An energy extremum principle for local action

Abstract

This work describes the solution in terms of the energy of the mixed boundary problem, formulated for the elastic body subjected to prescribed boundary displacements field. The extremum theorems herein proved are particular corollaries of the classical reciprocity theorems. Let us consider a part, ∂VP, of the unconstrained boundary containing point P on which a displacement field up shall be prescribed. The displacement is produced by tractions acting on a part, ∂VQ, of the unconstrained boundary containing a point Q and disjoined from ∂VP. The strain energy of the body in this elastic state is greater than the strain energy produced by boundary forces acting on ∂VP and creating the same displacement field uP there. The lower bound theorem herein proved gives a quantification of Boussinesque's local perturbation principle and a measure of the strain energy related to local action. The theorem applies both for structures and solids.