The free boundary for elastic-plastic torsion problems

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 $K = \left \{ {w \in H_0^1\left ( Q \right ), \left | {\nabla w} \right | \leqslant 1 } \right \}$ 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 $E = \left \{ {x \in Q; \left | {\nabla u\left ( x \right )} \right | < 1} \right \}, P = \left \{ {x \in Q; \left | {\nabla u\left ( x \right )} \right | = 1} \right \}$ 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.