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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The free boundary for elastic-plastic torsion problems

Authors: Luis A. Caffarelli and Avner Friedman
Journal: Trans. Amer. Math. Soc. 252 (1979), 65-97
MSC: Primary 35J20; Secondary 35R35, 73Cxx
MathSciNet review: 534111
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Abstract: Consider the variational inequality: find $ u\, \in \,K$ such that $ \int_Q {\nabla u\, \cdot \,\nabla \left( {v\, - \,u} \right)} \, \geqslant \,\mu \,\int_Q {\left( {v\, - \,u} \right)} \,\left( {\mu \, > \,0} \right)$ for any $ v\, \in \,K$, where $ K\, = \,\left\{ {w\, \in \,H_0^1\left( Q \right),\,\left\vert {\nabla w} \right\vert\, \leqslant \,1\,} \right\}$ and Q is a simply connected domain whose boundary is piecewise $ {C^3}$. The solution u represents the stress function in a torsion problem of an elastic bar with cross section Q; the sets $ E\, = \,\left\{ {x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\ve... ...x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\vert = \,1} \right\}$ are the elastic and plastic subsets of Q. The ridge R of Q is, by definition, the set of points in Q where dist $ \left( {x,\,\partial Q} \right)$ is not $ {C^{1,1}}$. The paper studies the location and shape of E, P and the free boundary $ \Gamma \, = \,\partial E\, \cap \,Q$. It is proved that the ridge is elastic and that E is contained in a $ \left( {c/\mu } \right)$-neighborhood of R, as $ \mu \, \to \,\infty \,\left( {c\, > \,0} \right)$. The behavior of E and P near the vertices of $ \partial Q$ is studied in detail, as well as the nature of $ \Gamma $ away from the vertices. Applications are given to special domains. The case where Q is multiply connected is also studied; in this case the definition of K is somewhat different. Some results on the ``upper plasticity'' and ``lower plasticity'' and on the behavior as $ \mu \, \to \,\infty $ are obtained.

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Keywords: Variational inequality, elastic set, plastic set, free boundary, reentrant corner, ridge
Article copyright: © Copyright 1979 American Mathematical Society