • • • < 4 °) 



Thus the effects diminish very rapidly as the distance from 

 the loaded area increases, and so far as rupture is concerned 

 interest centres in the material close to the loaded area. The 

 determination of what happens close to the loaded area is in 

 general rather a delicate operation, especially at points 

 situated close to its boundary. 



At any point on the axis of z 



W = z 2 + r' 2 , 



where r' is the distance of the element da from the centre of 

 the loaded area, and so 



dR/dz = z/R. 



Thus from the last of equations (37) 



2wnA _CC£_j _ ( a ' z • 27rr'dr f 



= 277{l-,?/(a' 2 + ^}. 



Thus in the limit when 2 = 0, i. e. at the centre of the 

 loaded area, 



A = T(l-2i7)/n (41) 



Now by symmetry it is clear that rr, <^>, zz are the 

 principal stresses at the centre, and also that 



fr = <j)(f). 



Thus 2rr+ zz = rr + $(/> + zz 



= (3m-n)A=2(l+?/)T. 



But by the surface conditions 



zl=T. 



Hence 

 and 



7r=(l + 2 v )T/2,\ 

 S=Q-^=(1-2,)T/2.J v " ; 



