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Angela Madeo

Publications and source records attributed to Angela Madeo.

At least 19 recordsLinked to original sources
arXiv2026

Breaking Scale Separation in Metamaterials' homogenization: Interface-Inertia-Enhanced Relaxed Micromorphic Model

Homogenized continuum models are widely used to describe wave propagation and band-gap behavior in mechanical metamaterials without explicitly resolving their microstructure. Their validity, however, typically relies on the classical separation of scales assumption, according to which the wavelength of the propagating disturbance is much larger than the characteristic size of the unit cell. In finite-size metamaterial samples and at higher frequencies, this assumption progressively breaks down, and the dynamic response becomes strongly influenced by the way the microstructure is truncated at the external boundaries. In this work we introduce a fundamentally new concept in the homogenized description of mechanical metamaterials: the inertial contribution of macroscopic interfaces. We show that different truncations of the same lattice generate boundaries with distinct mass distributions, which lead to measurable differences in the dynamic response of finite-sized specimens. To capture this complex mechanism in a homogenized framework, we extend the relaxed micromorphic model by introducing a kinetic surface energy defined on the boundary of the considered body. This generates an additional inertial term in the boundary conditions that can be seen as the homogenized counterpart of the interface inertia produced by the truncation of the microstructure. As a result, the homogenized model can now distinguish between finite-sized specimens that share identical bulk properties but differ only in the configuration of their interfaces. The proposed formulation preserves the variational structure of the relaxed micromorphic model while enabling the continuum to reproduce boundary-dependent responses observed in fully resolved simulations, particularly in frequency regimes ....... See the PDF for the full abstract.

math.NA↗
arXiv2026

Unveiling the key role of Interfaces in the Design of finite-sized Metamaterial Structures

This paper investigates the influence of interfaces on the performance of finite-sized mechanical metamaterial structures for vibration damping applications. The metamaterial structures are designed in a sandwich configuration in which two homogeneous plates are connected to a metamaterial array. We test four different arrays that are obtained from the same metamaterial by differently cutting the metamaterial's unit cell at the metamaterial/plate interface. When the four unit cells are periodically repeated in space, they create the same infinitely large metamaterial with an identical mechanical response. In finite-sized structures, however, the different interfaces between the metamaterial array and the plates~--~called ``material interfaces''~--~and between the metamaterial and the air~--~called ``free interfaces''~--~strongly affect the specimen's vibration transmission characteristics. Using experimental measurements and validated finite-element (FE) models, we demonstrate a significant influence of the different types of interfaces on the global responses and local displacement fields of the structures. We also demonstrate the presence of a vibroacoustic coupling in the structures which also depends on the type of metamaterial/plate interfaces. Furthermore, we explore optimization strategies for enhancing the vibration damping performance of the metamaterial structures considering not only the metamaterial array but also the adjacent structures (the homogeneous plates). A comparison with benchmark cases illustrates the optimization potential that the interfaces' design offers for the vibration damping capability of finite-sized metamaterial structures. We show that optimizing the type of targeted interfaces can shift a metamaterial's response from underperforming to significantly outperforming compared to classical solutions for noise and vibration damping in civil engineering.

physics.app-ph↗
arXiv2025

Static size-effects meet the dynamic scattering properties of finite-sized mechanical metamaterials: a relaxed micromorphic study with parameter identification via two-stage static-dynamic optimization

Mechanical metamaterials exhibit size-effects when a few unit-cells are subjected to static loading because no clear micro-macro scale separation holds and the characteristic length of the deformation becomes comparable to the unit-cell size. These size-effects typically manifest themselves as a strengthening of the response in a form summarized as "smaller is stiffer". Moreover, the dynamical behavior of mechanical metamaterials is very remarkable, featuring unique phenomena such as dispersive behavior and band-gaps where elastic waves cannot propagate over specific frequency ranges. In these frequency ranges, the wavelength becomes gradually comparable to the unit-cell size, giving rise to microstructure related phenomena which become particularly visible in the reflection/transmission patterns where an incident wave hits the metamaterial's interfaces. This raises the question of whether the static size-effects and dynamic reflection/transmission patterns are correlated. In this work, we investigate the interaction of the static size-effects and the dynamic scattering response of mechanical metamaterials by employing the relaxed micromorphic model. We introduce a two-stage optimization procedure to identify the material parameters. In the first stage, the static material parameters are identified by exploiting the static size-effects through a least squares fitting procedure based on the total energy. The dynamic parameters are determined in the second stage by fitting the dispersion curves of the relaxed micromorphic model to those of the fully discretized microstructure. At this second stage, we assess the results obtained by fitting the dispersion curves in one and in two propagation directions, for both the relaxed micromorphic model (RMM) with curvature and its reduced counterpart (RRMM) without curvature. The full abstract is presented in the paper.

