A thin-walled spherical pressure vessel having an inner...

A thin-walled spherical pressure vessel having an inner radius $r$ and thickness $t$ is subjected to an internal pressure $p .$ Show that the increase in the volume within the vessel is $\Delta V=\left(2 p \pi r^{4} / E t\right)(1-\nu) .$ Use a small-strain analysis.

Key Concepts

Thin-Walled Pressure Vessel Theory

This concept involves the analysis of structures whose wall thickness is much less than their radius, allowing the assumption that the stresses are uniformly distributed across the thickness. It simplifies the stress analysis by using membrane theory and is essential when deriving relationships between applied pressure, induced stress, and resulting deformations in the vessel.

Small-Strain Analysis

Small-strain analysis assumes that deformations are sufficiently small such that linear approximations of strain and stress are valid. This simplification allows the neglect of higher-order effects, making the derivation of deformation relationships, such as the increase in volume due to pressure, mathematically manageable and accurate within the limits of elasticity.

Hooke's Law and Elasticity

Hooke's Law forms the fundamental relationship between stress and strain in elastic materials through Young's modulus. This concept is critical in determining how a material deforms under an applied load. In the context of pressure vessels, it provides the basis for quantifying the linear deformation response due to the internal pressure.

Poisson's Ratio

Poisson's ratio quantifies the degree of transverse strain relative to axial strain when a material is stretched or compressed. It is key in understanding how a material's lateral dimensions change with an applied load, influencing the volumetric expansion or contraction of structures such as pressure vessels.

Geometric Relations of Spherical Structures

The spherical geometry of the pressure vessel ensures symmetry in stress and strain, which simplifies the mathematical treatment of deformation. By linking the radial expansion to the change in volume through geometrical relationships, one can derive expressions that relate internal pressure, material properties, and the overall volumetric increase in the vessel.

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