The free boundary for elastic-plastic torsion problems

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