Pr Jean-Francois Hassan Ganghoffer

Lakiss

Keywords: Homogenization, Cauchy, micro-inertia, second gradient, dynamic, acoustic, vibration, Periodic laminates, composites

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Résumé -Abstract

Titre de la thèse en français : Réponse dynamique de milieux à microstructure par des formulations à gradient d'ordre supérieur obtenues par des méthodes d'homogénéisation.

Résumé

Une approche dynamique discrète (DDM) est proposée dans le contexte de la mécanique des poutres pour calculer les caractéristiques de dispersion des structures périodiques. Cette démarche permet de calculer les caractéristiques de dispersion de milieux périodiques unidimensionnels et bidimensionnels. Il est montré qu'un développement d'ordre supérieur suffisamment élevé des forces et des moments d'éléments structuraux est nécessaire pour décrire avec précision les modes de propagation d'ordre supérieur. Ces résultats montrent dans l'ensemble que les calculs des caractéristiques de dispersion de systèmes structurels périodiques peuvent être abordés avec une bonne précision par la dynamique des éléments discrets.

Les comportements non classiques peuvent être capturés non seulement par une expansion d'ordre supérieur mais aussi par des formulations à gradient supérieur. Nous calculons ainsi les paramètres constitutifs macroscopiques jusqu'au deuxième gradient du déplacement en utilisant deux formulations différentes, soit selon une méthode d'homogénéisation dynamique à gradient supérieur (DHGE) prenant en compte les effets de micro-inertie, ou alternativement selon le principe de Hamilton. Nous analysons ensuite la sensibilité des termes constitutifs du second gradient aux paramètres microstructuraux pour des matériaux composites à microstructure périodique de type laminés. En plus, on montre que les modèles du deuxième gradient formulés à partir de l'énergie interne totale en tenant compte des termes de gradient d'ordre supérieur donnent la meilleure description du propagation d'onde à travers ces milieux.

On analyse les contributions d'ordre supérieur et de microinertie sur le comportement mécanique de structures composites en utilisant une méthode d'homogénéisation dynamique d'ordre supérieur qui intègre les effets de microinertie. Nous calculons la réponse effective statique longitudinale à gradient d'ordre supérieur, en quantifiant la différence relative par rapport à la formulation classique de type Cauchy qui repose sur le premier gradient du déplacement. Nous analysons ensuite les propriétés de propagation d'ondes longitudinales en termes de fréquence propre de composites, en tenant compte de la contribution de la microinertie. La longueur interne joue un rôle crucial dans les contributions de microinertie avec un effet substantiel pour les faibles valeurs de longueur interne, et qui correspond à une large gamme de matériaux utilisés en ingénierie des structures.

La méthode d'homogénéisation développée montre un effet de taille important pour les modules élastiques homogénéisés d'ordre supérieur. Par conséquent, nous développons une formulation indépendante de la taille qui est basée sur des termes de correction liée aux moment quadratique. Dans ce contexte, on analyse l'influence des termes de correction sur le comportement statique et dynamique de composites à inclusion.

Mots clés : Homogénéisation, Cauchy, micro-inertie, second gradient, dynamique, acoustique, vibration, stratifiés périodiques, composites. Titre de la thèse en anglais : Homogenization-based higher-gradient dynamical response of micro-structured media.

Abstract

A discrete dynamic approach (DDM) is developed in the context of beam mechanics to calculate the dispersion characteristics of periodic structures. Subsequently, based on this dynamical beam formulation, we calculate the dispersion characteristics of one-dimensional and two-dimensional periodic media. A sufficiently high order development of the forces and moments of the structural elements is necessary to accurately describe the propagation modes of higher order. These results show that the calculations of the dispersion characteristics of structural systems can be approached with good accuracy by the dynamics of the discrete elements.

Besides, non-classical behaviors can be captured not only by higher order expansion but also by higher gradient formulations. To that scope, we develop a higher gradient dynamic homogenization method with micro-inertia effects. Using this formulation, we compute the macroscopic constitutive parameters up to the second gradient, using two distinct approaches, namely Hamilton's principle and a total internal energy formulation. We analyze the sensitivity of the second gradient constitutive terms on the inner material and geometric parameters for the case of composite materials made of a periodic, layered microstructure. Moreover, we show that the formulations based on the total internal energy taking into account higher order gradient terms give the best description of wave propagation through the composite.

We analyze the higher order and micro-inertia contributions on the mechanical behavior of composite structures by calculating the effective static and dynamic properties of composite beams using a higher order dynamic homogenization method. We compute the effective longitudinal static response with higher order gradient, by quantifying the relative difference compared to the classical formulation of Cauchy type, which is based on the first gradient of displacement. We then analyze the propagation properties of longitudinal waves in terms of the natural frequency of composite structural elements, taking into account the contribution of micro-inertia. The internal length plays a crucial role in the contributions of micro-inertia, which is particularly significant for low internal length values, therefore for a wide range of materials used in structural engineering. The developed method shows an important size effect for the higher gradients, and to remove these effects correction terms have been incorporated which are related to the quadratic moment of inertia. We analyze in this context the influence for first gradient (green), first gradient with higher order micro-inertia effects (yellow) in [START_REF] Bacigalupo | Second-gradient homogenized model for wave propagation in heterogeneous periodic media[END_REF]) and the exact solution (red) provided by (Sun et al.,1968 3.7: Normalized phase velocity curves for longitudinal wave (a) and for transverse wave (b) obtained by the DHGE method with first and second gradient constitutive parameters (blue and red respectively). The exact solution is presented in black (Sun et al., 1968), while the yellow color curves correspond to the first gradient constitutive formulation with higher gradient inner micro-inertia terms provided in (Bacigalupo & Gambarotta, 2014) Table 2.1 : Geometric and mechanical characteristics of the beam. ....................................... 52 Table 2.2 : Frequency values that satisfy the separation of scales as a function of the number of unit-cells considered within the basic period of the structural system. ................................... 69 Table 3.1: Geometric and mechanical layers properties the laminate layers. .......................... 97 Table 3. 5.1 : Relative difference between the homogenized second gradient constitutive parameters of the 2D composite structure in Fig. 1a using HGE without the correction terms and the analytical results of (Li, 2011) 5.2 : Relative difference between of the homogenized second gradient constitutive parameters of the 2D composite structure in Fig. 5.1a using HGE with corrections terms and the analytical results of (Li, 2011)

List of Figures

List of Tables

Résumé de la Thèse en Français

Titre de la thèse en français : Réponse dynamique de milieux à microstructure par des formulations à gradient d'ordre supérieur obtenues par des méthodes d'homogénéisation.

Les modèles micromécaniques prédictifs visant à comprendre l'impact de la géométrie et des propriétés du renfort et de la matrice sur la réponse de composites à l'échelle macroscopique font l'objet de développements significatifs, la détermination expérimentale des propriétés mécaniques des composites étant délicate. Par conséquent, des approches prédictives d'inspiration micromécanique s'avèrent très utiles afin de lier le comportement aux différentes échelles et de construire une loi de comportement à l'échelle d'un milieu continu effectif, grâce auxquelles les propriétés équivalentes du milieu continu de substitution peuvent être liées aux paramètres microstructuraux géométriques et mécaniques des constituants individuels du composite. En outre, la conception optimale de matériaux architecturés et de composites nécessite de comprendre de façon quantitative la relation entre l'architecture interne et les propriétés physiques, et de formuler les approches théoriques et numériques adéquates pour évaluer ces propriétés en prenant en compte les effets de microstructure dans des conditions tant statique que dynamique.

La mécanique des milieux continus généralisés peut être définie comme une théorie des milieux continus où la mécanique des milieux continus classiques développée par Cauchy est insuffisante pour la description des phénomènes physiques (Kröner, 1968). Afin de conserver une description de type milieu continu, il faut adopter une modélisation de type milieux continus enrichis pour modéliser le milieu homogène équivalent résultant du processus d'homogénéisation. Les techniques d'homogénéisation classiques rencontrent des limites lorsque la longueur d'onde du chargement ou de la déformation deviennent comparable à la taille de la microstructure ou de celle d'un volume élémentaire représentatif de cette dernière (Trinh et al., 2012). L'objectif de l'homogénéisation vers des milieux continus généralisés est de remédier à ces limitations et d'étendre la plage de validité de l'approche de type milieu continu au-delà de l'hypothèse stricte de la séparation d'échelle (Mindlin, 1964;Eringen et Suhubi, 1964). L'incorporation de l'échelle microstructurale dans les formulations de type milieux continus enrichis implique la relaxation de l'hypothèse de localité des interactions et l'introduction d'une interaction spatiale à grande distance et d'échelles de longueur internes pour tenir compte de l'influence des points voisins dans la définition des variables cinématique, statiques et la formulation des équations constitutives. Il existe essentiellement deux façons de généraliser la définition classique d'un continuum, classées en milieux continus d'ordre supérieur et milieux à gradient supérieur. Les milieux d'ordre supérieur impliquent des degrés de liberté supplémentaires comme le milieu de Cosserat développé par les frères Cosserat (Cosserat et Cosserat, 1909) et le milieu micromorphe (Eringen et Suhubi, 1964;Mindlin, 1964;Germain, 1973). Dans l'élasticité de Cosserat encore appelée micropolaire (Cosserat et Cosserat, 1909), des rotations locales sont ajoutées comme degrés de liberté. Dans les milieux à gradient d'ordre supérieur, on garde les mêmes degrés de liberté mais des gradients d'ordre supérieur du champ de déplacement ou de la déformation complètent le jeu de degrés de liberté.

L'objectif général de la thèse est de formuler des méthodes d'homogénéisation dynamiques de milieux hétérogènes en prenant en compte l'impact de la microstructure sous-jacente par des formulations enrichies de type milieux continus du second gradient pour la description des interactions spatiales à longue portée ainsi que les effets de micro-inertie.

Dans cette thèse, nous déterminons dans une première étape le développement asymptotique d'ordre supérieur des forces et des moments dans les éléments structuraux de type poutres pour des sollicitations dynamiques (Ayad et al., 2020a). Par la suite, nous utilisons les résultats obtenus pour calculer les caractéristiques de dispersion de milieux périodiques unidimensionnels et bidimensionnels. L'équilibre de la cellule irréductible d'un milieu périodique permet d'obtenir la structure fondamentale du diagramme de dispersion aux basses fréquences, décrivant le comportement acoustique de la structure au-delà de la limite de la théorie non dispersive de Cauchy. Nous montrons que la dynamique discrète permet d'accéder aux modes de fréquence plus élevés pour un domaine incluant plusieurs cellules, ces plages de fréquence dispersive n'étant pas accessibles par des formulations classiques. Ces analyses sont ensuite étendues pour des structures bidimensionnelles (Fig. R1) afin d'en décrire les caractéristiques de dispersion. (Sun et al.,1968).

compte des termes de gradient d'ordre supérieur et de la micro-inertie donnent une meilleure description des diagrammes de fréquence et des vitesses de phase et de groupe (Fig. R5). Nous analysons ensuite les propriétés de propagation d'ondes longitudinales en termes de fréquence propre des structures composites, en tenant compte de la contribution de la microinertie. Des expressions analytiques sont obtenues qui relient les paramètres constitutifs effectifs du matériau composite à ses caractéristiques de vibration à l'échelle macroscopique.

Un aspect novateur réside dans l'établissement de la preuve que les contributions de microinertie peuvent contrebalancer l'effet des termes du second gradient sur les fréquences propres de la structure, leur importance relative dépendant du mode de propagation considéré. La longueur interne joue un rôle crucial dans les contributions de micro-inertie, avec un effet substantiel pour des valeurs de longueur interne faibles, qui seront ainsi observés pour une large gamme de matériaux utilisés en ingénierie des structures (Fig. R7). La fréquence de dispersion obtenue en utilisant la formulation à gradient supérieur dépend des propriétés de la microstructure; cette dépendance est plus importante pour le mode transversal que pour le mode longitudinal pour les propriétés des composites considérées. Les méthodes développées et les résultats obtenus serviront de référence pour le calcul des propriétés statiques et dynamiques des structures composites et de matériaux architecturés au sens large, fournissant une base théorique et numérique pour la quantification de la sensibilité des propriétés mécaniques effectives enrichies aux paramètres microstructuraux.

Chapter 1 Introduction

Chapter 1: General Introduction

Overview

The development of additive manufacturing has allowed the engineering of a new class of materials that obtain their effective mechanical attributes by their inner topological design rather than their chemical composition, and generate exceptional material performances far beyond those of the base material. Artificial materials of this kind deserve the epithet "architectured materials or metamaterials" (with a base material being metal, polymer or ceramic), considering their potential to yield effective static and dynamic attributes that can well extend the design space of the base material used [START_REF] Cui | Metamaterials: Theory, Design and Applications[END_REF][START_REF] Vaezi | A review on 3D micro-additive manufacturing technologies[END_REF].

Architectured materials define a new class of artificial materials with exceptional static and dynamic characteristics typically not encountered in natural materials (Reda et al., 2018). The mechanical response of periodic architectured materials can be analyzed at the macroscopic level, on the scale of a unit cell that is multiplied to build a regular (periodic) lattice. They are used in a wide range of engineering applications, such as vibration control, energy absorption (Karathanasopoulos et al., 2017) and chiral structures (tetrachiral lattice, right), [START_REF] Karathanasopoulos | Mechanics of beams made from chiral metamaterials: Tuning deflections through normalshear strain couplings[END_REF].

A lot of attention has been given in the last years to the static behavior of architectured materials. In particular, an emphasis has been posed on the design of 2D unit cells with tailored stiffness attributes. Designing materials of controlled mechanical properties has opened new possibilities in a series of engineering applications, such as in morphing and mechanical engineering, wave propagation and vibration [START_REF] Xu | Design of lattice structures with controlled anisotropy[END_REF], Cheng et al., 2018).

Architectured materials designed with a quasi-periodic inner topology have witnessed a rapid expansion in the recent period; the complex internal topology of architectured materials prohibits the use of direct numerical simulations or full-field computations of macrostructures made of architectured materials, which become numerically too expensive. Whereas, the mechanical response of periodic architectured materials can be analyzed at the unit cell scale.

For the computation of the effective mechanical properties of architectured media, homogenization methods have been commonly employed. The determination of a homogenized material behavior in the static regime constitutes the first step towards the characterization of its wave propagation characteristics [START_REF] Reda | Micropolar dissipative models for the analysis of 2D dispersive waves in periodic lattices[END_REF]Bacigalupo and Gambarotta, 2014). The latter can be obtained by computing the material's dynamic attributes at the scale of their representative building blocks [START_REF] Lachenal | Review of morphing concepts and materials for wind turbine blade applications[END_REF].

Artificial materials demonstrate not only uncommon static properties, but also dynamic mechanical attributes. A primal design objective has been to obtain inner material architectures that favor the development of bandgap frequency regions. Different lattice configurations have been shown to provide wave propagation isolation characteristics. More specifically, triangular and hexagonal honeycomb lattices have been identified to exhibit isolation properties (Karathanasopoulos et al., 2017), while tetrachiral honeycombs have been shown to permit a veering of certain modes. The latter have found direct applications in both passive and active vibration control (Bacigalupo and Gambarotta, 2017). Recently, architectured materials that make use of inner folding mechanisms have attracted researchers' attention for their ability to control elastic waves (Bacigalupo and Gambarotta, 2017).

The use of resonant elements has been associated to material designs of enhanced dynamic properties. More specifically, resonant internal units have been related to acoustic band gaps [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF][START_REF] Pennec | Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate[END_REF][START_REF] Liu | A review on auxetic structures and polymeric materials[END_REF] along with a tuning capability at low frequencies [START_REF] Bigoni | Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization[END_REF]. Their wave propagation characteristics are highly sensitive to their inner material topology, with different inner architectures that exhibit utterly different dynamical properties [START_REF] Bacigalupo | Auxetic anti-tetrachiral materials: Equivalent elastic properties and frequency band-gaps[END_REF].

In the realm of composite engineering, the development of predictive micromechanical models is both appealing and necessary in order to bridge the scales and to provide a constitutive law at a continuum scale, whereby the effective continuum properties can be evaluated from the geometrical and microstructural parameters of the individual constituents of the composite. The presence of a strong contrast of properties between the reinforcement and the surrounding matrix entails the development of strong strain gradients within the composite, leading very often to the need of modeling the composite as an enriched continuum in the framework of the mechanics of generalized continua (like a strain gradient or a micromorphic substitution medium). Homogenization theory has become over the last four decades a fundamental tool to analyze the overall behavior of highly heterogeneous media such as composites. It most often relies on the determination of a representative elementary volume (RVE), which implicitly assumes that the structure of the RVE is found almost everywhere in the structure, which amounts in fact to a disguised pseudo-periodicity.

For this reason, periodic homogenization plays a privileged role as a theory to account for the overall homogenized behavior of multiphase materials like composites. Initiated by the works of (Bensoussan et al., 1978;Sanchez-Palencia, 1980), several authors analyzed more recently the influence of higher-order terms in the asymptotic expansions of the fields on the macroscopic behavior of linear elastic composites (Gambin and Kröner, 1989;Boutin, 1996;Triantafyllidis and Bardenhagen, 1996;Smyshlyaev and Cherednichenko, 2000;Tran et al., 2012). Furthermore, designing architected and composite materials in an optimal way requires understanding the relation between the internal architecture and physical properties, and suitable ways to evaluate these properties. In classical continuum mechanics (Truesdell and Noll, 1965), all higher order displacement gradients are neglected and only the first order displacement gradient is involved in measuring the deformations of a body. Such kinematic framework turns out not to be rich enough to describe a variety of important mechanical and physical phenomena (Auffray, 2010). Especially, the classical continuum mechanics proves to be intrinsically size independent, as reported by (Truesdell and Noll, 1992). Particularly, size effects and a non-local behavior may arise due to the discrete nature of matter at a sufficiently small-scale level.

Higher order static homogenization

Classical homogenization techniques limits to large wavelength of the loading or deformation field when it is comparable with the typical microstructure size. In situations where strict scale separation assumption fails, the classical Cauchy theory proves unable to predict the overall behavior; particularly, size effects and even non-local behaviors may arise due to the discrete nature of matter at sufficiently small scales level. The lack of a material length scale parameter in the classical continuum elasticity theory leads however to an inaccurate description of the macroscopic structural behaviors (Goda and Ganghoffer, 2015).

Two distinct strategies emerged to remedy the lack of scale separation in the modeling of homogenized media: the first strategy is phenomenological in nature and uses generalized continuum mechanics. The incorporation of the microstructural scale within enhanced continuum formulations involves the relaxation of the local action hypothesis and the introduction of a spatial interaction and typical length scales to account for the influence of neighboring points in the formulation of constitutive equations.

The mechanics of generalized continua can be defined as the mechanics of matter for circumstances in which the conventional continuum mechanics cannot offer a promising approach (Kröner, 1968). In order to keep a continuum description, one has to consider a generalized continuum to model the substitution material resulting from the homogenization process. The objective of homogenization towards generalized continua is to remedy the classical homogenization techniques limitations and to extend the range of validity of the continuum approach beyond the strict assumption of the scale separation (Mindlin, 1964;Eringen and Suhubi, 1964). There are essentially two ways to generalize the classical definition of a continuum, classified into higher-order and higher-gradient continua. Higher-order continua involves additional degrees of freedom like the Cosserat medium promoted by the Cosserat brothers (Cosserat and Cosserat, 1909) and the micromorphic medium (Eringen and Suhubi, 1964;Mindlin, 1964;Germain, 1973). In Cosserat elasticity (Cosserat and Cosserat, 1909), local rotations are added as degrees of freedom. In higher-gradient continua, one keeps the same degrees of freedom but higher gradients of the displacement field are added as arguments of the strain energy density. Amongst the possible a priori choices of the effective substitution medium mentioned previously (Cosserat, micromorphic, strain-gradient …), the strain gradient elastic continuum introduced in (Mindlin, 1964) and (Mindlin & Eshel, 1968) offers several advantages as underlined in (Yvonnet et al., 2020) since it is rich enough to represent emergent properties without having recourse to the micromorphic medium that has many more inherent parameters (Auffray et al., 2015) and such a model can be constructed using asymptotic homogenization (Boutin, 1996;[START_REF] Bacigalupo | Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits[END_REF]Gambarotta, 2010, 2014;Tran et al., 2012).

A large body of works has been devoted to strain gradient homogenization in the literature since more than 40 years (Ben-Amoz, 1976; Forest and Sab, 1998;Forest, 1998;Forest et al., 2001;Trinh et al., 2012;Tran et al., 2012;Rahali et al., 2015;Bacigalupo and Gambarotta, 2010) for composites with periodic or random microstructures (Goda & Ganghoffer, 2016), auxetic architectured materials (Bacigalupo & Gambarotta, 2014) and random fibrous media (Berkache et al., 2017). Different approaches have been employed in the literature for the derivation of the effective mechanical properties of strain gradient models of heterogeneous microstructures.

Periodic homogenization methods relying on the asymptotic expansion of the searched solution versus a small parameter representative of the relative scale of the unit cell (with respect to the macroscale of the overall structure) have been developed in (Bensoussan et al., 1978;Sanchez-Palencia, 1980) considering the main proponents. Expansion of the series up to the second order leads to the formulation of an effective strain gradient continuum whereby the local solution depends on the macroscopic strain gradient (Gambin and Kröner, 1989;Boutin, 1996;Triantafyllidis and Bardenhagen, 1996;Smyshlyaev and Cherednichenko, 2000;Tran, 2012). The strain gradient elastic moduli are computed by solving in a sequential manner two unit cell boundary value problems, a classical one leading to the Cauchy moduli and a higher order BVP associated with the macroscopic strain gradient as a kinematic source term.

Pure asymptotic expansion methods (Boutin, 1996;Smyshlyaev and Cherednichenko, 2000) are sometimes combined with a Fast Fourier Transform FFT method to compute the effective properties (Tran et al., 2012;Li and Zhang, 2013). A quite common strategy to compute the strain gradient moduli is to use suitable quadratic boundary conditions over the RVE as the minimum kinematic loading capturing a gradient of the strain at the macroscopic level, an approach followed by many authors (Gologanu et al., 1997;Forest, 1998;Forest et al., 2001;Kanit et al., 2003;Geers et al., 2001;Kouznetsova et al., 2004;Auffray et al., 2010;Barboura and Li, 2018). This method extends the classical kinematic uniform boundary conditions employed in Cauchy type homogenization (Huet, 1990). The method however suffers from two major flaws, since the effective strain gradient moduli remain non-zero for homogeneous materials or under conditions of scale separation (Forest and Trinh, 2011).

Corrections to these spurious effects were recently proposed (Yvonnet et al., 2020), where they added a body force field to the quadratic boundary conditions together with a correction of the localization tensor to get a convergence effective strain gradient modulus for big number of cells. Very recently, (Monchiet et al., 2020) made a correction to the employed quadratic boundary conditions, thereby getting rid of the strain gradient effects (the effective medium remains of Cauchy type). Furthermore, the computed strain gradient effects converge asymptotically with the number of periods. Note that this asymptotic homogenization method avoids the spurious effect since the source term in the second order unit cell problem vanishes for a homogeneous medium. Moreover, another type of correction was provided in (Li, 2011) where second gradient moduli not only vanish for homogenous media but are independent of periods number; the correction terms used the first gradient moduli and the quadratic inertia of the structure. Furthermore, the author provides an analytical solution of the higher gradient constitutive parameters for 1D and 2D composites structures.

Impact of higher order homogenization on static and dynamic behavior

The descriptions of microstructural effects analyzed in wave propagation are most of the time phenomenological in nature, following either the line of micromorphic models (Berezovski et al., 2013) or second gradient continuum models (Fafalis et al., 2012). The phenomenological nature of these models entails that the material parameters involved in the wave propagation formulations need to be calibrated to experiments, and consequently such models are not predictive. The large number of intrinsic parameters to be identified requires a somewhat difficult measurement protocol; this issue is addressed especially in (Polyzos et al., 2015;Iliopoulos et al., 2016;Réthoré et al., 2015). In order to circumvent this drawback, multiscale methods have been developed to link the dispersive aspects of wave propagation to the microstructure of the material. Amongst these, homogenization techniques derive the dynamical macroscopic behavior by upscaling the microscopic one, so obtaining a more coarse description of the dispersive wave propagation features, but still reflecting in a predictive manner the impact of the underlying material microstructure.

It is experimentally proven that materials with a heterogeneous microstructure exhibit a dispersive wave propagation behavior, that is, each wavenumber travels with a different phase velocity (Erofeyev, 2003). However, homogenization models based on the classical, Cauchytype elasticity theory are not able to reproduce different aspects of the physical behavior of heterogeneous media, such as wave dispersion.

The impact of the heterogeneous microstructure on the wave propagation characteristics of materials was first recognized and analyzed by Mindlin (1964), who proposed several nonlocal continuum models to capture the dispersion of planar waves. Since the early works in the 1960s by Toupin (1962) and Koiter (1969), there have been several literature contributions to the topic, recently summarized in (Askes and Aifantis, 2009). Generalized continuum theories offer an attractive alternative for capturing dynamic behaviors overlooked by classical elasticity, especially dispersion relations (Lombardo and Askes, 2012). In order to study low frequency wave propagation in effective continuum media, enriched second gradient elasticity descriptions have been developed which account for microstructural effects descriptive of the dispersive wave behavior (Aifantis, 1998). Different theories based on an enrichment of the classical elasticity framework by higher order gradients have been proposed in the past to overcome deficiencies of classical elasticity (see Aifantis, 1992Aifantis, ,1998 Askes and Metrikine, 2002). A detailed comparison between the dispersive characteristics of various simplified models of gradient elasticity can be found in (Askes et al., 2008). Accordingly, differences between classical elasticity considerations and formulations that account for the role of higher order inner material kinematics or finite strains can be found in (Reda et al. 2018, Karathanasopoulos et al., 2019). These homogenization methods encounters limitations in describing the wave dispersion on all the structure Brillouin zone as shown in the figure below. Homogenization in the time domain has also been proposed by several authors, see e.g. (Chen and Fish, 2000), considering both fast and slow time scales. In many studies, gradient elasticity theories have been derived from the continualization of the response of a discrete lattice consisting of discrete masses and springs, see for example the early work of Mindlin (1972). One important challenge of gradient elasticity theories is a physical interpretation of the involved length-scale parameters as a function of the given microstructure (Askes and Aifantis, 2011).

Resorting to computational approaches, finite element computations over the unit cell with periodic Bloch type boundary conditions have been performed to model acoustic waves scattering and to evaluate the band structure (Hussein, 2009;Huang, 2011) Bisegna & Caruso, 2011). Higher gradient shows a huge influence on the eigenfrequency values as described in (Altan et al.,1996) with respect to classic continuum.

Motivation of the thesis

The general objective of the thesis is to develop higher gradient homogenization methods to predict the response of periodic structures and composites covering successively the static, vibratory and acoustic responses at a macroscopic level of description, intermediate between the microscopic and structural scales.

Higher gradient formulation has been applied to different composite structures and different architecture materials. In this thesis, we analyze the impact of higher gradient formulation on the macroscopic response; moreover, we relate this impact to the microstructure properties.

The need to employ such enriched formulations going beyond Cauchy elasticity adopting in a first step discrete models of structural systems is highlighted in the first part of the thesis.

Dispersion characteristics of periodic structural systems using higher order beam dynamics

Lattice configurations have been shown to provide special wave propagation characteristics. Moreover, wave motion in triangular and hexagonal honeycomb lattices exhibits characteristic pass and stop band as shown in (Karathanasopoulos et al., 2017).

Floquet-Bloch principle has been used to explore the dispersion characteristics of periodic structures (Phani et al;2006) which is based on the mass and stiffness matrix of the structure, by considering the structure as a discrete system.

In this thesis, we derive higher order discrete mechanics methods (DDM) using the internal forces and moments and we show that this method is not limited to Cauchy linear dispersion behavior.

Furthermore, we show the size dependence of this method, where studying two or three unit-cell structure will increase the degree of freedom and obtain the optical dispersion curves.

Strain gradient homogenization based on dynamic higher gradient strain energy and effects on wave propagation

In this thesis, we develop a higher gradient dynamic homogenization method with microinertia effects. To that scope, we compute the macroscopic constitutive parameters up to the second gradient, using two distinct approaches, namely the Hamilton's principle and the total strain energy formulation.

Using the derived formulation, we calculate the dispersion characteristics of the composite materials with a periodic, layered microstructure, where we show that second gradient macroscopic constitutive formulations derived using the dynamic higher gradient strain energy (DHGE) outperform first gradient approaches, better approximating the wave propagation characteristics of the effective medium.

Strain gradient homogenization with micro inertia effects for vibration analyses

We compute the complete set of the effective static and dynamic properties of composite beam structures using a dynamic higher gradient energy homogenization method (DHGE), which incorporates micro-inertia effects.

