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 such that for any , where and *Q* is a simply connected domain whose boundary is piecewise . The solution *u* represents the stress function in a torsion problem of an elastic bar with cross section *Q*; the sets are the elastic and plastic subsets of *Q*. The ridge *R* of *Q* is, by definition, the set of points in *Q* where dist is not . The paper studies the location and shape of *E*, *P* and the free boundary . It is proved that the ridge is elastic and that *E* is contained in a -neighborhood of *R*, as . The behavior of *E* and *P* near the vertices of is studied in detail, as well as the nature of 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 are obtained.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1979-0534111-0

Keywords:
Variational inequality,
elastic set,
plastic set,
free boundary,
reentrant corner,
ridge

Article copyright:
© Copyright 1979
American Mathematical Society