The Compatibility Constraint in Linear Elasticity

Abstract

It is shown that the St. Venant-Beltrami equation of compatibility can be regarded as an internal constraint on the admissible strains, a constraint maintained by reactive stress fields that are collectively characterized in a global manner. It is also shown that such reactive stresses have an important role in a formal integration of the mixed boundary-value problem of linear elasticity. The main tools — Beltrami’s map, Donati’s theorem, and Cesàro’s formula — have been available for a century or so; they are here used in a form as close to original as possible.