33 research outputs found
Complete band gaps including non-local effects occur only in the relaxed micromorphic model
In this paper we substantiate the claim implicitly made in previous works
that the relaxed micromorphic model is the only linear, isotropic, reversibly
elastic, nonlocal generalized continuum model able to describe complete
band-gaps on a phenomenological level. To this end, we recapitulate the
response of the standard Mindlin-Eringen micromorphic model with the full
micro-distortion gradient of P, the relaxed micromorphic model depending only
on the Curl P of the micro-distortion P, and a variant of the standard
micromorphic model in which the curvature depends only on the divergence Div P
of the micro distortion. The Div-model has size-effects but the dispersion
analysis for plane waves shows the incapability of that model to even produce a
partial band gap. Combining the curvature to depend quadratically on Div P and
Curl P shows that such a model is similar to the standard Mindlin-Eringen model
which can eventually show only a partial band gap
Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: new perspectives towards meta-structural 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 an 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
meta-structural design.Comment: The paper has been already accepted in Mathematics and Mechanics of
Solids as it i
Milieux continus généralisés : Application aux grandes transformations des renforts de composites quasi-inextensibles
Dered materials in the simplest and more effective way. However, there are some cases in which the considered materials are heterogeneous even at relatively large scales and, as a consequence, the effect of microstructure on the overall mechanical behavior of the medium cannot be neglected. In such situations, Cauchy continuum theory may not be useful to fully describe the mechanical behavior of considered materials. It is in fact well known that such continuum theory is not able to catch significant phenomena related to concentrations of stress and strain and to specific deformation patterns in which high gradients of deformation occur and which are, in turn, connected to particular phenomena which take place at lower scales. Generalized continuum theories may be good candidates to model such micro-structured materials in a more appropriate way since they are able to account for the description of the macroscopic manifestation of the presence of microstructure in a rather simplified way.
The present manuscript is organized as follows: In ch.1 a general description of fibrous composite reinforcements is given. In ch.2 some fundamental issues concerning classical continuum mechanical models are recalled. In ch.3 we start analyzing some discrete and continuum models for the description of the mechanical behavior of 2D woven composites. At this stage of the manuscript, we want to show how some discrete numerical simulations allowed us to unveil some very special deformation modes related to the effect of the local bending of fibers on the overall macroscopic deformation of fibrous composite reinforcements. Such discrete simulations showed rather clearly that microscopic bending of the fibers cannot be neglected when considering the deformation of fibrous composite reinforcements. For this reason, we subsequently introduced a continuum model which is able to account for such microstructure-related effects by means of second gradient terms appearing in the strain energy density. In ch.4 we reduce the general continuum mechanical framework introduced in ch.2 to the particular case of 2D continua. In ch.5 we introduce a strong kinematical hypothesis on the admissible deformations, assuming that the yarns composing the woven reinforcements are inextensible.La microstructure des matériaux constitue un outil essentiel pour optimiser les propriétés mécaniques des structures et ainsi améliorer leurs performances. Les modèles de Cauchy ne sont pas toujours adaptés à la description de la réponse dynamique de certains matériaux microstructurés montrant des comportements mécaniques exotiques. Les théories de milieux continus généralisés peuvent être de bonnes candidates pour modéliser ces matériaux d’une façon plus précise et plus réaliste, aussi bien en statique qu’en dynamique, puisqu’elles peuvent décrire, même d’une façon simplifiée, la manifestation macroscopique de la présence d’une microstructure. Ce manuscrit est organisé comme suit : - Dans le chapitre 1 nous introduisons les aspects généraux de la mécanique des renforts fibreux.- Dans le chapitre 2 nous rappelons certains concepts fondamentaux concernant la mécanique des milieux continus classiques. De plus, nous introduisons les théories de deuxième gradient à l’aide du Principe des Travaux Virtuels.