math.NA↗
arXiv2025

Time domain analysis of microstructured materials through the reduced relaxed micromorphic model

Microstructured materials, such as architected metamaterials and phononic crystals, exhibit complex wave propagation phenomena due to their internal structure. While full-scale numerical simulations can capture these effects, they are computationally demanding, especially in time-domain analyses. To overcome this limitation, effective continuum models have been developed to approximate the macroscopic behavior of these materials while retaining key microscale effects. In this work we investigate the time-domain dynamic response of microstructured materials and focus on their effective micromorphic counterparts. We compare direct numerical simulations of discrete microstructures with predictions from micromorphic models to assess their accuracy in capturing transient wave phenomena. Our findings provide new insights into the applicability and limitations of micromorphic models in time-dependent analyses, contributing to the development of improved predictive tools for metamaterial design and engineering applications.

physics.comp-ph↗
arXiv2025

Effective interface forces to model boundary effects in a finite-size metamaterial through the reduced relaxed micromorphic model

We use the reduced relaxed micromorphic model (RRMM) to capture the effective "bulk" dynamical response of finite size metamaterial specimens made out of a Labyrinthine unit cell. We show that for small finite-size specimens, boundary effects can play a major role, so that the RRMM needs an enrichment to capture the metamaterial's bulk response, as well as the boundary effects. A benchmark test is introduced to show that different metamaterial/ homogeneous material interfaces can drive completely different responses even if the bulk metamaterial remains the same. We show with no remaining doubts that the concept of "interface forces" must necessarily be introduced if one wants to model finite-size metamaterials in a homogenized framework.

physics.app-ph↗
arXiv2024

Design and experimental validation of a finite-size labyrinthine metamaterial for vibro-acoustics: enabling upscaling towards large-scale structures

In this paper, we present the design and experimental validation of a labyrinthine metamaterial for vibro-acoustic applications. Based on a 2D unit cell, different designs of finite-size metamaterial specimens in a sandwich configuration including two plates are proposed. The design phase includes an optimisation based on Bloch-Floquet analysis with the aims of maximising the band gap and extruding the specimens in the third dimension while keeping the absorption properties almost unaffected. By manufacturing and experimentally testing finite-sized specimens, we assess their capacity to mitigate vibrations in vibro-impact tests. The experiments confirm a band gap in the low- to mid-frequency range. Numerical models are employed to validate the experiments and to examine additional vibro-acoustic load cases. The metamaterial's performances are compared to benchmark solutions, usually employed for noise and vibration mitigation, showing a comparable efficacy in the band gap region. To eventually improve the metamaterial's performance, we optimise its interaction with the air and test different types of connections between the metamaterial and the homogeneous plates. This finally leads to metamaterial samples largely exceeding the benchmark performances in the band gap region and reveal the potential of interfaces for performance optimisation of composed structures.

physics.app-ph↗
arXiv2023

Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple

We derive the Green's functions (concentrated force and couple in an infinite space) for the isotropic planar relaxed micromorphic model. Since the relaxed micromorphic model particularises into the microstretch, Cosserat (micropolar), couple-stress, and linear elasticity model for certain choices of material parameters, we recover the fundamental solutions in all these cases.

math.AP↗
arXiv2023

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.

physics.app-ph↗
arXiv2023

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.

