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◦ 2 Relaxed micromorphic modelling of finite-size metamaterials for the classical Cauchy model, and for the relaxed micromorphic model, where we set The Neumann boundary condition for the classical Cauchy model are 2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation) 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 [7] We write “m” for “micro” and “M” for “macro” for the corresponding elastic parameters to shorten the following expressions. 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.

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Modeling Finite-Size Metamaterials with Relaxed Micromorphic Theory

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