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, φθ 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 < eπ. 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 < eπ), which provides an upper bound for E(0) in terms of data (φ and φθ) 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.

On Saint-Venant's principle in a poroelastic arch-like region

“…We will proceed here to obtain an appropriate bound for E(0). To this end, we follow an idea developed by Flavin and Gleeson [21] and consider two arbitrary smooth couples (1) , (1) , (2) , (2) satisfying the same boundary conditions like ( , ) and introduce the following scalar product:…”

“…The spatial behavior of solutions of the biharmonic equation for a homogeneous isotropic elastic material occupying an arch-like region was recently studied by Flavin [20], Flavin and Gleeson [21], Chiriţȃ [22], and D'Apice and Chiriţȃ [23,24].…”

On Saint-Venant's Principle for a Homogeneous Elastic Arch-Like Region

“…A cross-sectional measure of the biharmonic solution is introduced and it is shown to decay exponentially with respect to , provided that the radius a and b of the arch-like region satisfy the condition b a e % . For the limiting case of an arch-like region with b a close to e % , the methods of [10,11] fail to establish exponential decay.…”

“…The spatial behaviour of solutions of the biharmonic equation for a homogeneous isotropic elastic material occupying an arch-like region was studied recently by Flavin [10] and Flavin and Gleeson [11]. A cross-sectional measure of the biharmonic solution is introduced and it is shown to decay exponentially with respect to , provided that the radius a and b of the arch-like region satisfy the condition b a e % .…”

End effects for a generalized biharmonic equation with applications to functionally graded materials

“…This last equation was treated previously by Flavin [3], Flavin and Gleeson [4] and Chiriţȃ [5] in concern with the case when a traction is applied on the edge θ = 0, the other three edges being free of charge. Decay estimates of solutions to the biharmonic equation in an inhomogeneous rectangular strip have been established, for instance, by [6][7][8] and [9].…”