We observe that within the static analysis case, higher-order effects require high internal length values and highly non-linear strain profile distributions for non-negligible higher-order effects to appear. We subsequently analyze the longitudinal, higher-gradient eigenfrequency properties of composite structural members, accounting for the role of micro-inertia contributions. Thereupon, we derive analytical expressions that relate the composite material's effective constitutive parameters with its macroscale vibration characteristics. We provide for the first time evidence that micro-inertia contributions can counteract the effect of secondgradient properties on the eigenfrequencies of the structure, with their relative significance depending on the mode of interest. Moreover, we show that the internal length plays a crucial role in the significance of micro-inertia contributions, with their effect to be substantial for low, rather than for high internal length values, thus for a wide range of materials used in engineering practice.

Wave propagation analysis of composites structures using independent size higher gradient formulation

We use a higher gradient energy formulation (HGE) to compute the complete set of effective properties, including the correction terms of the higher gradient moduli involving the quadratic inertia tensor, which has the benefit to generate size independent higher gradient constitutive parameters. The static homogenized properties of inclusion-based composites are computed together with their internal lengths as an application of the proposed homogenization method, highlighting the importance of the correction terms. We subsequently analyze the variation of these parameters for different microstructural mechanical and geometrical parameters, showing that the mechanical effect is limited to low contrast, whereas the volume fraction of reinforcement has a stronger effect on internal lengths. We provide as a novel aspect a study of the influence of correction terms on the wave propagation features of composite structures, showing a decrease of the dispersive frequency due to the correction. In addition, we show that frequencies computed depend on the propagation mode of interest, with the shear frequency mode being more affected than the longitudinal mode irrespective of the formulation employed.

Organization of the thesis

The thesis aims to handle the lack of space and time scale separation within periodic microstructured media (architected materials, composites) by resourcing to generalized continua. This motivation serves as a starting point in chapter 2, where we develop a discrete dynamic method (DDM) to evaluate the dispersion diagrams of 1D and 2D architected materials described with beams as structural elements. The performed analyses show the limitations of Cauchy elasticity in describing the dynamics of simple structural systems, calling for the consideration of higher spatial gradients in their modeling.

Chapter 3 advances novel homogenization methods towards effective strain gradient medium for composite materials, based on a higher gradient strain energy formulation. A systematic sensitivity of microstructural parameters is performed to highlight the factors responsible for strain gradient effects at the macroscopic level. We analyze the influence of higher gradient formulation on the dispersion characteristics through a layered microstructure.

Apart from the wave propagation, we study the impact of higher gradient and micro-inertia terms calculated using the dynamical higher gradient energy formulation (DHGE) on the resonance response in chapter 4. Moreover, we calculate the correction terms that eliminate the higher gradient terms for homogenous structures and provide independent size moduli in chapter 5. We analyze the impact of these terms on the wave propagation behavior through these composites for different microstructure properties. Each chapter includes a thorough discussion of the employed method and obtained results. Finally, we conclude with an overview of the main thrust of the thesis, a critical analysis of the proposed methods and results, a synthetic overview of ongoing works, and proposals for future developments.

Chapter 2 Dispersion characteristics of periodic structural systems using higher order beam element dynamics

Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics

Summary

In the current chapter, we elaborate a beam mechanics based, discrete dynamics approach for the computation of the dispersion characteristics of periodic structures. To that scope, we compute the higher order, asymptotic expansion of the forces and moments developed within beam structural elements upon dynamic loads. Thereafter, we employ the obtained results to compute the dispersion characteristics of one and two-dimensional periodic media. In the onedimensional space, we demonstrate that single unit-cell equilibrium can provide the fundamental, low frequency band diagram structure, which can be approximated by nondispersive Cauchy media formulations. However, we show that the discrete dynamics method can access the higher frequency modes by considering multiple unit-cell systems for the dynamic equilibrium, frequency ranges that cannot be accessed by simplified formulations. We extend the analysis in the two-dimensional space computing, the dispersion attributes of square lattice structures. Thereupon, we demonstrate that the discrete dynamics dispersion results well compare to the once obtained using Bloch theorem computations. We show that a high order expansion of the structures' inner element forces and moments is required for the higher wave propagation modes to be accurately represented, contrary to the shear and the longitudinal mode which can be captured using a lower, fourth-order expansion of its inner dynamic forces and moments. The provided results can serve as a reference analysis for the computation of the dispersion characteristics of periodic structural systems with the use of discrete element dynamics.

Introduction

In Elastodynamic homogenization approaches have been developed for both periodic and random heterogeneous media [START_REF] Willis | The overall elastic response of composite materials[END_REF][START_REF] Willis | Variational principles and operator equations for electromagnetic waves in inhomogeneous media[END_REF][START_REF] Willis | Continuum Micromechanics[END_REF][START_REF] Nassar | On asymptotic elastodynamic homogenization approaches for periodic mediaJ[END_REF]. In contrast to asymptotic approaches, the elastodynamic homogenization formulation is exact and does not make any scale separation hypothesis. As such, the resulting effective continuum is nonlocal in space and time. However, its formulation is rather unfavorable for numerical implementation. Asymptotic elastodynamic homogenization strategies are in general nonexact, with their accuracy to depend on the order of expansion of the local fields versus the underlying small scale mechanical parameters. Τhe macroscopic fields together with the effective homogenized continuum behavior are obtained as a volume averaging process over a representative unit cell. A key assumption typically made is the one of scale separation that is considering long wavelengths and low frequencies for the propagating waves within the effective dynamic continuum [START_REF] Parnell | Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. I -Theory[END_REF]).

However, it has been shown that frequency has an impact on wave propagation through photonic crystals and metamaterials. Furthermore, the homogenized model of the periodic structure is not unique. Contrariwise, it depends on both the microstructural behavior it aims to describe and on the frequency of the incident waves. This interdependence is expressed in the dispersion diagram, which relates the frequency of propagating waves to the wave number.

The dispersion diagram provides information on the phase and group velocity magnitudes and on the possible existence of bandgaps for a particular microstructural design [START_REF] Guenneau | Homogenization Techniques for Periodic Structures[END_REF].

Wave propagation has been extensively analyzed within the context of artificial materials, as of their potential to create amongst others wave tuning and isolating properties, which are typically not encountered for conventional materials (Deymier, Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics

Page | 50

The dynamics of periodic discrete structures have been modeled in the literature using beam mechanics in the low frequency regime for which a homogenized continuous equivalent description can be obtained under the condition of scale separation [START_REF] Caillerie | Homogenisation of periodic trusses[END_REF][START_REF] Hans | Dynamics of discrete framed structures: a unified homogenized description[END_REF][START_REF] Rallu | High frequency reduced models for repetitive beams[END_REF]. The validity of the method is accordingly limited below the diffraction frequency limit. However, when moving to the high frequency regime for which the wavelength becomes comparable to the unit cell size as shown in Fig. 2.1, a different formulation is required which goes beyond the assumption of scale separation, which is no more applicable. In higher frequencies the periodicity of the system needs to be satisfied for the scale separation to be also valid. The high frequency behavior of composite structures has been first analyzed in (Boutin et al., 2012(Boutin et al., & 2014)). The authors employ an asymptotic multiscale method to address the modeling of large scale modulations of high frequency mechanical waves. The developed theoretical formulation has allowed for the identification of the physical behavior of the composite structure in frequency ranges that are not accessible by the methods typically used for low frequency analysis.

However, up to now, no higher order discrete mechanics methods have been presented, which can inherently account for the effect of the propagating frequency on the considered media. What is more, the potential of dynamic discrete beam mechanics formulations to capture higher frequency ranges, which are not accessible for low frequency approaches has not been studied. In this chapter, we elaborate the higher order asymptotic expansion of the inner forces and moments developed within beam elements upon dynamic loads, which we employ to compute the dispersion characteristics of one and two-dimensional architectured materials. The elaborated discrete dynamics methods (DDM) incorporates the higher order frequency of the propagating waves for the first time, thus information on the wave characteristics of the media, in its very formulation. As such it can be used to characterize both low and high frequency propagation phenomena. In Section 2.2, we compute and provide the higher order, frequencydependent asymptotic expansion forms for the normal, transverse forces and bending moments, making use of the dynamic equilibrium equations and of the beam constitutive law. We subsequently calculate the dispersion characteristics of one-dimensional periodic structures, providing insights into the frequency scale-separation domains and in the effects of the frequency magnitude on the dispersion diagram characteristics (Section 2.3). We extend the analysis to two-dimensional architectured materials space, computing the band diagram structure of a square lattice, which we compare with the ones obtained by the Floquet-Bloch theorem predictions (Section 2.4). We comment on the main findings of the current work in Section 2.5 and conclude in Section 2.6.

Structural dynamics of discrete and continuous systems

Dispersion characteristics of discrete systems based on the asymptotic description of their inner forces and moments

In the current section, we express the forces and moments acting on beam element structure under dynamic loads. Each element is considered to consist of two mass-less end nodes, as schematically depicted in Fig. 2.2a. The kinematics of each beam are defined by the two nodal displacement components and by the node rotation (the rotation axis being perpendicular to the element plane, Fig. 2.2a).

Figure 2.2 : Forces and moments acting at the end nodes of each beam element (a). A single unit-cell structure in a one dimensional domain without transverse forces and moments (a) and its two neighboring cell representation (c).

The elements have density  , length o l , area A and elastic modulus E . Their geometric and mechanical parameters are summarized in Table 2.1.

      (2.1)
Using the definitions of Eq. (2.1), we obtain through their asymptotic, frequency-based expansion as elaborated in Appendix 1.A, the following expressions for the forces , NT and moment M acting on each element (O represents the starting node and E represent the extremity node, and the beam is oriented from the left to the right):

        24 5 00 1 2 1 2 1 2 1 2 1 2 0 0 1 2 2 1 ( , ) ( , ) - - (2 ) - ( 8
7 ) ( ) 6 ( , ) ( , ) O E ll EA N u u N u u u u u u u u o l la N u u N u u             (2.2) 4 4 0 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 8 8 0 10 0 1 2 1 2 0 1 2 1 2 2 1 2 1 ( ) ( ) ( -) ( ) - ( 26
9 ) - ( 22
13 ) - 12 2 ( , , , ) ( , , , ) ( ) . () ( ) - (65. -61. ) (( ) ) 240 ( , , , ) (-,-, , ) O E l l l l v v v v EI e f T v v T v v l ll l gv hv o l m T v v T v v                                         (2.3)         4 4 0 0 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 8 8 10 0 0 1 2 0 0 1 2 0 1 2 1 2 2 1 2 1 ( ) ( ) ( -) (2 ) - 22 13 - 4 -3 6 ( , , , ) ( , , , ) 3 -( ) -( ) -. (( ) )) ( , , , ) ( , ,-,-) O E l l l l v v v v EI M v v M v v n n l l pv qv l l r s o l M v v M v v                                      (2.4)
In Eqs. (2.2-2.4), the coefficients a-s stand for numerical constants arising from the asymptotic expansion of the forces and moments, which are summarized in Appendix 1.A.

The variables  and  stand accordingly for the compression and bending parameters defined as

  1/2 2 / wE   and   1/4 2 / Aw EI  
, w being the frequency of the propagating waves.

The asymptotic expansion of the transverse forces and moments in Eqs. (2.3) and (2.4) is carried out up to the 8 th order, while the corresponding normal force expression up to the 4 th order (which correspond to the same order of frequency. For the propagating waves to be characterized, the force and moment equilibrium needs to be formed and its non-trivial solution with respect to the frequency needs to be obtained. In Sections 2.3 and 2.4, we elaborate on the dispersion attributes of one-dimensional and two-dimensional systems.

Dispersion characteristics of 1D media based on Cauchy mechanics

The dispersion relation for an effective Cauchy continuum constitutes a first order approximation of the effective behavior of the medium. In the one-dimensional space the dynamic equilibrium equation along with the materials constitutive behavior are described as follows:

  22 22 , ( , ) ( , ) ( , )
.

d x t u x t u x t u x t dx x E t         (2.5)
Where in Eq. (2.5), E stands for the material's Young's modulus and  for its density.

Inserting the harmonic plane wave solution (

() i wt k n n u Ue  
) in Eq. (2.5), we obtain the dispersion relation (w-k) in non-normalized and normalized form (

2 0 / w E l   ) accordingly,
as follows:

2 0 / , / w E E w E k k kl wl      (2.6)

Dispersion Diagrams for 1D periodic media

In the current section, we analyze the dispersion behavior of one-dimensional periodic media that are simulated as discrete systems (Section 2.2.1). Thereupon, we discuss the limits of validity of the homogenized Cauchy continuum consideration, while we address the issue of scale separation and frequency dependence of the solution. We study, one-cell, two-cell and three-cell systems.

One-Cell periodic system Ω1

In this section we analyze the case of 1D systems described by a sole unit-cell. The cell is described as a beam of length 0 ll  , with a horizontal displacement u . Since, we solely analyze its one-dimensional behavior, the media acts in traction and compression, so that the sole force resultant acting is the longitudinal force N defined by Eq. (2.2). We next write the equilibrium of an internal node n, by considering a set of two neighboring cells as shown in Fig. 2.2c. Using Eq. (2.2) up to its fourth order expansion, we derive the equilibrium of the n node as the following expression for the normalized frequency 0 / X w w  :

4 1 -1 -1 1 2 -1 1 - 1
OE n n n n n n n n n n n n n N u u N u u u u u X u u u X u u u                   (2.7)
Where in Eq. (2.7)   Xwincludes the frequency of the propagating waves, while n defines the index of the nodes along the structure (Fig. 2.2c). Inserting the harmonic plane wave solution 0 0 0

( ( 1
) ) ( ) ( ( 1)
)

11 ,,

i wt k n l i wt k n l i wt k n l n n n u Ue u Ue u Ue               
, Eq. (2.7) simplifies into the following dispersion relation of order O(w 4 ):

42 0 0 0 7 15 1 cos( ) 15 cos( ) 45 45 cos( ) 0 82 X kl X kl kl                    (2.8)
The solution of Eq. (2.8) leads to two identical real solutions (double root) upon which the dispersion relation of the structure is computed. In Fig. 2.3 we depict the root of Eq. (2.8) along with the one predicted for the homogenized Cauchy continuum in Eq. (2.6).

(-( 1) ) 1 0, i k n l n n w u U e U u        (2.9)
This low frequency domain corresponds to low wave numbers (Fig. 2.3) and it is nondispersive. Therefore, two successive nodes admit identical displacements, so that the periodicity and the separation of scales in the (w-k) space are satisfied. In this range, the Cauchy predictions are sufficient to describe the wave propagation characteristics of the media. Beyond this limit, thus at higher frequencies, the Cauchy medium is no longer sufficient to describe the effective behavior of the system, as the latter becomes dispersive. We subsequently determine the scale separation condition in terms of both the frequency w and wavenumber k, based on the dispersion relation of Fig. 2.3.

Accounting for the effect of the wave frequency on the periodicity attributes of the medium

We extend the previous analysis which neglects the propagating wave frequency, beyond the non-dispersive region of the dispersion diagram of Fig.

Re(

) Re( ( ) )

n i w t i i w t n n u U e U e i          (2.10)
Eq. (2.10) can be simplified discerning two cases for the order of the exponent n. In particular:

if 2 nq 
(even integer) then the displacement at the corresponding node is 2 0

( 1) cos( 3) ( 1) sin( 3)

q q u U w t    
q q u U w t     
As a result, when the frequency   0 3 ww  of the propagating wave is taken into consideration, the assumption of the displacement periodicity on the nodes is lost. A total of four successive cells is required for the displacements of the initial node to be recovered, thus the periodicity now holds for a set of four unit cells:

10 sin( 3 ) u U w t  , 20 cos( 3 ) u U w t    , 30 sin( 3 ) u U w t    , 40 cos( 3 ) u U w t  (2.11)
The displacements of the successive nodes defined in Eq. (2.11) are shown in Fig. 2.4, while the displacement of the 5 th node, which is identical to the one of the first node is provided in the middle. , the displacement of the 5 th node is identical to that of the first node. As a result, the periodicity condition at the limit of the non-dispersive region is four cells -rather than one cell-and cannot be satisfied for two neighboring nodes. The same applies to different wave number values with a non-negligible w. Indicatively, for a wave number double the one previously considered, that is for 0 kl   , corresponding to a frequency 0 23 ww  (Eq. (2.8)), the following set of nodal displacements for the two unit cells system are obtained:

-0

Re( ) Re( ) (-1) (-1) cos( 23

) iwt i n iwt n n n u U e U e U w t           (2.12)
Eq. (2.12) corresponds to a two-cell periodicity of the nodal displacement components. The previous analysis highlights the fact that for a Cauchy medium periodicity is not satisfied for any wave number and frequency that are away from the origin of the dispersion curve of Fig.

2.3.

Two-cell systems Ω2

We subsequently consider a system comprised of two equal unit-cells (Fig. 2.5a), which contains an internal node v in addition to the two extremity nodes (Fig. 2.5a). The length of the period becomes accordingly 0 2 ll , while the system's minimum repetition consists of a total of 4 unit-cells (Fig. 2.5b).

Figure 2.5 : A basic system of two-cells with an internal node v considered as the basic material unit (a). Two adjacent material units comprising a total of 4 unit-cells (b).

We initially express the displacement of the internal node of Fig. 2.5a as a function of its end nodes , OE uu. To that scope we write the balance of forces for the internal node v of Fig.

2.5a.

For the equilibrium, we make use of the second order approximation of the normal forces of Eq. (2.2), upon which we obtain:

              22 2 2 2 ( , ) ( , ) 0 66 3 -3 0 42 4 3 - OE EO OE EO N v u N u v u X u X X v X u u v X               (2.13)
Eq. (2.13) can be used to analyze the limiting states of the two-cell system kinematics. In particular, the internal displacement noted v tends to infinity for the resonance wave frequency 0 3 ww . For lower wave frequency values w, the motion of the internal mass v is in the same direction as to the motion of the reference nodes (variables O u and E u ), as the denominator of Eq. (2.13) is positive. For 0 3 ww  the internal node is out of phase with the respect to the motion of the extremity nodes of the two-cell system. Eq. (2.13) can be used to write the longitudinal forces of the extremity nodes of the two-cell system of Fig.

          4 2 4 2 2 0 12 36 -7 60 -36 ( , ) -( , ) 24 3 - OE E EE u X X u X X N v u N u v E A lX          (2.14)           4 2 4 2 2 0 7 -60 36 --12 -36 ( , ) 24 3 -
OE O O u X X u X X N u v EA lX    (2.15)
Eqs. (2.14) and (2.15) can be subsequently used to compute the equilibrium at the node n of two adjacent material units (Fig. 2.5b), each of them comprised of two basic unit-cell systems, as follows:

4 4 2 2 2 1 -1 4 2 1 60 36 6 18 ( , ) -( , ) 0 - - - - 7 7 14 7 7 6 18 - - - 0 14 7 7 OE n n n n n X N u v N v u u X X u X X uX                 (2.16)
Inserting the harmonic plane wave solution 0 (

,

i wt k n l n u Ue        0 ( 2 ) i wt k n l n u Ue      0 ( ( 1) 2 ) 1 i wt k n l n u Ue       
into Eq. (2.16) allows to compute the dispersion characteristics of the two-cell system, as the solution of the following equation:

      42 0 0 0 7 -cos(2 ) -60 -12cos(2 ) -36 cos(2 ) -1 0 kl X kl X kl  (2.17)
The latter are presented in Fig. We subsequently analyze the periodicity of the system in high wavelength region where 0 0 kl  . By writing the continuity equations between the neighboring cells (Eq. (2.16)), without replacing the displacement of the internal nodes by their values in terms of the extremity nodes. Thereupon, using the equilibrium of the internal node (Eq. (2.13)), we formulate the following system of equations:

  21 22 22 22 ( , ) ( , ) 0 ( , ) -( , ) 0 -1- 1- 0 63 . -12 0 0 1- -1- 36 OE EO OE nn N v u u N u u v N u u v v N v v u u XX u XX v XX                                     (2.18)
Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics

Page | 61

For the first solution ( 0 w  ) of Eq. (2.18), the periodicity of neighboring nodes is satisfied, as we have shown in Eq. (2.9). At higher frequencies, the second solution of Eq. (2.18). This indicates that the displacement of the internal nodes is in phase opposition with the displacement of the external nodes. Therefore, the periodicity is lost on Ω1, but is kept on Ω2, so that the separation of scales is satisfied over the period Ω2 in the high wavelength region   0 0 kl  . As a result, moving to the two-cells system extends the range of validity of the separation of scales to higher frequencies. In the sequel, we consider the three-cell period Ω3 to analyze whether the choice of a larger period allows to further extend the range of frequencies for which scales can be separated.

Three-cells system Ω3

We subsequently consider the period Ω3 composed of three cells of identical length, the driving external mode kinematic variables are O u and E u for the principal cell with internal variables 1 v and 2 v as schematically depicted in Fig. 2.7a:

Figure 2.7 : Three-cells system Ω3 with two internal nodes 1 v and 2 v considered as the basic material unit (a). Two adjacent material units comprised of a total of 6 unit-cells (b).

We first express the displacement of the two internal nodes, writing the equilibrium equations for the nodes 1 v and 2 v of Fig. 2.7a. Inserting the expression of the dynamic longitudinal forces defined in Eq. (2.2) up to the second order, we obtain:

                2 2 2 1 2 1 1 2 0 2 2 2 2 1 2 1 2 0 . ( , ) ( , ) 0 - 6 4 -12 6 0 6. ( , ) ( , ) 0 - 6 6 4 -12 0 6 D O F E EA N v v N v u u X v X v X l EA N v v N v u u X v X v X l                          (2.19)
Solving the system of Eqs. (2.19) allows us to express the internal variables versus the driving external variables, namely the displacements of the extremity nodes E u and O u , as follows:

              2 2 2 2 2 2 1 2 2 2 2 2 3 -4 6 -3 - - 6 6 -4 12 42 , 6 6 15 -6 - 15 ( -6)( -) 5 5 O E O E X X u X u X u X u X v v X X X X                                 (2.20)
The frequencies (2.20) along with the equilibrium equation of the common node between the two cells leads to the following dispersion relation:

      6
n n n n n N u v N u v u X X X u X X u X X X                (2.21)
After inserting the harmonic planar wave Ansatz into Eq. ( 21), we obtain a solution of order 6 w (three real roots), which is defined in a normalized X form as follows:

        6 4 2 0 0 0 0 52 2 cos(3 ) - 684 36 cos(3 ) 1728 216 cos(3 ) -432 1 cos(3 ) 0 X kl X kl X kl kl       (2.22)
The solution of Eq. ). The first branch is the acoustic branch passing through the origin while the second and third branches are optical branches, which can be no longer described by Cauchy continuum mechanics. Note that the group velocity changes its sign when moving across the transition points of Fig. 2.8.

In the high wavelength region ( c L , Fig. 2.1) for which 0 0 kl  , periodicity is obtained for a single unit cell, thus already in the Ω1 analysis space (Section 2.3.1), so that two neighboring nodes have the same displacements. In the non-zero frequency region   0 X  , the displacements of the internal variables are expressed versus the displacements of the driving variables 1 n u  and n u , as follows:

              2 2 2 2 2 2 1 1 1 2 2 2 2 2 3 -4 6 -3 - - 6 6 -4 12 42 , 6 6 15
-6 -15 ( -6)( -) 5 5

n n n n n n X X u X u X u X u X v v X X X X                                   (2.23)
By developing these expressions at low frequencies ( 0 X  ), we obtain:

    1 1 2 1 11 2 , 2 33 n n n n n n n n v u u u v u u u        (2.24)
Thus, at low frequencies, the periodicity for a three-cell system is maintained, as a consequence of the one-cell periodicity in this frequency domain. At higher frequencies, taking the resonance frequency 0 6 ww  based on the dispersion relation (2.22), the displacements are unbounded according to Eq. (2.23). In order to study the periodicity of a three-cell system, we impose identical displacements over neighboring super cells:

1 1 3 2 4 , nn u u u v v v v      
. We write the equilibrium equations for the internal nodes, and the equation of continuity between the two neighboring cells, condensed in the following system:

1 2 1 1 2 1 2 2 3 1 2 2 2 2 2 2 1 2 2 2 2 ( , ) ( , ) 0 ( , ) ( , ) 0 0 ( , ) ( , ) 0 0
-1- 2 1- -1- 6 3 6 0 -1- -1- 2 1- . 0 6 3 0 2 1- -1- -1- 3 6 6 n n n n N v v N v u u N v v N v u u N u u v N u v v X X X u X X X v v X X X                                                                         2 22 -6 0 XX  (2.25)
This leads to distinguish two cases, for 0 X  , the internal nodes have the same displacements 12 v v u  , so that the periodicity is recovered for the single unit cell system Ω1.

The frequency 0 6 ww  is a double root of Eq. (2.25), so that leads to the single equation 12 0 u v v    , which admits an infinity of solutions represented by a basis of two vectors, so that the periodicity is not satisfied on Ω1, neither in Ω2 (since 1 vu  and 2 vu  ) but instead it is satisfied in Ω3. Therefore the separation of scales is now fulfilled on Ω3. As a result, in the wavenumber zone which fulfills the condition 0 0 kl  , we shall distinguish different behaviors depending on the frequency of interest, which corresponds to the three branches of the dispersion diagram (Fig. 2.8). We subsequently extend the discrete dynamical analysis to the two-dimensional situation.

Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics Page | 65

Dynamical response of a 2D network material

In the current section, we compute the dispersion characteristics of two-dimensional media subjected to an incident horizontal wave with wave vector 0 ( ) cos( ) / k w n l  using the discrete force and moment asymptotic expansions of Section 2.2. To that scope, we employ a square lattice with identical horizontal and vertical elements as depicted in Fig. 2.9a. Each beam element is assigned a length 0 ll  , density  and modulus E, while the beam thickness is set to 0 0.1 tl  . Each node has three degrees of freedom, namely two in-plane displacement components (u, v) and a rotation , as elaborated in Section 2.3. In order to study the dispersion properties of the periodic 2D network, we need to consider all neighboring elements attached to the lattice's principal node, as depicted in Fig. 2.9b (in black). For the square lattice in Fig. 2.9a, a total of 4 elements are attached to this central node (Fig. 2.9b).

Figure 2.9 : Square lattice along with its basic elements (dark grey) in each material direction (a). The principal node 1, along with the four elements required for the description of its discrete equilibrium (b). The equilibrium of the normal, shear and moments for the principal node 1 is pictured in (c).

In order to compute the systems dispersion characteristics, we write the horizontal and vertical force equilibrium equations for the principal node 1 (Fig. 2.9b), as well as the moment equilibrium for the same node. The corresponding forces and moments are schematically depicted in Fig. 2.9c. We number the displacement components of each node   ,, i i i uv , as depicted in Fig. 2.9b, upon which we derive:

1 2 1 4 1 3 1 3 1 5 5 1 ( , ) ( , ) -(-,-, , ) -(-,-, , ) 0 N u u N u u T u u T u u      (2.26) 1 3 1 5 1 4 4 1 1 2 1 2 ( , ) ( , ) -(-,-, , ) ( , , , ) 0 N v v N v v T v v T v v        (2.27) 1 2 1 2 1 3 1 3 1 4 1 4 1 5 1 5 ( , , , ) (--, , ) -( , ,-,-) -(--,-,-) 0 M v v M u u M v v M u u          (2.28)
The

      1 0 2 0 -( - - )
, , , ,

xy ll n n n i t n k n k u v u v e   
, allows us to reduce the set of equations to 3 making use of the transformation matrix T (Phani et al., 2006) as follows:

    1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 1 1 1 -. . -. . -. . -. . -. . -. . -. . -. . -. . -. . -. . -. . . , , 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 y y x x y y x x y y x x T I k l I k l I k l I k l I k l I k l I k l I k l T I k l I k l I k l I k l q T q q u v u v u v u v u v q u v
                  (2.29)
The transformed 3x3 matrix with unknown displacement components q of the principal node 1 (Fig. 2.9b) is used to compute the dispersion characteristics of the lattice structure (Fig.

2.9a

), setting its determinant to zero. A total of three roots arise, equal to the number of degrees of freedom of the principal node, corresponding to the shear, longitudinal and torsional mode.

For comparison purposes, we evaluate the dispersion diagram using the Bloch's theorem (Phani et al., 2006), using the basic unit cell depicted in Fig. 2.9b. Given the three degrees of freedom per node, a total of 15 degrees of freedom (named q ) are necessary for the construction of the unit-cell's total mass M and stiffness K matrices (Fig. 2.9b) as in the discrete case, with relevant details for the stiffness and mass matrix assembly process provided in

  2 -0 K w M q  (2.30)   2 -0 T T K w M Tq  (2.31)
Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics

Page | 67

We can reduce the set of equations defined in (2.30), using the periodicity transformation matrix T of Eq. (2.29) to a system of order 3 equal to the number of degrees of freedom of the primal node (1).

