Spatial Decay Estimates for the Biharmonic Equation in Plane Polars with Applications to Plane Elasticity

Abstract

The present paper considers an isotropic and homogeneous elastic body occupying the arch-like region a ≤ r ≤ b, 0 ≤ θ ≤ α, where (r, θ) denote plane polar coordinates. The arch-like body is in equilibrium under an (in plane) self-equilibrated load on the edge r = a, while the other three edges r = b, θ = 0 and θ = α are traction-free and the body forces are absent. An appropriate measure is defined in terms of the Airy stress function ϕ, provided that the opening angle of the arch-like region is lower than 2 . Then the spatial behavior of the solution is studied and a clear relationship is established with Saint-Venant's principle on such regions. In fact, for a bounded arch-like region it is shown that the measure decays at least algebraically with respect to r, while for an unbounded region our result reveals a relationship with the classical Phragmèn—Lindelöf theorem.