- Dans le chapitre 3 nous nous proposons de présenter une première modélisation des renforts fibreux de composites en mettant en place des modèles numériques discrets. Cette modélisation discrète permet de rendre compte de certains effets de la microstructure des renforts fibreux sur leur comportement macroscopique global. En particulier, il sera montré que la flexion locale des mèches à l’échelle mesoscopique a un effet non-négligeable sur le comportement macroscopique global de ces matériaux. Dans un deuxième moment nous introduisons une modélisation continue de deuxième gradient pour la description des mêmes matériaux et nous montrons que les termes d’ordre supérieur permettent une description satisfaisante des effets de flexion locale sur-cités.- Dans le chapitre 4 on particularise le cadre général de la mécanique des milieux continus introduit dans le chapitre 2 au cas particulier des milieux continus 2D. On mettra un accent fort sur l’interprétation géométrique des mesures de déformation de deuxième gradient qui seront directement reliées aux courbures dans le plan de certaines lignes matérielles. Ces lignes matérielles seront ensuite interprétées dans les chapitres suivantes comme décrivant les mèches des renforts fibreux de composites qu’on se propose d’étudier.- Dans le chapitre 5 nous introduisons une hypothèse cinématique forte sur les déformations admissibles, en supposant que les mèches du renfort considéré sont inextensibles. Cette hypothèse nous permettra de construire un modèle simplifié de premier gradient pour le comportement des renforts de composites 2D qui est encore représentatif de leur comportement mécanique. Une méthode numérique permettant de montrer certaines solutions concernant le cas du bias extension test est codée en Mathematica et les résultats obtenus sont discutés
Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures
In this paper, we show that the transient waveforms arising from several
localised pulses in a micro-structured material can be reproduced by a
corresponding generalised continuum of the relaxed micromorphic type.
Specifically, we compare the dynamic response of a bounded micro-structured
material to that of bounded continua with special kinematic properties: (i) the
relaxed micromorphic continuum and (ii) an equivalent Cauchy linear elastic
continuum. We show that, while the Cauchy theory is able to describe the
overall behaviour of the metastructure only at low frequencies, the relaxed
micromorphic model goes far beyond by giving a correct description of the pulse
propagation in the frequency band-gap and at frequencies intersecting the
optical branches. In addition, we observe a computational time reduction
associated with the use of the relaxed micromorphic continuum, compared to the
sensible computational time needed to perform a transient computation in a
micro-structured domain
Relaxed micromorphic broadband scattering for finite-size meta-structures -- a detailed development
The conception of new metamaterials showing unorthodox behaviors with respect
to elastic wavepropagation has become possible in recent years thanks to
powerful dynamical homogenization techniques. Such methods effectively allow to
describe the behavior of an infinite medium generated by periodically
architectured base materials. Nevertheless, when it comes to the study of the
scattering properties of finite-sized structures, dealing with the correct
boundary conditions at the macroscopicscale becomes challenging. In this paper,
we show how finite-domain boundary value problems canbe set-up in the framework
of enriched continuum mechanics (relaxed micromorphic model) by imposing
continuity of macroscopic displacement and of generalized traction when
non-local effects areneglected.The case of a metamaterial slab of finite width
is presented, its scattering properties are studied viaa semi-analytical
solution of the relaxed micromorphic model and compared to numerical
simulationsencoding all details of the selected microstructure. The reflection
coefficient obtained via the twomethods is presented as a function of the
frequency and of the direction of propagation of the incidentwave. We find
excellent agreement for a large range of frequencies going from the long-wave
limitto frequencies beyond the first band-gap and for angles of incidence
ranging from normal to nearparallel incidence. The case of a semi-infinite
metamaterial is also presented and is seen to be areliable measure of the
average behavior of the finite metastructure. A tremendous gain in termsof
computational time is obtained when using the relaxed micromorphic model for
the study of theconsidered metastructure
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
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, while
it does in isotropic linear elasticity