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The conclusion that metamaterials could have negative mass is a consequence of improper constitutive characterisation

David Cichra, Vít Průša, K. R. Rajagopal, Casey Rodriguez, Martin Vejvoda

TL;DR

This paper addresses the claim that metamaterials can exhibit frequency-dependent or negative effective mass. It shows that such conclusions arise from using an inappropriate constitutive relation and proposes an effective constitutive framework in the form of rate-type relations that preserve Newton's second law with a constant positive mass. Through a single mass-in-mass system and a one-dimensional lattice, the authors demonstrate that the same dynamics can be captured without negative mass by replacing the standard Hookean spring with a rate-type constitutive relation, yielding an operator $ ext{L}(x_1)$ that acts on displacement. Extending to a continuous model, this rate-type approach preserves energy and provides a consistent, 3D extendable description of metamaterials, avoiding paradoxes tied to frequency-dependent mass and offering a robust design paradigm based on constitutive engineering. Overall, the work reframes metamaterial analysis around effective constitutive relations rather than effective mass, ensuring physically sensible, energy-conserving descriptions.

Abstract

The concept of "effective mass" is frequently used for the simplification of complex lumped parameter systems (discrete dynamical systems) as well as materials that have complicated microstructural features. From the perspective of wave propagation, it is claimed that for some bodies described as metamaterials, the corresponding "effective mass" can be frequency dependent, negative or it may not even be a scalar quantity. The procedure has even led some authors to suggest that Newton's second law needs to be modified within the context of classical continuum mechanics. Such absurd physical conclusions are a consequence of appealing to the notion of "effective mass" with a preconception for the constitutive structure of the metamaterial and using a correct mathematical procedure. We show that such unreasonable physical conclusions would not arise if we were to use the appropriate "effective constitutive relation" for the metamaterial, rather than use the concept of "effective mass" with an incorrect predetermined constitutive relation.

The conclusion that metamaterials could have negative mass is a consequence of improper constitutive characterisation

TL;DR

This paper addresses the claim that metamaterials can exhibit frequency-dependent or negative effective mass. It shows that such conclusions arise from using an inappropriate constitutive relation and proposes an effective constitutive framework in the form of rate-type relations that preserve Newton's second law with a constant positive mass. Through a single mass-in-mass system and a one-dimensional lattice, the authors demonstrate that the same dynamics can be captured without negative mass by replacing the standard Hookean spring with a rate-type constitutive relation, yielding an operator that acts on displacement. Extending to a continuous model, this rate-type approach preserves energy and provides a consistent, 3D extendable description of metamaterials, avoiding paradoxes tied to frequency-dependent mass and offering a robust design paradigm based on constitutive engineering. Overall, the work reframes metamaterial analysis around effective constitutive relations rather than effective mass, ensuring physically sensible, energy-conserving descriptions.

Abstract

The concept of "effective mass" is frequently used for the simplification of complex lumped parameter systems (discrete dynamical systems) as well as materials that have complicated microstructural features. From the perspective of wave propagation, it is claimed that for some bodies described as metamaterials, the corresponding "effective mass" can be frequency dependent, negative or it may not even be a scalar quantity. The procedure has even led some authors to suggest that Newton's second law needs to be modified within the context of classical continuum mechanics. Such absurd physical conclusions are a consequence of appealing to the notion of "effective mass" with a preconception for the constitutive structure of the metamaterial and using a correct mathematical procedure. We show that such unreasonable physical conclusions would not arise if we were to use the appropriate "effective constitutive relation" for the metamaterial, rather than use the concept of "effective mass" with an incorrect predetermined constitutive relation.
Paper Structure (11 sections, 76 equations, 2 figures)

This paper contains 11 sections, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model systems
  3. Single mass-in-mass system
  4. Standard approach
  5. Approach based on rate-type constitutive relation
  6. One dimensional mass-in-mass lattice system
  7. Dispersion relation for mass-in-mass lattice system
  8. Standard approach
  9. Approach based on a rate-type constitutive relation
  10. Continuous model
  11. Conclusion

Figures (2)

  • Figure 1: Mass-in-mass model.
  • Figure 2: Infinite mass-in-mass one-dimensional lattice structure.