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Thin cylindrical shell under internal pressure: ratio of longitudinal strain to volumetric strain For a thin cylinder (diameter d, thickness t) under internal pressure p, with hoop and longitudinal stresses as in thin-shell theory, express (longitudinal strain)/(volumetric strain) in terms of Poisson’s ratio ν. Choose the correct formula.

Difficulty: Medium

(1 − 2ν) / (5 − 4ν)
(1 − ν) / (5 − 2ν)
(1 − 2ν) / (1 + ν)
(1 − ν) / (3 − 2ν)

Correct Answer: (1 − 2ν) / (5 − 4ν)

Explanation:

Given / Thin-shell stresses
Hoop stress: σ_h = p d / (2 t)Longitudinal stress: σ_l = p d / (4 t) = σ_h/2

Strains (Hooke’s law with Poisson effect)
Longitudinal strain: ε_l = (1/E)(σ_l − ν σ_h)Hoop strain: ε_h = (1/E)(σ_h − ν σ_l)

Volumetric strain for thin cylinder
ΔV/V ≈ ε_l + 2 ε_h

Step-by-step
Let σ_l = S ⇒ σ_h = 2Sε_l = (S/E)(1 − 2ν)ε_h = (S/E)(2 − ν)Volumetric strain: ε_v = ε_l + 2ε_h = (S/E)[(1 − 2ν) + 2(2 − ν)] = (S/E)(5 − 4ν)Ratio ε_l/ε_v = (1 − 2ν)/(5 − 4ν)

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