Piecewise Rigidity
نویسندگان
چکیده
In this paper we provide a Liouville type theorem in the framework of fracture mechanics, and more precisely in the theory of SBV deformations for cracked bodies. We prove the following rigidity result: if u ∈ SBV (Ω, RN ) is a deformation of Ω whose associated crack Ju has finite energy in the sense of Griffith’s theory (i.e., HN−1(Ju) <∞), and whose approximate gradient ∇u is almost everywhere a rotation, then u is a collection of an at most countable family of rigid motions. In other words, the cracked body does not store elastic energy if and only if all its connected components are deformed through rigid motions. In particular, global rigidity can fail only if the crack disconnects the body.
منابع مشابه
Smooth Cocycle Rigidity for Expanding Maps, and an Application to Mostow Rigidity
We consider the Livsic cocycle equation, with values in compact Lie groups, and dynamics given by a piecewise-smooth expanding Markov map. We prove that any measurable solution to this equation must necessarily have a smooth version. We deduce a smooth rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps, and apply this to give a variation on the proof of Mostow's ...
متن کاملPiecewise monotone maps without periodic points : Rigidity , measures and complexity
We consider piecewise monotone maps of the interval with zero entropy or no periodic points. First, we give a rigid model for these maps: the interval translations mappings, possibly with ips. It follows, e.g., that the complexity of a piecewise monotone map of the interval is at most polynomial if and only if this map has a nite number of periodic points up to monotone equivalence. Second, we ...
متن کاملMarked Length Rigidity for Fuchsian Buildings
We consider finite 2-complexes X that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT(-1) metrics on X which are piecewise hyperbolic, and satisfy a natural non-singularity condition at vertices are marked length spectrum rigid within certain classes of negatively curved, piecewis...
متن کاملNonlinearity, Quasisymmetry, Differentiability, And Rigidity in One-Dimensional Dynamics
In this article, we review some of our research in the study of one-dimensional dynamical systems, in particular, some technique and results in the study of the smooth classification of certain one-dimensional maps. The main results which we review are that if the conjugacy between two mixing and nice quasihyperbolic maps is differentiable at one point with uniform bound, then it is piecewise s...
متن کامل