Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$

Marco Caroccia, Nicolas Van Goethem

TL;DR

The paper studies the space $\mathrm{BD}_{\mathrm{dev}}$ of functions with bounded deviatoric deformation, identifying a sharp rigidity structure for maps whose deviatoric part lies in the wave cone and constructing an explicit kernel projection to enable iterative blow-up arguments for relaxation and homogenization.A fourth-order annihilator is computed to characterize the wave cone, allowing a precise description of the singular parts of $\mathcal{E}_d u$ and the decomposition into absolutely continuous, jump, and Cantor pieces with explicit polar data.The main contributions are (i) a rigidity theorem for $\mathrm{BD}_{\mathrm{dev}}$ maps with constant polar in the wave cone, (ii) an explicit projection $\mathcal{R}_K$ onto $\mathrm{Ker}({\mathcal{E}}_d)$, and (iii) a framework enabling integral representation and homogenization for energies that depend on $u$ as well as $\mathcal{E}_d u$, with applications to material science.These results extend the BD theory to deviatoric settings, overcoming invariance issues and enabling an iterative blow-up approach for relaxation problems and materials modeling where shear-dominated effects are central.

Abstract

We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed.

Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$

TL;DR

The paper studies the space $\mathrm{BD}_{\mathrm{dev}}$ of functions with bounded deviatoric deformation, identifying a sharp rigidity structure for maps whose deviatoric part lies in the wave cone and constructing an explicit kernel projection to enable iterative blow-up arguments for relaxation and homogenization.A fourth-order annihilator is computed to characterize the wave cone, allowing a precise description of the singular parts of $\mathcal{E}_d u$ and the decomposition into absolutely continuous, jump, and Cantor pieces with explicit polar data.The main contributions are (i) a rigidity theorem for $\mathrm{BD}_{\mathrm{dev}}$ maps with constant polar in the wave cone, (ii) an explicit projection $\mathcal{R}_K$ onto $\mathrm{Ker}({\mathcal{E}}_d)$, and (iii) a framework enabling integral representation and homogenization for energies that depend on $u$ as well as $\mathcal{E}_d u$, with applications to material science.These results extend the BD theory to deviatoric settings, overcoming invariance issues and enabling an iterative blow-up approach for relaxation problems and materials modeling where shear-dominated effects are central.

Abstract

We provide a structural analysis of the space of functions of bounded deviatoric deformation, , which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for -maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on as well as . Our approach overcomes several difficulties as compared to the case, in particular due to the lack of invariance of under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed.

Paper Structure

This paper contains 30 sections, 27 theorems, 386 equations.

Table of Contents

  1. Introduction
  2. The space of bounded deviatoric deformations
  3. Energy minimization problems in Materials science
  4. Rigidity result
  5. Kernel projection
  6. The role of Theorems \ref{['MainTheoremINTRO']} and \ref{['MainTheoremINTRO2']} in homogenization
  7. Fine properties
  8. Strategy of the proof for the rigidity
  9. Organization of the paper
  10. Acknowledgments
  11. Preliminaries and main result
  12. General notations
  13. Maps of bounded deformation
  14. Maps of bounded deviatoric deformation
  15. Poincaré-Sobolev and compactness
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $n\geq 3$. There exists an annihilator ${\mathcal{A}}$ for ${\mathcal{E}}_d$ of order $4$ for which it holds Moreover, for any $u\in \mathrm{BD}_{\mathrm{dev}}(K)$ satisfying for some $a,b\in \mathbb R^n$, $\mu\in \mathcal{M}(K;\mathbb R^+)$, one of the following two cases holds:

Theorems & Definitions (63)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4: Poincaré-Sobolev inequality
  • Theorem 2.5: Compactness Theorem
  • ...and 53 more