In Fig. 2.10a, we provide the dispersion characteristics of the square lattice using a low order approximation of the forces and moments entering Eqs. (2.2-2.4) in Eqs. (2.26-2.28).

More specifically, in Fig. 2.10a, the normal force is expanded up to the 2 nd order, while the shear force and moments up to the 4th order (which correspond to the same order of frequency).

Contrariwise, in Fig. 2.10b, we provide the result upon a higher order expansion of the forces and moments (4th order for the normal force and 8 th order for the shear forces and moments).

The Bloch theorem results are depicted in Figs. 2.10a and 2.10b with dashed lines.

.9b using the dynamic expression of the inner forces and moments up to the fourth order for the inner shear and moments along with the Bloch theorem solution (dashed lines) (a). Dispersion curves of same lattice for a higher order expansion (8 th order) of the inner shear and moments (continuous lines) along with the Bloch theorem solution (dashed lines) (b).

Fig. 2.10 suggests that the effect of the order of the asymptotic expansion is more prominent for the accuracy of the higher modes and more specifically for the torsional mode.

In particular, while the Bloch theorem formulation predicts a normalized frequency value of ). However, using its higher, height order formulation a value of 1.19 is obtained, which differs less than 5% higher than the Bloch solution predictions. The same tendency is observed on the Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics Page | 68 longitudinal mode, where the Bloch theorem solution gives a normalized frequency value of 0.6, while the fourth order DDM solution overestimates its value to 0.7, a difference that becomes practically negligible for the case where higher order expansions are used, as in Fig. 10b. The wave number of interest accordingly affects the accuracy of the discrete dynamics solution, so that for low wavelengths 0 kl  , the low order expansion of the inner dynamic forces and moments suffices to accurately capture the Bloch results for the first two modes (shear and longitudinal mode of Fig. 2.10a). However, for the higher torsional modes, differences which are up to 15% with respect to the Bloch wave solution are obtained using the higher order expansion of the lattices' inner forces and moments, with the actual difference value to depend on the propagating wave number.

Discussion

The analysis of Section 2.3, indicates that the acoustic, low frequency branch of onedimensional periodic systems can be well captured, using a single nodal dynamic equilibrium of the structural system. In the low frequency range, the beam discrete dynamics method yields dispersion attributes which can be well compared to the ones computed upon a simplified effective Cauchy medium formulation (Fig. 2.3). In such a space, no additional information is obtained for the system dynamics with respect to classical approaches. However, when multiscale equilibria are used, considering internal dynamic degrees of freedom for the system analyzed, higher frequency optical branches can be captured (Figs.

Table 2.2 : Frequency values that satisfy the separation of scales as a function of the number of unit-cells considered within the basic period of the structural system.

Period

Scale separation frequency values

0 w X w     One unit cell Ω1 0 X  Two unit cells Ω2 0 X  , 23 X  Three unit cells Ω3 0 X  , , 6 X  
The discrete dynamics asymptotic expansion method has been further employed to compute the band diagram structure of architectured square lattice materials. It has been shown that the low order (up to the fourth order, Appendix 1.A) dynamic asymptotic expansion of the lattice's inner forces and moments well suffices to capture only the low frequency modes (shear and longitudinal mode) of the structure (Fig. 2.10a). However, for its higher modes to be sufficiently represented, a higher order expansion of its internal dynamic shear forces and moments is required (Section 2.4). In particular, the discrete dynamics results of Fig. 2.10b suggest that the difference of the Bloch results and the discrete beam dynamics computations differ by at most 15% for all wave numbers for the highest torsional mode when an 8 th order approximation of the structure's internal forces and moments is used; a value that becomes almost an order of magnitude higher for low order asymptotic approximations of the structures' inner dynamics (Fig. 2.10a). Note however that the shear and longitudinal modes converge to the Bloch results in the long wavelength regime

    00 0 arccos / 2 kl kl    
, already for a low order dynamic expansion of the structures' inner forces and moments (Fig. 2.10a).

Conclusion

Overall, in the current chapter, a higher order, discrete dynamics method has been elaborated to compute the dispersion attributes of periodic structures. The proposed methodology has been based on the higher order dynamic asymptotic expansion of the structure's inner forces and moments. Thereupon, analytical expressions for the normal, shear and bending moments developed within beam element structures have been provided, as a function of the propagating frequency. The methodology has been employed to compute the Chapter 2: Dispersion characteristics of periodic structural systems using higher order beam element dynamics Page | 70 dispersion characteristics of one and two-dimensional periodic media. It has been shown that within the one-dimensional space, the discrete dynamics approach can capture, not only the acoustic branch, but also higher, optical frequency branches of the dispersion diagram, which are not accessible for methods restrained to low frequencies, such as Cauchy media formulations. What is more, the method inherently describes the dependence of the wavelength on the propagating frequency (w(k)), separating the frequency regions for which optical, rather than acoustic branches are expected. In the two-dimensional space, it has been demonstrated that the eighth order asymptotic expansion of the system dynamics yields high accuracy solutions for the lower order modes of architectured lattice materials, with the discrete dynamics dispersion predictions to deteriorate for the higher order modes of the Floquet-Bloch spectrum. This work can provide the basis for the dynamic, discrete element-based computation of the dispersion characteristics of periodic material structures and constitutes a reference in the development of enhanced formulations of higher accuracy within and beyond the upper limits of the Floquet-Bloch theorem of two and three-dimensional media.

Chapter 3 On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects In the current chapter, we develop a higher gradient dynamic homogenization method with micro-inertia effects. To that scope, we compute the macroscopic constitutive parameters up to the second gradient, using two distinct approaches, namely the Hamilton's principle and the total internal energy formulation. Thereupon, we analyze the sensitivity of the second gradient constitutive terms on the inner material and geometric parameters for the case of composite materials with a periodic, layered microstructure. The results suggest that the significance of the second gradient terms highly depend on the differences in the geometric and material properties of the underlying microstructural phases, as well as on the deformation mode of interest. What is more, the wave propagation study indicates that different higher gradient results are obtained, depending on the formulation used and on the wavenumber range of interest. In particular, Hamilton based, second gradient models perform poorly in the low wavenumber range, compared to first gradient or to analytic, exact solutions. Contrariwise, second gradient macroscopic constitutive formulations derived using the total higher gradient internal energy outperform first gradient approaches, better approximating the wave propagation characteristics of the effective medium, as the corresponding comparison in between the frequency diagrams and phase and group velocities suggest.

Introduction

It is experimentally proven that materials with a heterogeneous microstructure exhibit a dispersive wave propagation behavior, that is, each wavenumber travels with a different phase velocity (Erofeyev, 2003). However, homogenization models based on the classical, Cauchytype elasticity theory are not able to reproduce different aspects of the physical behavior of heterogeneous media, such as wave dispersion.

The impact of the heterogeneous microstructure on the wave propagation characteristics of materials was first recognized and analyzed by Mindlin (1964) In order to circumvent this drawback, multiscale methods have been developed to link the dispersive aspects of wave propagation to the microstructure of the material. Amongst these, homogenization techniques derive the dynamical macroscopic behavior by upscaling the microscopic one, so obtaining a more coarse description of the dispersive wave propagation features, but still reflecting in a predictive manner the impact of the underlying material microstructure.

Amongst the employed micromechanical methods, asymptotic homogenization is quite Since the aforementioned approaches restrict to the low frequency range, they cannot describe the high frequency features encountered for microstructured periodic materials, like the presence of band gaps or negative refraction. High frequency behaviors were investigated in both discrete materials and composites with a high contrast of their elastic properties using asymptotic expansion methods (Andrianov et al., 2008(Andrianov et al., , 2011;;Craster et al., 2010;Zhikov (2000), Smyshlyaev (2009), Auriault and Boutin, 2012). First strain gradient formulations such as the ones elaborated in Mindlin (Mindlin 1964[START_REF] Mindlin | Second gradient of strain and surface-tension in linear elasticity[END_REF] have further interest due to the possibility of associating them to higher order inertia terms, where the kinetic energy depends on the velocity gradient. These effects allow to better describe strain singularities at sharp crack tips and to capture size effects within the dynamic material behavior (dispersion relations) manifested by real materials such as porous materials, polymer foams (see Mindlin, 1964;

Askes and Aifantis,2011). In the same spirit, a continuum model of Cauchy type including micro-inertia terms in the kinetic and potential energy was proposed in Wang and Sun (2002) and [START_REF] Sun | Modeling Heterogeneous Media with Microstructures of Different Scales[END_REF] with applications to the computation of the dispersion diagram for laminates which is compared with the exact solution for laminated media (Sun et al., 1968) obtained by the elasticity theory. The exact solution has been proved experimentally in (Robinson and al., 1974), where the existence of propagating bands and gaps in layered media was experimentally demonstrated.

A homogenization method for the analysis of wave propagation in materials with a periodic microstructure has been developed in [START_REF] Bacigalupo | Computational two-scale homogenization of periodic masonry: characteristic lengths and dispersive waves[END_REF]. The method is based on suitable micro-fluctuation functions which represent the material microstructure, incorporated in a variational-asymptotic approach. This approach has been recently extended in (Bacigalupo and Gambarotta, 2014a,b) to a higher order asymptotic expansion of the inner inertia terms, based on the mathematical convergence of the asymptotic solution (Smyshlyaev and Cherednichenko, 2000), though with first order macroscopic constitutive parameters.

However, up to now, the significance of higher order macroscopic terms in the static and wave propagation analysis of microstructured media with micro-inertia effects has not been analyzed. As a result, the corresponding macroscopic higher gradient internal length parameters have not been evaluated, while their effect on the wave propagation attributes of microstructured media remains unknown.

In this chapter, we develop a dynamic second gradient constitutive formulation for the analysis of the static and dynamic properties of heterogeneous media with micro-inertia effects.

To that scope, we make use of the Hill-Mandel principle to relate macroscopic to microscopic parameters (Section 3.2), which we use as a basis for the derivation of the macroscopic equation of motion using separately the Hamilton's principle (Section 3.2.2) and the higher gradient description of the media's total internal energy (Section 3.2.3). We subsequently identify the micromechanical constants relating micro-scale attributes with macro-scale mechanics in the case of longitudinal and transverse deformations of layered two-phase composite materials (Sections 3.3.1). Thereafter, we analyze the significance of the higher gradient constitutive parameters as a function of the geometric and material properties of the underlying microstructure in the static case (Section 3.3.2). We extend our analysis in the wave propagation regime, computing phase and group velocities and comparing the Hamilton and the higher gradient energy method results with already available literature results in Sections 3.3.3.1 and 3.3.3.2 accordingly. We discuss on the obtained results as a function of the method and wave number interest region in Section 3.4 and conclude in Section 3.5.

Homogenization methods

Second gradient mechanics and Hill-Mandel principle

We consider a representative volume element (RVE) of a heterogeneous medium occupying a domain Ω. The RVE is assumed to be small compared to the overall dimensions of the medium. We write the dynamic equilibrium of motion, as follows:

, ij j i u   (3.1) micro-displacement.
j i ij ij ij ij ji j i ijk ij k ij k j k i k ij ij ij ijk ij k U U E d E XX U U Kd X X X X d S x                                                      (3.2)
Where the bracket notation . used in Eq. (3.2) denotes volume averages over the RVE of volume  . The micro and macro scale position vectors are denoted with labels x and X respectively, while the lower and upper scale notations u and U are used for the micro and macro displacement components. We write the micro-displacement on the domain boundary as a function of the first and of the second order strain gradient terms of Eq. (3.2), as follows (Trinh et al., 2012):

1 on 2 i j ij ijk j k u x E K x x    (3.3)
Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 81

We subsequently compute the difference between the average of the microscopic energy and macroscopic strain energy by writing the following identity resulting from (3.3):

  1 .0 2 i j ij ijk j k mi mi m u x E K x x n ds            (3.4)
Using the divergence theorem, we simplify Eq. (3.4) (the calculation is detailed in Appendix 2.A), relating macroscopic and averaged microscopic energy terms with microinertia components as follows:

1 2 ij ij ijk ijk ij ij i i ij j ijk j k E K S u u E x K x x                (3.5)
The linearity of the problem allows to introduce the localization operators relating the microscopic displacement ( , ) i u X x and strain  

, ij Xx 
to the macro-strain components, as follows: A are the first order, the coupling between the first and the second order stiffness tensors respectively, while ij  and ijk S stand accordingly for the Cauchy and for the hyperstress tensor.

        , ( ) ( ) , ( ) ( ) ( ) ( )
E K i ijk jk ijkl jkl E K ij ijkl kl ijklm klm u X x H x E X H x K X X x A x E X A x K X      (3.
In the sequel, we use Eq. (3.5) along with the Hamilton's principle to derive the dynamic equation of motion accounting for micro inertia terms.

Hamilton principle based second gradient mechanics

In order to derive the macro equation of motion we use Hamilton's principle, as follows:

  , , ii Vt Vt T P dVdt Q U dSdt      (3.8)
Where V in Eq. (3.8) stands for the volume, i Q for the traction exerted on the boundary of the volume V , and P and T are the deformation and kinetic energy densities respectively. The deformation energy density is defined as:

11 , 22 ij ij i i P T u u     (3.9)
Using Eq. (3.5) we write the expressions of deformation energy as a function of the macroscopic strains and of the micro-inertia components, as follows:

    11 22 11 , , , , , 22 ij ij ijk ijk i i ij j ijk j k P E K S P P u u E x K x x P E E K K P E E E                    (3.10)
Where ' P represents the deformation energy part related to micro-inertia terms .Accordingly, we write the kinetic energy part as a function of the macro-displacement components and mixed macroscopic displacement and microscopic fluctuation displacement term ˆi u terms contained in T  , as follows:

11 22 i i i i T u u U U T     (3.11)
Where T  in Eq. (3.11) relates to microscopic and higher gradient strain terms, as shown in the sequel. In particular, in order to derive an explicit form of T  , we decompose the local displacement in the RVE into a displacement linear in the imposed macroscopic strain 0 i U as in Eq. (3.3) and a microscopic fluctuation term ˆi u as follows :

0 0 ˆ 1 2 i i i i i ij j ijk j k u U u x u U E x K x x x          (3.12)
By substituting the displacement field of Eq. (3.12) in the kinetic energy expression of Eq.

(3.11) we get the macro and the micro part of the kinetic energy, as follows:

     0 0 0 1 1 ˆˆ2 2 i i i i i i T U U u U u u       (3.13)
The first part in the kinetic energy definition, Eq. (3.13), defines the macroscopic kinetic energy and the second part the coupling between the macroscopic and microscopic components, while the last term stands for purely microscopic energy contributions. Note that, according to the micromechanics assumption, tensors ij E and ijk K are constant over the RVE.

As a result, the average density of the kinetic energy of the region

                1 1 1 ˆˆˆ2 2 2 11 ˆˆˆˆ, 22
i i i ij j i ijk j k i i i ij j i ijk j k i i ij ijk T U U u E x u K x x u u T u E x u K x x u u T E K                 (3.14)
We next substitute into Eq. (3.8) the expression of deformation and kinetic energy to obtain the macro equation of motion:

,

1 1 1 0 2 2 2 i i ij ij ijk ijk Vt U U T E K S P dVdt                (3.15)
We simplify Eq. (3.15) using integration by parts of each component as detailed in Eqs.

(2.A.3-2.A.18) of the Appendix 2.A. Thereupon, we obtain the equation of motion which contains surface   S and volume  

V integrals as follows: 2 , , 2
2 2 2 2 ij j i ijk jk i i i j ij j ij i i i j ij j k ijk j k ij P P U V S U dV U dV U dV X E X t E T P P U dV U dV U dV X t E X X t K X X K                                                                                                                  1 2 2 3 4 4 4 4 1 4 2 5 5 i j k ijk a b a b c d d a b i i T U dV X X t K S S S S S S S S S S S U U dV                                                     (3.16)
Where the scalars i S therein are surface integrals, detailed in the Appendix 2.A.

Separating the volume integrals from the surface integrals   S in Eq. (3.16), we obtain the following macroscopic equation of motion:

,, ij j ijk kj i i S F U      (3.17)
Where in Eq. (3.17) the Cauchy and the hyperstress tensors are calculated in the static situation (following the Hamilton principle formulation [Wang and Sun, 2006]), as follows:

1 1 .. 2 2 1 1 2 2 E E E E E K K E ij krij krmn mndl rsdl rsmn mnij dl klij klmn mnrst klrst klmn mnij rst E K K E K K K K ijk
rsdl rsmn mnijk rsijk rsmn mndl dl pfijk pfmn mnrst dlrst dlmn mnijk rst

A C A A C A E A C A A C A K S A C A A C A E A C A A C A K          (3.18)
In the Hamilton formulation of Eq. (3.17), the dynamic part is contained in the effective body force term i F defined as follows (it does not depend on the constitutive law):

2 2 2 2 i j ij ij ij k ijk ijk ijk P P T P T P F X E t E t E X t K t K K                                                                                                              (3.19)
The micro-inertia force of Eq. (3. 19) is calculated as a function of the displacement and strain localization operators introduced in Eq. (3.6) detailed in the Appendix 2.A.3 as follows:

    2 1 1 1 2 2 2 2 1 1 2 2 4 E E m K E r i mn imn j mij n i j n rsk irsk j rij s k i s j k j K E p K K r rs rijk s irs j k i s j k rst rijk s t irst j k i s j k t j k F E H x H x x x K H x H x x x x x X E H x H x x x x x K H x x H x x x x x x XX                                                 (3.20)
In the sequel, we derive the dynamic equilibrium based on the total higher gradient potential energy in the presence of micro-inertia effects.

Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 85

Dynamic higher gradient energy formulation of the constitutive law

In the Dynamic Higher Gradient Energy (DHGE) formulation, we compute the effective dynamic constitutive law by deriving the expression of the internal energy. Inserting the localization tensors of Eq. (3.6) back into Eq. (3.5) enables to write the total internal energy as a function of the macro-strain parameters in Eq. (3.21), as follows:

  int 1 2. ( , ) . . 2 . . . . ij ij ijk ijk ij ij i i ij j ijk j k E E E K K E K K ijkl
W E K E K S u u E x K x x A E C A E A E C A K A K C A E A K C A K H E H E H                        1 2 1 2 E K E E jk irst rst ijk jk iq q ijk jk iqr q r K E K K K K ijkl jkl iqr qr ijkl jkl irst rst ijkl jkl iq q ijkl jkl iqr q r E H K H E E x H E K x x H K H E H K H K H K E x H K K x x                   (3.21)
The first order macro stress tensor and the second gradient tensor are obtained by deriving the expression of the energy (Eq. (3.21)) with respect to the macro-deformation of the first and of the second gradient respectively (the derivations are detailed in Appendix 2.B):

      int 2 int 2 ( , , , ) ( , , , ) 1 1 .. 2 2 2 dyn ij ij ij E E E E E K K E dl krij krmn mndl rsdl rsmn mnij rst klij klmn mnrst klrst klmn mnij E E E E E E dl qdl qij kij kdl idl j dij l W E K E K W E K E K E t E E A C A A C A K A C A A C A E H H H H H x H x                            int 2 int 2 1 2 2 ( , , , ) ( , , , ) 1 1 1 1 . 2 2 2 2 K E K E K E rst qrst qij krst kij irst j rij s t ijk ijk ijk E K K E K K K rsdl rsmn mnijk rsijk rsmn mndl dl pfijk pfmn mnrst dlrst dlmn K H H H H H x H x x W E K E K W E K E K S K t K A C A A C A E A C A A C                       . 1 . 2 2 1 1 . 2 2 2 K mnijk rst E K E K E K qdl qijk rdl rijk idl k j dijk l dl K K K K K K lrst lijk drst dijk rijk s t irst j k rst AK H H H H H x x H x E H H H H H x x H x x K                       (3.22)
We note that the dynamic part in the macroscopic constitutive expressions in Eq. (3.22) arises from the derivation of the micro-inertia terms with respect to the tensors E and K .

Considering a plane harmonic wave solution   1 () 0

I k X wt U U e   
 , the second time derivative of the strain and strain gradient are expressed as follows:

2 E E w K K        (3.23)
Where w in Eq. (3.23) is the frequency of the propagating waves. Using Eq. (3.22), we express the homogenized dynamic constitutive tensors in Eq. (3.7), as follows (Appendix 2.B):

    hom, 2 hom, 2 1 .. 2 2 1 1 2 2 2 E E E E qdl qij kij kdl dyn E E E E ijdl krij krmn mndl rsdl rsmn mnij E E idl j dij l K E K E qrst qij krst kij dyn E K K E ijrst klij klmn mnrst klrst klmn mnij K E irst j rij H H H H C A C A A C A w H x H x H H H H B A C A A C A w H x H                     hom, 2 hom, hom, 1 1 1 2 2 2 2 1 1 2 2 st E K E K qdl qijk rdl rijk dyn E K K E dyn ijkdl rsdl rsmn mnijk rsijk rsmn mndl dlijk E K idl k j dijk l dyn K K K K ijkrst pfijk pfmn mnrst dlrst dlmn mnijk xx H H H H D A C A A C A w B H x x H x A A C A A C A                               2 1 1 2 2 2 K K K K lrst lijk drst dijk K K rijk s t irst j k H H H H w H x x H x x               
(3.24)

Results

Composite laminated media

We apply the previously elaborated methodologies to the case of composite materials with a periodic, layered microstructure. To that scope, we define below the attributes of the unit-cell structure, its total length set to 1 Lm  . Fig. 3.2 summarizes the geometric parameters characterizing the unit-cell, as well as the material parameters for both two phases. The micro-displacement field is represented with superscripts for each medium and subscripts for the direction of each displacement component, as follows: , uu uu uu

            
(3.25)

Identification of micromechanical constants upon normal deformations

We subsequently determine the expressions of the localization operators of Eq. (3.6) for the case of normal macroscopic deformations. In such a deformation mode, the macro displacement field is along the X direction, so that it holds:

1 1 11 111 ( ) 0, 0 U f X E K     (3.26)
The micro-displacements follow the definition of Eq. (3.25) and (3.6), where the coefficients of the localization tensors are denoted with i a and i b for the first and second layer successively, as follows:

  .

E E x u x H H a a x a a x a K K E E x A A a a a x K K E u x H H b b x b b x b K                                                              2
EK E x K E E x A A b b b x K K                                  (3.
            11 11
( ) 2 ( ) 2 . ( ) 2 ( ) 2 .
E x x a a a x K E x x b b b x K                                         (3.28)
Where  in Eq. (3.28) defines the first Lame coefficient, equal to   for the first and second gradient macro strains. We use the interface continuity condition for the displacements, upon which we obtain two equations. Accordingly, using the boundary conditions for the displacement field, we retrieve four additional equilibrium equations, noting that first and second gradient terms ij E and ijk K can be treated independent. The body forces are null within the cell, so that the continuity of stresses and stress gradients provides us with a total of three additional equations (two and one respectively). The boundary and interface conditions are summarized as follows: Remark: note that we impose the quadratic displacement only to the RVE boundary and not in the entire volume; this entails that the average (in the volume sense) of the fluctuating displacement does vanish only on the RVE boundary but not necessarily within the whole volume. We use the condition that the average microscopic strains ij  and their gradient ()

(0) (0) ( ) , ( ) 22 (0) (0), ( 0
) (0) uu dd u d E d K u d E d K xx                      (3.
k ij  
are equal to the macroscopic strain ij E and strain gradient ijk K components respectively similarly to (Boutin, 1996), so that the fluctuation of the strain and strain gradient has zero mean values.

Note that Eq. (3.30) provides solely one additional dependent condition, as the average of micro-strains leads to a condition that is a linear combination of the ones contained in Eq.

                                             1 2 1 1 2 2 2 2 1 2 0 1 1 1 2 1 2 2 1 1 2 1 1 1 2 2 2 2 1 1 1 2 1 1 1 2 2 1 2 1 1 1 2 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 1 2 2 1 2 1 2 1 1 1 2
                                                                                                               2 2 1 2 2 22 4 1 1 2 1 2 2 1 2 1 1 2 2 1 1 1 2 0 1 1 1 2 1 2 2 1 1 2 1 2 2 2 2 2 2 1 1 1 2 1 1 1 2 2 1 2 1 1 1 2 2 2 2 2 1 1 2 1 2 2 1 1 1 2 3 , 2 2 2 2 22 2 , , 2 2 2 2 2 2 2 2 , 2 2
                                                                                       2 2 2 2 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 2 11 4 1 1 2 1 2 2 2 2 , 2 2 2 2 2 2 Ld d d d d d d dd L b dd                         
(3.31)

Identification of micromechanical constants upon transverse deformations

In the transverse deformation case, the displacement field is parallel to the direction of the layers, so that 2

1 21 211 ( ) 0, 0 U f X E K     .
Writing the micro-displacements as a function of the macro-strain in each layer through the localization operators of Eq. (3.6), we obtain:

( ) . . 2
( ) . . 2
( ) ( ) 2 EK EK E EE x u x H H a a x a a x a KK EE x u x H H b b x b b x b KK x u x u x Ax E C K H                                                 21 1211 211
.

K E H K      (3.32)
Eq. (3.32) contains a total of 12 unknowns. The micro-strains field and its gradient are computed using Eq. (3.32), as follows:

        1 1 21 1 1 1 2 1 21
1 1 , 2 2 1 ( , ) ( ) ( ) 2 EK E E uu x x x x A A K xx E a A a a x Cx K a K CK uu x x x x A A xx                                                         21
K E K E b A b b x Cx K b K CK                         (3.33)
Since the first and second gradient micro-strains in Eq. (3.33) ,

,

E E E E E E K K H H H H A A A A   (3.34)
Using Eq. (3.33), we compute the micro-stress developed in each of the inner material phases, as follows: ( , ) 2 ( , ) 2 ( , ) 2 ( , ) 2

    11
x x x x a E a K a x K AE Cx K x x x x b E b K b x K AE Cx K                       (3.35)
Where the first and second gradient macro-strain components entering Eq. ( average of the microscopic first gradient deformations and the average of the microscopic second gradient deformations to be equal to the corresponding macroscopic ones, as follows:

UU E E E XX E K K K X               (3.
(0) (0) ( ) , ( ) 22 (0) (0), (0) (0)
,,

uu dd u d E d K u d E d K xx E K K                   (3.37)
Solving the system of Eq. (3.37) using Eq. (3.33)-(3.36), we obtain the following constants i a and i b for the transverse deformation mode:

2 1 2 1 2 1 2 1 2 2 1 2 1 1 2 1 2 0 1 1 2 2 1 1 2 2 1 2 2 2 2 1 1 1 2 2 1 2 2 1 1 2 1 2 2 2 1 2 2 1 3 2 2 2 2 1 1 1 2 1 1 3 ( -)( -3 ) ( -) ((- 3 ) ( -2 )) , ( ) ( ) 2 2 2 2 , ( ) (2 )
d d d d L d d L d L d Ld a a L d d L d d d d d d d d d d Ld d L a dd d L d d d a                                                        2 1 2 1 2 2 1 2 2 1 2 1 2 1 2 1 2 4 0 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 2 2 2 1 1 1 2 2 1 2 2 2 1 1 1 2 2 1 1 2 1 2 2 2 2 2( ) 2 ( -)( -3 ) , , ( ) ( ) ( ) ((- 3 )
( -2 )) , ( )
2 2 2 d d L d dd L d d d d L a b d d L d d d d L d L d Ld b dd d d d d d d Ld d L b                                                          2 2 1 2 2 1 2 2 1 3 2 2 2 2 2 2 2 1 1 1 2 1 2 1 1 2 1 1 2 1 3 2 1 2 2 1 1 1 2 4 1 2 2 1 2 , ( ) (2 ) 2 , 2(
)

2 2 , , 0 (
)

dd dd d L d d d d L d d b dd L L d d b A C d d L                                  (3.38)

Homogenized higher gradient static effective properties

We subsequently analyze the higher gradient static effective properties of the layered media. The latter are independent of the method used for their derivation, as it can be observed by comparing the Hamilton principle expanded in section 3.2.2 (Eqs. (3.17-3.18)) and the internal energy method in section 3.2.3 (Eqs. (3.22)-(3.24)), 0 w  ).

Longitudinal mode

We compute the higher gradient constitutive properties of the layered composite of Section 3.3.1 for the case of identical elastic material properties  

1 2 1 2 ,           , namely
Lamé coefficient and shear modulus values used in the elastic law ) is recorded in Fig. 3.4a as a function of the elastic modulus ratio  . The results indicate that the effect of second gradient terms increase when the contrast in the material attributes of the microstructural phases increases.

    1,2 1,2 11 1,2 1,2 11
( ) 2

Transverse mode

For the case of transverse deformations, we calculate the homogenized constitutive tensors for equal micro-scale material properties using Eq. (3.18)

C B D L AL                                (3.41)
We depict the dependence of the coupling and second gradient stiffness term of Eq. Thereupon, the first gradient is increasingly insufficient to describe the behavior of the medium as the inner contrast of the micro-scale material properties increases, whereas the second gradient significance increases.

