248 
PROFESSOR G. H. DARWIN ON THE DISTRIBUTION OF STRAIN 
the total amount of crumpling at any depth, we require to find the integral effect 
taken from t' to t, which is greater than t\ 
The integral stretching from consolidation to time t is 
K = 
Y-v- 
6a :t dv 
c dx 
dv 
Now near the surface v is nearly equal to v 0 + x — ; 
hence 
K=e( Y-v 0 )- 
dv 
dx 
6ld 
x + — 
c 
But 
Hence 
dv 
V 
dx ( 7 Tuty 
t - 
Y 
( met)* 
near the surface. 
K = e(V — v 0 ) — 
eY 
( 7 r/c)* [ft 
x b/cft' 
7k “r ~ 
( 12 ) 
At the time t', given by t' — cx/Qk, 
K=e(Y-v 0 )~ 
eY 
( 7 TfC)* 
1 /6/c\r 
x* ( j + ~ 
= e(V-v 0 )~ 
eV 
(7r/c ) 4 
2 ( 7 )^. • (13) 
This gives the total stretching from the time of consolidation until the surface of 
no strain has got down to x. 
If, therefore, we subtract (13) from (12), we get the total stretching between the 
time t' and the time t, and the result is obviously— 
K — - 
eY 
(met)* 
,’6/eA^ A 6 ict 
x — 2 ( — j x> -j- — 
This is clearly 
K = 
dv 
dx 
x 
2 - I - 1 
\ G J _ 
(14) 
This expression is essentially negative, and therefore the total effect from t' to t is 
a crumpling, as was foreseen. 
This integral crumpling vanishes at the same place as does dKjdt , that is to 
say, when 
6/d 
x — 
