Gabriele Barbagallo

First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model

In this paper, we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulus μc, still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic length Lc measuring the non-locality of the phononic crystal is of the order of 13 of the diameter of its fluid-filled holes.

Real wave propagation in the isotropic-relaxed micromorphic model

For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

Bias extension test on an unbalanced woven composite reinforcement: Experiments and modeling via a second-gradient continuum approach

The classical continuum models used for the woven fabrics do not fully describe the whole set of phenomena that occur during the testing of those materials. This incompleteness is partially due to the absence of energy terms related to some microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. To account for the most fundamental microstructure-related deformation mechanisms occurring in unbalanced interlocks, a second-gradient, hyperelastic, initially orthotropic continuum model is proposed. A constitutive expression for the strain energy density is introduced to account for (a) in-plane shear deformations, (b) highly different bending stiffnesses in the warp and weft directions, and (c) fictive elongations in the warp and weft directions which eventually describe the relative sliding of the yarns. Numerical simulations which are able to reproduce the experimental behavior of unbalanced carbon interlocks subjected to a bias extension test are presented. In particular, the proposed model captures the macroscopic asymmetric S-shaped deformation of the specimen, as well as the main features of the associated deformation patterns of the yarns at the mesoscopic scale.

On the role of micro-inertia in enriched continuum mechanics

In this paper, the role of gradient micro-inertia terms η¯∥ ∇u,t∥2 and free micro-inertia terms η∥P,t∥2 is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the term η¯∥ ∇u,t∥2 alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the term η∥P,t∥2 alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia η¯∥ ∇u,t∥2, in the sense of Cartan–Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia η¯∥ ∇u,t∥2 on more classical enriched models such as the Mindlin–Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin–Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.

Modelling the deep drawing of a 3D woven fabric with a second gradient model

Experimental testing on dry woven fabrics exhibits a complex set of evidences that are difficult to completely describe using classical continuum models. The aim of this paper is to show how the introduction of energy terms related to the micro-deformation mechanisms of the fabric, in particular to the bending stiffness of the yarns, helps in the modelling of the mechanical behaviour of this kind of materials. To this aim, a second gradient, hyperelastic, initially orthotropic continuum theory is proposed to model fibrous composite interlocks at finite strains. In particular, the present work explores the relationship between the onset of wrinkling during the simulation of the deep drawing of a woven fabric and the use of a second gradient model. It is shown that the introduction of second gradient terms accounting for the description of in-plane and out-of-plane bending rigidities decreases the onset of wrinkles during the simulation of deep drawing. In this work, a quadratic energy, roughly proportional to the square of the curvature of the fibres, is presented and implemented in the simulations. This simple constitutive assumption allows the effects of the second gradient energy on both the wrinkling description and the numerical stability of the model to be clearly shown. The results obtained in second gradient simulations are descriptive of the experimental evidence of deep drawing whose description is targeted in this work. The present paper provides additional evidence of the fact that first gradient continuum theories alone cannot be considered fully descriptive of the behaviour of dry woven composite reinforcements. On the other hand, the proposed second gradient model for fibrous composite reinforcements opens the way both to the more accurate simulation of complex forming processes and to the possibility of controlling the onset of wrinkles.

A Review on Wave Propagation Modeling in Band-Gap Metamaterials via Enriched Continuum Models

In the present contribution we show that the relaxed micromorphic model is the only non-local continuum model which is able to account for the description of band-gaps in metamaterials for which the kinetic energy accounts separately for micro and macro-motions without considering a micro-macro coupling. Moreover, we show that when adding a gradient inertia term which indeed allows for the description of the coupling of the vibrations of the microstructure to the macroscopic motion of the unit cell, other enriched continuum models of the micromorphic type may allow the description of the onset of band-gaps. Nevertheless, the relaxed micromorphic model proves to be yet the most effective enriched continuum model which is able to describe multiple band-gaps in non-local metamaterials.

Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design

In the present paper, the material parameters of the isotropic relaxed micromorphic model derived for a specific metamaterial in a previous contribution are used to model its transmission properties. Specifically, the reflection and transmission coefficients at an interface between a homogeneous solid and the chosen metamaterial are analyzed by using both the relaxed micromorphic model and a direct FEM implementation of the detailed microstructure. The obtained results show excellent agreement between the transmission spectra derived via our enriched continuum model and those issued by the direct FEM simulation. Such excellent agreement validates the indirect measure of the material parameters and opens the way towards an efficient metastructural design.