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On Exotic Materials in 3D Linear Elasticity with High Symmetry Classes

Nicolas Auffray, Guangjin Mou, Boris Desmorat

Abstract

An anisotropic elastic material is referred to as exotic when, under specific loadings, its mechanical response exhibits a higher degree of symmetry than that prescribed by its intrinsic material symmetry. Such materials, which may be regarded as lying, conceptually and functionally, between two distinct symmetry classes, are of significant practical relevance. They enable the tailored design of metamaterials capable of reconciling otherwise incompatible mechanical requirements; for example, achieving directional isotropy of the Young's modulus in an intrinsically anisotropic medium. This work focuses on the systematic classification of exotic structures within the framework of three-dimensional linear elasticity. An exhaustive classification is carried out, leading to the enumeration of 18 exotic structures corresponding to symmetry classes higher than orthotropy. Representative examples of exotic elastic behaviours are analysed in detail.

On Exotic Materials in 3D Linear Elasticity with High Symmetry Classes

Abstract

An anisotropic elastic material is referred to as exotic when, under specific loadings, its mechanical response exhibits a higher degree of symmetry than that prescribed by its intrinsic material symmetry. Such materials, which may be regarded as lying, conceptually and functionally, between two distinct symmetry classes, are of significant practical relevance. They enable the tailored design of metamaterials capable of reconciling otherwise incompatible mechanical requirements; for example, achieving directional isotropy of the Young's modulus in an intrinsically anisotropic medium. This work focuses on the systematic classification of exotic structures within the framework of three-dimensional linear elasticity. An exhaustive classification is carried out, leading to the enumeration of 18 exotic structures corresponding to symmetry classes higher than orthotropy. Representative examples of exotic elastic behaviours are analysed in detail.
Paper Structure (18 sections, 11 theorems, 62 equations, 7 figures, 1 table)

This paper contains 18 sections, 11 theorems, 62 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction and motivation
  2. The space of elasticity tensors
  3. Transformation of an elasticity tensor
  4. Harmonic structure of $\mathbb{E}\mathrm{la}$
  5. Explicit harmonic decompositions
  6. From symmetry classes to exotic structures
  7. Definition of symmetry classes
  8. Symmetry classes of $\mathbb{E}\mathrm{la}$
  9. Exotic structures for symmetry classes higher than orthotropic
  10. Particular cases of 3D Transverse Isotropic (TI) exotic materials
  11. Uncoupled Transverse Isotropy (UTI)
  12. Transverse Isotropy with isotropic deviatoric elasticity (IDTI)
  13. Transverse Isotrop with Isotropic Young's modulus (IYTI)
  14. Proof of the main results
  15. Proof of Proposition \ref{['prop1']}
  16. ...and 3 more sections

Key Result

Proposition 2.1

The tensor $\underset{\approx}{\mathrm{C}}\in\mathbb{E}\mathrm{la}$ admits the uniquely defined Clebsch-Gordan Harmonic Decomposition associated with the use of deviatoric projector $\underset{\approx}{\mathrm{J}}=\underset{\approx}{\mathrm{I}}-\frac{1}{3}\underset{\sim}{\mathrm{1}}\otimes\underset{ in which $\left(\alpha, \beta ,\underset{\sim}{\mathrm{h}}^{a},\underset{\sim}{\mathrm{h}}^{b},\und

Figures (7)

  • Figure 1: The structure of symmetry classes of $\mathbb{E}\mathrm{la}$ as a partially ordered set (poset).
  • Figure 2: The geometric structure represented as a sequence of simplices.
  • Figure 3: The generic structure of a triclinic elasticity tensor.
  • Figure 4: Lattice of exotic structures
  • Figure 5: Location of the UTI exotic structure (highlighted in bold) within the transition map
  • ...and 2 more figures

Theorems & Definitions (19)

  • Definition 1.1: Exotic materials
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2: Clips operations between $\mathrm{SO}(3)$-closed subgroups, olive2019effective
  • Definition 3.1: Generic structure and exotic structures
  • Proposition 3.2
  • Theorem 3.3: Number of exotic structures in $\mathbb{E}\mathrm{la}$ for symmetry class greater than orthotropic
  • Theorem 3.4: Characterisation of the cubic class of $\underset{\approx}{\mathrm{C}}\in\mathbb{E}\mathrm{la}$
  • Example 4.1.1
  • ...and 9 more