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 V  V  V  dx   dx dy   dy dz   dz  
1 0
  dx    dy   dz 
 dxdydz  dxdydz  1   1  1   dxdydz .
 dx   dy   dz 
Taking into account that the relative elongation in the direction
of the principal stresses
dx dy dz
  ;   ;   , we obtain:
3
2
1
dx dy dz
V  V                   .
0 1 2 3 1 2 1 3 2 3 1 2 3
In real structural materials the relative elongations are

usually measured in thousands or ten thousand parts. Therefore,
their products can be neglected

V  V      . (3.26)
0 1 2 3
The relative change in volume (volumetric deformation)

V
        . (3.27)
2
1
3
V
0
Substituting the generalized Hooke's law (3.25) into formula
(3.27), we obtain:
1 2
       . (3.28)
E 1 2 3
Expression (3.28) is called voluminous Hooke's law.
We see that the change of volume depends only on the sum of
the principal stress and not on their relationship.
Consider the case of a comprehensive hydrostatic tensile of the
material, when        . Then, according to the formula
1 2 3
(3.28)we will have
 
  3   21    , (3.29)
E K

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