How Unit Cell Size Affects Dispersion in Metamaterials | HackerNoon

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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◦

3 Dispersion curves

We assume a plane strain[8] time harmonic ansatz for the displacement u and the micro-distortion tensor P

Substituting the ansatz (3.1) in the equilibrium equations (2.6), we obtain the homogeneous algebraic linear system

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.

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

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.

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.