Discontinuous Dependency of Plate Strain Energy on Boundary Contour
Publication: Journal of Engineering Mechanics
Volume 116, Issue 12
Abstract
This note shows that the strain energy of a plate suffers a finite jump when the boundary contour of the plate is subject to a change, however small, from a smooth curve to a piecewise linear curve, if either the displacement or the slope, both of which are prescribed, is homogeneous at every point along the plate boundary. The proof is based on an observation that the plate strain energy depends linearly on Poisson's ratio when the lateral load, the boundary displacement and slope are specified. It also makes use of a previous result that the Gaussian curvature term in plate strain energy is a discontinuous functional of the boundary contour, and extends the theorem to that of the plate strain energy itself. This note suggests that this singular behavior may result from a discontinuous dependency of the solution to plate equation on small changes of the domain of existence.
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References
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Ahmed, S. S., and Dey, S. S. (1988). “The circle‐polygon paradox in the light of the boundary‐element method.” Comput. Struct., 29(5), 919–922.
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Babuska, I. (1962). “The theory of small changes in the domain of existence in the theory of partial differential equations and its applications.” Differential equations and their applications: Proc. Conf. Prague, Academic Press, New York, N.Y., 13–26.
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Rhee, H. C., and Atluri, S. N. (1986). “Polygon‐circle paradox in the finite element analysis of bending of a simply supported plate.” Comput. Struct., 22(4), 553–558.
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Zhu, C., and DiMaggio, F. L. (1989). “Contribution of Gaussian curvature to strain energy of plates.” J. Engrg. Mech., ASCE, 115(7), 1434–1440.
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Copyright © 1990 ASCE.
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Published online: Dec 1, 1990
Published in print: Dec 1990
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