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A characterization of Generalized functions of Bounded Deformation

Antonin Chambolle, Vito Crismale

TL;DR

The paper proves that $GBD(\\Omega)$, the space used to model crack growth in elasticity via symmetrized gradients and jump sets, can be completely characterized by a finite set of slicing directions. It introduces a finite direction set $V=\{e_i\}_{i=1}^d \cup \{e_i+e_j\}_{1\le i<j\le d}$ and shows that finiteness of the directional measures $\\Lambda_u^ξ$ for all $ξ \in V$ implies membership in $GBD(\\Omega)$, yielding a global Radon measure $\\Lambda_u$ controlling all directions. The main constructive step provides approximations $u^\\varepsilon$ in $SBV^∞$ with jump sets confined to cube-grid boundaries and a uniform energy bound $\\int_{\\Omega_\\varepsilon} |e(u^\\varepsilon)| dx + \\mathcal{H}^{d-1}(\\partial^* \\mathcal{B}^\\varepsilon \\cap \\Omega_\\varepsilon) \\\le C \\\Lambda_u^V$, with $u^\\varepsilon \to u$ in measure. Consequences include the equivalence $GBV^{\\mathcal{E}}(\\Omega;\\R^d)=GBD(\\Omega)$ and GSBD^p compactness results, enabling finite-direction discretization frameworks for fracture energies and related nonlocal approximations. The results provide a practical, direction-local criterion for identifying $GBD$ functions and facilitate robust discretization and compactness analyses in fracture mechanics.

Abstract

We show that Dal Maso's GBD space, introduced for tackling crack growth in linearized elasticity, can be defined by simple conditions in a finite number of directions of slicing.

A characterization of Generalized functions of Bounded Deformation

TL;DR

The paper proves that , the space used to model crack growth in elasticity via symmetrized gradients and jump sets, can be completely characterized by a finite set of slicing directions. It introduces a finite direction set and shows that finiteness of the directional measures for all implies membership in , yielding a global Radon measure controlling all directions. The main constructive step provides approximations in with jump sets confined to cube-grid boundaries and a uniform energy bound , with in measure. Consequences include the equivalence and GSBD^p compactness results, enabling finite-direction discretization frameworks for fracture energies and related nonlocal approximations. The results provide a practical, direction-local criterion for identifying functions and facilitate robust discretization and compactness analyses in fracture mechanics.

Abstract

We show that Dal Maso's GBD space, introduced for tackling crack growth in linearized elasticity, can be defined by simple conditions in a finite number of directions of slicing.

Paper Structure

This paper contains 4 sections, 11 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Discrete energy estimate
  4. Convergence of Discretizations

Key Result

Theorem 1

Let $u \in L^0(\Omega; \mathbb{R}^d)$ and assume there is an orthonormal basis $(e_i)_{i=1}^d$ of $\mathbb{R}^d$ such that for any $\varepsilon>0$ it holds that $\xi\in V := \{e_i:{i=1,\dots,d}\} \cup \{e_i+e_j\colon {1\le i<j\le d}\}$, and that for $\mathcal{H}^{d-1}$-a.e. $z\in\xi^\perp$, $u^\xi_ Then there exist a dimensional constant $C>0$ and a family $(u^\varepsilon)_{\varepsilon>0}\subset

Theorems & Definitions (24)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Proposition 2
  • Corollary 4
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Corollary \ref{['cor:GBDnew']}
  • proof : Proof of Corollary \ref{['cor:comp']}
  • ...and 14 more