Question

Volumetric strain produced in sphere is how much time the strain in its diameter

Answer 1

Let a sphere of diameter x and volume of sphere is V.

from formula of volume of sphere ,
V = 4/3 π(x/2)³ [ as we mentioned x is diameter so, radius of sphere = (x/2)]

so, V = 1/6 πx³ ......(1)

differentiating both sides,

dV = (1/6) × π (3x²) dx

or, dV/V = (π/2)x² dx/V

or, dV/V = (π/2)x²dx/(1/6 πx³) [ from equation (1)]

or, dV/V = (6 x²dx)/(2 x³)

or, dV/V = 3( dx/x )

hence, it is clear that, volumetric strain. produced in sphere is three times the strain in its diameter.

Answer 2

Volumetric strain:

It is defined as the ratio of change in volume to the initial volume of the body. It is equal to the sum of strain in all directions.

If equal normal forces are acting on all faces of the cube, so strain will be equal in all directions

(ϵx = ϵy = ϵz).

ϵv = ϵx + ϵy + ϵz

ϵv = 3ϵx

Hence, Volumetric strain is 3 times the strain in all direction.