Chap. 9] SEISMIC METHODS 443 



Since the elementary body must be in equilibrium against rotation, the 

 tangential stresses in each pair referred to above are equal to one another. 

 (a) Normal stresses. Under the influence of the normal stresses Xx , 

 Yy , and Zj , three sides of the element of volume suffer the displacements 

 u in the x direction, v in the y direction, and w in the z direction. Then 

 the deformations referred to unit length, or the specific strains, are du/dx, 

 dv/dy, and dw/dz. The change in volume AV, resulting from the deforma- 

 tions in the three directions, is equal to the sum of the specific strains: 



^J^^e = ^ + ^ + ^-:^. (9-1) 



V dx dy dz 



If is negative, reduction of volume or compression takes place; if 

 positive, extension or dilation. The change in volume is accompanied »by 

 a change in shape; when the length is extended, the section is reduced. 

 In this type of deformation, the angles are preserved. Since the strains are 

 proportional to the stresses, the relation between normal stress and the 

 corresponding strain in the x direction may be written^ 



^ = e.X., (^2) 



dx 



where c is the dilation coefficient. A more customary definition of this 

 coefficient may be arrived at by writing 



1 = ?f = 11^ 

 t du M ' 



dx I 



where P/S is the load per unit area, S is area, and 1/t is Young's modulus: 



E = ^.1, (9-3) 



that is, load per unit area divided by relative elongation. 



The expansion du/dx in the x direction produces a reduction of section 

 in the y-z plane. Assuming the reductions in width, dv/dy and dw/dz, 

 to be equal and to be a fraction of and proportional to the elongation, 

 we have 



^ = ^J? = ^^. (9-4a) 



dy dz dx 



du 



ax 



^ / 



The factor a is called Poisson's ratio. From eq. (9-4a), <^ = -^ / 



»» Assuming, for the moment, that Xx alone is effective. The complete expres- 

 sions are given in eq. (9-6). 



