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Transactions of the American Mathematical Society

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The free boundary for elastic-plastic torsion problems

Authors: Avner Friedman and Gianni A. Pozzi
Journal: Trans. Amer. Math. Soc. 257 (1980), 411-425
MSC: Primary 35R35; Secondary 49A29, 73C99, 73K99
MathSciNet review: 552267
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Abstract: Consider the variational inequality: Find $ u\, \in\, K$ such that $ \int_Q {\nabla u \cdot\, \nabla\, (\upsilon\, -\, u)\, \geqslant\, \mu\, \int_Q\, {(\upsilon\, -\, u)\,(\mu\, >\, 0)} } $ for any $ \upsilon\, \in\, K$, where $ K\, =\, \{ w\, \in\, H_0^1(Q);\,\left\vert {\nabla\, w} \right\vert\, \leqslant\, 1\} $ and Q is a 2-dimensional simply connected domain in $ {R^2}$ with piecewise $ {C^3}$ boundary. The solution u represents the stress function in a torsion problem of an elastic-plastic bar with cross section Q. The sets $ E\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, <\, 1\} $, $ P\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, =\, 1\} $ are the elastic and plastic sets respectively. The purpose of this paper is to study the free boundary $ \partial E\, \cap\, Q$; more specifically, an estimate is derived on the number of points of local maximum of the free boundary.

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Article copyright: © Copyright 1980 American Mathematical Society

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