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

Table of Links

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◦

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

While keeping the geometry and the material unaltered, the dispersion properties of a microstructured isotropic Cauchy material are inversely proportional to the size of its unit cell, meaning that halving the size of the unit cell will double the frequency response for each value of the length k of the wavevector, which also changes with the same inverse proportionality since it represents the spatial periodicity of the structure. This can be easily retrieved by performing standard Bloch-Floquet analysis.