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Authors:

(1) Jendrik Voss, Institute for Structural Mechanics and Dynamics, Technical University Dortmund and a Corresponding Author (jendrik.voss@tu-dortmund.de);

(2) Gianluca Rizzi, Institute for Structural Mechanics and Dynamics, Technical University Dortmund;

(3) Patrizio Neff, Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen;

(4) Angela Madeo, Institute for Structural Mechanics and Dynamics, Technical University Dortmund.

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Abstract and 1. Introduction

1.1 A Polyethylene-based metamaterial for acoustic control

2 Relaxed micromorphic modelling of finite-size metamaterials

2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation)

3 Dispersion curves

4 New considerations on the relaxed micromorphic parameters

4.1 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s bulk material properties

4.2 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s size

4.3 Relaxed micromorphic cut-offs

5 Fitting of the relaxed micromorphic parameters: the particular case of vanishing curvature (without Curl P and Curl P˙)

5.1 Asymptotes

5.2 Fitting

5.3 Discussion

6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P)

6.1 Asymptotes and 6.2 Fitting

6.3 Discussion

7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes

7.2 Fitting

7.3 Discussion

8 Summary of the obtained results

9 Conclusion and perspectives, Acknowledgements, and References

A Most general 4th order tensor belonging to the tetragonal symmetry class

B Coefficients for the dispersion curves without Curl P

C Coefficients for the dispersion curves with P

D Coefficients for the dispersion curves with P◦

7.3 Discussion


Figure 6: Dispersion curves ω(k) for 0 degrees (left) and 45 degrees (right) with pressure curves colored in yellow and shear in blue. The dots are the points computed with Comsol Multiphysics® while the smooth curves show the analytical expression of the dispersion curves for the relaxed micromorphic model. The value of the curve’s horizontal asymptotes are also shown with dashed lines.


This paper is available on arxiv under CC BY 4.0 DEED license.


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