Reduced relaxed micromorphic modeling of harmonically loaded metamaterial plates: investigating boundary effects in finite-size structures

In this paper, we propose an approach for describing wave propagation in finite-size microstructured metamaterials using a reduced relaxed micromorphic model. This method introduces an additional kinematic field with respect to the classical Cauchy continua, allowing to capture the effects of the underlying microstructure with a homogeneous model. We show that the reduced relaxed micromorphic model is not only effective for studying infinite-size metamaterials, but also efficient for numerical simulations and analysis on specimens of finite size. This makes it an essential tool for designing and optimising metamaterials structures with specific wave propagation properties. The proposed model's efficiency is assessed through numerical simulations for finite-size benchmark problems, and shows a good agreement for a wide range of frequencies. The possibility of producing the same macroscopic metamaterial with different but equivalent unit cell "cuts" is also analysed, showing that, even close to the boundary, the reduced relaxed micromorphic model is capable of giving accurate responses for the considered loading and boundary conditions.Comment: 16 pages, 15 figure

Tracking defect-induced ferromagnetism in GaN:Gd

We report on the magnetic properties of GaN:Gd layers grown by molecular beam epitaxy (MBE). A poor reproducibility with respect to the magnetic properties is found in these samples. Our results show strong indications that defects with a concentration of the order of 10^19 cm^-3 might play an important role for the magnetic properties. Positron annihilation spectroscopy does not support the suggested connection between the ferromagnetism and the Ga vacancy in GaN:Gd. Oxygen co-doping of GaN:Gd promotes ferromagnetism at room temperature and points to a role of oxygen for mediating ferromagnetic interactions in Gd doped GaN

Towards the conception of complex engineering meta-structures: relaxed-micromorphic modelling of mechanical diodes

In this paper we show that an enriched continuum model of the micromorphic type (Relaxed Micromorphic Model) can be safely used to model metamaterials' response in view of their use for meta-structural design. We focus on the fact that the reduced model's structure, coupled with the introduction of well-posed interface conditions, allows us to easily test different combinations of metamaterials' and classical-materials bricks, so that we can eventually end-up with the conception of a meta-structure acting as a mechanical diode for low/medium frequencies and as a total screen for higher frequencies. Thanks to the reduced model's structure, we are also able to optimize this meta-structure so that the diode-behaviour is enhanced for both "pressure" and "shear" incident waves and for all possible angles of incidence.Comment: 19 pages, 18 figures (43 pictures). arXiv admin note: substantial text overlap with arXiv:2007.1494

Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach

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

Remarks on wave propagation in an acoustic metamaterial modeled as a relaxed micromorphic continuum

In order to describe elastic waves propagation in metamaterials, i.e. solids with heterogeneities or microstructure, it is necessary to consider non-local or higher-order models. The relaxed micromorphic model (RMM) proposed here can describe these effects as a continuous material with enriched kinematics. We present a new unit cell giving rise to a metamaterial for acoustic application. The microstructure is engineered to show a band-gap in the low acoustic regime (600-2000 Hz) for which waves cannot propagate through the material. We concentrate on the size effects to make full advantage of the particularly beneficial structure that the model provides. The RMM material parameters are fitted using a new algorithm relying on cutoffs and asymptotes (obtained via a Bloch-Floquet analysis). In particular, by enhancing the kinetic energy of the model with a new inertia term, we enable decreasing curves (modes with negative group velocity)

Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size effects

From frequency-dependent models to frequency-independent enriched continua for mechanical metamaterials

Mechanical metamaterials have recently gathered increasing attention for their uncommon mechanical responses enabling unprecedented applications for elastic wave control. To model the mechanical response of large metamaterials' samples made up of base unit cells, so-called homogenization or upscaling techniques come into play trying to establish an equivalent continuum model describing these macroscopic metamaterials' characteristics. A common approach is to assume a priori that the target continuum model is a classical linear Cauchy continuum featuring the macroscopic displacement as the only kinematical field. This implies that the parameters of such continuum models (density and/or elasticity tensors) must be considered to be frequency-dependent to capture the complex metamaterials' response in the frequency domain. These frequency-dependent models can be useful to describe some of the aforementioned macroscopic metamaterials' properties, yet, they suffer some drawbacks such as featuring negative masses and/or elastic coefficients in some frequency ranges. More than being counter-intuitive, this implies that the considered Cauchy continuum is not positive-definite for all the considered frequencies. In this paper, we present a procedure, based on the definition of extra kinematical variables (with respect to displacement alone) and the use of the inverse Fourier transform in time, to convert a frequency-dependent model into an enriched continuum model of the micromorphic type. All the parameters of the associated enriched model are constant (i.e., frequency-independent) and the model itself remains positive-definite for all the considered frequency ranges. The response of the frequency-dependent model and the associated micromorphic model coincide in the frequency domain, in particular when looking at the dispersion curves.Comment: 37 pages, 10 figure