A correspondence between compressible and incompressible plane elastostatics

Abstract

It is shown that a heretofore seemingly unnoticed correspondence (or analogy) exists between the traction boundary value problem for compressible media and the displacement boundary value problem for incompressible media occupying the same domain, in plane, isotropic, linear elastostatics. The Airy stress function, which satisfies equilibrium identically, has to be biharmonic in order to satisfy the compatibility condition in a compressible body. Correspondingly, a displacement potential function, which satisfies the incompressibility condition identically, has to be biharmonic in order to satisfy equilibrium. Since Stokes flow is governed by identical relations as incompressible plane elasticity, if displacement is interpreted as velocity and the shear modulus as dynamic viscosity, the correspondence extends to that between compressible elasticity and Stokes flow for boundary value problems indicated above. This analogy provides a rare example of a constrained system which is equivalent to the same system without the constraint.