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

Department of Mathematical Physics, National University of Ireland, Galway, Ireland

B. Gleeson

Department of Mathematical Physics, National University of Ireland, Galway, Ireland

Solutions of the biharmonic equation are considered in the curvilinear rectangular region 0 , a r b in the presence of boundary conditions = _r = 0 on the edges r = a, r = b, = = 0 on the edge = , (r, ) denoting plane polar coordinates, a, b, (< 2) being constants; non-null boundary conditions are envisaged on the other edge = 0, involving the specification of , thereon. An energy-like measure E() of the solution in the region between arbitrary and = is defined, and is proven to be positive definite provided that b/a < e . It is established that E() / E(0) decays (at least) exponentially with respect to , under the aforementioned restriction on b/a. Additionally, a principle of the Dirichlet type is established (again provided b/a < e ), which provides an upper bound for E(0) in terms of data ( and ) prescribed on the edge = 0. When combined with the earlier result we obtain an explicit upper decay estimate for E(). 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 = 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 = 0, carrying a self-equilibrated load, and a smooth (traction-free) curve.

Key Words: Spatial decay • Saint-Venant principle • two-dimensional elastostatics • curvilinear rectangle

Mathematics and Mechanics of Solids, Vol. 8, No. 4, 337-348 (2003)
DOI: 10.1177/10812865030084001


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