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Korn type Inequalities for Objective Structures

Bernd Schmidt, Martin Steinbach

TL;DR

The paper develops a discrete Korn-type framework for objective structures, generalizing lattice rigidity to particle systems acted on by discrete Euclidean groups. It introduces three local seminorms, $\|\cdot\|_{\mathcal{R}}$, $\|\cdot\|_{\mathcal{R},0}$, and $\|\cdot\|_{\mathcal{R},0,0}$, and proves their equivalence under suitable richness conditions, yielding a full discrete Korn inequality for bulk objective structures and an intrinsic rigidity interpretation in extended dimensions. It also shows that the weaker and stronger seminorms need not be equivalent in general, via explicit 1D chain examples, while stronger norms like $\|\nabla_{\mathcal{R}}\cdot\|_2$ provide a robust discrete $H^1$-type control. The results connect continuum elasticity ideas with atomistic stability analyses, enabling energy estimates, stability criteria, and potential numerical algorithms for generalized, nonperiodic structures such as nanotubes and other objective materials. The framework uses a Fourier-analytic treatment on discrete Euclidean groups and leverages a Turán-type minimax lemma to control skew-symmetric components, culminating in a set of intrinsic and extrinsic rigidity characterizations that underpin stability analyses in high-dimensional or buckling-prone configurations.

Abstract

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.

Korn type Inequalities for Objective Structures

TL;DR

The paper develops a discrete Korn-type framework for objective structures, generalizing lattice rigidity to particle systems acted on by discrete Euclidean groups. It introduces three local seminorms, , , and , and proves their equivalence under suitable richness conditions, yielding a full discrete Korn inequality for bulk objective structures and an intrinsic rigidity interpretation in extended dimensions. It also shows that the weaker and stronger seminorms need not be equivalent in general, via explicit 1D chain examples, while stronger norms like provide a robust discrete -type control. The results connect continuum elasticity ideas with atomistic stability analyses, enabling energy estimates, stability criteria, and potential numerical algorithms for generalized, nonperiodic structures such as nanotubes and other objective materials. The framework uses a Fourier-analytic treatment on discrete Euclidean groups and leverages a Turán-type minimax lemma to control skew-symmetric components, culminating in a set of intrinsic and extrinsic rigidity characterizations that underpin stability analyses in high-dimensional or buckling-prone configurations.

Abstract

We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.
Paper Structure (11 sections, 26 theorems, 201 equations, 1 figure)

This paper contains 11 sections, 26 theorems, 201 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Objective structures
  3. Discrete subgroups of the Euclidean group
  4. Orbits of discrete subgroups of the Euclidean group
  5. A discrete Korn type inequality
  6. Deformations and local rigidity seminorms
  7. Equivalence of local rigidity seminorms
  8. Intrinsic seminorms and their equivalence to local seminorms
  9. Seminorm kernels
  10. Stronger seminorms
  11. Two basic examples in real and Fourier space

Key Result

Theorem 2.1

Let $\mathcal{G}<\mathop{\mathrm{E}}\nolimits(d)$ be discrete, $d\in{\mathbb N}$. There exist $d_1,d_2\in{\mathbb N}_0$ such that $d=d_1+d_2$, a $d_2$-dimensional space group $\mathcal{S}$ and a discrete group $\mathcal{G}'<\operatorname{{\mathrm O}}(d_1)\oplus \mathcal{S}$ such that $\mathcal{G}$ i

Figures (1)

  • Figure 1: $\mathcal{G}_1\cdot x_{1,0}$ (left) and $\mathcal{G}_2\cdot x_{2,0}$ (right)

Theorems & Definitions (77)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: The Plancherel formula
  • Lemma 2.5
  • Definition 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 67 more