New considerations on the relaxed micromorphic parameters

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 New considerations on the relaxed micromorphic parameters

In this section, we draw some useful considerations about the consistency of the relaxed micromporpic model with respect to a change of unit cell’s size and of the material properties of the base material. The model’s consistency is checked against a standard Bloch-Floquet analysis of the wave propagation performed using the unit cell described in Section 1.1 with built in periodic Bloch-Floquet boundary conditions from Comsol Multiphysics®.

The following two connections between the properties of the unit cell and the behaviour of the dispersion curves can be drawn:

• The dispersion curves scale proportionally in ω with respect to the speed of the wave of the bulk material composing the unit cell;

• The dispersion curves scale inversely in both ω and k with respect to the size of the unit cell.

Both results are useful to avoid repeating the time-consuming fitting procedure when changing the size of the cell and the base material’s properties while keeping the unit cell’s geometry unchanged.