How Symmetric and Skew-Symmetric Tensors Interact
by LabyrinthineMarch 23rd, 2025
Too Long; Didn't Read
This section examines the general form of 4th order tensors in tetragonal symmetry, Voigt notation representations, and the impact of symmetric/skew-symmetric tensor components on wave dispersion.
Table of Links
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)
4 New considerations on the relaxed micromorphic parameters
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
6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P)
6.1 Asymptotes and 6.2 Fitting
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◦
A Most general 4th order tensor belonging to the tetragonal symmetry class
Considering the following quadratic form
where L is a 4th order tensor and D is a 2nd order one, the most general form of L if it belongs to the tetragonal symmetry class written in Voigt notation is
where the order of the element of the vector associated with the quadratic form A.1 is
If we now split the tensor D in its symmetric and skew-symmetric part, the corresponding vector in Voigt notation are
Because of the class of symmetry considered, it is necessary to take into account a mixed constitutive matrix that couples the symmetric and skew-symmetric part of D in order to build back the quadratic form Y
B Coefficients for the dispersion curves without Curl P
C Coefficients for the dispersion curves with Curl P
This paper is available on arxiv under CC BY 4.0 DEED license.
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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