The circular disc made of linear elastic incompressible material and the 'bathyscaphe lesson'

TL;DR

This paper analyzes a linear elastic incompressible circular disc under boundary loads using complex variable analysis, revealing non-trivial solutions despite inextensibility constraints and discussing implications for design and theory.

Contribution

It introduces a novel analysis of an incompressible elastic disc with boundary inextensibility, highlighting the paradoxical solutions and their physical interpretation.

Findings

Non-trivial solutions exist despite inextensibility constraints.

Linearized theory can represent valid stress distributions within rigid systems.

Numerical analysis confirms the approximations and reveals critical conditions.

Abstract

A linear elastic circular disc is analyzed under a self-equilibrated system of loads applied along its boundary. A distinctive feature of the investigation, conducted using complex variable analysis, is the assumption that the material is incompressible (in its linearized approximation), rendering the governing equations formally identical to those of Stokes flow in viscous fluids. After deriving a general solution to the problem, an isoperimetric constraint is introduced at the boundary to enforce inextensibility. This effect can be physically realized, for example, by attaching an inextensible elastic rod with negligible bending stiffness to the perimeter. Although the combined imposition of material incompressibility and boundary inextensibility theoretically prevents any deformation of the disc, it is shown that the problem still admits non-trivial solutions. This apparent paradox is resolved by recognizing the approximations inherent in the linearized theory, as confirmed by a geometrically nonlinear numerical analysis. Nonetheless, the linear solution retains significance: it may represent a valid stress distribution within a rigid system and can identify critical conditions of interest for design applications.