Thin cylindrical shell under internal pressure\nThe volumetric strain of a thin cylinder is expressed in terms of hoop strain ε1 and longitudinal strain ε2 as:

Introduction / Context:
For thin cylindrical pressure vessels, internal pressure produces hoop and longitudinal stresses, leading to corresponding strains. The overall volume change (volumetric strain) is the sum of normal strains along three mutually perpendicular directions.

Given Data / Assumptions:

• Thin cylinder with diameter much larger than thickness (membrane theory).
• Principal strains: hoop (circumferential) ε1, longitudinal (axial) ε2.
• Radial strain is negligible in thin-shell theory compared to hoop and longitudinal components.

Concept / Approach:
Volumetric strain in small-deformation theory equals the algebraic sum of strains in three orthogonal directions. A thin cylinder has two identical circumferential directions around the axis (hoop in two orthogonal tangential directions) and one longitudinal direction along the axis.

Step-by-Step Solution:
Volumetric strain, ε_v ≈ ε_x + ε_y + ε_z.For a cylinder: two circumferential (hoop) directions → ε_x = ε_y = ε1; one longitudinal direction → ε_z = ε2.Therefore, ε_v = ε1 + ε1 + ε2 = 2 ε1 + ε2.

Verification / Alternative check:
Using stress/strain relations, ε1 and ε2 can be computed from hoop and longitudinal stresses. Substituting those into the sum still yields ε_v = 2 ε1 + ε2.

Why Other Options Are Wrong:
Expressions with minus signs would imply volume decreases with positive tensile strains, which contradicts basic strain superposition; ε1 + ε2 ignores the second circumferential direction.

Common Pitfalls:
Counting only one hoop direction; including radial strain in thin-shell problems where it is negligible.

Final Answer:
2 ε1 + ε2