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.
Mechanical Engineering
Strength of Materials
Difficulty: Medium
Choose an option
-
A(1 − 2ν) / (5 − 4ν)
-
B(1 − ν) / (5 − 2ν)
-
C(1 − 2ν) / (1 + ν)
-
D(1 − ν) / (3 − 2ν)
Answer
Correct Answer: (1 − 2ν) / (5 − 4ν)
Explanation
Given / Thin-shell stressesHoop stress: σh = p d / (2 t)Longitudinal stress: σl = p d / (4 t) = σh/2
Strains (Hooke’s law with Poisson effect)Longitudinal strain: εl = (1/E)(σl − ν σh)Hoop strain: εh = (1/E)(σh − ν σl)
Volumetric strain for thin cylinderΔV/V ≈ εl + 2 εh
Step-by-stepLet σl = S ⇒ σh = 2Sεl = (S/E)(1 − 2ν)εh = (S/E)(2 − ν)Volumetric strain: εv = εl + 2εh = (S/E)[(1 − 2ν) + 2(2 − ν)] = (S/E)(5 − 4ν)Ratio εl/εv = (1 − 2ν)/(5 − 4ν)
Final Answer(1 − 2ν) / (5 − 4ν)