On Saint-Venant's Principle for a Curvilinear Rectangle in Linear Elastostatics

Abstract

Solutions of the biharmonic equation are considered in the curvilinear rectangular region 0 ≤ θ ≤ α, a ≤ r ≤ b in the presence of boundary conditions φ = φ _r = 0 on the edges r = a, r = b, φ = φθ = 0 on the edge θ = α, (r, θ) denoting plane polar coordinates, a, b, α(< 2π) being constants; non-null boundary conditions are envisaged on the other edge θ = 0, involving the specification of φ, φ_θ thereon. An energy-like measure E(θ) of the solution in the region between arbitrary θ and θ = α is defined, and is proven to be positive definite provided that b/a < e ^π. It is established that E(θ) / E(0) decays (at least) exponentially with respect to θ, under the aforementioned restriction on b/a. Additionally, a principle of the Dirichlet type is established (again provided b/a < e ^π), which provides an upper bound for E(0) in terms of data (φ and φ_θ) prescribed on the edge θ = 0. When combined with the earlier result we obtain an explicit upper decay estimate for E(θ). The estimate can be regarded as a version of Saint-Venant's principle for a curvilinear strip, in the context of two-dimensional (homogeneous isotropic) elastostatics, the edge θ = 0 being subjected to a self-equilibrated (in-plane) load, the remainder of the boundary being traction-free. The Saint-Venant estimate continues to hold, mutatis mutandis, for any simply connected, two-dimensional domain, whose boundary consists of a straight line θ = 0, carrying a self-equilibrated load, and a smooth (traction-free) curve.