A thin cylindrical shell of diameter (d) and thickness (t) is subjected to an internal pressure (P). The ratio of longitudinal strain to volumetric strain is

(where, \(\frac{1}{m}\) = Poisson's ratio)

Concept:

We use strain relations in a thin cylindrical shell to determine the ratio of longitudinal strain to volumetric strain under internal pressure.

Given:

• Diameter of the cylinder, \( d \)
• Thickness of the cylinder, \( t \)
• Internal pressure, \( P \)
• Poisson's ratio, \( \frac{1}{m} \)

Step 1: Calculate Hoop Stress and Longitudinal Stress

Hoop stress, \( \sigma_h = \frac{P d}{2 t} \)

Longitudinal stress, \( \sigma_l = \frac{P d}{4 t} \)

Step 2: Calculate Longitudinal Strain

Longitudinal strain, \( \epsilon_l = \frac{\sigma_l}{E} - \frac{1}{m} \cdot \frac{\sigma_h}{E} \)

Substituting the stresses:

\( \epsilon_l = \frac{P d}{4 t E} \left(1 - \frac{2}{m}\right) \)

Step 3: Calculate Hoop Strain

Hoop strain, \( \epsilon_h = \frac{\sigma_h}{E} - \frac{1}{m} \cdot \frac{\sigma_l}{E} \)

Substituting the stresses:

\( \epsilon_h = \frac{P d}{4 t E} \left(2 - \frac{1}{m}\right) \)

Step 4: Calculate Volumetric Strain

Volumetric strain, \( \epsilon_v = \epsilon_l + 2 \epsilon_h \)

Substituting the strains:

\( \epsilon_v = \frac{P d}{4 t E} \left(1 - \frac{2}{m} + 4 - \frac{2}{m}\right) = \frac{P d}{4 t E} \left(5 - \frac{4}{m}\right) \)

Step 5: Determine the Ratio of Longitudinal Strain to Volumetric Strain

Ratio, \( \frac{\epsilon_l}{\epsilon_v} = \frac{\frac{P d}{4 t E} \left(1 - \frac{2}{m}\right)}{\frac{P d}{4 t E} \left(5 - \frac{4}{m}\right)} = \frac{m - 2}{5m - 4} \)