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The Quarterly Journal of Mechanics and Applied Mathematics 1992 45(4):555-566; doi:10.1093/qjmam/45.4.555
© 1992 by Oxford University Press
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CONVEXITY CONSIDERATIONS FOR THE BIHARMONIC EQUATION IN PLANE POLARS WITH APPLICATIONS TO ELASTICITY

J. N. FLAVIN

( Department of Mathematical Physics, University College Galway )

Solutions of the biharmonic equation are considered in the arch-like region 0 < {theta} < {alpha}, a < r < b in the presence of boundary conditions {varphi} = {varphi}, = 0 on the edges r = a, r = b ((r, {theta}) denoting plane polar coordinates). A cross-sectional measure F({theta}) of the solution is considered and is proved to be convex in {theta} for b/a ≤ exp {pi}. If, additionally, the condition {varphi} = {varphi}{theta}= {theta} 0 obtains on the edge {theta} = {alpha}, F({theta}) satisfies an enhanced inequality(generalized convexity); upper bounds for F({theta}) in terms of suitable data follow.

The analysis is relevant to an elastic arch-like strip in a state of plane stress, all of whose edges are free except the edge {theta} = 0 which is subjected to a self-equilibrated load. An upper estimate is obtained for F({theta})—a cross-sectional measure of stress—which decays exponentially with respect to {theta}. This may be viewed as an expression of Saint-Venant's principle for the context in question.


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