Wave propagation in layered, periodic media

In the current section, we use the second gradient macroscopic dynamic equation of motion with micro-inertia effects to study the wave propagation characteristics of periodic, composite layered media. For the study, we use both the Hamilton formulation of Section 3.2.2 and the higher gradient energy method of Section 3.2.3 upon longitudinal and transverse deformations.

We determine the dispersion characteristics of the propagating longitudinal and shear waves, as function of the propagating wave numbers   k . In particular, we compute the phase velocity   c , describing the velocity at which the phase of any given frequency   w travels within the Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 97 medium and the group velocity   g v , which describes the velocity of the envelope shape of the wave amplitudes. The phase and group velocity are defined as follows:

w c k  , g w v k    (3.43)
For the present study, we consider different material properties for each of the phases within the unit-cell structure of the periodic medium. The material properties of both phases are listed in Table 3.1. U stands for the displacement amplitude, k for the wave number and c for the phase velocity. The wave number is described in a normalized form kL   , while i stands for the imaginary number, such that 2 1 i  . The subscript d is used to denote the direction of propagation, taking the values one for the longitudinal and two for the transverse.

Hamilton principle up to second gradient in dynamic state

In the current section, we use the second gradient Hamilton principle formulation of Section 3.2.2 to derive the macro equation of motion and the wave propagation characteristics of the laminated media, using the material properties in we note that the coupling terms in the micro-inertia force cancel out. We substitute the expressions of the first, second order stress and micro inertia forces back into Eq. (3.17) and using the homogenized properties of the media (Eqs. (3.45-3.47)), we obtain the macroequations of motion both for the longitudinal and the transverse mode, as follows:

      2 2 3 1 1 1 1 1 1 11 11 1 1 1 1 1 2 2 3 2 2 4 1 11 1 1 1 1 11 11 1 11 1 1 11 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 2 2 4 1 2 2 E E K E d d dd dd d dd dd K E K K d dd dd d dd dd E E d dd dd F E H x H x x K H x H x x X E H x H x x K H x H x x XX E H x H x x X                                                         2
          33 2 2 2 4 2 2 1 7 4 2 7 3 5 7 6 7 H +H I-H +H +H c +H c + c 0 d d d d d d H H H H H            (3.48)
Restricting the kinematics to the first gradient, we obtain the following macro-equations of motion: We depict in Fig. 3.5 the dispersion characteristics of the laminated media using Eqs. (3.48) and (3.49) for the longitudinal mode in Fig. 3.5a and for the transverse mode in Fig. 3.5b. The second gradient Hamilton principle based results are shown with a cyan color, while the first gradient results in green. The latter have been already presented in (Wang and Sun, 2002). We further compare the results with the ones provided in [START_REF] Bacigalupo | Second-gradient homogenized model for wave propagation in heterogeneous periodic media[END_REF] depicted in yellow, and with the exact solution of (Sun et al., 1968) in black. We make use of the periodicity of the medium along the X direction to obtain a better approximation of the dispersion curves, translating the wave vector for k   by 2 . The results in Fig. 3.5 are normalized with respect to the phase velocities at zero frequency for both modes, for first gradient (green), first gradient with higher order micro-inertia effects (yellow) in [START_REF] Bacigalupo | Second-gradient homogenized model for wave propagation in heterogeneous periodic media[END_REF]) and the exact solution (red) provided by (Sun et al.,1968) (Appendix 2.C).

    3
The results of Fig. 3.5 show that the extension of Hamiltonian principle into second gradient doesn't provide an improvement with respect to first gradient results, yielding phase velocities which deviate from the exact solution to a greater extent than first order methods for wave number values close to kL   . However, all methods converge to the exact solution predictions for higher wave number values.

In the next section, we develop the dynamic higher gradient energy homogenization method (DHGE) as an alternative approach for the computation of the effective behavior of the composite medium.

Higher gradient dynamic homogenization method

In the current section we use the DHGE method developed in Section 3. -

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 hom, 2 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 22 1 2 1 1 1 1 hom,
A C A A C A H H H H C w A C A A C A H x H x C C c A C A B                              11 1 11 1 1 11 1 1 1 1 1 11 2 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 22 3 4 1 1 1 1 1 11 1 1 1 1 hom, 11 1 1 2 2 - 1 2 K E K E K E K dd dd dd dd d dd dd d d K E K E K E dd dd d d d d d d dd dd d d d d E K E d d d d d d dd dd d dyn dd H H H H A C A w H x H x x A C A A C A C C c A C A A C A D                                      1 11 1 11 1 11 2 1 1 1 11 1 1 11 1 1 1 1 1 1 1 1 1 11 22 3 4 1 11 1 1 1 11 1 1 1 1 1 11 hom, 11 11
1 11 1 1 1 1 1 2 2 2 - 1 2 1 2 K E K E K d dd dd dd dd E K K E E K dd dd d d d d d d dd dd d d d d K K E K d d d d d d dd dd d d dyn dd K dd dd d

H H H H w H x x H x A C A A C A C C c A C A A C A

A A C A                         
K K K K dd dd dd dd K K K K K dd dd d d d d d d d d d H H H H w H x x H x x A C A C C c                    (3.

51)

Remark: we checked that the dynamical second gradient moduli vanish for a homogeneous laminate.

      33 2 2 2 2 2 4 hom 2 2 1 2 7 5 6 7 7 1 (C -C c ) C +(C -C ) C - c 0 d d d d cC d           (3.54)
Restricting the kinematics to the first gradient we obtain the following macro-equation of motion: . The near zero slope applies for the second gradient homogenization method of Section 3.2.3, while all other approaches lead to a more prominent discontinuity point for a normalized wavenumber value of kL   . The continuous frequency plots at kL   require a zero group velocity at the same point, as indicated by the black curves of Fig. 3.8c and 3.8d.

      33 2 2 2 hom 2 2 1 2 7 7 1 (C -C c ) C - c 0 dd C d        (3.
The dynamic, second gradient constitutive formulation is closer to the minimum group velocity for both the longitudinal and the transverse mode, with first gradient formulations to predict notably higher values (Figs. 3.8c, 3.8d).

Discussion

The results of Section 3.3 suggest that the significance of higher gradient constitutive terms in the presence of micro-inertia effects depends both on the geometric and on the material properties of the underlying microstructure. In particular, the coupling and second gradient terms increase upon increasing difference of the relative fraction ( 12 / dd ) of the phases Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 106 contained within the unit-cell architecture, as shown in Fig. 3.3. What is more, the evolution of the second gradient characteristic lengths indicate that higher order effects become more prominent upon increasing differences in the moduli of the phases contained within the unit cell ( ) for the longitudinal mode (Fig. 3.4a) and for the transverse mode (Fig. 3.4b), with the most prominent changes to take place upon geometric rather than material differences in the underlying microstructural phases.

The second gradient effects depend, not only on the geometric and material parameters of the underlying unit-cell architecture, but also on the propagating wave-number of interest.

This can be observed in Fig. 3.7, where the addition of second gradient effects improves the consistency of the computed results (red curve in Fig. 3.7a) for wavenumbers close to kL   with respect to the Floquet-Bloch, elasticity-based solution. The differences observed for moderate wavenumbers vanish in the higher wavenumber regime of 2 kL   , where all formulations yield the same predictions with the one provided by the elasticity solution (Appendix 2.C).

The higher gradient constitutive modelling results are considerably sensitive to the formulation used for their derivation. In particular, using the Hamilton's principle which involves the static Cauchy and hyperstress tensors (the second gradient formulation in Section 3.2.2) leads to phase velocity predictions which substantially differ from the ones predicted by the elasticity-based, Floquet-Bloch results or from first gradient constitutive modelling approximations in the wavenumber region close to kL   , both for the longitudinal (Fig. 3.5a) and for the transverse mode (Fig. 3.5b). The recorded differences can be primarily attributed to the fact that the dynamic, micro-inertia terms of Eq. The dynamic, second gradient constitutive modeling results obtained based on the Hamilton's principle in Section 3.2.2, fundamentally differ from the ones obtained with the use of the dynamic higher gradient energy method of Section 3.2.3, for which the Hill-Mandel expression is directly used to compute the total internal energy stored in the medium (Section 3.2.3). Thereupon, the derived macro-stress and hyperstress are frequency dependent (Section 3.2.3) and not static as in the Hamilton principle based derivations (Section 3.2.2). In the energy based method, the micro-inertia effects related to each of the constitutive tensors can be directly assessed, contrary to the Hamilton formulation, where the constitutive terms are static and not dynamic, with the micro-inertia effects to be assimilated in the i F term of Eq.

(3.19).

The DHGE method yields high consistency results throughout the entire wavenumber range, as it can be observed from the dispersion curves of Fig. 3.7 and from the frequency and group velocity curves of Fig. 3.8. More specifically, first order gradient based methods lead to an inflection point at kL   (Fig. 3.7), a discontinuity which is not present for the Floquet-Bloch solution or nearly disappears for the higher gradient based homogenization method results in Fig. 3.7 (red). For normalized wavenumbers equal to kL   , the frequency does not depend on position -as the Floquet-Bloch results suggest -, so that a transition from planar to standing waves takes place (frequency depends only on time and not on position). This is expressed in the frequency plots of Fig. 3.8a and 3.8b, where the first gradient constitutive results (yellow and blue curves) do not provide continuous frequencies, while their corresponding group velocities are far from zero. Contrariwise, the second gradient formulation elaborated in Section 3.2.3 yields a nearly continuous frequency curve slope, with a group velocity value that approaches zero at the maximum frequency point (Fig. 3.8c, 3.8d).

Conclusion

Overall, in the current chapter, a dynamic higher gradient constitutive model with microinertia effects has been developed for the computation of the static and dynamic attributes of heterogeneous materials with an inner microstructure. The method has been used to compute the macroscopic static and dynamic second gradient constitutive parameters of two-phase composite layered materials, using both the Hamilton's principle and the total internal higher gradient energy. The sensitivity of the second gradient constitutive parameters has been analyzed as a function of the inner material and geometric microstructural parameters for the case of layered two-phase composite structure. It has been noted that the significance of second gradient terms increases, upon increasing differences in the volumetric ratio and in the material properties of the microstructural phases for both longitudinal and transverse deformation patterns. What is more, evidence has been provided that the incorporation of higher gradient macroscopic constitutive parameters can well describe the wave propagation characteristics of the micro-structured media when formulated using the Hill-Mandel equality in the context of its total, dynamic higher gradient deformation energy. The latter yields high accuracy results

for the propagating phase and group velocities over the entire wave-number space, outperforming first order constitutive formulations.

Chapter 4: Higher-gradient and micro-inertia contributions on the mechanical response of composite beam structures

Summary

In the current work, we study the role of higher-order and micro-inertia contributions on the mechanical behavior of composite structures. To that scope, we compute the complete set of the effective static and dynamic properties of composite beam structures using a higher-order dynamic homogenization method which incorporates micro-inertia effects. We consider different inner composite element designs, with material constituents that are of relevance for current engineering practice. Thereupon, we compute the effective static longitudinal highergradient response, quantifying the relative difference with respect to the commonly employed, Cauchy-mechanics formulation. We observe that within the static analysis range, higher-order effects require high internal length values and highly non-linear strain profile distributions for non-negligible higher-order effects to appear. We subsequently analyze the longitudinal, higher-gradient eigenfrequency properties of composite structural members, accounting for the role of micro-inertia contributions. Thereupon, we derive analytical expressions that relate the composite material's effective constitutive parameters with its macroscale vibration characteristics. We provide for the first-time evidence that micro-inertia contributions can counteract the effect of second-gradient properties on the eigenfrequencies of the structure, with their relative significance to depend on the mode of interest. What is more, we show that the internal length plays a crucial role in the significance of micro-inertia contributions, with their effect to be substantial for low, rather than for high internal length values, thus for a wide range of materials used in engineering practice.

Introduction

In the case of materials with a non-homogeneous microstructure, the classical, first order formulation of linear elasticity does not typically suffice to fully describe the effective macroscopic material behavior (Mindlin & Eshel, 1968). Higher gradient formulations are required, which take into account not only the local strain of the material, but also the variation of the strain field, thus nonlocal information [START_REF] Lim | A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation[END_REF].

A wide range of enhanced continuum theories has been up to now developed to circumvent the limitations of classical, first order models. The formulation of the generalized continua has been based on either top-down or bottom-up approaches. In the first case, the macroscale energy functions are assumed to contain higher order contributions that relate to inner microstructural parameters [START_REF] Engelbrecht | Waves in microstructured materials and dispersion[END_REF]Berezovski et al., 2013;[START_REF] Dingreville | Multiresolution modeling of the dynamic loading of metal matrix composites[END_REF][START_REF] Dingreville | Wave propagation and dispersion in elasto-plastic microstructured materials[END_REF]. The latter need to be calibrated on available macroscopic data, such as experimental non-destructive tests [START_REF] Dontsov | A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion[END_REF][START_REF] Engelbrecht | Reflections on mathematical models of deformation waves in elastic microstructured solids[END_REF], but cannot distinguish between the different microstructural contributions as of their very nature. In the bottom-up approach, a continualization or a homogenization scheme needs to be applied. For the former, the higher-order interactions between the different microstructural phases have to be appropriately described (Metrikine & Askes, 2002, Pichugin et al., 2008); so that choices for the discretization of the higher-order parameters need to be made The modeling approach is crucial both for the accurate characterization of the effective static macroscale properties and for the sufficient representation of its wave propagation characteristics (Chen & Fish,2001;Reda et al., 2018). In particular, the incorporation of higher gradient contributions can eliminate singularities or discontinuities that may appear for first order formulations [START_REF] Exadaktylos | Surface instability in gradient elasticity with surface energy[END_REF][START_REF] Chang | Wave propagation in granular rod using high-gradient theory[END_REF]. What is more, higherorder models can capture wave dispersion phenomena over a wide range of wave numbers and more importantly in the high wave number region, where typically low-order methods are insufficient [START_REF] Bennett | Elasticity theories with higher-order gradients of inertia and stiffness for the modelling of wave dispersion in laminates[END_REF] ). In particular, evidence has been provided that the incorporation of higher-order terms leads to increased natural frequency values, with the value of the internal length to be crucial for the relative differences recorded [START_REF] Li | Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory[END_REF]. Analytical solutions for the natural frequencies of straingradient rod structures verify the previous conclusions (Altan et al.,1996). It should be noted that higher-gradient terms can affect both the primal and the higher order modes, with the higher order frequencies to be more sensitive to gradient effects (Altan et al.,1996).

Furthermore, in the case of forced-vibrations, it has been noted that higher-gradient contributions may not-only modify the form of the time-displacement curves, but also the maximum values of the recorded deformations [START_REF] Tsepoura | Static and dynamic analysis of a gradient-elastic bar in tension[END_REF]. Analogous conclusions have been reported for the role of higher-gradient strain contributions on the vibration response of plate structures [START_REF] Yin | Vibration analysis of microscale plates based on modified couple stress theory[END_REF].

Up to now, the higher-gradient strain constitutive parameters have been characterized

either through asymptotic homogenization approaches or postulated to lie within a certain range. However, no exact second gradient homogenization method has been used to characterize the static and dynamic response of micro-structured materials upon a direct correlation of micro and macro-scale parameters. What is more, the effect of micro-inertia has been typically disregarded. As a result, the significance of micro-inertia contributions on the dynamic characteristics of structural members remains non-quantified along with its relative importance with respect to the eigenmode of interest.

In the current work, we make use of a higher-gradient homogenization scheme, which avoids the need to postulate the form of the non-local interactions, while it accounts for the Chapter 4: Higher-gradient and micro-inertia contributions on the mechanical response of composite beam structures Page | 117 effect of micro-inertia contributions. In Section 4.2, we elaborate on the computation of the effective first and second-gradient static and dynamic properties of the materials used for the design of the composite representative unit-cell hereby studied. Subsequently, we compute the static response of a beam structure architectured with different inner microstructural material phases, quantifying the effect of higher-gradient stiffness attributes (Section 4.3.1). Thereafter, we compute the eigenfrequencies of the composite beam structure using the Cauchy mechanics assumptions which we compare with the use of dedicated finite element models (Section 4.3.2.1). We extend the Cauchy mechanics framework to the higher-gradient and micro-inertia analysis space, providing analytical solutions for the dependence of the eigenfrequencies on the first, second-gradient and micro-inertia parameters (Section 4.3.2.2). We subsequently compute the eigenfrequency properties for the second-gradient and micro-inertia constitutive modeling case which we compare with the Cauchy mechanics predictions for different modes and for different inner material designs within (Section 4.3.2.2) and beyond (Section 4.3.3) the range of commonly used engineering materials. We discuss the obtained results in Section 4.4

and conclude in Section 4.5.

Materials and methods

Composite beams

We consider a composite laminated beam structure of length L (Fig. For the unit-cell design, different material combinations are considered, with material constituents typically used in composite structures [START_REF] Mallick | Fibre-Reinforced Composites -Materials, Manufacturing and Design[END_REF]. More specifically, carbon, epoxy and glass-based unit-cell material designs are considered, their elastic moduli ( E ), density (  ) and lame coefficient values (

        / 1 / 1 2 , / 2 1 EE           
) summarized in Table 4.1: [START_REF] Mallick | Fibre-Reinforced Composites -Materials, Manufacturing and Design[END_REF].

Higher gradient and micro-inertia constitutive material modeling

In the current work, we base our developments on the generalized Hill-Mandel lemma to express the energy stored in the deformed domain for an adopted strain gradient effective continuum (Chapter 3), as follows:

1 2 ij ij ijk ijk ij ij i i ij j ijk j k E K S u u E x K x x             (4.1)
Where in Eq. (4.1),  ij and ijk S represent the homogenized stress and hyperstress tensors while ij E and ijk K are their conjugate first and second gradient strain tensors. Accordingly,

  , , 1
2 ij i j j i uu   
and ij  stand respectively for the micro-strain and micro-stress developed within the microscale of the material. The dynamic energy part contains the micro-inertia contributions generated by the microscale inertia terms. Thereupon, the density  and the micro-displacement field i u appear.

For the derivation of the homogenized stress and hyperstress constitutive behavior based on Eq. (4.1), the expression of the material's total deformation energy is used, as elaborated in (Chapter 3). In the corresponding constitutive formulation, the homogenized stress and hyperstress tensors include both the static and dynamic, time-dependent strain components denoted with dl E and rst K accordingly for the first and for the second gradient strain tensors.

The complete set of constitutive equations writes in terms of the dynamical stress tensor expressions:

 

EK C C B B EK E S D D A K E                                   
      ( ) ( ) hom,
E i E i ii dyn E E E Ei ii AA CC CC H H H x AA B D C B D C                                               () 11
Ki K i E i K i E i K i K i ii H H H x H x x A C A A                           (4.3)
Where in Eq. ( 4.3), i  and i  are the Lame constants of each material phase contained in the unit-cell design with the subscript  

dyn dyn EK C C C C EK E S C C C K E                                    (4.4)
The stiffness coefficients 1 i C are calculated based on the unit-cell constituent material properties of Table 4.1, with the localization operator expressions defined by Eq. (3.A2) and (3.A5). For the computations to follow, equal phase thickness properties ( 12dd  ) are employed.

Static and dynamic equilibrium of the homogenized structure

The dynamic equilibrium equation of motion along the longitudinal axis of the composite beam structure in Fig. 4.1b writes as follows:

  2 hom 11 111 2 XX U S t         (4.5)
Where in Eq. (4.5), hom

 stands for the homogenized density of the beam microstructure.

Using the stress and hyperstress definitions of Eq. (4.4) and after substitution into the dynamic equilibrium Eq. (4.5), we obtain the following dynamical equilibrium equation:

2 2 2 4 2 1 1 1 hom 1 2 5 2 2 2 4 2 U U U U C C C X t X X t                  (4.6)
Where in Eq. (4.6), the coupling stiffness terms 1

Results

Static response

In the current section, we compute the longitudinal static response of a composite beam structure (Fig. 4.1a) based on the higher-gradient equilibrium definition of Eq. (4.8). For the computation of the Cauchy and higher-gradient stiffness parameters ( We note that the internal length values obtained using the homogenization method of Section 4.2.2 lie in close agreement to the ones provided in [START_REF] Tan | Homogenized Gradient Elasticity Model for Plane Wave Propagation in Bilaminate Composites[END_REF], where a 0.193mm, 0.163mm and 0.15mm value has been accordingly computed for the Borosilicate Glass-Silicon Carbide, E-Glass-Epoxy and Carbon-Epoxy unit-cell designs.

We compute the higher-gradient static structural response by solving Eq. (4.8), which requires a total of four boundary conditions, as of its fourth-order nature. We employ two classical, one non-classical and a symmetry boundary condition for the strain at the middle of the beam structure. In particular, we set the displacement at the start equal to zero    

0 0 u 
and the stress at the other end equal to the area-normalized applied force  

    1 0 0 1 // P A C   . 0 (0) 0, ( ) / (0) , ( / 2) 0 SG u L P A L           (4.9)
Solving the equilibrium of Eq. (4.8) with the boundary conditions of Eq. (4.9), we directly obtain the strain and displacement field along the length of the beam structure. In Fig. 4.2, we provide the strain and displacement profile for the case of a microscale material design with Borosilicate Glass and Silicon Carbide, with an internal length value of 4.2. The difference in the end-tip displacement is in the order of 4%, a value which reflects the maximum difference for all practice-relevant composite material designs of Table 4.2. In particular, a tip-displacement difference of 3.7% is computed for the E-Glass and Epoxy unit-cell material design of Table 4.2 and a 3.1% for the Epoxy and Carbon unit-cell material design. We note that analogous results are obtained for the case of a variable beam length, where the higher-gradient end-tip displacement of a beam structure with a length ) as a reference (also depicted in Fig. 4.2), we observe that as we increase the internal length values, the difference in the higher-gradient and Cauchy mechanics predictions of the end-tip displacement of the structure are increasing. In particular, the difference in the end-tip displacement between the higher-gradient and the Cauchy mechanics predictions become 13% (green curve) and 18.4%

Micro-inertia effects on the higher-gradient, eigenfrequency properties

In the sequel, we analyze the eigenfrequency properties of the composite structure accounting both for micro-inertia and higher-gradient effects on its dynamic characteristics.

For completeness, in Section 4.3.2.1, we compute the eigenfrequency properties in the simplified, Cauchy-mechanics case, where micro-inertia and higher-gradient effects are neglected. Subsequently, in Section 4.3.2.2, we quantify the effect of micro-inertia contributions on the effective dynamic properties for composite materials used in praxis, while in Section 4.3.3 we carry out a general-case, sensitivity study on the internal-length and relative density values.

Cauchy mechanics eigenfrequency attributes

We hereby compute the eigenfrequency properties of the composite beam structure of Fig. 1a using the dynamic Cauchy mechanics formulation of Eq. (4.7). For the eigenfrequency computation, we insert the general waveform solution () 0 ( , )

i kX t U X t U e   
in the differential equilibrium of Eq. (4.7), where k stands for the propagating wavenumber and  for the corresponding frequency. After simplifications, we obtain:

1 22 1 hom C k       (4.10)
The static displacement field profile   Z X is obtained using the method of separation of variables for the macro-displacement field  

, U X t . The time-dependent response   T t is thus decoupled from the spatial displacement profile, as follows:

( , ) ( ) ( ) U X t Z X T t  (4.11)
Inserting the definition of Eq. (4.11) to the dynamic equilibrium of Eq. (4.7), we obtain for the time-dependent and the following two equations:

1 22 1 hom ( ) ( ) ( ) C T t T t k T t              (4.12a)       1 2 1 11 0 C Z X k C Z X       (4.12b)
The general solution form of Eq. (4.12b) writes:

12 ( ) ( ) cos( ) ( ) sin( ) Z X R k k X R k k X       (4.13)
Where in Eq. (4.13) the constant parameters 1

Mat Mat Mat c c c Mat Mat Mat c c c Mat Ma c c C C C Hz Hz Hz L L L C C C Hz Hz Hz L L L C Hz L                                            1 1 3 6 3 6 1 1 3 hom hom 2 3 1.6 10 , 2.4 10 t Mat c C C Hz Hz L L            (4.15)
We compare the computed eigenfrequencies of Eq. The eigenfrequency values obtained by the finite element based eigenfrequency analysis match the results obtained by the use of the homogenized structural properties. In particular, the relative difference lies below 1% for all micro-scale material attributes of Table 4.2 for all eigenfrequency values of Eq. (4.15). We below extend the eigenfrequency analysis to quantify the effect of higher-gradient and micro-inertia contributions..

Micro-inertia effects on the higher-gradient dynamic attributes

We subsequently analyze the effect of micro-inertia contributions on the longitudinal eigenfrequencies of different composite beam designs in the presence of higher-gradient effects. To that scope, we consider the complete form of the dynamic equilibrium equation provided in Eq. (4.6), for which we obtain the following expression for the eigenfrequency values  , proceeding in a complete analogy to Section 4.3.2.1:

1 2 1 22 51 2 1 hom 2 C k C k kC         (4.16)
The eigenfrequency definition of Eq. (4.16) depends, not only on the static stiffness term attribute 1 1

C as in the Cauchy mechanics formulation case (Eq. (4.10)), but also on the secondgradient stiffness 1 5

C and on the micro-inertia coefficient C term. Applying the separation of variables process (Eq. (4.11)) for the dynamic equilibrium Eq. (4.6), we obtain the following set of decoupled space and time differential equations in complete analogy to Section 4.3.2.1:

1 2 1 22 51 2 1 hom 2 ( ) ( ) ( ) C k C T t T t k T t kC                (4.17a)     1 1 1 2 hom 2 5 1 2 0 Z C Z C C Z               (4.17b)
Eq. (4.17b) contains not only the forth-order, higher-gradient term related to the stiffness coefficient 1 5

C , but also the effect of the micro-inertia term

1 2
C encompassed in the modified coefficient of the second order spatial derivative. Its effect is frequency-dependent, as it can be directly observed out of the multiplying frequency term. The general solution of the differential Eq. (4.17b) takes the form:

1 2 3 4 ( ) ( ) ( ) ( ) cos( ) ( ) sin( ) XX Z X C k e C k e C k k X C k k X                    (4.18)
Where in Eq. (4.18),

1 hom 1 2 hom 15 1 1 2 1 hom 2 5 5     C C k C C k C  
is a constant parameter that depends on all effective material parameters entering Eq. (4.6), as well as on the propagating wavenumber k . The constants ()  i Ck are computed for the case of zero end-displacements and displacement gradients (

' (0 ) (0 ) 0 Z L Z L  ),
          22 sin 2 cos( ) 2 1 0 L L L L LL kL e e k k kL e e k e e                                (4.19)
After the computation of the roots of Eq. (4.19), the eigenfrequencies can be directly computed using Eq. (4.16), which can be re-written in a non-dimensional form as follows:

1 2 1 51 2 1 hom 11 2 11 hom hom C k C kL L kC CC                (4.20)
For the eigenfrequencies' computation, the complete set of effective material parameters needs to be calculated. The effective relative density hom  and micro-inertia

1 2
C parameters for each of the composite material designs of Table 4 Table 4.4 indicates that the eigenfrequency values computed with the incorporation of second-gradient effects are higher than the ones obtained by the classical, Cauchy-mechanics based formulation. The difference for the Borosilicate Glass-Silicon Carbide material case is in the order of 4-6% for the second-gradient modeling formulation, with the actual value to depend on the mode of interest, upon a uniquely increasing relative difference for the higher modes. The corresponding relative difference for the case where micro-inertia contributions are accounted for is lower compared to the second-gradient modeling case. The numerical result reflects the definition of Eq. (4.20), where the micro-inertia terms contribute to the denominator of the frequency expression.

   c   SG    % / SG c c      SG I      % / SG I c c       1  3.
For the higher modes, the second-gradient formulation with micro-inertia contributions provides results which are closer to the Cauchy mechanics predictions, with the purely secondgradient formulation to substantially overestimate the eigenfrequency values. In Table 4.5, we provide the corresponding results for the case of a composite material design with a lower internal length value   0.185 e l mm  and a lower relative density value. Table 4.5 provides relative differences in the eigenfrequency values that increase upon increasing mode for the second-gradient modeling formulation, as in the case of the composite material design of Table 4.4, though lower in magnitude, as of the lower internal length value.

Table 4.5 : Eigenfrequency values of the composite E-Glass and Epoxy (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with microinertia contributions (SG+μI) modeling formulation.

Mode

Cauchy SG SG I    c   SG    % / SG c c      SG I      % / SG I c c       1  3.
However, when micro-inertia effects are considered in the eigenfrequency computation, the eigenfrequency values computed are closer to the Cauchy-mechanics predictions.