physics.app-ph↗
arXiv2022

Multi-element metamaterial's design through the relaxed micromorphic model

Exploring the dynamical response of mechanical metamaterials has gathered increasing attention in the last decades, enabling the design of microstructures exotically interacting with elastic waves (focusing, channeling, band-gaps, negative refraction, cloaking, and many more). Yet, the application and use of such metamaterials in engineering practice is still deficient due to the lack of effective models unveiling metamaterials' interactions with more classical materials at finite scales. In this paper, we show that the relaxed micromorphic model can bring an answer to this open problem and can be effectively used to explore and optimize metamaterials' structures consisting of metamaterials' and classical materials' bricks of finite size. We investigate two examples, namely a double-shield structure that can be used to widen the frequency range for which the internal region can be protected and a multiple-shield structure that optimizes both the screening of the regions internal to the single shields and of the zones exterior to the shields themselves. The exploration of these complex meta-structures has been enabled by the finite element implementation of the relaxed micromorphic model that predicts their response at a fraction of the computational cost when compared to classical simulations.

physics.class-ph↗
arXiv2022

Cosserat micropolar elasticity: classical Eringen vs. dislocation form

In this paper we do a comparative presentation of the linear isotropic Cosserat elastic model from two perspectives: the classical Mindlin-Eringen-Nowacki description in terms of a microrotation vector and a new formulation in terms of a skew-symmetric matrix and a curvature energy in dislocation form. We provide the reader with an alternative representation of the energy for the isotropic Cosserat model to ease the comparison with the relaxed micromorphic model and the geometrically nonlinear Cosserat elastic model.

math-ph↗
arXiv2022

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 $\text{Curl}\dot{P}$ 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.

physics.app-ph↗
arXiv2021

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

We consider the classical Mindlin-Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field P. In effect, P is now only required to match the tangential derivative of the classical displacement u which is known at the Dirichlet-part of the boundary. We derive the full boundary condition, in adding the missing Neumann condition on the Dirichlet-part. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part. Finally, we show that compared to classical Dirichlet conditions on u and P, the new boundary condition modifies the interpretation of the constitutive parameters.

math.AP↗
arXiv2021

Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach

In this paper, a coherent boundary value problem to model metamaterials' behavior based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary conditions, thus disclosing the possibility of exploring the scattering patterns of finite-size metamaterials' specimens. Thanks to the simplified model's structure (few frequency- and angle-independent parameters), we are able to unveil the scattering metamaterial's response for a wide range of frequencies and angles of propagation of the incident wave. These results are an important stepping stone towards the conception of more complex large-scale meta-structures that can control elastic waves and recover energy.

physics.app-ph↗
arXiv2021

Boundary and interface conditions in the relaxed micromorphic model: exploring finite-size metastructures for elastic wave control

In this paper, we establish well-posed boundary and interface conditions for the relaxed micromorphic model that are able to unveil the scattering response of fully finite-size metamaterials' samples. The resulting relaxed micromorphic boundary value problem is implemented in finite element simulations describing the scattering of a square metamaterial's sample whose side counts 9 unit cells. The results are validated against a direct finite element simulation encoding all the details of the underlying metamaterial's microstructure. The relaxed micromorphic model can recover the scattering metamaterial's behavior for a wide range of frequencies and for all possible angles of incidence, thus showing that it is suitable to describe dynamic anisotropy. Finally, thanks to the model's computational performances, we can design a metastructure combining metamaterials and classical materials in such a way that it acts as a protection device while providing energy focusing in specific collection points. These results open important perspectives for the short-term design of sustainable structures that can control elastic waves and recover energy.

physics.app-ph↗
arXiv2021

Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, the method allows to prove the existence and uniqueness of a subsonic solution of the secular equation, a problem which remains unsolved in almost all generalised linear theories of elastic materials. Since the results are suitable to be used for numerical implementations, we propose two numerical algorithms which are viable for any elastic material. Explicit numerical calculations are made for alumunium-epoxy in the context of the Cosserat model. Since the novel form of the secular equation for isotropic elastic material has not been explicitly derived elsewhere, we establish it in this paper, too.

math.AP↗
arXiv2021

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

We solve the St.Venant torsion problem for an infinite cylindrical rod whose behaviour is described by a family of isotropic generalized continua, including the relaxed micromorphic and classical micromorphic model. The results can be used to determine the material parameters of these models. Special attention is given to the possible nonphysical stiffness singularity for a vanishing rod diameter, since slender specimens are in general described as stiffer.

math.CA↗