1. Introduction. We describe two methods, relevant to the study of Saint-Venant's principle, for the derivation of decay estimates in a linear isotropic homogeneous elastic nonprismatic cylinder loaded by prescribed end displacements and with fixed curved lateral surface. The results and associated calculations are expressed in terms of integrals taken over plane cross sections of the cylinder rather than averages over partial volumes as in many previous discussions. (See, for example, Toupin [25], Oleinik and Yosifian [23], and Fichera [6,7]; other references to this and related issues may be found in the comprehensive survey by Horgan and Knowles [15].) A notable exception is the investigation by Biollay [1] who examines the three-dimensional semi-infinite prismatic beam with lateral sides held fixed and data specified over the base of the cylinder. (We also mention in this respect the study by Knowles [19] of the semi-infinite plane strip with lateral sides stress-free.) An exponentially decreasing decay rate is obtained which, of course, is a common feature of most studies of

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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.