While for the first mode the eigenfrequency values for the second-gradient and secondgradient with micro-inertia formulations are rather close, the corresponding difference for the third mode becomes substantial. In particular, neglecting the micro-inertia effects leads to a substantial overestimation of the third eigenfrequency with respect to the higher-gradient with micro-inertia dynamic model (SG+μI). More specifically, a nearly three times higher relative eigenfrequency difference is obtained for the third mode (5.52 vs 1.9% in Table 4.5).

In Table 4.6, we summarize the eigenfrequency values for the third composite material combination of Table 4.3, with the lowest internal length value   0.157 e l mm  and the most lightweight material design of Table 4.3. Table 4.6 reveals the same pattern with the ones observed for the inner composite material designs of Tables 4.4 and 4.5. In particular, the second-gradient formulation yields higher eigenfrequency values with respect to the Cauchy-mechanics predictions for all modes, with the relative differences to increase upon increasing mode value. However, in the highergradient with micro-inertia contributions case (SG+μI), an inverse pattern is obtained. More specifically, while the SG+μI modeling formulation yields an eigenfrequency value that is very close to the Cauchy mechanics predictions for the third mode, the use of a merely secondgradient formulation overestimates the eigenfrequency value by an additional 4% (4.5% in Table 4.6).

Table 4.6 : Eigenfrequency values of the composite Carbon and Epoxy (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with microinertia effects (SG+μI).

Mode

Cauchy

SG

SG I    c   SG    % / SG c c      SG I      % / SG I c c       1  3.

Sensitivity of the dynamic properties on the micro-inertia contributions for different second-gradient material designs

The results of Section 4.

Discussion

The selection of the proper modeling formulation is of crucial importance for the accurate characterization of the mechanical response of multiscale structures. From a constitutive modeling perspective, the internal length is sufficient to assess the importance of highergradient stiffness contributions on the static macroscale mechanical response of beam structures. However, for the dynamic structural properties, not only the complete spectrum of the available static-related attributes need to be accounted for, but also all dynamics-related characteristics, micro-inertia contributions included.

The bottom-up static analysis of Section 4.3.1 indicates that the relative importance of higher-gradient contributions on the longitudinal static response is rather low for typical composite element designs with low internal length values (   0...0.25 e l mm  , Fig. 4.2). In particular, using the utterly non-linear strain profile distribution of Fig. 4.2a corresponding to a highly inhomogeneous element strain evolution, the difference of the higher-gradient end-tip displacement solution from the Cauchy mechanics results is below 5%. Substantial differences can occur ( 10%  ) only for internal length values that way exceed ( 0.5 e l mm  ) the range computed for the composite material designs of Table 4.1 as in Fig. 4.3. The framework provided in Section 4.2 allows for a direct, multiscale computation of the relevant highergradient parameters for micro-scale material constituents beyond the ones summarized in Table 4.1, providing a generic means of quantifying the magnitude and significance of the microscale parameters.

While the effective static displacements of Figs. 4.2 and 4.3 are reduced upon the incorporation of the higher-gradient stiffness contributions, the corresponding dynamic eigenfrequency attributes are increased (Sections 4.3.2, 4.3.3). Considering a purely secondgradient constitutive modeling formulation, without the effect of micro-inertia parameters leads solely to higher eigenfrequency values for all structural modes, an effect that can be explained using Eq. (4.16) upon a zero value of the micro-inertia coefficient term ( 1 2 0 C  in Eq. (4.16)). However, when micro-inertia effects are accounted for, the structural eigenfrequency values can be either closer to or quasi-identical to the Cauchy mechanics predictions, depending on the mode of interest and on the internal length value. The

Chapter 5

Wave Propagation in 2D composites Structure using Higher Gradient Size Independent Homogenization

Introduction

Predictive micromechanical models aiming at understanding the impact of the geometry and properties of reinforcement and matrix on the macro scale composite response witness a significant development, as the experimental determination of the mechanical properties of composites is delicate due to the number of independent moduli that have to be measured in 3D situation. Hence, predictive approaches prove convenient in order to bridge the scales and to provide a constitutive law at a continuum scale, whereby the equivalent continuum properties can be related to both the geometrical and mechanical microstructural parameters of the composite constituents. Furthermore, designing such kind of materials in an optimal way requires understanding the relation between the internal architecture and physical properties, and suitable ways to evaluate these properties.

The mechanics of generalized continua can be defined as the mechanics of matter for circumstances in which the conventional continuum mechanics cannot offer a promising approach (Kröner, 1968). In order to keep a continuum description, one has to consider a generalized continuum to model the substitution material resulting from the homogenization process. As known, classical homogenization techniques limited to small loading or deformation field wavelength comparing to the microstructure size (Trinh et al., 2012). The objective of homogenization towards generalized continua is to remedy to these limitations and to extend the range of validity of the continuum approach beyond the strict assumption of the scale separation (Mindlin, 1964;Eringen and Suhubi, 1964). The incorporation of the microstructural scale within enhanced continuum formulations involves the relaxation of the local action hypothesis and the introduction of a spatial interaction and typical length scales to account for the influence of neighboring points in the formulation of constitutive equations.

There are essentially two ways to generalize the classical definition of a continuum, classified into higher-order and higher-gradient continua. Higher-order continua involves additional degrees of freedom like the Cosserat medium promoted by the Cosserat brothers (Cosserat and Cosserat, 1909) and the micromorphic medium (Eringen and Suhubi, 1964;Mindlin, 1964;Germain, 1973). In Cosserat elasticity (Cosserat and Cosserat, 1909) local rotations are added as degrees of freedom. In higher-gradient continua, one keeps the same degrees of freedom but to higher gradients of the displacement field are added as arguments of the strain energy density.

In classical continuum mechanics (Truesdell and Noll, 1965), all higher order displacement gradients are neglected and only the first order displacement gradient is involved in measuring the deformations of a body. Such kinematic framework turns out not to be rich enough to describe a variety of important mechanical and physical phenomena (Auffray, 2010).

Especially, the classical continuum mechanics proves to be intrinsically size independent, as reported by in (Truesdell and Noll, 1992). Particularly, size effects and a non-local behavior may arise due to the discrete nature of matter at a sufficiently small-scale level. The lack of a material length scale parameter in the classical continuum elasticity theory leads to an inaccurate description of the structural behaviors with microstructure (Goda et al., 2015).

The development of additive manufacturing has indeed allowed for the engineering of a new class of materials that obtain their effective mechanical attributes by their inner topological design rather than from their chemical composition, and generate exceptional material performances far beyond those of the base material (Cheng et al., 2018). Artificial materials of the kind deserve the wording architectured materials or metamaterials, as of their potential to yield effective static and dynamic attributes that can well extend the design space of the base material used. Architectured materials designed with a quasi-periodic inner topology have witnessed a rapid expansion in the recent period, the complex internal topology of architectured materials prohibits the use of direct numerical simulations or full-field computations of macrostructures made of architectured materials, which become numerically too expensive. While, the mechanical response of periodic architectured materials can be analyzed at the macroscopic level at the scale of a unit cell that is repeated to build a regular lattice.

Amongst the possible a priori choices of the effective substitution medium mentioned previously (Cosserat, micromorphic, strain-gradient …). The strain gradient elastic continuum introduced in (Mindlin, 1964) and (Mindlin & Eshel, 1968) offers several advantages as underlined in (Yvonnet et al., 2020). Since it is rich enough to represent emergent properties without having recourse to the micromorphic medium that has many more inherent parameters (Auffray et al., 2015) and such a model can be constructed using asymptotic homogenization (Boutin, 1996;[START_REF] Bacigalupo | Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits[END_REF]Gambarotta, 2010, 2014;Tran et al., 2012).

A large body of works has been devoted to strain gradient homogenization in the literature since more than 40 years (Ben-Amoz, 1976; Forest and Sab, 1998;Forest, 1998;Forest et al., 2001;Trinh et al., 2012;Tran et al., 2012;Rahali et al., 2015;Bacigalupo and Gambarotta, 2010). For both composites with periodic or random microstructures (Goda & Ganghoffer, 2016), auxetic architectured materials (Bacigalupo & Gambarotta, 2014) or random fibrous media (Berkache et al., 2017). Different approaches have been employed in the literature for the derivation of the effective mechanical properties of strain gradient models of heterogeneous microstructures.

Periodic homogenization methods rely on the asymptotic expansion of the searched solution versus a small parameter representative of the relative scale of the unit cell (with respect to the macroscale of the overall structure   /L l  

) have been developed in (Bensoussan et al., 1978;Sanchez-Palencia, 1980) considering the main proponents.

Expansion of the series up to second order leads to the formulation of an effective strain gradient continuum whereby the local solution depends on the macroscopic strain gradient (Gambin and Kröner, 1989;Boutin, 1996;Triantafyllidis and Bardenhagen, 1996;Smyshlyaev and Cherednichenko, 2000;Tran, 2012). The strain gradient elastic moduli are computed by solving in a sequential manner two unit cell boundary value problems, a classical one leading to the Cauchy moduli and a higher order BVP associated with the macroscopic strain gradient as a kinematic source term.

Pure asymptotic expansion methods (Boutin, 1996;Smyshlyaev and Cherednichenko, 2000) are sometimes combined with a Fast Fourier Transform FFT method to compute the effective properties (Tran et al., 2012;Li and Zhang, 2013). A quite common strategy to compute the strain gradient moduli is to use suitable quadratic boundary conditions over the RVE as the minimum kinematic loading capturing a gradient of the strain at the macroscopic level. An approach followed by many authors (Gologanu et al., 1997;Forest, 1998 Barboura and Li, 2018). This method extends the classical kinematic uniform boundary conditions employed in Cauchy type homogenization (Huet, 1990). The method however suffers from two major flaws, since the effective strain gradient moduli remain non-zero for homogeneous materials or under conditions of scale separation (Forest and Trinh, 2011).

Corrections to these spurious effects were recently proposed (Yvonnet et al., 2020), where they added a body force field to the quadratic boundary conditions together with a correction of the localization tensor to get a convergence effective strain gradient moduli for big number of cells. Very recently, (Monchiet et al., 2020) made a correction to the employed quadratic boundary conditions, thereby getting rid of the strain gradient effects (the effective medium remains of Cauchy type). Furthermore, the computed strain gradient effects converge asymptotically with the number of periods. Note that this asymptotic homogenization method avoid the spurious effect since the source term in the second order unit cell problem vanishes for a homogeneous medium. Moreover, another type of correction provided in (Li, 2011) where second gradient moduli are independent of periods number, the correction terms used the first gradient moduli and the quadratic inertia of the structure. Furthermore, the author provides an analytical solution of the higher gradient constitutive parameters for 1D and 2D composites structures.

The homogenization method is crucial for both the static analysis of composites and for its wave dispersion characteristics. Classical elasticity based on Cauchy continuum mechanics assumptions lead to linear dispersion relations (Chapter 2), so it proves unable to describe the dispersion relations occurring in microstructured materials. On the other hand, enriched continua like strain gradient medium prove able to capture the dispersive nature of propagating waves, as shown e.g. in (Rosi & Auffray, 2016). What is more, the validity range of higher gradient formulations is broader than that of classical homogenization formulations (Rosi & Auffray, 2016).

The main objective and novel aspect of this chapter is the setting up of a homogenization method towards strain gradient effective media, as an extension of the methodology used in 1D heterogeneous media (Chapter 3), relying on variational principles articulated with Hill macrohomogeneity lemma. As will be evidenced, the method is able to deliver effective strain gradient moduli that are size independent and is thus able to capture effective properties that are intrinsic to the microstructure, and their influence on the wave dispersion characteristics are analyzed in function of the microstructure contrast between the composite constituents.

In this work, we propose theoretical developments for the derivation of the effective strain gradient properties of composite materials including corrections terms accounting for higher gradient terms, and wave propagation analysis using the homogenized properties in Section 5.2.2. In Section 5.3, we elaborate on the computation of the effective second gradient effective moduli for 2D composites structures, which we compare to analytical results (Section 5.3.2.1), while in Section 5.3.2.2, we analyze the size effects on the high gradient effective moduli with and without corrections terms. Furthermore, we analyze the variation of the higher gradient internal length (a second order tensor) for different contrasts of microstructure volume fraction and moduli, and asses the importance of correction terms (Section 5.3.2.3). In addition to the static analysis, we derive the wave propagation of the composite structure for classic continua, which we compare with the use of dedicated finite element models (Section 5.3.3.2). We subsequently compute the wave propagation properties for such composites using the higher gradient moduli with the corrections terms in Section 5.3.3.3, which we compare to Cauchy mechanics prediction for different microstructure properties in Section 5.3.3.4. We discuss the obtained results in Section 5.4 and conclude in Section 5.5.

The following notations are used throughout the present work. The gradient of a scalar or tensor field with respect to the position is denoted with the nabla operator and the position as a subscript, viz.

 

X EX 

represents the gradient of the tensor field   EX with respect to X . The transpose of a tensor is written with a superscript notation, for instance T B . The symmetrized dyadic product is denoted s  . The dot product therein represents the internal product in the space of Cartesian tensors. The simple, double and triple contractions of tensors are written :, respectively. The vector product of two vectors or tensors , UVis denoted UV  .

Materials and methods

Theoretical formulation

We impose a macroscopic strain   EX and a strain gradient   KXto a unit cell, the displacement will be decomposed to a quadratic form linear to the imposed parameters, and a fluctuation displacement () i uy, nil on the unit cell boundaries. Therefore, the displacement will be as follows:

  1 : ( ) 2 i ij j ijk j k i u E y K y y u y      (5.1)
Where ijk K is the second gradient of macro-displacement   UX tensor, here defined as the second displacement gradient corresponding to Mindlin (Mindlin, 1964) form I of strain gradient elasticity. Similar to first gradient homogenization, the fluctuating part of the displacement corrects for the existing discrepancies between the macroscopic imposed displacement and the real displacement field of the media.

Deriving the micro-displacement in Eq. (5.1), we get the micro-strain as function of the macroscopic strains :

      , E X K X as follows: 1 () 2 ij ij ijk k ij E K y y     (5.2) Wherein   ,, 1 ( ) ( ) 2
ij ijqt qt ijqtw qtw ij ijk ijkqt qt ijkqtw qtw ijk W C E B K E W S D E A K K            (5.4)
An extended minimization principle over all periodic fluctuations holds for the strain energy in Eq. (5.3) using the micro-strain expression, Eq. (5.2), so that one can write: The linearity of the fluctuating displacement with respect to the kinematic loading in Eq.

              1 W
(

w C E B K E E K y H y E H K C y d E E K y H y E H K E C C H C H C H H                                                   l n n l E wqt Y K E E K qtw ijqt w ijkl y kqtw w mnqt y mij mnkl y mij y kqtw Y d K C y C H y C H C H H d                         (5.9)
The same holds for the second order macroscopic stress tensor, so we derive the energy from Eq. ( 5.3) with respect to the second gradient imposed macroscopic-strain ijk K as mentioned in Eq. (5.4)

S D E A K K E K y H y E H K C y d K E K y H y E H K E C y y C H C H C                                                    n d d n n d K E y mijk y wqt Y K K K K qtw ijqt k w k ijsd y sqtw w mnqt y mijk mnsd y mijk y sqtw Y H H d K C y y y C H y C H C H H d               (5.10)
By identification, we simply get the first and the second gradient elasticity tensors hom C and hom A respectively, and the coupling between them encapsulated in tensors hom B and hom

D

as the following integrals:

                         
C C C H C H C H C H H d B C y C H y C H C H H d D C y y C H C H                                              hom hom n d d n n d K K E mijk mnwd y mijk y wqt qtijk Y K K K K ijkqtw ijqt k w k ijsd y sqtw w mnqt y mijk mnsd y mijk y sqtw Y C H H d B A C y y y C H y C H C H H d                 (5.11)
Using these effective tensors, we further write the tensor form of the stress and hyper-stress accounting for their symmetric (Auffray et al., 2010)

C C C E C C C E C C C E                              (5.
A A A A A A S S A A A A A A S A A A S S S                      hom hom
K K K K K K A A                                           (5.12b)
We calculate the correction terms of the second gradient as described in Appendix 4.B.

We identify the first and second gradient corrected constitutive tensors as follows: (5.14)

Homogenization modeling

In this work, we implement the previous computation of the effective first and second gradient moduli in the 2D case using the finite element code FreeFem, where we use the BVP for first and second gradient from Eqs. (5.8a, 5.8b) in order to compute the displacement within the unit cell domain. Each elasticity tensor is written in 2D case using the general formula derived in Eq. (5.11) as function of the materials parameters and the localization variables   , EK HH . In the sequel, we describe the methodology used in the finite element study to calculate the homogenized properties of the heterogeneous structures for the longitudinal case, and then other cases will be derived using the same methodology.

First gradient modeling

We start imposing a longitudinal macroscopic strain to the unit-cell    The weak form of the BVP1 for the first gradient formulation writes from Eq. (8a) as follows:

                              
E H E H E E H E v E H E H E H E v d H E H E v v                                        (5.16)
Where  

Second gradient modeling

Extending previous writings to the second order formulation, we express the micro-strain from Eq. (5.2) as function of the second gradient longitudinal displacement localizators 111

K K K i x y i x x x K K K i x y i y y y K K i y x x x y y K x H K H K K x H K v K x H K H K H K v d H K H K v v                                                 (5.18)
Solving the BVP in Eq. (5.18) on the same mesh generated for the first gradient unit-cell problem in section 5. ADaccordingly, as described in Eq. (5.11). The computed effective properties form the basis for the analysis of wave propagation features in heterogeneous materials.

The next section is devoted to the study of wave propagation within 2D composites on the basis of the formulated strain gradient effective continuum.

5.2.3Wave dispersion for 2D structures

For the homogenized 2D continuum model, the equations of motion using the derived second gradient formulation express for the longitudinal and transverse parts as the two following partial differential equations along the  

S S S S X X X X X X X X X X V S S S S X X X X X X X X X X U                                                                    (5.19)
Here, U and V are the horizontal and vertical components of the acceleration vector, the overhead dot denoting the time derivative. The homogenized effective density therein is given by hom

i i i ii A / A      
 where   ii ,A  are the density and surface for each constituent within the heterogeneous structure (each constituent having a homogeneous density and properties), which follows the mixture rules, and the homogenized stress and hyper-stress are calculated using Eq. (5.12a) and (5.12b).

In order to obtain the dispersion relation for the propagating waves through the structure, we consider a harmonic wave propagating along an axis in the infinite planar second gradient effective medium, so the generalized displacement field with components , UV at a point

        12 1 2 0 1 2 0 , , xy I t k X k X U X X U e V X X V          (5.20) Where   00 ,
UV is the displacement amplitude,   , y x k k are the wave vector components,  is the frequency, and I is the imaginary number such that 2 1 I  . The wave vector is a complex number: its real part represents the attenuation in the X1-X2 plane, and its imaginary part defines the phase constants. In this study, we consider the wave vector without attenuation.

Substituting the ansatz of Eq. (5.20) in the equation of motion Eq. (5.19) delivers the following algebraic equation:

  0 12 0 , , 0 U D k k V      (5.21)
The dispersion relation will be derived by imposing a unique solution to the last characteristics equation (5.21) as the frequency  as function of the wave vector components   , xy kk .

Application to inclusion based composite materials

Overview of inclusion based composite models

Inclusion architectures are considered as the most significant feature in composite design due to their numerous practical applications and they spread across many industrial areas. They provide an excellent compromise between lightness and stiffness, and often for a low cost of production. Introducing such inclusions where they are needed is an efficient method of using inclusions to improve composite performances. Mechanical properties of composites containing inclusions have become a topic of great interest in recent years because of the manifest technological importance of such materials. Their excellent mechanical properties are largely because their microstructure can be adjusted to achieve controlled mechanical properties at the mesoscopic or macroscopic levels. In view of exploiting the full potential of these solids, their constitutive mechanical response has clearly to be entirely understood and modeled based on the available microstructural information.

Based on loading conditions and structural requirements, shape of inclusions in a composite may vary: it can be for instance particles, continuous fibers, and short fibers. In the last years, composites composed of inclusions with various shapes have become popular in a wide range of applications, especially through fabrication techniques such as injection molding, bulk molding compound, and three-dimensional weaving and braiding. Since inclusions reinforced composites made of a base material with stronger inclusions are attractive due to their increased usage in structural and noise control engineering, the need for understanding of the composites are growing up. The assessment of the intrinsic mechanical properties of such composites is a critical issue since experimental measurements may be difficult and costly. We address in this section the issue of constructing first and second order grade effective constitutive models as substitutes of the initial structure of such composites reinforced with inclusions.

There is a long history in the evaluation of the effective elastic properties of such heterogeneous materials, which remains one of the most tangible problems of micromechanics.

A wide diversity of approaches have been developed to predict the overall mechanical responses of predicting the effective elastic properties of composite materials containing inclusion, based on the knowledge of the mechanical properties, geometries, and distributions of their inherent phases [START_REF] Torquato | Random Heterogeneous Materials, corrected[END_REF][START_REF] Milton | The Theory of Composites[END_REF], and (Goda et al., 2016). Starting from the pioneering papers of Eshelby who introduced the celebrated Eshelby's tensor (Eshelby, 1957) for isotropic ellipsoidal inclusions, analytical predictions of the effective properties of such inclusion-based composites have been provided in the dilute regime. [START_REF] Mori | Average stress in matrix and average energy of materials with misfitting inclusions[END_REF] method that proves efficient for the study of the overall behavior of composite materials containing non-dilute concentration of reinforcements. This scheme further enjoys the property [START_REF] Weng | The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds[END_REF]1992) that the elastic moduli of identical shaped multiphase composites predicted by the Mori-Tanaka mean-field theory systematically coincide with Hashin-Shtrikman's bounds [START_REF] Hashin | A variational approach to the theory of the elastic behaviour of multiphase materials[END_REF]. Computational homogenization for reinforced composites define a third set of methods to evaluate the effective properties from the work of (Guedes and Kikuchi, 1990) devoted to linear elasticity problems.

Finite element methods constitute a powerful technique widely used to evaluate the local stress and strain states for complex microstructures, see (Goda et al., 2016) and references therein. Based on the equivalent strain energy approach, based on the relationship established between the strain energy of the considered microstructures and that of the homogenized equivalent model, FE analyses have been developed over the past years to compute the first and second order grade continuum models equivalent to such reinforced composites. In this method, boundary conditions based on combinations of imposed quadratic displacements and nil tractions applied along the outer boundaries of an identified representative volume element are formulated (Goda et al., 2016), leading to the determination of both classical and strain gradient effective constitutive elastic moduli.

In the current section, we use the second gradient modeling described in section 5.2.2 for the higher gradient energy (HGE) formulation derived in section 5.2.1 for 2D composites structures in order to compute their static properties. In Section 5.3.2.1, we compare the provided results of the higher gradient constitutive parameters with previous studies for the same composites structure. Subsequently, in Section 5.3.2.2, we study the impact of correction terms defined in Eq. (5.13) on the size effects as for 1*1, 2*2 and 3*3 unit-cells. For completeness, we analyze the effect of microstructure contrast on the internal length parameters in Section 5.3.2.3.

Static properties of 2D composite microstructures

We consider a unit size  

Validation with analytical second gradient formulation

We use the second gradient formulation described in Section 5.2.1, to compute the higher gradient elastic properties of the composite structure as in Fig. 5.1a for the case where the second phase is a hole, which we compare to the analytical higher gradient constitutive parameters (Li, 2011) in Table 5.1.

In view of the tetragonal symmetry class of the chosen RVE in Fig. 5.1a as discussed in (Auffray et al., 2015), the higher gradient elasticity tensor will have the following structure: .

.

ijklmn A A A A A A Sym A A A A A A A Sym A A A A A A                     
(5.22)

The analytical higher gradient constitutive parameters are calculated in (Li, 2011) Using the analytical higher gradient constitutive parameters from Eqs. (5.23a, b) for the 2D structure in Fig. 5.1a, we calculate the relative difference between the higher gradient homogenized properties without correction terms (SG) using the static higher gradient energy formulation (HGE in short) as described in section 5.2.1, and the analytical solution provided in (Li, 2011) as in Table 5.1. (Li, 2011), with the maximum relative difference less than 1.35% for the homogenized higher gradient parameters.

Using the higher gradient formulation described in Section 5.2.1 and the correction terms from Eq. (5.13), we calculate the relative difference between the homogenized properties calculated using the HGE including correction terms (SG*), and the analytical solution provided in (Li, 2011) as in Table 5.2. From the comparison provided in Table 5.2, the maximum relative difference between the HGE formulation described in Section 5.2.1 and the analytical solution of (Li, 2011) is less than 3.5% for the homogenized higher gradient parameters including the correction terms.

Sensitivity of the homogenized properties to size effects

In this part, we analyze the size effects on the higher gradient constitutive parameters. To that scope, we study the variation of the higher gradient constitutive terms with and without the correction terms as described in Eq. (5.13) for different RVE sizes.

For the representative unit-cell of the limited plate of Fig. 5.2a, we choose one, two or three unit-cells as in Fig. 5.2b, 5.2c, and 5.2d accordingly. We note that 1 l mm  is the dimension of one unit-cell. The composite is made of a linearly elastic matrix material with the following properties   verified that Cauchy tensor is size independent as mentioned in (Truesdell and Noll, 1992).

We show the variation of second gradient effective properties without any correction in Fig. 5.3a, and with the correction provided in Eq. (5.13) in Fig. 5.3b as follows:

Sensitivity of internal length parameters to the microstructure contrast

We evaluate the impact of higher gradient constitutive parameters on the macroscopic behavior of the structure by studying the internal length parameters () le mm  as defined in Eq. continua like strain gradient medium prove able to capture the dispersive nature of propagating waves, as shown e.g. in (Rosi & Auffray, 2016). What is more, the validity range of higher gradient formulations is broader than that of classical homogenization formulations (Rosi & Auffray, 2016).

In this part, we study the wave propagation through the square matrix with circular inclusion of Fig. . Firstly, we use COMSOL to calculate the dispersion characteristics of the structure (Section 5.3.3.1). Then using the homogenized properties calculated using the HGE formulation from Section 5.2.1, We know that the validity range of homogenization-based dispersion relation will be restricted to the low wavenumber as discussed in (Rosi & Auffray, 2016). We compare COMSOL results with first gradient solution in Section 5.3.3.2. For completeness, in Section 5.3.3.3, we calculate the dispersion relations using the higher gradient formulation with and without the correction terms of Eq. (5.13), while in Section 5.3.3.4; we study the sensitivity of microstructure geometrical and mechanical contrast on the difference between first and second gradient with and without the correction terms.

COMSOL dispersion characteristics

We generate the following domain and FE mesh of the composite unit cell (Fig. 5.1b), showing the displacement shape for the longitudinal and the transverse mode (using COMSOL) as follows:

First gradient dispersion analysis

Starting with the classical Cauchy type homogenization relying on the homogenization method described in Section 5.2.1 restricted to the first gradient, we calculate the homogenized Cauchy tensor as defined in Eq. (5.12a) for the composite structures (Fig.

C C C C C C MPa C C C               (5.24)
We calculate the dispersion relations using the methodology described in Section 5.2.3 but restricting to the first order approximation in Eq. (5.19) for the first two modes, and we compare this later with COMSOL results of Fig. the one provided from COMSOL computations at small wavenumber; however, the first gradient is providing a non-dispersive relation.

Second gradient dispersion analysis

We subsequently analyze the effect of the higher gradient constitutive parameters on the wave dispersion characteristics. To that scope, we calculate the second gradient elasticity tensor from Eq. (5.11), without correction terms using the second gradient formulation described in Section 5.

A A A A A A Sym A A A A A A A Sym A A                 m hom 221122 hom hom hom 111122 221122 122122 8.7 0 0 0 2.5 N A A A A                          (5.25)
We calculate next the second gradient constitutive tensor including the correction terms as defined in Eq. (5.13) of the 2D matrix with the circular inclusion of Fig. As Fig. 5.9 shows, consideration of higher gradient terms for the longitudinal and transverse modes shows dispersive properties, which is not the case for the first gradient linear dispersion characteristics (in Blue). The relative difference between the higher gradient formulations without correction terms (Red) and the first gradient response (Blue) for the transverse mode (lower band) and for the longitudinal mode (upper mode) are 27% and 22% accordingly at 1 0.6 k m   , while introducing the correction terms (Green) this difference will decrease to 6% and 4% for the transverse and longitudinal modes accordingly.

A A A A A A Sym A A A A A A A Sym A                m*
N A A A A A                           (

Sensitivity of the dispersive frequency to the microstructure properties

We subsequently analyze the effect of microstructure geometrical and mechanical contrasts on the relative difference between the first (C) and the higher gradient approximations without and with corrections (SG and SG* accordingly) at for the transverse and longitudinal modes. For the transverse mode (Fig. 10a), the % of dispersive frequency will vary by 127% with the radius for the higher gradient behavior without the correction terms (SG), while when considering the correction terms (SG*) the variation with radius will decrease to a maximum of 72%. Meanwhile, the longitudinal mode variation is less important than for the first mode, while it is 27% for the SG approximation and 17% for SG*.

