What is volumetric strain in thin cylinders?

Short Answer:

Volumetric strain in thin cylinders is the ratio of the change in volume of the cylinder to its original volume when it is subjected to internal pressure. In simple words, it shows how much the total volume of a thin-walled pressure vessel changes due to the combined effect of hoop and longitudinal stresses.

In a thin cylinder, internal pressure causes an increase in diameter, length, and volume. The volumetric strain is the sum of the linear strains in the circumferential, longitudinal, and radial directions. It is given by:

where  = volumetric strain,  = internal pressure,  = diameter,  = wall thickness,  = Young’s modulus, and  = Poisson’s ratio.

Detailed Explanation:

Volumetric Strain in Thin Cylinders

When a thin cylindrical pressure vessel is subjected to internal pressure, it expands slightly due to the stresses developed in its wall. The internal pressure acts equally in all directions, producing three types of deformations:

  1. Increase in diameter (circumferential direction).
  2. Increase in length (longitudinal direction).
  3. Increase in volume (overall change).

The volumetric strain is the ratio of the change in volume () to the original volume () of the cylinder. It is a measure of how much the volume of the cylinder changes under pressure and depends on both hoop stress and longitudinal stress.

Mathematically,

Derivation of Volumetric Strain

Let the thin cylindrical vessel have:

  • Internal diameter =
  • Length =
  • Wall thickness =
  • Internal pressure =
  • Material properties: Young’s modulus = , Poisson’s ratio = .

Due to internal pressure, two main stresses are induced in the wall:

  1. Hoop stress, 
  2. Longitudinal stress, 

These stresses cause corresponding strains in the material.

  1. Hoop (circumferential) strain

The circumferential or hoop strain is given by:

Substituting the values of stresses:

 

This represents the strain along the circumference or the change in diameter of the cylinder.

  1. Longitudinal strain

Similarly, the longitudinal strain is given by:

Substitute the stress values again:

 

This represents the strain along the length of the cylinder.

  1. Volumetric strain

Now, the total volumetric strain (e_v) is the sum of the linear strains in three mutually perpendicular directions — circumferential, longitudinal, and radial.
For thin cylinders, the radial strain is very small compared to the other two and is often neglected. Therefore,

(Here,  appears twice because the hoop strain acts in two perpendicular circumferential directions.)

Substitute the values of  and :

Simplifying,

 

This is the final expression for the volumetric strain in a thin cylinder under internal pressure.

Interpretation of Volumetric Strain Formula

From the derived expression:

  1. Direct Proportionality:
    The volumetric strain increases with internal pressure () and diameter ().
  2. Inverse Proportionality:
    It decreases with higher wall thickness () and higher modulus of elasticity ().
  3. Effect of Poisson’s Ratio ():
    As the material’s Poisson’s ratio increases, the volumetric strain decreases slightly because the lateral contraction offsets the expansion.

Simplified Approximation

In some cases, a simplified form of volumetric strain is used for easy calculation:

This form gives approximately the same result and highlights the dependence of volumetric strain on material and geometry.

Applications of Volumetric Strain in Thin Cylinders

  1. Pressure Vessel Design:
    Helps in determining the expansion of the cylinder when subjected to internal pressure.
  2. Hydraulic and Pneumatic Systems:
    Used to calculate how much fluid volume a pressure vessel or pipe can accommodate under operating pressure.
  3. Boilers and Gas Cylinders:
    Important in determining allowable deformation to avoid excessive expansion that may lead to rupture.
  4. Aerospace and Automotive Applications:
    Ensures that fuel or air tanks withstand pressure without large changes in volume.
  5. Testing and Calibration:
    Used in experimental setups to measure material behavior under internal pressure conditions.

Practical Example

If a thin cylinder with:

  • Internal diameter ,
  • Wall thickness ,
  • Internal pressure ,
  • Material having  and ,

Then,

 

This means the total volume of the cylinder increases by 0.0144% under the given pressure.

Conclusion

The volumetric strain in a thin cylinder is the ratio of the change in its volume to the original volume due to internal pressure. It results from the combined effects of hoop and longitudinal strains. The final expression is:

It shows that volumetric strain increases with internal pressure and diameter but decreases with wall thickness and Young’s modulus. Understanding volumetric strain helps engineers design safe and efficient pressure vessels that can withstand internal pressures without excessive deformation.