Volumetric Strain of the Space inside Shell:

To determine the volumetric strain (ratio of change in volume to the original volume) the expression for the volume of the sphere shall be considered.

Capacity/volume of the sphere, V = π d ³ /6

Enlarged volume, V + δV  = (π/6) (d + δ d )³

Where δd is the equivalent change in diameter.

Change in volume, δV  = (π /6 )(d + δ d )3  - (π/6) d 3

Expanding and neglecting second order terms and on simplifying, we get

δV  = (π d ²/2 )δ d

∴ Volumetric strain,

ε _v  = δV/V=  (π d ²/  2) δd / (π d ²/  6) d        

ε  = δV/V= 3× (δd/d)

Thus,

ε_v = 3 × εh

or,        volumetric strain = 3 × hoop strain

 which gives the expression,

ε _v  = 3 pd (1 - v)/ 4tE