As Figs. 5.10c and 5.10d show, the % of dispersive frequency changes by 50% and 25% for the transverse mode using SG and SG* formulations versus the values of elasticity ratio between the matrix and the inclusion. Meanwhile, for the longitudinal mode it changes by 8% highlight that the internal length is more sensitive to the inclusion radius (R), over a wide range of variation of the elasticity ratio. For completeness, it has been shown that including higher gradient with correction terms for small geometric and mechanical contrast has no impact on the macroscopic static or dynamic behavior, whereby the classical first gradient is sufficient to describe the composite macroscopic behavior.

Moreover, the wave propagation analysis of Section 5.3.3.4 shows that the microstructural geometrical and mechanical parameters impact as a consequence the maximum frequency occurring at the wavenumber 0.6(1/ ) km  , these dependency being related to the mode number. Fig. 5.10a and 5.10b show that for high volume fraction of inclusions (high inclusion radius), the maximum difference between the first and second gradient dispersive frequency is increasing for the second gradient formulation with or without correction terms (SG*, SG) but this difference shows a smaller variation for the corrected formulation (SG*). Moreover, the variation of the dispersive frequency is increasing with the elasticity ratio  

/

EEand it shows an asymptotic convergence; this variation is less pronounced for the longitudinal mode or when the correction term is incorporated into the analysis. Fig. 5.10 also shows that this variation will depend on the mode under consideration: the transverse mode experiences a more significant variation than the longitudinal mode, the difference between modes resulting from the transverse internal length 33 le taking a higher value than the longitudinal length 11 le as analyzed in section 5.3.2.3.

Higher strain gradient effects can be obtained using architectured materials like

S K S K S K S K S K S K                                                         () N      (5.27)
Using these homogenized properties, we calculate the higher gradient internal length using Eq. (5.14) to get   These internal lengths shows the important of higher gradient constitutive parameters comparing to Cauchy behavior for the pantographic structure (Fig. 5.11).

Conclusion

Overall, the current work has analyzed for the first time the role of correction terms and microstructure mechanical and geometrical contrast of the individual constituents of composites (inclusion and matrix) on their static and dynamic response. In the static situation, we have shown that higher gradient internal length characteristics decrease when adding corrections terms which provides size-independent higher gradient constitutive parameters.

The internal length parameters are strongly dependent on the geometrical microstructure properties (the fraction of inclusion), whereas the microstructure mechanical properties (elasticity ratio between the matrix and the inclusion) have a lesser impact on the internal lengths which are converging at high contrast of elastic moduli. The sensitivity of the internal lengths to the microstructure properties is significant for the size-dependent higher gradient formulation, while adding corrections terms makes it less sensitive. Furthermore, the dispersive frequency obtained using the higher gradient formulation depends on the microstructure properties; this dependency is more prominent for the transverse mode than the longitudinal mode for the considered composite properties. We aspire that the provided results serve as a reference for the computation of the static and dynamic properties of composite structures and architectured materials in the broad sense specially for the ones which provides a prominent higher gradients constitutive behavior (as the discussed pantograph), providing a fundamental theoretical and numerical basis for the assessment of the relative significance of higher-gradient material parameters.

Conclusions and Perspectives

A higher gradient static and dynamic formulation has been developed to evaluate the classic and higher gradient constitutive parameters of heterogeneous structures, based on a dynamical homogenization method. To that scope, we made use of the Hill extended macrohomogeneity condition relating the macroscopic and microscopic strain energies. The calculated higher gradient constitutive parameters prove to be size-dependent, with strain gradient effective moduli increasing in a nonlinear manner with the unit cell number, but converging for a few repetitions of the unit cell. The incorporation into the formulation of correction terms based on the quadratic inertia tensor provides size-independent higher gradient constitutive parameters. This formulation has been used to analyze the static and dynamic mechanical response of composite structures.

As to dynamical aspects, the performed analyses evidence that a first gradient type behavior is not sufficient to describe the wave propagation features, since it is limited to a nondispersive behavior. To the contrary, higher order formulations have the ability to describe the dispersive behavior encountered in such microstructures materials. To that scope, we elaborate a discrete dynamics method (DDM) to compute the dispersion attributes of periodic structures, with a proposed methodology based on the higher order dynamic asymptotic expansion of the structure's inner forces and moments; the higher order terms of the proposed formulation provide a dispersive frequency of propagating waves.

Another modeling strategy followed in the thesis consists in incorporating higher gradient dynamic macroscopic constitutive parameters, which proves able to model the wave propagation characteristics of microstructured media. The dispersion frequency proves to be strongly dependent on the microscopic mechanical and geometrical contrast through the influence of the higher gradient constitutive parameters. This dependency is related to the chosen formulation of size effects, where the largest frequency values stand for the sizeindependent higher gradient formulation. As a matter of completeness, it has been shown that for small geometric and mechanical contrasts the classical Cauchy mechanics type formulation is sufficient to describe the composite macroscopic behavior. Secondly, we show that higher gradient contributions in statics depend on the selected nonclassical boundary conditions, whereas higher-order effects require highly non-linear strain profile distributions. While higher gradient constitutive formulation predicts much higher eigenfrequencies of the heterogeneous structure compared to Cauchy response, the consideration of micro-inertia contributions may counteract the effect of higher-gradient stiffness terms. Moreover, our analyses show that higher gradient effects mostly appear in the low range of internal length values that is relevant for a wide range of commonly used engineering materials. Applications of this methodology have been done to the dynamical study of vibrating laminated heterogeneous beams.

Several open questions need to be investigated before higher order homogenization should be considered an established method. What are the adequate boundary conditions that should be used? As we know, dispersion properties homogenization-based analyses are limited to small wavenumbers. As such, it remains an open question how to analyze higher gradient contribution for large wavenumbers.

In terms of perspectives, we have established a platform of models and simulation tools for the higher gradient homogenization of composites structures. These tools can be used to determine the higher order effective static and dynamical response of architectured materials, like the pantographic structure, which shows a prominent higher gradient contribution.

Different applications can be envisaged on such structures using the tools provided in this thesis to study their vibration and acoustic properties. Some work has been recently initiated in this direction with the group of Pr. Francesco dell'Isola (Department of Structural and Geotechnical Engineering, Rome, Italy) in order to investigate from an experimental point of view the higher gradient constitutive behavior for pantographic structure and to compare it with the homogenized model.

Furthermore, an extension of the developed finite element model to 3D is planned to calculate the homogenized properties of different architecture materials. An application on non-centrosymmetric structures can be done using the developed formulation to analyze the coupling tensors, which form the acoustical gyrotropic tensor that is responsible for the socalled acoustical activity.

The 3D model can be used to analyze all resonance modes for architecture materials, and to find the geometric and mechanical properties, which can increase the first eigenfrequency values for isolation purpose. Numerical simulations have to be combined with experimental investigations to validate the choice of the more adequate homogenization formulation and dynamical constitutive model.

( )

.cos( . ) .sin( . )

u x c x c x   (1.A.4)
In which the coefficient   It then holds:

  2 1 0 1 0 1 0 2 1 0 0 -.cos( . ) ( ) - -.
.sin( . ) . .cos( . ) sin( . ) .sin( . ).sin( . ) -cos( . )

.cos( . ). .cos( . ) sin( . )

u u l N x EA u x x l EA u x l u x u l x l                    (1.A.7)
We can now identify the values of forces at the extremities x=0 and x=l0:

T y EI c e c e c i e c i e

                                      (1.A.12)
In which parameter . In order to determine the coefficients of Eq. (1.A.12) we adopt the following boundary conditions:

0 0 0 0 0 0 0 0 1 1 2 3 4 -- 0 2 1 2 3 4 1 1 2 3 4 -- 0 2 1 2 3 4 (0) ( ) . . . . ( 0
                                    (1.A.13)
After solving this system and inserting the constants in the previous expression of the transverse force and moment we get:

      0 0 1 0 0 2 2 1 2 1 2 0 0 0 0 1 0 0 0 0 2 3 1 2 1 2
sin( ).sinh( ) . (cos( ) cosh( )).

.

( , , , )

. cosh( ).sin( ) cos( ).sinh( ) . (1 cos( ).cosh( ))

1 . sin( ) sinh( ) .

. ( , , , )

(1 cos

l l v l l v EI M v v l l l l l l l l EI T v v                                                        0 0 0 0 1 1 0 0 2 0 0 0 0 2 0 0
sin( ).cosh( ) sinh( ).cos( ) .

. sin( ) sinh( ) .

. sin( ).sinh( ) ( ).cosh( ))

.(cos( ) cosh( ))

l l l l v l l v l l l l l l                                         (1.A.14)
We assume that the bending wave length f  is much larger than the beam length, thus the compression coefficient   0 l  is very small, so forces can be developed in power of   0 l  using asymptotic expansion: Using the divergence theorem we get the new form of this integral which is equal to zero (using the assumption of displacement on the boundaries): We start to calculate using the integration by parts of each integral: We note that that the effective stress can be defined from the strain energy density () wE:

l l l l v v v v EI T v v l ll l v v o l                                   (1.
l l l l v v v v EI l v v l l l ol                                                (
    , 1 1
E K x K x u x E K x x d E K x K x K x K x u u E x u K x x u E K                                                      1 1 2 2 1 2 1 2 ijk k ji ijk j ki i i ij i j ijk j k i ij ij ij ij ijk ijk i i ij i j ijk j k i ij ij ijk ijk ij ij i i ij j ijk j k x K x u u E u x K x x u E K S u u E u x K x x u E K S u u E x K x x                                      (
, , , 1 , , 1 , ( , ) 2 1 2
ij ij i j j i ij ij ij ij ij i j ij ij ij ij i j ij i j ij j i ij j i w E K E U U w wdVdt wdVdt E dVdt E dVdt dVdt U dVdt E Integral E dVdt U dVdt U n dVdt U dVdt S U dVdt                                                       (2.A.5)
Similarly, the hyperstress tensor derives from the same strain energy density as:

  , ijk ijk w E K S K    (2.A.6)
Then, it holds due to Euler theorem for functions homogeneous of degree two:

1 2 2 ijk ijk ijk ijk Z K Z Z K S K      (2.A.7)
This entails the following variation (we note that in the following variation we will neglect the time constant because we will derive the final equation with respect to t):

      2 , , , , , , , , 2 , , 2 ,
                                               , 2 , , , 2 , , 2 2
P P U n dSdt U dVdt S U dVdt E X E X E                                                                                         (
                                                                                               2
P P P Integral E dVdt E E dVdt K K X K PP U n dSdt U dVdt X K X X K P S U dVd X X K                                                                            
                                                                         i U dVdt   
                                                                                                Vdt (2.A.16) 5 , b ij ij ij ijk ijk ijk T ij i j ijk k ijk T ij k ijk T T T Integral E dVdt E E dVdt K K t K TT E U dVdt t K X t K T U n dVd X t K                                                                                      
                                                                                                                 1

2.A.3 Simplification of Fi

In order to simplify the micro-inertia force which is in the last equation of motion  

T H E H K E x K x x E x H E H K E x K x x K x x H E H K E x K x x H E H K E x K                                      
C A E A E C A K A K C A E A K C A K H E H E H E H K H E E x H E K x x H K H E           1 2 K K K K ijkl jkl irst rst ijkl jkl iq q ijkl jkl iqr q r H K H K H K E x H K K x x               (2.B.1)
The macroscopic stress is the variation of the internal energy with respect to the macro- The dispersion of the transverse wave propagating in the direction normal to the layering will be obtained by replacing in Eq. (2.C.1)  

W E K E K W E K E K E t E A C A E A C A E A C A K A K C A H E H H E x H                                1

Appendix 3.A: Identification of micromechanical constants upon normal deformations

We hereby identify the micro-mechanical constants relating micro and macro-scale displacements and strains for the case of longitudinal deformations at the macro-scale of the structure. To that scope, we introduce displacement and strain localization operators, which we fully characterize with the use of boundary and interface conditions. In Fig.

( ) 2 ( ) 2 .

E x x a a a x K E x x b b b x K                                         (3.A.3)
Where  in Eq. (3.A.3) defines the first Lame coefficient. We use the interface continuity condition for the displacement, noting that first and second gradient terms ij E and ijk K can be treated as independent variables. The body forces are null within the cell, so we also use the continuity of stresses and stress gradients. As an additional condition, we use the average of the microscopic strains for the first and second gradient to be equal to the corresponding macroscopic strains, so that a sufficient number of equations is derived to determine all constants i a and i b . These conditions are summarized in Eq. (3.A.4), as follows:

                                             
                                                                                                                    a dd
                                                                                           2 2 2 2 1 2 1 2 1 1 1 2 1 2 1 1 2 1 2 2 1 1 4 1 1 2 1 2 2 2 , 2 2
K K K K K K                               (4.A.2)
We not that ijk K is symetric to the last two index.

The hyperstress is expressed versus the strain gradient tensor and the coupling tensor with the first gradient term as follows:

K K K K K A A A A A A K                               (4.A.4)
Restricting to the second order we obtain the following displacement and strain at microscale as function of macroscopic parameters as follows: Where E and K are the first and the second macroscopic strain gradient.

We evaluate using Eq. (B2) the following integral over the entire domain V:


          / 1 / 1 2 , / 2 1 EE           
.

Figure R.

  Figure R.1: Treillis carré avec ses éléments répétitifs de base (gris foncé) (a). Le noeud principal 1, ainsi que les quatre éléments nécessaires à la description de son équilibre discret (b). L'équilibre de la force normale, du cisaillement et des moments pour le noeud principal 1 est représenté en (c). .................................................................................................................. 17Figure R.2 : Courbes de dispersion pour une structure en treillis carré de la Fig. R1b en utilisant l'expression dynamique des forces et des moments intérieurs jusqu'au quatrième ordre pour le cisaillement et la solution issue du théorème de Bloch (lignes pointillées) (a). Courbes de dispersion du même réseau pour une expansion d'ordre supérieur (8ème ordre) du cisaillement interne et des moments (lignes continues) et solution du théorème de Bloch (lignes pointillées) (b). ....................................................................................................................................... 18 Figure R.3 : Géométrie périodique de matériaux composites stratifiés (à gauche) et les paramètres de la cellule unitaire (à droite). ............................................................................ 19 Figure R.4 : Courbes de vitesse de phase obtenues par le principe de Hamilton jusqu'au second ordre (Cyan), pour le premier gradient (vert), premier gradient avec effets de micro-inertie d'ordre supérieur (jaune) dans (Bacigalupo et Gambarotta, 2014b) et la solution exacte (rouge) fournie par (Sun et al.,1968). ................................................................................................ 19 Figure R.5: Fréquence des ondes longitudinales (a) et transversales (b) et vitesse de groupe pour les ondes longitudinales (c) et transverses (d) pour les deux méthodes d'homogénéisation dynamique en fonction de la valeur du nombre d'onde. ......................................................... 20 Figure R.6 : Déformation et déplacement le long d'une poutre en utilisant la formulation du second gradient pour une déformation fixé à l'extrémité de la poutre 0

1 :

 1 Figure R.1: Treillis carré avec ses éléments répétitifs de base (gris foncé) (a). Le noeud principal 1, ainsi que les quatre éléments nécessaires à la description de son équilibre discret (b). L'équilibre de la force normale, du cisaillement et des moments pour le noeud principal 1 est représenté en (c). .................................................................................................................. 17Figure R.2 : Courbes de dispersion pour une structure en treillis carré de la Fig. R1b en utilisant l'expression dynamique des forces et des moments intérieurs jusqu'au quatrième ordre pour le cisaillement et la solution issue du théorème de Bloch (lignes pointillées) (a). Courbes de dispersion du même réseau pour une expansion d'ordre supérieur (8ème ordre) du cisaillement interne et des moments (lignes continues) et solution du théorème de Bloch (lignes pointillées) (b). ....................................................................................................................................... 18 Figure R.3 : Géométrie périodique de matériaux composites stratifiés (à gauche) et les paramètres de la cellule unitaire (à droite). ............................................................................ 19 Figure R.4 : Courbes de vitesse de phase obtenues par le principe de Hamilton jusqu'au second ordre (Cyan), pour le premier gradient (vert), premier gradient avec effets de micro-inertie d'ordre supérieur (jaune) dans (Bacigalupo et Gambarotta, 2014b) et la solution exacte (rouge) fournie par (Sun et al.,1968). ................................................................................................ 19 Figure R.5: Fréquence des ondes longitudinales (a) et transversales (b) et vitesse de groupe pour les ondes longitudinales (c) et transverses (d) pour les deux méthodes d'homogénéisation dynamique en fonction de la valeur du nombre d'onde. ......................................................... 20 Figure R.6 : Déformation et déplacement le long d'une poutre en utilisant la formulation du second gradient pour une déformation fixé à l'extrémité de la poutre 0

Figure R. 2 :SG

 2 Figure R.1: Treillis carré avec ses éléments répétitifs de base (gris foncé) (a). Le noeud principal 1, ainsi que les quatre éléments nécessaires à la description de son équilibre discret (b). L'équilibre de la force normale, du cisaillement et des moments pour le noeud principal 1 est représenté en (c). .................................................................................................................. 17Figure R.2 : Courbes de dispersion pour une structure en treillis carré de la Fig. R1b en utilisant l'expression dynamique des forces et des moments intérieurs jusqu'au quatrième ordre pour le cisaillement et la solution issue du théorème de Bloch (lignes pointillées) (a). Courbes de dispersion du même réseau pour une expansion d'ordre supérieur (8ème ordre) du cisaillement interne et des moments (lignes continues) et solution du théorème de Bloch (lignes pointillées) (b). ....................................................................................................................................... 18 Figure R.3 : Géométrie périodique de matériaux composites stratifiés (à gauche) et les paramètres de la cellule unitaire (à droite). ............................................................................ 19 Figure R.4 : Courbes de vitesse de phase obtenues par le principe de Hamilton jusqu'au second ordre (Cyan), pour le premier gradient (vert), premier gradient avec effets de micro-inertie d'ordre supérieur (jaune) dans (Bacigalupo et Gambarotta, 2014b) et la solution exacte (rouge) fournie par (Sun et al.,1968). ................................................................................................ 19 Figure R.5: Fréquence des ondes longitudinales (a) et transversales (b) et vitesse de groupe pour les ondes longitudinales (c) et transverses (d) pour les deux méthodes d'homogénéisation dynamique en fonction de la valeur du nombre d'onde. ......................................................... 20 Figure R.6 : Déformation et déplacement le long d'une poutre en utilisant la formulation du second gradient pour une déformation fixé à l'extrémité de la poutre 0 SG   (Vert) et la

Figure 2 . 1 : 50 Figure 2 . 2 : 51 Figure 2 . 3 : 55 Figure 2 . 4 : 58 Figure 2 . 6 : 65 Figure 2 . 10 : 67 Figure 3 . 1 : 80 Figure 3 . 2 : 87 Figure 3 . 3 : 95 Figure 3 . 4 : 96 Figure 3 . 5 :

 2150225123552458266521067318032873395349635 Figure 2.1 : Propagating wave lengths Lc within repetitive network materials with repetitive unit cells of size l0. Scale separation applies when 0 c Ll  . ................................................. 50 Figure 2.2 : Forces and moments acting at the end nodes of each beam element (a). A single unit-cell structure in a one dimensional domain without transverse forces and moments (a) and its two neighboring cell representation (c). ........................................................................... 51 Figure 2.3 : Longitudinal dispersion curve obtained for the Cauchy effective continuum (in blue) and by the dynamic discrete method (in Red). .............................................................. 55 Figure 2.4 : Displacement of a series of 5 nodes in the 1D space, when the wave frequency is accounted for in the periodic solution. .................................................................................. 57 Figure 2.5 : A basic system of two-cells with an internal node v considered as the basic material unit (a). Two adjacent material units comprising a total of 4 unit-cells (b). .......................... 58 Figure 2.6: Dispersion curves for a two-cell system that includes an optical branch (blue line) and an acoustic branch (red line). .......................................................................................... 60 Figure 2.7 : Three-cells system Ω3 with two internal nodes 1 v and 2 v considered as the basic material unit (a). Two adjacent material units comprised of a total of 6 unit-cells (b). .......... 61 Figure 2.8 : Dispersion curve of the three-cells system including an acoustic (blue) and two optimal branches (green and red). ......................................................................................... 63 Figure 2.9 : Square lattice along with its basic elements (dark grey) in each material direction (a). The principal node 1, along with the four elements required for the description of its discrete equilibrium (b). The equilibrium of the normal, shear and moments for the principal node 1 is pictured in (c). ...................................................................................................................... 65 Figure 2.10 : Dispersion curves for a square lattice structure of Fig. 2.9b using the dynamic expression of the inner forces and moments up to the fourth order for the inner shear and moments along with the Bloch theorem solution (dashed lines) (a). Dispersion curves of same lattice for a higher order expansion (8 th order) of the inner shear and moments (continuous lines) along with the Bloch theorem solution (dashed lines) (b). ..................................................... 67 Figure 3.1: Homogenization concept from micro to macro model ......................................... 80 Figure 3.2: Periodic layered composite material geometry (left) and its unit cell material parameters (right). ................................................................................................................ 87 Figure 3.3: Homogenized second gradient and coupling tensors in function of layers thickness ratio d1/d2 for longitudinal (a) and for transverse (b). ............................................................ 95 Figure 3.4 : Second gradient characteristic length as a function of the elastic modulus ratio  for the case of longitudinal and b) for transverse deformations (b) for different thickness ratio values. .................................................................................................................................. 96 Figure 3.5: Phase velocity curves obtained by Hamilton principle up to second order (Cyan), for first gradient (green), first gradient with higher order micro-inertia effects (yellow) in (Bacigalupo & Gambarotta, 2014b) and the exact solution (red) provided by (Sun et al.,1968) (Appendix 2.C). .................................................................................................................. 100 Figure 3.6 : Variation of dynamic second gradient characteristic length as a function of the normalized wavenumber for different values of the phase velocity for the case of longitudinal (a) and transverse (b) deformation mode. ............................................................................ 102 Figure 3.7: Normalized phase velocity curves for longitudinal wave (a) and for transverse wave (b) obtained by the DHGE method with first and second gradient constitutive parameters (blue and red respectively). The exact solution is presented in black (Sun et al., 1968), while the yellow color curves correspond to the first gradient constitutive formulation with higher gradient inner micro-inertia terms provided in (Bacigalupo & Gambarotta, 2014). .............. 104 Figure 3.8: Frequency of the longitudinal (a) and transverse waves (b) and group velocity for the longitudinal (c) and transverse waves (d) for both dynamic homogenization methods as a function of the propagating wave number value. ................................................................. 105

100 Figure 3 . 6 :

 10036 ) (Appendix 2.C). .................................................................................................................. Variation of dynamic second gradient characteristic length as a function of the normalized wavenumber for different values of the phase velocity for the case of longitudinal (a) and transverse (b) deformation mode. ............................................................................ 102 Figure

  . .............. 104 Figure 3.8: Frequency of the longitudinal (a) and transverse waves (b) and group velocity for the longitudinal (c) and transverse waves (d) for both dynamic homogenization methods as a function of the propagating wave number value. ................................................................. 105

Figure 4 . 1 :Figure 4 . 6 :Figure 5 . 4 :Figure 5 . 5 :Figure 5 . 6 :Figure 5 . 8 :Figure 5 . 9 :Figure 5 . 10 :

 41465455565859510 Figure 4.1: A two-phase composite beam structure (a) with effective Cauchy (C), Second-Gradient (SG) and micro-inertia (𝜇𝐼) properties (b) which depend on its micro-scale material parameters. Upon loading, the obtained static (c) and dynamic (d) characteristics directly depend on the modeling formulation employed for the assessment of its structural response at the effective continuum scale. ............................................................................................. 118 Figure 4.2 : Strain and displacement along the beam using the second gradient formulation for 0 SG   (Red), and the linear solution for 0 SG   (Black) for a loading

2 :

 2 Constant values obtained by the Hamilton principle based on second gradient mechanics ............................................................................................................................. 99Table 3.3: Constants value obtained by the DHGE method. ................................................ 103 Table 4.1 : Mechanical properties of the materials used in the design of the microstructure of the composite beam of Fig. 4.1a. ........................................................................................ 118 Table 4.2 : Homogenized stiffness coefficients and internal length values for different Carbon and Glass-based unit-cell material designs using the material properties of Table 4.1 and the definitions of Eq. (4.3). ....................................................................................................... 122 Table 4.3 : Homogenized static and dynamic properties of the composite unit-cell material designs of Tables 1 and 2 for equal micro-scale layer thickness i d in Fig. 4.1a. .................. 129 Table 4.4 : Eigenfrequency values of the composite Silicon Carbide-Borosilicate Glass (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with micro-inertia (SG+μI) modeling formulation. ..................................................................... 130 Table 4.5 :Eigenfrequency values of the composite E-Glass and Epoxy (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with micro-inertia contributions (SG+μI) modeling formulation. ..................................................................... 131 Table 4.6 : Eigenfrequency values of the composite Carbon and Epoxy (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with micro-inertia effects (SG+μI). .................................................................................................................. 131 Table

  . ............................................................................... 157 Table

  ....................................................................................... 157

Figure R. 1 :

 1 Figure R.1: Treillis carré avec ses éléments répétitifs de base (gris foncé) (a). Le noeud principal 1, ainsi que les quatre éléments nécessaires à la description de son équilibre discret (b). L'équilibre de la force normale, du cisaillement et des moments pour le noeud principal 1 est représenté en (c). Les résultats de dispersion dynamique discrète sont validés par comparaison avec les prédictions du théorème de Bloch (Fig. R2) (Phani et al., 2006). Un développement d'ordre supérieur suffisamment élevé des forces et des moments des éléments structuraux est nécessaire pour décrire avec précision les modes de propagation d'ordre supérieur. A l'inverse, nous montrons que les modes de propagation longitudinaux et transverses peuvent être capturés en utilisant une expansion du quatrième ordre des forces et moments dynamiques. Ces résultats montrent que le calcul des caractéristiques de dispersion de systèmes structurels périodiques peut être abordés avec une bonne précision par la dynamique d'éléments discrets de type poutres.

Figure R. 2 :

 2 Figure R.2 : Courbes de dispersion pour une structure en treillis carré de la Fig. R1b en utilisant l'expression dynamique des forces et des moments intérieurs jusqu'au quatrième ordre pour le cisaillement et la solution issue du théorème de Bloch (lignes pointillées) (a). Courbes de dispersion du même réseau pour une expansion d'ordre supérieur (8ème ordre) du cisaillement interne et des moments (lignes continues) et solution du théorème de Bloch (lignes pointillées) (b).Au-delà de la méthode discrète développée, qui permet d'accéder à des propriétés acoustiques dispersives, nous développons dans cette thèse des méthodes d'homogénéisation dynamiques à gradient supérieur avec des effets de micro-inertie qui décrit le comportement au-delà des prédictions du milieu effectif de Cauchy[START_REF] Ayad | On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects[END_REF]. Dans cette optique, nous calculons les paramètres constitutifs macroscopiques jusqu'au second gradient en utilisant deux formulations, celle de l'énergie interne totale et le principe de Hamilton. Nous analysons ensuite la sensibilité des termes constitutifs du second gradient aux paramètres microstructuraux pour des matériaux composites à microstructure périodique de type laminés (Fig.R3).

Figure R. 3 :

 3 Figure R.3 : Géométrie périodique de matériaux composites stratifiés (à gauche) et les paramètres de la cellule unitaire (à droite). Les résultats suggèrent que l'importance des termes du second gradient dépend fortement du contraste de propriétés géométriques et mécaniques des constituants, ainsi que du mode de déformation, longitudinal ou transverse. De plus, l'étude de propagation d'ondes indique que les deux formulations utilisées conduisent à des diagrammes de dispersion qui diffèrent selon la plage de variation du nombres d'onde. En particulier, les modèles du second gradient formulés à partir du principe de Hamilton décrivent mal la plage de faibles nombre d'ondes par rapport au premier gradient ou aux solutions analytiques (Fig. R4).

Figure R. 4 :

 4 Figure R.4 : Courbes de vitesse de phase obtenues par le principe de Hamilton jusqu'au second ordre (Cyan), pour le premier gradient (vert), premier gradient avec effets de microinertie d'ordre supérieur (jaune) dans (Bacigalupo et Gambarotta, 2014b) et la solution exacte (rouge) fournie par(Sun et al.,1968).

Figure R. 5 :

 5 Figure R.5: Fréquence des ondes longitudinales (a) et transversales (b) et vitesse de groupe pour les ondes longitudinales (c) et transverses (d) pour les deux méthodes d'homogénéisation dynamique en fonction de la valeur du nombre d'onde. Le rôle des contributions d'ordre supérieur et de la micro-inertie sur le comportement mécanique de structures composites est analysé en calculant les propriétés statiques et dynamiques effectives de poutres composites en utilisant une méthode d'homogénéisation dynamique d'ordre supérieur qui intègre les effets de micro-inertie. Nous calculons la réponse effective statique longitudinale à gradient d'ordre supérieur, en quantifiant la différence relative par rapport à la formulation classique de type Cauchy qui repose sur la prise en compte du seul premier gradient du déplacement (Fig. R6).

