CONVEXITY CONSIDERATIONS FOR THE BIHARMONIC EQUATION IN PLANE POLARS WITH APPLICATIONS TO ELASTICITY
James N. Flavin · The Quarterly Journal of Mechanics and Applied Mathematics · 1992
Solutions of the biharmonic equation are considered in the arch-like region 0 < θ < α, a < r < b in the presence of boundary conditions φ = φ, = 0 on the edges r = a, r = b ((r, θ) denoting plane polar coordinates). A cross-sectional measure F(θ) of the solution is considered and is proved to be convex in θ for b/a ≤ exp π. If, additionally, the condition φ = φθ= θ 0 obtains on the edge θ = α, F(θ) satisfies an enhanced inequality(generalized convexity); upper bounds for F(θ) in terms of suitable data follow. The analysis is relevant to an elastic arch-like strip in a state of plane stress, all of whose edges are free except the edge θ = 0 which is subjected to a self-equilibrated load. An upper estimate is obtained for F(θ)—a cross-sectional measure of stress—which decays exponentially with respect to θ. This may be viewed as an expression of Saint-Venant's principle for the context in question.