Authors:
(1) Jendrik Voss, Institute for Structural Mechanics and Dynamics, Technical University Dortmund and a Corresponding Author (jendrik.voss@tu-dortmund.de);
(2) Gianluca Rizzi, Institute for Structural Mechanics and Dynamics, Technical University Dortmund;
(3) Patrizio Neff, Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen;
(4) Angela Madeo, Institute for Structural Mechanics and Dynamics, Technical University Dortmund. Authors: Authors: (1) Jendrik Voss, Institute for Structural Mechanics and Dynamics, Technical University Dortmund and a Corresponding Author (jendrik.voss@tu-dortmund.de); (2) Gianluca Rizzi, Institute for Structural Mechanics and Dynamics, Technical University Dortmund; (3) Patrizio Neff, Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen; (4) Angela Madeo, Institute for Structural Mechanics and Dynamics, Technical University Dortmund. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 1.1 A Polyethylene-based metamaterial for acoustic control 1.1 A Polyethylene-based metamaterial for acoustic control 2 Relaxed micromorphic modelling of finite-size metamaterials 2 Relaxed micromorphic modelling of finite-size metamaterials 2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation) 2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation) 3 Dispersion curves 3 Dispersion curves 4 New considerations on the relaxed micromorphic parameters 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 4.1 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 4.2 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s size 4.3 Relaxed micromorphic cut-offs 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 Fitting of the relaxed micromorphic parameters: the particular case of vanishing curvature (without Curl P and Curl P˙) 5.1 Asymptotes 5.1 Asymptotes 5.2 Fitting 5.2 Fitting 5.3 Discussion 5.3 Discussion 6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P) 6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P) 6.1 Asymptotes and 6.2 Fitting 6.1 Asymptotes and 6.2 Fitting 6.3 Discussion 6.3 Discussion 7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes 7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes 7.2 Fitting 7.2 Fitting 7.3 Discussion 7.3 Discussion 8 Summary of the obtained results 8 Summary of the obtained results 9 Conclusion and perspectives, Acknowledgements, and References 9 Conclusion and perspectives, Acknowledgements, and References A Most general 4th order tensor belonging to the tetragonal symmetry class A Most general 4th order tensor belonging to the tetragonal symmetry class B Coefficients for the dispersion curves without Curl P B Coefficients for the dispersion curves without Curl P C Coefficients for the dispersion curves with P C Coefficients for the dispersion curves with P D Coefficients for the dispersion curves with P D Coefficients for the dispersion curves with P Abstract We present an inertia-augmented relaxed micromorphic model that enriches the relaxed micromorphic model previously introduced by the authors via a term Curl P˙ in the kinetic energy density. This enriched model allows us to obtain a good overall fitting of the dispersion curves while introducing the new possibility of describing modes with negative group velocity that are known to trigger negative refraction effects. The inertia-augmented model also allows for more freedom on the values of the asymptotes corresponding to the cut-offs. In the previous version of the relaxed micromorphic model, the asymptote of one curve (pressure or shear) is always bounded by the cut-off of the following curve of the same type. This constraint does not hold anymore in the enhanced version of the model. While the obtained curves’ fitting is of good quality overall, a perfect quantitative agreement must still be reached for very small wavelengths that are close to the size of the unit cell. 1 Introduction Metamaterials are materials whose mechanical properties go beyond those of classical materials thanks to their heterogeneous microstructure. They can show unusual static/dynamic responses such as negative Poisson’s ratio [27], twist or bend in response to being pushed or pulled [18, 36], band-gaps [28, 46, 8, 13], cloaking [11, 31], focusing [20, 16], channeling [24, 45], negative refraction [52, 25, 47], etc. The working frequency of each metamaterial strongly depends on the characteristic size and the geometry of the underlying unit cell, as well as on the choice of the base material. In this paper, we present a labyrinthine metamaterial that, thanks to the use of a polymeric based material and an optimized distribution of mass inside the unit cell (see Figure 1), gives rise to a wide acoustic band-gap with characteristic unit cell’s size of the order of centimeters. The direct finite element modeling of structures build up with this labyrinthine metamaterial is unfeasible due to the extremely tight meshing that would be needed to correctly cover the narrow strips of material inside each unit cell. It is thus apparent the need for a homogenized model to use this type of very promising metamaterials in actual engineering designs. Various homogenization techniques have been developed with the purpose of providing rigorous predictions of the macroscopic metamaterial’s mechanical response when the properties of the base materials and their spatial distribution are known. These homogenization approaches have been shown to be useful in describing the overall behavior of metamaterials in the static and quasi-static regimes [6, 40, 3, 30, 21, 48, 35, 9, 12, 44, 29, 19, 22] as well as, more recently, in the dynamic regime [5, 14, 10, 15, 4, 23, 49, 50, 51, 42, 41, 43]. However, these models are often unsuited to deal with finitesize metamaterials, because they are based on upscaling techniques valid for unbounded media. Because of that, finite-size metamaterials’ structures are mostly investigated via Finite Element simulations which are performed using directly the microstructured material, e.g. [26]. The downside of this approach is that the computational cost quickly becomes unsustainable (especially for unit cells as the one presented in this paper), although the propagation patterns obtained are very accurate. This heavily limits the possibility of exploring large-scale or very convoluted geometric meta-structures. To overcome this problem and open up the possibility of designing complex meta-structures using the metamaterial presented in this paper as a basic building block, we propose to use an inertia-augmented relaxed micromorphic model. This model is based on the relaxed micromorphic model that we previously established [34, 32, 17, 1, 2] and has been augmented with a new inertia term accounting for coupled space-time derivatives of the micro-distortion tensor. The relaxed micromorphic model has extensively proven its efficacy in describing the broadband behavior of many infinite and finite-size metamaterials [1, 2, 37, 38, 39] and is extended in this paper so as to be able to account for negative group velocity which was not the case before. We will show that the proposed model is able to describe well the labyrinthine metamaterial’s response for a large range of frequencies (going beyond the first band-gap) and wave numbers (approaching the size of the unit cell) and for all directions of propagation with a limited number of frequency- and scale-independent constitutive parameters. The new inertia-augmented term will be shown to trigger modes with negative group velocities that are known to be associated with negative refraction phenomena. The results presented in this paper will allow us to shortly present new designs of finite-size labyrinthine metamaterials’ structures that can control elastic energy in the acoustic regime for eventual subsequent re-use. 1.1 A Polyethylene-based metamaterial for acoustic control In this section, we present a new unit cell’s design that gives rise to a metamaterial for acoustic control. This unit cell is designed to achieve a band-gap at relatively low frequencies (600−2000 Hz) so that application for acoustic control can be targeted. The unit cell considered is made out of polyethylene, cf. Table 1. Compared to Aluminium or Titanium, that we used for the metamaterials studied in [37, 38, 39], Polyethylene gives rise to lower wave speeds, thus allowing band-gap phenomena to appear at lower frequencies. A further lowering of the band-gap is obtained through the adoption of a labyrinth-type geometry, cf. Figure 1. This structure presents a tetragonal symmetry and thus features a reduced number of parameters with respect to a fully anisotropic system. The circular center of the unit cell is connected by thin bars allowing the heavier center to move easily, thus giving rise to local resonance phenomena of relatively low frequencies while additionally providing a very soft macro-material behaviour. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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