Figure R. 6 :SG

 6 Figure R.6 : Déformation et déplacement le long d'une poutre en utilisant la formulation du second gradient pour une déformation fixé à l'extrémité de la poutre 0 SG   (Vert) et la

Figure R. 7 :

 7 Figure R.7 : Différence relative des valeurs de fréquence propre calculées pour le deuxième gradient (SG) et le deuxième gradient avec micro-inertie (SG + μI) ((a) et (b) respectivement) pour les trois premiers modes propres pour différentes valeurs de la longueur interne e l .Nous appliquons ensuite la méthodes d'homogénéisation à des composites en 2D en prenant en compte l'effet de taille par le biais des modules mécanique du comportement à gradients supérieurs. Dans cette optique, des termes de correction des modules du second gradient sont intégrés dans la formulation, ces derniers s'annulant pour des structures homogènes et étant indépendant de la taille du volume élémentaire représentatif(Li, 2011), corrigeant ainsi les déficiences des modèles du second gradient de la littérature. On considère une matrice comportant une inclusion circulaire centrale; afin de quantifier l'importance des effets du second gradient, on calcule les longueurs caractéristiques en fonction des contrastes de paramètres géométrique et mécanique de la structure (Fig.R8).

Figure R. 8 :

 8 Figure R.8: Variation des longueurs internes

Figure R. 9 :

 9 Figure R.9: Courbes de dispersion en utilisant une approximation classique (Bleu), une approximation du second gradient sans termes de correction (Rouge) et avec des termes de correction (Vert) pour le mode de cisaillement (bande inférieure) et le mode longitudinal (bande supérieure) dans la première partie de la zone de Brillouin de la matrice. On remarque que les gradients supérieurs apportent de la dispersion au comportements acoustique de la structure ; la différence entre les formulations à gradients supérieurs et la théorie classique est analysée pour différentes fractions de l'inclusion et différents rapport d'élasticité entre la matrice et l'inclusion en Fig. R10.

Figure R. 10 :

 10 Figure R.10 : Différence relative maximale des courbes de dispersion entre la formulation du second gradient pour

(

  Gambarotta, 2014). Different strategies can be followed to reach properties beyond those of conventional materials, including adjusting the degree of anisotropy, the average density by tuning the inner microstructural design, or designing microstructures with noncentrosymmetric or chiral features to activate interactions between different deformation modes and thereby access new effective properties. Examples of architected materials designs prone to strong anisotropy or exhibiting chiral effects are given in Fig. 1.1.

Figure 1 . 1 :

 11 Figure 1.1: Architected materials with anisotropic designs (left),(Karathanasopoulos et al.,

Figure 1 . 2 :

 12 Figure 1.2: Dispersion analysis on a hexachiral material with superposed domains of validity of the continuum models (Rosi & Auffray, 2016). Amongst the employed micromechanical methods, asymptotic homogenization is quite appealing and has developed importantly in the field of wave propagation over the last decade (Fish and Chen, 2001; Guenneau et al., 2013; Bacigalupo and Gambarotta, 2010, 2014). Boutin and Auriault (1993). Extending the approach developed by Bensoussan et al., 1978; Sanchez-Palencia, 1980, and Bakhvalov and Panasenko, 1984) have proposed the long wave propagation in periodic media based on homogenization. The asymptotic expansion of the displacement field includes terms up to the third order to describe wave scattering. The high frequency behavior of composite structures has been first analyzed in (Boutin et al., 2012, 2014). The authors employ an asymptotic multiscale method to address large scale modulations of high frequency mechanical waves. The developed theoretical formulation allows the identification of the physical behavior of composite structures in frequency ranges that are not accessible by the methods typically used for low frequency analysis.

Figure 2 . 1 :

 21 Figure 2.1 : Propagating wave lengths Lc within repetitive network materials with repetitive unit cells of size l0. Scale separation applies when 0 c Ll  .

Figure 2 . 3 :

 23 Figure 2.3 : Longitudinal dispersion curve obtained for the Cauchy effective continuum (in blue) and by the dynamic discrete method (in Red).

Fig. 2 .

 2 Fig. 2.3 suggests a separation of the two solutions obtained by the Cauchy continuum and by the DDM after a certain wavenumber magnitude or accordingly frequency value. In particular, in the low frequency regime, the band diagram corresponding to wavelengths that are very large compared to the unit cell size (Fig. 2.1), the Cauchy and the dynamic discrete solution of (2.8) coincide. More specifically, noting that

2 . 3 .

 23 To that scope, we employ as a limiting point for our analysis the wavenumber 0 beyond which the Cauchy and discrete solution deviate, as shown in Fig.2.3. For this value, the longitudinal displacement of an intermediate node n (Fig.2.2c)

  if 21 nq  (odd integer) then the displacement at the corresponding node is 2 1 0

Figure 2 . 4 :

 24 Figure 2.4 : Displacement of a series of 5 nodes in the 1D space, when the wave frequency is accounted for in the periodic solution.

Fig. 2 .

 2 Fig. 2.4 indicates that when the frequency of the propagating waves is accounted for in the system's periodic solution   0 3 ww , the displacement of the 5 th node is identical to that of

  2.6 in the interval of the Brillouin zone (Brillouin & Parodi,

Figure 2 . 6 : 2 . 3 .

 2623 Figure 2.6: Dispersion curves for a two-cell system that includes an optical branch (blue line) and an acoustic branch (red line).

  starting point of the descending optical branch on Fig.2.6), the previous linear system of equations writes 3 3 0 uv    for both equations of motion in Eq.

  the denominator of Eq. (2.20) to zero and are the resonance frequencies of the system, for which the displacement tends to infinity. Applying the continuity of forces on node n between two neighboring cells Ω3 (Fig 2.7b) and using Eq.

( 2 .

 2 22) provides the dispersion relation of the 3-cells system, which is plotted in the Brillouin zone 0

Figure 2 . 8 :

 28 Figure 2.8 : Dispersion curve of the three-cells system including an acoustic (blue) and twooptimal branches (green and red).

(

  Phani et al., 2006; Reda et al., 2018; Karathanasopoulos et al., 2019). The dynamic equilibrium under the action of zero external forces then writes as follows:

Figure 2 . 10 :

 210 Figure 2.10 : Dispersion curves for a square lattice structure of Fig. 2.9b using the dynamic expression of the inner forces and moments up to the fourth order for the inner shear and moments along with the Bloch theorem solution (dashed lines) (a). Dispersion curves of same lattice for a higher order expansion (8 th order) of the inner shear and moments (continuous lines) along with the Bloch theorem solution (dashed lines) (b).

1 .

 1 14 for the extreme wave number values of mode, the low order discrete dynamics asymptotic solution is a clear multiple of it (Fig 2.10a

Chapter 3 :

 3 On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Summary

  appealing and has developed importantly in the field of wave propagation over the last decade, (seeFish and Chen., 2001; Guenneau et al., 2013; Bacigalupo and Gambarotta, 2010, 2012, 2014 a,b ).Boutin and Auriault (1993), extending the approach developed byBensoussan et al. (1978), Sanchez-Palencia (1980) andBakhvalov and Panasenko (1984), have proposed the long wave propagation in periodic media based on homogenization. The asymptotic expansion of the displacement field includes terms up to the third order to describe wave scattering.Homogenization in the time domain has also been proposed by several authors, see e.g. (Chen and Fish, 2000; Vivar-Pérez et al., 2009), considering both fast and slow time scales. In many studies, gradient elasticity theories have been derived from the continualisation of the response of a discrete lattice consisting of discrete masses and springs, see for example the early work ofMindlin (1972). Note that one of the biggest challenges of gradient elasticity theories is a physical interpretation of the involved length-scale parameters as a function of the given microstructure(Askes and Aifantis, 2011).Resorting to computational approaches, finite element computations over the unit cell with periodic Bloch boundary conditions have been performed to model acoustic waves scattering and to evaluate the band structure(Hussein, 2009; Huang, 2011). As an alternative strategy for the solution of the Floquet-Bloch problem, plane wave expansions of the displacement vector, elastic moduli and mass are done in Fourier series with respect to the periodicity vectors (Nemat-Nasser et al., 2011-2012).

Figure 3 . 1 :

 31 Figure 3.1: Homogenization concept from micro to macro model

  Therefore, it follows from (3.13) the expansion of the macro and micro kinetic energy terms successively as:

Figure 3 . 2 :

 32 Figure 3.2: Periodic layered composite material geometry (left) and its unit cell material parameters (right).

  27) Eq. (3.27) contains a total of 10 unknown parameters i a and i b . The micro-stress developed in each inner material phase is given as a function of the micro-strains of Eq. (3.27), as follows:

29 )Chapter 3 :

 293 On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 89The 9 equations in the set of Eqs. (3.29) do however not suffice to calculate the 10 coefficients appearing in Eq.(3.27). As an additional condition, we require the average of the microscopic deformations for the first and second gradient to be equal to the corresponding macroscopic deformations, as follows:

( 3 .

 3 29). Solving the system of Eqs. (3.29, 3.30) with the use of Eq. (3.27) and (3.28), we obtain the following expressions of the microscopic coefficients i a and i b :

36 )

 36 In order to calculate the constants i a and i b , we use the continuity conditions for the displacement at the interface, which provide two equations; the boundary conditions for the displacements provide 4 equations and the continuity of stress and stress gradients at the interface provide two and one equilibrium equation respectively. What is more, we use the Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 92

  the constants of the displacement vectors in each layer, we use Eq. (3.18) or the static part of Eq. (3.24) to calculate the homogenized tensors of first gradient elasticity, hom C (in MPa ), the coupling tensor hom hom , BD (in MPa m  ) and the second gradient tensor hom A (in 2 MPa m  ) respectively, as a function of the geometric attributes of the unit-cell. In Eq. (3.39), we provide the corresponding expressions as a function of the layers thickness ratio 12 / dd   and of the total unit-cell length L of the cell, as follows: identical properties, the set of Eqs. (3.39) provides a first gradient tensor hom 1111 C which is identical to the constitutive tensor of both layers. In Fig. 3.3a, we depict the homogenized tensors as a function of the geometrical ratio 12 / dd   for equal material moduli in each inner material phase. What is more, we explore the sensitivity of the second gradient terms on the discrepancies of the inner material properties in the layered composite medium for the case of equal inner layer thickness   12 dd  . Thereupon, we calculate the characteristic second gradient length 111 () e lm as a function of the elastic modulus ratio

3.4a. Chapter 3 : 40 )

 340 On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects The characteristic length of Eq. (3.40) for equal layers thickness and for the first layer being 1.5 times the thickness of the second (

( 3 .

 3 41) in Fig.3.3b.

Figure 3 . 3 :

 33 Figure 3.3: Homogenized second gradient and coupling tensors in function of layers thickness ratio d1/d2 for longitudinal (a) and for transverse (b).

Fig. 3 .

 3 Fig. 3.3 indicates that upon increasing geometric discrepancies of the layers within the unitcell, the higher gradient contributions become more significant. We subsequently analyze the sensitivity of the second gradient terms on the moduli ratio value

Figure 3 . 4 :

 34 Figure 3.4 : Second gradient characteristic length as a function of the elastic modulus ratio  for the case of longitudinal and b) for transverse deformations (b) for different thickness ratio values.

Fig. 3 .

 3 Fig. 3.4b suggests that as the contrast of the microscale material properties increases, the corresponding significance of second gradient effects increases for the transverse propagation mode. Thereupon, the first gradient is increasingly insufficient to describe the behavior of the

  For the computation of the wave propagation attributes, a harmonic wave propagation Ansatz is considered, with a displacement d U being a function of the propagating wave numbers and frequency components, as follows:

Figure 3 . 5 :

 35 Figure 3.5: Phase velocity curves obtained by Hamilton principle up to second order (Cyan),

2 . 3 1 d

 231 to study the wave propagation characteristics of laminated media upon both longitudinal and transverse deformations. Using the dynamic first and second gradient stress tensors evaluated in Section 3.2.3 (Eq. (3.22)) we compute the dispersion characteristics of the heterogeneous laminated medium when imposing a longitudinal harmonic wave normal to the layering. In this part we use the homogenized expressions of the dynamic tensors in function of the frequency, Eq. (3.24), for both longitudinal    and transverse   2 d  modes as follows:

The constants 1 d C to 6 dC

 16 entering Eq.(3.51) are summarized in Table3.3 for the case of the material properties listed in Table3.1. Using the expressions of Eq. (3.51), we compute the sensitivity of the second gradient characteristics lengths   kL for different values of the phase velocity, for the case of longitudinal (Fig. 3.6a, 11 / cc ) and transverse deformations (Fig. 3.6b, 22 / cc ). Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 102

Figure 3 . 6 :

 36 Figure 3.6 : Variation of dynamic second gradient characteristic length as a function of the normalized wavenumber for different values of the phase velocity for the case of longitudinal (a) and transverse (b) deformation mode.

 2 Fig. 3 .

 23 Fig. 3.7b the ones corresponding to the transverse mode.

Figure 3 . 7 :

 37 Figure 3.7: Normalized phase velocity curves for longitudinal wave (a) and for transverse wave (b) obtained by the DHGE method with first and second gradient constitutive parameters (blue and red respectively). The exact solution is presented in black(Sun et al., 1968), while the yellow color curves correspond to the first gradient constitutive formulation with higher gradient inner micro-inertia terms provided in(Bacigalupo & Gambarotta, 2014).

Fig. 3 ./Chapter 3 :

 33 Fig. 3.7 indicates that the DHGE method provides a very good agreement with the exact solution (black) over the entire range of normalized wave number values both for the longitudinal and for the transverse mode. The first gradient results (blue) are very close to the first gradient constitutive modeling results with higher gradient micro-inertia terms (yellow curves) in the left plot of Fig. 3.7a.We subsequently compute the normalized frequency

Figure 3 . 8 :

 38 Figure 3.8: Frequency of the longitudinal (a) and transverse waves (b) and group velocity for the longitudinal (c) and transverse waves (d) for both dynamic homogenization methods as a function of the propagating wave number value.

Figures 3 .

 3 Figures 3.8a and 3.8b indicate that the exact solution predicts a maximum frequency value for a propagating wave number kL   , with a continuous frequency gradient, so that '( ) '( ) ww   

Chapter 3 :

 3 (3.5) contribute solely to the potential energy part P and not to the kinetic energy part T  of Eq.(3.11), in which only higher strain gradient terms appear (Eqs. (3.11)-(3.13)). The overestimation of the phase velocity for Hamilton-based methods in the wavenumber region kL   is already visible for first gradient constitutive modeling results (green curves in Fig. 3.5), an observation made in previous contributions (Wang and Sun 2002, Sun and Huang 2007). In this wavenumber space and upon the Hamilton formulation, the Floquet-Bloch results are best approximated by higher asymptotic expansion contributions of the micro-inertia terms with the macroscopic constitutive parameters retained to their first-order approximation (Bacigalupo & Gambarotta, 2014b). However, the dispersion curves retain a discontinuous rate of change at kL   , contrary to the continuous elasticity theory predictions. On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 107

(

  Hermitian, Element-Free Galerkin formulations). By that means, certain postulations for the micro-scale mechanical behavior are introduced (Polyzos & Fotiadis, 2012). Accordingly, homogenization schemes have been mainly based on asymptotic formulations, which rely on the expansion of the spatial and temporal parameters to fast and slow varying components (Andrianov et al., 2008; Brito-Santana et al., 2015; Hui & Oskay, 2014). Thereupon, different models have been elaborated and used to characterize the dispersion properties of media with an underlying periodic, layered microstructure (Wautier & Guzina, 2015; Bennett et al., 2007; De Domenico & Askes, 2016).

2  with thickness 1 d and 2 d(

 22 4.1a), with a twophase unit-cell material design of length l . The unit-cell contains materials with different elastic moduli values 1 E and 2 E as well as density values 1  and Fig. 4.1a). The static and dynamic characteristics of the structure (Fig. 4.1b) related to the Cauchy (C), second-gradient (SG) and micro-inertia ( I  ) properties are computed using the framework elaborated in Section 4.2.2. The dependence of the effective static (Fig. 4.1c) and dynamic (Fig. 4.1d) structural properties on the microstructural design parameters and modeling order are analyzed in Section 4.3.

Figure 4 . 1 :

 41 Figure 4.1: A two-phase composite beam structure (a) with effective Cauchy (C), Second-Gradient (SG) and micro-inertia (𝜇𝐼) properties (b) which depend on its micro-scale material parameters. Upon loading, the obtained static (c) and dynamic (d) characteristics directly depend on the modeling formulation employed for the assessment of its structural response at the effective continuum scale.

( 4 . 2 )

 42 Where in Eq. (4.2), the coefficients hom ijdl C stand for the first order constitute stiffness component, while the coefficients hom ijrst B and hom ijkdl D for the coupling stiffness terms between the first and second order stress and kinematic parameters, which are non-zero for non centrosymmetric microstructures. The coefficients hom ijkrst A represent the purely second gradient constitutive moduli. For the calculation of each constitutive stiffness parameter of Eq. (4.2), localization operators need to be employed, relating micro-displacements and micro-strains with the corresponding macro-strain and strain gradient tensors dl E and rst K .The localization operators relating the micro-displacements to the strain components are denoted with E H and K H respectively, while the localization operators relating micro and macro strains are E A and K A accordingly. The constants included in the displacement and strain localization operators (Eq. 3.A2) are identified using the boundary and continuity conditions for the first and the second gradient terms (Eq. 3.A4). They are functions of the individual material parameters contained in the different material phases of the unit-cell structure (Eq. 3.A5), as elaborated in Appendix 3.A. For the case of longitudinal loading, the stiffness coefficients of Eq. (4.2) express in a simplified form as a function of the localization operators as follows:

3 C and 1 4 C 1 5, the stiffness coefficient 1 2 C

 3412 entering Eq. (4.4) cancel out. In Eq. (4.6), the root of the ratio of the second to first gradient coefficients characterizes the micro-inertia contribution part, represented by the second term on the left hand side of Eq. (4.6). We note that Eq. (4.6) simplifies to the classical Cauchy mechanics description of the medium, once the micro-inertia and second gradient contribution terms are neglected, as follows:

1 1C and 1 5 C 1 C and 1 5 C

 1515 accordingly), we make use of the constitutive expressions of Eq. (4.3). Different unit-cell material designs are employed relevant to engineering praxis (Tan & Poh, 2018). In particular, Borosilicate Glass and Silicon Carbide, E-Glass and epoxy, as well as Carbon and Epoxy unit-cell designs are used in equal volumetric proportions within the unit-cell structure (Tan & Poh, 2018). Each of them leads to different first and second gradient ( 1 accordingly) stiffness coefficients, as well as effective internal length values e l , as summarized in

  set the rate of strain change at the middle of the beam structure equal to zero value at the start of the beam equal to a non-classical, non-Cauchymechanics value SG  that can be in general different from the constant strain profile corresponding to the Cauchy-mechanics solution of the problem

  properties summarized in Table4.2). For the strain-related end boundary condition, we employ the limiting case of a zero end-strain   0 SG   , leading to a highly non-linear strain profile evolution, as depicted in Fig.4.2a.

Figure 4 . 2 :

 42 Figure 4.2 : Strain and displacement along the beam using the second gradient formulation for

Fig. 4 .

 4 Fig. 4.2 highlights a strong boundary layer effect occurring at both domain edges, since the local strains for the boundary condition

2 .

 2 7% and 2% with respect to the Cauchy mechanics solution for Borosilicate Glass and Silicon Carbide. In Fig. 4.3, we analyze the sensitivity of the longitudinal static response on the internal length value of the material, employing e l values that way exceed the range of internal length values computed for the common engineering material designs ofTable 4.2.

Figure 4 . 3 :

 43 Figure 4.3 : Macroscopic strain (a) and displacement (b) profiles for a total of four internal length values, starting from the Cauchy-mechanics case (

(

  blue curve) for the artificial material designs with internal length values of 0

R and 2 RChapter 4 :

 24 can be directly determined by the boundary conditions of the structure. For the case of zero end displacements ( (0 ) 0 ZL  ), we determine the values of k which give a unique solution of the constants 1 Higher-gradient and micro-inertia contributions on the mechanical response of composite beam structures Page | 127 Where  kL  in Eq. (4.14) stands for the normalized wavenumber. We insert the roots of Eq. (4.14) into the frequency expression of Eq. (4.10) to obtain the eigenfrequency values for the different modes. In Eq. (4.15), we provide the first three eigenmodes for all unit-cell from

  (4.15) with the ones obtained out of the dynamic, finite element analysis (Abaqus) of the composite beam structure for the case of a micro-scale design composed by Borosilicate Glass and Silicon Carbide, as summarized in Table4.1, using the same boundary conditions of Section 4.3.2.1. We provide the mesh and the displacement profiles for the first three eigenmodes along with the corresponding frequency values in Fig.4.4:

Figure 4 . 4 :

 44 Figure 4.4: Mesh (a) and longitudinal displacement profiles for the first three eigenfrequency values (b, c and d accordingly) along with the corresponding eigenfrequency value for the Borosilicate Glass and Silicon Carbide unit-cell design of Table 4.2.

1 2C

 1 We subsequently compute the first three eigenfrequencies of the composite beam structure of Fig.4.1b (L=10mm), considering the different unit-cell material property combinations of Tables 4.2 and 4.3. In Table4.4, we provide the normalized eigenfrequency values for a unit-cell structure composed of Borosilicate Glass and Silicon Carbide. For each mode, the Cauchy mechanics eigenfrequency ( C  , Section 4.3.2.1) is computed, along with the second-gradient eigenfrequency value ( SG  ) for which the micro-inertia effects are disregarded ( is set to zero in Eq. (4.20)) and the eigenfrequency value where both secondgradient and micro-inertia contributions are considered (

1 C 2 and 4 . 3 . 2 C

 12432 3.2.2 provide evidence that the role of micro-inertia on the effective structural dynamic properties depends both on the magnitude of the internal length parameter, as well as on the mode of interest. We subsequently analyze the sensitivity of the three primal longitudinal eigenfrequency values on the value of the internal length parameter e l . To that scope, we use the Cauchy mechanics stiffness 1 and relative density value hom  computed for the composite Silicon Carbide and Borosilicate Glass material case in Tables 4.In Fig. 4.5, we compute the sensitivity of the eigenfrequency values of Eq. (4.16) on the micro-inertia contributions 1 , for the three primal structural modes for different values of the internal length e l .

Figure 4 . 5 : 2 C 0 C

 4520 Figure 4.5 : Relative difference of the eigenfrequency values computed for the secondgradient (SG) and the second-gradient with micro-inertia (SG+μI) formulation (subplots (a) and (b) accordingly) for the first three eigenmodes for different values of the internal length

Figure 4 . 6 :

 46 Figure 4.6 : Relative difference of the SG

Fig. 4 .

 4 Fig. 4.6 indicates that the differences observed in between the second-gradient and microinertia formulation (SG+μI) and the Cauchy-mechanics (C) results depend, not only on the value of the internal length parameter e l , but also on the ratio of the density values of the material phases contained in the microstructure of the material. Figs. 4.6a, 4.6b and 4.6c reveal the nature of the dependence for a given internal length value: Upon increasing ratios of the density values, the difference to the Cauchy mechanics predictions increase. The effect appears in the low range of e l values   0.25 e l mm  and is more prominent for low values of the 2 1 / 

  strain fluctuation. The strain energy of the effective strain gradient continuum,   W, M EK , writes as a bilinear form of the kinematic measures, adopting in the sequel a small strains formulation (so energies are expressible as quadratic functions of their kinematic arguments): elasticity tensor of the material at the microscale, hom ijklmn A therein the sixth order tensor of strain gradient moduli, and hom ijklm B a fifth order tensor of coupling moduli between first and second gradient effects; the presence in Eq. (5.3) of the sum   hom hom,T ijklm ijklm BB  is a consequence of the existence of a strain energy potential which guarantees the symmetry of the effective overall stiffness matrix, by Schwarz relation.The macroscopic strain gradient energy density entails the following strain gradient constitutive law at the macroscopic level, relating the average Cauchy stress ij  and hyperstress ijk S to the strain and strain gradient tensors

5 )

 5 The stationarity condition of the functional on the right-hand side of (5.5) delivers as a necessary condition a BVP to be satisfied by the optimal fluctuation associated to the real displacement field (in the absence of body forces) as follows:

  moments of inertia are components of the second order tensor importance of second gradient constitutive parameters by the internal length parameters, elaborated as the ratio of the second gradient constitutive parameters to the classical Cauchy tensor. We use the internal length formula from(Goda & Ganghoffer, 2016) with simplified Voigt notation as follows, in which the free Greek indices vary from 1 to 3:

,

  xy vv are the test function components. Using FreeFem, we carry out the unit cell problems of Eq. (5.16) on the composite structure (Fig. 5.1), we generate a triangular mesh with 15628 triangles and 8115 vertices. Solving the first gradient unit-cell problem leads to identify the first gradient longitudinal localizators 11 E i H , which are inserted in Eq. (5.11) to evaluate the rigidity tensors of first gradient and coupling between first and second gradient  

  .(5.7), by imposing a longitudinal strain gradient to the unit cell  expression of the micro-strain in Eq. (5.8b), we write the weak form of BVP 2 for the second gradient as follows:

  circular hole or an inclusion with radius R at the matrix center as in Fig. 5.1. According to (Auffray et al., 2015) coupling tensors will be negligible for such centrosymmetric structures, so we focus in this study on the first and second gradient constitutive tensors.

Figure 5 . 1 :

 51 Figure 5.1: Centrosymmetric matrix and central hole (a) and reinforcement material (b).

Figure 5 . 2 :

 52 Figure 5.2: Square pattern (a) with three different square unit-cell choices (b, c, d and e) of dimension l , 2l ,3l and 4l accordingly.

Figure 5 . 3 :

 53 Figure 5.3 : Second gradient constitutive moduli variation for different RVE sizes without correction terms hom ijklmn A

Fig. 5 .

 5 Fig. 5.3a shows that second gradient effective moduli without the correction are diverging for large RVE sizes. Contrariwise, when adding the correction terms of Eq. (5.13), the second gradient constitutive moduli are independent of the RVE size as shown in Fig. 5.3b.

( 5 . 12 /. 5 .

 5125 14), elaborated as the ratio of second gradient to first gradient equivalent moduli. In the sequel, we analyze the influence of inclusion radius R, elasticity ratio of the matrix to the inclusion  EEon the internal length parameters, using the size dependent higher gradient constitutive parameters hom ijklmn A as defined in Eq.(5.11). To that scope, we use the developed formulation in Section 5.2.1 to compute the first and the second gradient constitutive parameters without correction for the composite structure in Fig.5.1b, for different values of inclusion radius R for  4c and 5.4d accordingly.

Figure 5 . 6 :

 56 Figure 5.6 : Mesh of the composite structure (a), and displacement shape for the transverse (b) and longitudinal (c) propagating modes.

Figure 5 . 7 :

 57 Figure 5.7 : Longitudinal (Green) and transverse (Blue) wave dispersion (b) through the 2D composite structure from Fig. 5.1b in Brillouin zone (a).

Fig. 5 .

 5 Fig. 5.7b shows the dispersion diagram for the shear (Blue) and longitudinal modes (Green) over the entire Brillouin zone (Fig. 5.7a) of the composite unit cell (Fig. 5.1b). According to (Rosi & Auffray, 2016), the homogenization-based dispersion curves are restrained in the low wavenumber range, therefore, we subsequently compute the dispersion curves in the range of wavenumbers



  5.7b restricting to small wave numbers and to the low frequency domain. Therefore, we consider the expression of the wavenumber in zone (OA) of Fig.5.7a   to get the following dispersion curves:

Figure 5 . 8 :

 58 Figure 5.8: Dispersion relation of the composite structure (Fig. 5.1b) using Cauchy approximation (Blue), and COMSOL with the diamond points for the shear (lower band) and longitudinal modes (upper band).

Fig. 5 .

 5 Fig. 5.8 shows a good agreement between the first gradient dispersion solution (blue) and

5 . 26 )Chapter 5 :

 5265 Wave Propagation in 2D composites Structure using Higher Gradient Size Independent Homogenization Page | 165 Using the second gradient tensors in Eq. (5.25, 5.26) we derive the dispersion relation proceeding in complete analogy to Section 5.3.3.2 but now incorporating the second gradient contribution to the wave dispersion in Eq. (5.19). In the figure below, we present the wave dispersion using both Cauchy first gradient approximation and the higher gradient behavior with and without the correction terms of Eq. (5.13) (SG and SG* accordingly) for small wave numbers in the interval

Figure 5 . 9 :

 59 Figure 5.9: Wave dispersion diagram using the first gradient theory (Blue), the higher gradient approximation without correction terms (Red) and with correction terms (Green)for the shear mode (lower band) and the longitudinal mode (upper band).

