Relaxed micromorphic cut-offs

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◦

4.3 Relaxed micromorphic cut-offs

The cut-offs of the dispersion curves play an important role in fitting the material parameters of the relaxed micromorphic model [33, 17, 37]. For the convenience of the reader, we show the calculations of the analytic expressions again. In the case k = 0, the dispersion relation (3.6) simplifies into

Equations (4.7) can be simplified as in Table 2 with

The values of theses cut-offs have been fixed according to Comsol Multiphysics®simulations as

and the values of last points from Comsol Multiphysics®are used to fix the asymptotes, cf. Table 3.