Fine-Tuning Acoustic and Optical Waves 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◦

5.1 Asymptotes

Instead of fitting dispersion curves pointwise, we focus on using the analytical expression of the cut-offs (k = 0) and of the asymptotes (k → ∞). The explicit expression of the cut-offs is already discussed in Section 4.3. We use a similar approach to calculate the asymptotes as well by considering the limit k → ∞ where only the terms with the highest order of k are important. Thus, we arrive at

In contrast to the analytical expression of the cut-offs, we are not able to simplify the asymptotes’ expression in a feasible way. This is mainly due to the fact that we must solve a third order polynomial while we only had two non-zero cut-offs each before.

Since the dispersion curves of the unit cell obtained via Bloch-Floquet analysis are by nature periodic, the limit for k → ∞ is per se meaningless when considering the Bloch-Floquet approach. The value for k = 1/Lc (with Lc the size of the unit cell) is the periodicity limit. On the other hand, this limit has of course meaning for a continuum model like the relaxed micromorphic model. To reconcile these two limits in the fitting procedure, we will impose that the limit k → ∞ for our continuum model will coincide with the periodicity limit of the Bloch-Floquet curves k = 1/Lc. This strategy allows us to preserve the width of the band-gap, cf. Figure 2.

5.2 Fitting

We start the fitting with the macroscopic apparent density ρ and values of the macro parameters κM, µM, µ∗ M, i.e. the material constants necessary for the classical homogenization of an infinite large micromorphic material. For an anisotropic Cauchy material, the speed of the acoustic waves is

As a second step, we use the analytical expressions for the cut-offs (4.8) and calculate

5.3 Discussion

The fitting shown in Figure 3 behaves well for all frequencies ω and wavenumber k for zero degrees of incidence but looses some precision for an incidence angle of 45◦ especially for higher values of k. This calls for a further generalization of the relaxed micromorphic model which will be object of following papers. In any case, the achieved overall precision already allows us to explore the dispersive metamaterial’s characteristics at a satisfactory level.

The absence of higher-order terms (Curl P and Curl P˙) caused the reduction to a single expression k(ω) describing all three dispersion curves in one, cf. equation (5.1). Thus for every frequency ω, there is exactly

one wavenumber k which may be imaginary if the term inside the root is negative. For the plots here, we only show k(ω) where the expression is real-valued and ignore imaginary k(ω) which arise in the band gap and for higher frequencies. Moreover, we cannot have two distinct wavenumbers with the same frequency which implies that all curves are monotonic. For every group of dispersion curves, e.g. the three pressure waves for 45◦ incidence, each individual curve is bounded by the others. Starting with the acoustic curves, their asymptote must be below the cut-off of the lower optic curve of the same type (pressure or shear) while the asymptote of the lower optic curves is bounded from above by the cut-off of the highest optic curve. In particular, self-intersection between two pressure or two shear curves is not possible with this simplified version of the relaxed micromorphic model.

On the other hand, we observe that for the numerical values from Comsol Multiphysics® the asymptote of the acoustic shear wave at 45◦ should be slightly higher than the cut-off of the lower optic curve with 581.95 Hz and 554.61 Hz, respectively. In addition, assuming that all micro parameters κm, µm, µ∗ m (see Table 6) are larger than their corresponding macro counterparts κM, µM, µ∗ M, we did not manage to generate decreasing dispersion curves as observed for the lower optic pressure wave for an angle of incidence of 45◦ . This effect can instead be achieved when considering mixed space-time derivatives on the micro-distortion tensor P, cf. Section 7.

When the relative positions of the curves allow to fit the cut-offs and the asymptotes of each curve separately (e.g. for an incidence angle of zero degrees shown here) the simplified version of the relaxed micromorphic model used does indeed shows very good results. We want to emphasize that we only used the limit cases k = 0 (cut-offs) and k → ∞ (asymptotes) for the fitting procedure but have an appreciable approximation for all values of k, ω.