444 



SEISMIC METHODS 



[Chap. 9 



or as generally written, 



= ^ / 

 d I 





(9-46) 



Poisson's ratio is the ratio of relative reduction of diameter d and relative 

 elongation. For many substances, a is in the neighborhood of \. Substi- 

 tuting (9-2) in (9-4a), 



bv _ dw 

 dy ~ dz 



= (TtXx 



(9-5) 



The original extension in the x direction given by eq. (9-2) is opposed 

 by a reduction of the x~z section due to the normal stress Y,, . By analogy 

 with (9-5), this reduction is (du/dx) = atYy . A further reduction takes 

 place because of the stress Zj , which is atZ^ . Therefore, the total specific 

 strain is 



Hence, 



— = tXx — <reYj, — (TtZz 

 dx 



du 



the strain in the x direction: — = t[Xx — aiYy + Zj)] 



dx 



dv 



the strain in the y direction: — = e[Y„ — <t{Zz + Xi)] 



dy 



and 



the strain in the z direction: — = t[Zz — a{Xx + Y„)]. 



dz 



Adding these three equations and considering (9-1), 



e -e[(l - 2(r)(X. + Y, +Z.)]. 



y (9-6) 



(9-7) 



Adding to the right side of the first equation of (9-6), +Xi<re and — Xx<re, 

 the specific strain in the x direction 



^ = e[X.(l + cr) - a(Xx + Y, + Z,)]. 

 dx 



Re-substituting eq. (9-7), 



^^ = e[x,(l+.) 



(1 



(70 "I 



