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Spatial Decay Estimates for the Biharmonic Equation in Plane Polars with Applications to Plane Elasticity

Stan Chiri

Faculty of Mathematics, Al. I. Cuza, University of Iai, Blvd. Carol I, No. 11, 700506 -Iai, Romania

Ciro D'Apice

Department of Information Engineering and Applied Mathematics (DMA), University of Salerno, 84084 Fisciano (SA), Italy

The present paper considers an isotropic and homogeneous elastic body occupying the arch-like region a r b, 0 , where (r, ) denote plane polar coordinates. The arch-like body is in equilibrium under an (in plane) self-equilibrated load on the edge r = a, while the other three edges r = b, = 0 and = are traction-free and the body forces are absent. An appropriate measure is defined in terms of the Airy stress function , provided that the opening angle of the arch-like region is lower than 2 . Then the spatial behavior of the solution is studied and a clear relationship is established with Saint-Venant's principle on such regions. In fact, for a bounded arch-like region it is shown that the measure decays at least algebraically with respect to r, while for an unbounded region our result reveals a relationship with the classical Phragmèn—Lindelöf theorem.

Key Words: Saint-Venant's principle • plane stress • biharmonic equation • arch-like region

This version was published on June 1, 2007

Mathematics and Mechanics of Solids, Vol. 12, No. 3, 343-357 (2007)
DOI: 10.1177/1081286506059747


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