On a decomposition of regular domains into John domains with uniform constants

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding partition $\Omega = \Omega_1 \cup \ldots \cup \Omega_N$ with $\sum_{j=1}^N \mathcal{H}^1(\partial \Omega_j \setminus \partial \Omega) \le \theta$ such that each component is a John domain with a John constant only depending on $\theta$. The result implies that many inequalities in Sobolev spaces such as Poincar\'e's or Korn's inequality hold on the partition of $\Omega$ for uniform constants, which are independent of $\Omega$

A Korn-Poincar\'e-type inequality for special functions of bounded deformation

We present a Korn-Poincar\'e-type inequality in a planar setting which is in the spirit of the Poincar\'e inequality in SBV due to De Giorgi, Carriero, Leaci. We show that for each function in SBD$^2$ one can find a modification which differs from the original displacement field only on a small set such that the distance of the modification from a suitable infinitesimal rigid motion can be controlled by an appropriate combination of the elastic and the surface energy. In particular, the result can be used to obtain compactness estimates for functions of bounded deformation

Functionals defined on piecewise rigid functions: Integral representation and Γ\Gamma-convergence

We analyze integral representation and $\Gamma$-convergence properties of functionals defined on \emph{piecewise rigid functions}, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'e et al. 1998, 2001) to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation

On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets

An analysis of crystal cleavage in the passage from atomistic models to continuum theory

We study the behavior of atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.Comment: The final publication is available at springerlink.co