On Saint-Venant’s Problem with Concentrated Loads

Abstract

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder Ω loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces \(\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)\), with q∈(2−ϵ 0,+∞) for some positive ϵ 0 depending on Ω. Hence \(\hat{ \boldsymbol { s}}\) can model force acting on ∂Ω, concentrated on sets of zero Lebesgue surface measure of ∂Ω. Moreover, if \(\hat{ \boldsymbol { s}}\) is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant’s principle.