  5.10. In Figs. 5.10a and 5.10b, we analyze the relative difference (% of the dispersive frequency) for different inclusion radius R with an elasticity ratio equal to 100 between the matrix and the inclusion   12 / 100 E E MPa  for both transverse and longitudinal modes accordingly, and for different elasticity ratio   5.10c and 5.10d for both transverse and longitudinal modes accordingly.

Figure 5 . 10 :

 510 Figure 5.10 : Maximum relative difference in the dispersion curves at

  pantographic structures (dell'Isola et al., 2019; Rahali et al, 2015) as the one shown in Fig. 5.11 , where the structures is made of a linear elastic isotropic material with an elasticity coefficient and Poisson's ratio   2500 , 0.3 E MPa v  accordingly. Using the higher gradient formulation described in section 5.2.1 we calculate the following first and second gradient homogenized tensors (Eq. (5.27)) of the pantographic structure on the following mesh:

Figure 5 . 11 :

 511 Figure 5.11: Pantographic structure with perpendicular diagonal.

1

 1 Second gradient from surface to volume integrationTo derive the energy equation which link the macro-energy with the micro-energy and displacement we start by the following integral:

( 2 .

 2 A.18), we use the localization tensors of displacement and for deformation to simplify the form of F i . Starting with the potential energy P :

( 2 .B. 3 )

 23 Appendix 2.C: Elasticity SolutionUsing the elasticity theory and Floquet's theorem(Sun et al., 1968) they derive the exact dispersion relation of the longitudinal wave propagating in the direction normal to the layering Eq. (2.C.1) as follows:

3 . 2

 32 we summarize the geometric and material parameters of the two material phases contained within the unit-cell of the microscale, further elaborating the definitions of Fig. 4.1a. ( )

( 3 .A. 4 )

 34 Solving the system of Eqs. (3.A.4) with the use of Eq. (3.A.2) and (3.A.3), one obtains after straightforward computations the following expressions for the microscopic coefficients i a and i b :

5 )

 5 We note that an opposite sign has been used for the boundary conditions of the displacement to the one provided in Chapter 3, which leads to physically accepted results in the problems considered. The coefficients of Eq. (3.A.5) fully define the stress and strain definitions of Eqs. (3.A.2) and (3.A.3) and thus the macroscale stiffness definitions of Eq.

.,

  We identify from Eq. (4.B5) the first and second gradient constitutive tensors as follows:  in Eq.(4.C.1) are the Lame coefficients, evaluated versus the Young modulus and Poisson's ratio as

especially the impact of the higher-order inertial terms (see Metrikine and Askes, 2002; Askes and Aifantis, 2009;

	; Askes and Aifantis
	2009; Reda et al. 2018) and plasticity (see Askes and Metrikine, 2002; Askes et al., 2008;
	Askes and Aifantis, 2011; Askes et al., 2012). Their formulation has aimed at describing both
	static and dynamic phenomena (Gourgiotis and Georgiadis, 2009; Gonella and Ruzzene, 2010;
	Gonella et al., 2011; Ostoja-Starzewki, 2002; Papargyri-Beskou, 2004; Askes and Aifantis,
	2011).
	Gradient elasticity models are useful to predict the wave dispersion characteristics in
	heterogeneous materials or discrete systems (see Mindlin, 1964; Muhlhaus and Oka, 1996;
	Gonella et al., 2011, Gonella and Ruzzene, 2010). A thorough analysis of the effects brought
	about by gradient elasticity models can be found in (Papargyri-Beskou et al., 2009; Fafalis et
	al., 2012),

Table 2 .

 2 1 : Geometric and mechanical characteristics of the beam.

			Element geometry			Mechanical Properties
		Area/Height		/ Ah		Density	 (kg/m 3 )
		Length			0 l			Young's modulus	E (MPa)
	Quadratic moment of inertia	 I Ah 	2 /12		Eigenfrequency	 00 / w E l 	2	
	dN dx	Au	,	dT dy		Av
	-,-du N EA M dx	EI	2 dv 2 dy



In order to calculate the dynamic asymptotic expressions of the longitudinal force N, transverse force T and moment M, we use the following equilibrium equations and the element's constitutive law, linking the beam statics (force resultants and moments) to the kinematic variables:

  normal N, shear forces T and moments M entering Eqs. (2.26-2.28) are written in their asymptotic expansion form, summarized in Section 2. Eqs. (2.26-2.28) contain a total of 15 unknowns. Accounting for the periodicity of the nodal displacement components (

	1 0 1 . . n l e n l e 2 0 2 . . 	)	in	the	horizontal	and	vertical	material	direction

  2.6 and 2.8). Such branches cannot be obtained by Cauchy or low frequency based mechanical formulations. Therefore, the frequency based, asymptotic expansion of the internal normal forces can allows for the description of higher-frequency dynamics, separating low and high frequency modes and characterizing the corresponding frequency transition points (Section 2.3.2, 2.3.3). The normalized scale separation frequency values in the high wavelength region, as elaborated in Sections 2.3.2 and 2.3.3 are summarized in Table 2.2.

  On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects Page | 77Askes, 2002; Askes and Aifantis, 2009; Askes et al., 2002). A detailed comparison between the dispersive characteristics of various simplified models of gradient elasticity can be found in(Askes et al., 2008). Accordingly, differences in between classical elasticity considerations and formulations that account for the role of higher order inner material kinematics or nonfinite strains can be found in[START_REF] Reda | Wave propagation characteristics of periodic structures accounting for the effect of their higher order inner material kinematics[END_REF], Karathanasopoulos et al., 2019, Reda et al.

	2018c).
	The descriptions of microstructural effects analyzed in wave propagation are most of
	the time phenomenological in nature, following either the line of micromorphic models (see
	Berezovski et al., 2013) or second gradient continuum models (see Fafalis et al., 2012). The
	Applications of gradient elasticity in dynamics have fostered extensive research
	(Ostoja-Starzewki, 2002; Papargyri-Beskou, 2004; Andrianov, 2010; Askes and Aifantis,
	2011). Gradient elasticity models are useful to predict the wave dispersion characteristics in
	heterogeneous or discrete systems (see Mindlin, 1964; Muhlhaus and Oka, 1996; Gonella et
	al., 2011, Gonella and Ruzzene, 2010, Alibert, 2015). A thorough analysis of the effects

, who proposed several nonlocal continuum models to capture the dispersion relation of planar waves. Since the early works in the 1960s by

Toupin (1962)

and

Koiter (1969)

, there has been several literature contributions to the topic, recently summarized in

(Askes and Aifantis, 2009)

. Generalized continuum theories offer an attractive alternative for capturing dynamic behaviors overlooked by classical elasticity, especially dispersion relations

(Lombardo and Askes, 2012)

. In order to study low frequency wave propagation in effective continuum media, enriched second gradient elasticity descriptions have been developed which account for microstructural effects descriptive of the dispersive wave behavior

(Aifantis, 1998)

. Gradient-enriched theories can sufficiently capture dynamic behaviors overlooked by classical elasticity, in which second gradient terms are neglected. Different theories based on an enrichment of the classical elasticity framework by higher order gradients have been proposed in the past to overcome deficiencies of classical elasticity (see

Aifantis, 1992Aifantis, ,1998

;Askes and Aifantis 2009, Reda et al. 2018a) and plasticity (see Askes et al., 2002; Askes et al., 2008; Askes and Aifantis, 2011; Askes et al.,2012). Their formulation has aimed at describing both static and dynamic phenomena (Gutkin and Aifantis, 1999; Gourgiotis and Georgiadis, 2009; Gonella and Ruzzene, 2010; Gonella et al., 2011). brought about by gradient elasticity models can be found in (Papargyri-Beskou et al., 2009; Fafalis et al., 2012), especially the impact of the higher-order inertial terms (see Metrikine and

Chapter 3: phenomenological nature of these models entails that the material parameters involved in the wave propagation formulations need to be calibrated to experiments, and consequently such models are not predictive. The large number of intrinsic parameters to be identified requires a somewhat difficult measurement protocol; this issue is addressed especially in (Polyzos et al., 2015; Iliopoulos et al., 2016; Réthoré et al., 2015).

  are symmetric tensors, we require that the localization tensors of Eq. (3.6) enjoy the following symmetry properties:

	212	221	112	121
	2121	1221	21211	12211

  or Eq. (3.24)  

													0 w  , as a function
	of the layers thickness ratio	 	12 / dd	, to be:
	hom 2121	2
	hom	hom				 51 2
	21211	21121		2		 1	2
	hom		79	4	16	3		6		2	16	7		2
	211211	24							 1	4

Table 3 .

 3 1: Geometric and mechanical layers properties the laminate layers.

	Properties	First layer	Second layer
	Thickness		50%		50%
	Poisson's ration		 	0.3	 	0.3
	Shear modulus	1  	80760 MPa	2  	8076 MPa
	Mass density

Table 3

 3

	.1. To that purpose, we compute

d

H are constants summarized in Table

3

.2. Using Eq.

(3.20), we calculate the homogenized macro-inertia forces on the laminated medium as follows:

  Equation of motion Eq. (3.49) coincides with the one derived in [Wang and Sun, 2007]. The constants entering Eqs. (3.48) and (3.49) and used in subsequent computations are summarized in Table 3.2.

	Chapter 3: On the role of second gradient constitutive parameters in the static and dynamic
	analysis of heterogeneous media with micro-inertia effects
	Table 3.2: Constant values obtained by the Hamilton principle based on second gradient
	mechanics
			Mode constants		Longitudinal wave	Transverse wave
			1 H d		336 2	/ MPa m	/ 1334 Mpa m
			2 H d		51392.7 2	/ Mpa m	29367.27	/ Mpa m
			3 H d		-2238 Mpa	-1279 Mpa
			4 H d		-2336 Mpa	-1334 Mpa
			5 H d		122.72	5 kg m /	122.72	5 kg m /
			6 H d		/ 13.7 kg m	5	/ 9.12 kg m	5
			H	7		1 m	1 m
	-H	2 d	2 2 c HH  7 5 7 +H + d 	H	2 c 0  	(3.49)
							Page | 99

Table 3 .

 3 3: Constants value obtained by the DHGE method.

	55)

  ; M. Şimşek, 2016).

	Apart from the sufficient description of the wave propagation characteristics of
	engineering materials, the design of structures with optimized vibration properties (Shimoda
	et al., 2016) requires artificial material designs with extra-ordinary attributes. It has been
	observed that certain lattice material designs allow for an increase of the basic eigenfrequency,
	reducing structural weights while retaining specific stiffness characteristics for cantilever and
	doubly supported artificial beam structures (Cheng et al., 2018). What is more, different regular
	and irregular lattice patterns have been shown to allow for a substantial shift of the primal
	eigenmode to a higher frequency value, isolating lower range frequencies (Syam et al., 2018;
	Andresen et al., 2020).

Higher gradients contributions have been shown to affect the structural vibration properties

(Altan et al.,1996; Salehian & Inman, 2008; Bisegna & Caruso, 2011

Table 4 .

 4 For the computation of the Lame coefficients provided in Table4.1, the Poisson's ratio value was set to 0.35, 0.2, 0.22, 0.33, 0.27 for the Silicon Carbide, Borosilicate Glass, E-Glass, Epoxy, and Carbon accordingly

	Material	E (GPa)	 (kg/m 3 )	 (GPa)	 (GPa)
	Silicon Carbide	400	3000	346	148
	Borosilicate Glass	60	2300	16.6	25
	E-Glass	72	2500	23	29.5
	Epoxy	3.4	1250	2.48	1.28
	Carbon	250	1800	115.5	98.4

1 : Mechanical properties of the materials used in the design of the microstructure of the composite beam of Fig. 4.1a.

Table 4 . 2 :

 42 Table 4.2 : Homogenized stiffness coefficients and internal length values for different Carbon and Glass-based unit-cell material designs using the material properties of Table 4.1 and the definitions of Eq. (4.3).

	Materials	Unit-cell	1 1 C		MPa		 12 5 C MPa mm 		  l mm e
	1	Borosilicate Glass-Silicon Carbide	120761.54	5088.52		0.205
	2	E-Glass-Epoxy	9491.49	326.86		0.186
	3	Carbon-Epoxy	9913.36	245.23		0.157

Table 4 .

 4

			1 as follows:
	1	1	1 1 hom	2.1 10	6	,	2	1	2	1 1 hom	4.2 10	6	,	3	1	3	1 1 hom	6.3 10	6
	1	2	1 1 hom	7 10	5	,	2	2		2	1 1 hom	1.4 10	6	,	3	2		3	1 1 hom	2.12 10	6
			1
	1	3	1 hom	8 10	5	,	2

  .2 are provided in Table 4.3. The computations are based on the mechanical definitions of Section 4.2.2.

	Table 4.3 : Homogenized static and dynamic properties of the composite unit-cell material
	designs of Tables 1 and 2 for equal micro-scale layer thickness i d in Fig. 4.1a.
	Material	Unit-cell	 2 tonne/mm 1 C			hom		tonne/mm	3	
	1	Borosilicate Glass	Silicon Carbide	10 1.57 10  				9 2.65 10  
	2	E-Glass	Epoxy

Table 4 .

 4 4 : Eigenfrequency values of the composite Silicon Carbide-Borosilicate Glass (Tables 4.2, 4.3) beam structure upon a Cauchy (C), Second-Gradient (SG) and Second-Gradient with micro-inertia (SG+μI) modeling formulation.

	Cauchy	SG	SG I
	Mode

  ; Forest et al., 2001; Kanit et al., 2003; Geers et al., 2001; Kouznetsova et al., 2004; Auffray et al., 2010;

  5.6), tensors  Inserting into Eq. (5.6) the fluctuating displacement as in Eq. (5.7), we get the 1 st and 2 nd gradient unit cell BVP's as the stationarity conditions of the functional on the left hand side of

															, EK kl	klm		, allows writing:
	i u	( ) y		H	( ) : EK  ijk jk ijkl y E H	  y	:	 K	jkl	(5.7)
				Wherein	    EK H x H x ,	in Eq. (5.7) stand for the displacement localization tensors
	relating the fluctuating displacements to the first and second gradient strain tensors () EX and
	() KX accordingly.
	Eq. (5.5):
		div	 c ijkl	( ) : y			kl E		K	klm	m y			y		H	( ) : EK kmn mn klmn  y E H	:		K	lmn		 		0
																										l
		1: BVP		div	 c ijkl	( ) : y		kl E			y		H	kmn E	( ) : y	mn E		 		st 0 1 gradient unit cell problem	(5.8a)
																										l
		BVP	2 :		div	 c ijkl	( ) : y		K	klm	m y			y		H	K klmn	:	 K	lmn		 		0 2 gradient unit cell pb nd	(5.8b)
																										l
				These unit-cell BVP's lead to the expressions of the micro-displacement, these latter being
	used to evaluate the homogenized properties of the unit-cell. Deriving the internal energy in
	Eq. (5.3) with respect to the imposed first gradient macro-strain	ij E as mentioned in Eq. (5.4),
	we get the first gradient macro-stress as follows:
	ij	=							hom ijqt		:	qt				hom ijqtw		qtw
						ij
			1						mn				mnp		p					n y		E mrv	( ) :	rv	K mrvd	:	rvd			:	mnqt	( ) :
			2		ij	Y			qt						qtw		w					y		E qwd	( ) :	wd	K qwdx	:	wdx		
																									t
																								E		E	E
																													
				qt			ijqt					ijwd		y					wqt	mnqt	y	mij	mnwd	y	mij	y
																		d							n	n	d

  to get the following expression of the hyperstress tensor in component

	form:
	ijk	=	w		hom ijkqt	:	qt	hom ijkqtw qtw
			ijk
	1			mn		mnp p		n y		E mrv	( ) :		rv	K mrvd	:	rvd			:	mnqt	( ) :
	2	ijk Y		qt		qtw w	t y			E qwd	( ) :	wd	K qwdx	:	wdx		
		qt	ijqt k	d k ijwd y	  E wqt		mnqt y	n	K   mijk		mnwd

  2.2.1, we calculate the second gradient displacement localizations tensors

	tensor and the coupling tensor 	hom ijkqtw	,	hom ijkqt	
	111 K i H over the unit cell, which are used to compute the homogenized second gradient rigidity

Table 5 .

 5 1: Relative difference between the homogenized second gradient constitutive parameters of the 2D composite structure in Fig.1ausing HGE without the correction terms and the analytical results of(Li, 2011).From the Table5.1 we see the very good accordance between the HGE and the analytical solution in

	hom ijklmn A		. MPa mm	2		HGE method	Analytical solution (Li, 2011)	% relative difference
		111111 A			10.23	10.36	1.25
		111221 A			4.38	4.44	1.35
		111122 A			0	0	0
		221221 A			10.33	10.36	0.3
		122122 A			2.95	2.96	0.3

Table 5 .

 5

	2: Relative difference between of the homogenized second gradient constitutive
	parameters of the 2D composite structure in Fig. 5.1a using HGE with corrections terms and
	the analytical results of (Li, 2011).
	hom* ijklmn A		. MPa mm	2		HGE method	Analytical solution (Li, 2011)	% relative difference
		hom* 111111 A			2.22	2.3	3.47
		hom* 111221 A			0.96	0.99	3
		hom* 111122 A			0	0	0
		hom* 221221 A			2.32	2.32	0
		hom* 122122 A			0.66	0.66	0

  5.1b, where the matrix made of linearly elastic material with an elasticity

	coefficient   E ,	Poisson's			ratio   	and	density    accordingly
		1 E		100	MPa	1  1 , 0.3, 		1000 / Kg m	3		,	and	a	circular	inclusion	with
		2 E		10	MPa	2  , 0.3, 	2		1000 / Kg m	3		and radius	R		0.4 mm

  A.15)

					1 ( -) 2	0 3	(2	1	2	) -	4 2520 0 ( )		22	1	 13 -2	4 2520 0 ( )	0		1 4 -3	2	
	1 2 1 2 M(v ,v ,θ ,θ )=	6	0	2	0 -( )	8	13 6096384	1	61 5080320	2	0 -( )	8	0	17 6350400	1	-6 -2.508 10 .	2
					10 (( ) )) 0

2 Hamiltonian principle up to second gradient

				2.A.2)
	2.A., Vt  1 2   1 1 2 2 i i ij ij ijk ijk U U T E K S P dVdt           		0	(2.A.3)

  ,

	Integral			wdVdt		wdVdt	ijk w K	ijk K dVdt	ijk S K dVdt ijk	ijk S U dVdt i jk
		ijk ijk S K dVdt	ijk S U	i j	k	dVdt S	ijk k	i j U dVdt
		ijk S U n dSdt S i j k	ijk k	i j U dVdt
	S	a	S	ijk k	i j U dVdt S	a		S	ijk k	i j U dVdt

  ,

																										(2.A.8)
																											S	ijk kj		i U dVdt
		S	a			S	ijk k		i U dVdt j			S	ijk kj		i U dVdt
		S	a			S	ijk k		i U n dSdt j			S	ijk kj		i U dVdt S 	a		S	b			S	ijk kj		i U dVdt
			3 Integral	1 2  i i U U dVdt   	 	i i U U dVdt	(2.A.9)
				 	i i U U	T 		3   U U dVdt S i i   	 	i i U U dVdt
			We next calculate the variation of the internal energy fluctuation:
	4 Integral ij P E      	ij E    	( , , ij ij ij ij P E E K K dVdt , ) ijk ijk ij ijk P P E E E K              K P ijk  P ij E        	ij E ij E dVdt   P ij E   		ij E		ijk P K   	ij E   	ijk P K   	ij E dVdt  	(2.A.10)
	4 Integral	a					ij P E	ij E dVdt				ij P E		, i j U dVdt	ij P E	U	i	,	j	dVdt	j	ij P E	i U dVdt
				P
								i		j												i		4	a	i
				ij											j				ij						j	ij

  2.A.11)

	4 Integral	b	ij P E	ij E dVdt	ij P E	ij E	T	ij P t E	ij E dVdt
	ij P t E	ij E	T	2 2 t	ij P E	ij E dVdt
	2 2 t	ij P E	, ij U dVdt		2 2 t	ij P E

  Then, inserting (2.A.4-2.A.17) back into (2.A.3) and deriving with respect to time we get the following macro-equation of motion:

												
							
		         t     i j k ijk T U dVdt X X t K                   	(2.A.17)
	S	5	bi j k ijk T U dVdt X X t K                        
			, ij j	i U V	S	, ijk jk	i U dV	X	j	ij P E	i U dV	X	j	2 2 t	ij P E	i U dV
			j X t E ij T		i U dV	2 X X j	k	2 2 t	ijk P K	i U dV		2 X X j	k	ij P K	i U dV

  Using the definition of internal energy, we expand it in function of the localization tensors:

	i F			X  	j	  	1 2	mn E					E imn j H x H x E mij n 		2 	m i	j n x x			1 2	K	rst		  	K irst j H x		1 2	E rij s t H x x			r i s j t x x x	  	  
		2 XX j   k      	E 	1 4 pq	2 rst K 	  	 K pijk q K rijk s t 1 2 H x H x x H x x E ipq j k K irst j k H x x       p i	q j k r i s j k t x x x x x x x    		     	(2.A.24)
	Appendix 2.B: Higher gradient dynamic homogenization method
	int W E K 2. ( , )			ij E	.   ij	. ijk ijk K S			ij ij  			i u u E x i ij j    		1 2	ijk j k K x x	  
				. A E E ijkl kl				  . . ijmn A K C . K ijklf klf	. A E E mnrs rs		K mnrst rst A K	
					. H E E ijk	jk			. H K K ijkl	jkl		.	  	. H E E iqr	qr		. H K K irst	rst		iq q E x		1 2	iqr q r K x x	  
			E ijkl AE kl	.		ijmn	.			E mnrs	.		rs				E ijkl	.	K kl ijmn mnrst rst	K ijklf	E klf ijmn mnrs rs	K ijklf	K klf ijmn mnrst rst
								E											E												E	K	E	1	E
							ijk jk			iqr qr							ijk jk	irst rst	ijk jk iq q	2	ijk jk iqr q r
								K											E
								ijkl					jkl			iqr qr
																															  	np n p xx	  
	 	1 2	ij im E E					j m x x				1 2	ijk im K E		m j k x x x		1 8	ijk K K	imp		j k m p x x x x	(2.A.23)
		1 2	jk mp E E			E ijk H H	E imp					mpf K E	jk		E ijk H H	K impf		1 2	jkl K K	mpf		K ijkl H H	K impf
		ij T tE       	   		 	im E				j m x x		1 2	K	imk		m j k x x x		E	mp		H	E kij	H	E kmp		K	mpf		E kij H H	K kmpf
		ijk T tK       	   		 	1 2	im E				m j k x x x		1 4	K	imp		m j k p x x x x		E	pf		E mpf H H	K mijk		K	mpf		K lijk H H	K lmpf
				After deriving the kinetic and potential energy, we insert these expressions into the last
	expression of Fi as the following:

  Where V is the RVE volume. Since the origin is situated at the center of the RVE    , it holds separating the macroscopic energy terms from Eq. (4.13):We note that V is the RVE surface in the present 2D case.Integrating the strain energy and using the last expression of the first gradient macroscopic energy across the entire domain (4.B.4) we get the following relation:

	hom ijkl C  ij kl dV  	hom ijpl C	 E K y E ij ijk k 	pl		 K y dV plm m
	VV
		hom ijpl VV ij pl V C E E     	hom m ijpl y C E K d ij plm		(4.B.3)
		hom k ijpl VV pl ijk y C E K d     	hom ijpl C y y K K d k m ijk plm	
		V 	hom ijpl VC E E ij pl		hom ijkl VV ij kl C dV   	hom ijpl C y y K K dV k m ijk plm
		hom ijpl VV V hom ij pl ijkl ij kl VC E E C dV      hom ijpl ijk plm VC K K   k m y y d   	(4.B.4)
	V 	wdV	 V C E E hom ijpl ij pl  1 2 V 	hom ijklmn ijk lmn A K K	
		 11  hom 22 ijklmn ijk lmn VV V hom ijpl ij pl V A K K C E E dV   hom ijpl ijk plm k m VC K K y y d           
		hom 11 22 ijklmn ijk lmn VV V hom ijpl ij pl A K K d C E E dV         hom ijpl ijk plm VC K K y km yd        
		hom 11 22 ijklmn ijk lmn V A K K dV  	hom ijpl C E E dV ij pl  hom ijpl ijk plm km V V C K K I dV     
		   hom hom ijk lmn ijklmn ijlm kn VV hom 11 22 ijpl ij pl C E E dV K K A C I dV     

, XX directions of a Cartesian

, XX is assumed in the following form:

/ EE,

Acknowledgments

Acknowledgments

First, I would like to send special and warm thanks to my parents and my family (My wife, my brothers, my sisters…). I wouldn't have been able to pursue this project without your support.

I would like to thank all the teaching stuff at the Lebanese University (Faculty of Engineering) and to renew my gratitude for Dr. Marwan Fahs and Dr. Daoud Baalbaki who encouraged me to continue in this field. A special thanks to my teachers Ibrahim Cheaitou and Hassan Chour, who advised me to carry on my dream by doing this Ph.D. I am also very grateful for the "Association de Specialisation et d'Orientation Scientifique" for giving me the

Conclusion

Overall, the current chapter has analyzed for the first time the role of micro-inertia and higher-gradient material parameters on the dynamic behavior of structural members in a combined manner. Evidence has been provided that the static response can be well-separated from the dynamic behavior where not-only higher-gradient, but also micro-inertia contributions appear. More specifically, it has been shown that while high internal length values are a prerequisite for non-negligible, higher-gradient effects to appear both in statics and dynamics, this is not the case for micro-inertia contributions. In particular, it has been demonstrated that micro-inertia terms can be significant in the low range of internal length values that is relevant for a wide range of commonly-used engineering materials. Thereupon, micro-inertia contributions can counteract the effect of higher-gradient stiffness terms, so that a mere consideration of higher-gradient terms may lead to a substantial overestimation of the computed eigenfrequencies, in particular for the higher-order modes. We aspire that the provided results serve as a reference for the computation of the longitudinal vibration properties, not only of composite structures, but also of general-form structural systems, providing a fundamental basis for the assessment of the relative significance of micro-inertia and higher-gradient material parameters.

Chapter 5: Wave Propagation in 2D composites Structure using Higher Gradient Size Independent Homogenization

Summary

In the current chapter, we study the role of higher gradient contributions on the static and wave propagation characteristics of composite structures. To that scope, we use a higher gradient energy formulation to compute the complete set of effective higher gradient properties.

We use both a unit-cell size-dependent and size-independent modeling formulation which is based on correction terms that include quadratic inertia tensor terms. Thereupon, we compute the homogenized properties of composites with internal inclusion for which we computed the internal lengths parameters for different unit cell material and geometric properties. We show that the higher gradient effect is limited for low material contrast and strongly depends on the volumetric fraction of the material phases included. We provide evidence that the selection of the higher gradient modeling formulation is crucial for the computed wave propagation characteristics. In particular, the dispersion attributes computed using size-independent models are close to Cauchy mechanics prediction for small volumetric inclusion, while size-dependent models are overestimating the computed frequencies by a substantial amount. What is more, we show that frequencies computed depend on the propagation mode of interest, with the shear frequency mode to be more affected than the longitudinal mode irrespective of the formulation employed. In the sequel, we use the higher gradient constitutive moduli size independent hom* ijklmn A which includes the correction terms of Eq. (5.13) to the same analysis we applied for the noncorrected higher gradient constitutive moduli in Fig. 5.4 as follows:

Chapter 5: Wave Propagation in 2D composites Structure using Higher Gradient Size Independent Homogenization Page | 161

Wave propagation analysis within 2D second gradient media

Homogenization methods are crucial for both the static analysis of composites and for their wave dispersion characteristics. Classical elasticity based on Cauchy continuum mechanics assumptions lead to linear dispersion relations (Chapter 2), so it proves unable to describe the dispersion relations occurring in microstructured materials. On the other hand, enriched and 4% for the SG and SG* formulations accordingly. We should note that for high values of the elasticity ratio  

/

EEthe % of dispersive frequency converges to a constant value.

Discussion

The developed strain gradient proves to be size independent effects as analyzed in Section 5.3.2.2, due to introduced corrections terms in Eq. (5.13). Higher gradient constitutive parameters take a lesser value with the introduced correction, as shown by the analytical expression of higher gradient constitutive parameter in Eq. (5.13). This effect can also been seen in the values of the internal length analyzed in section 5.3.2.3: indeed, the internal length of the higher gradient constitutive size dependent parameters in Fig. 5.4 reaches bigger magnitude than the size independent higher gradients parameters (Fig. 5.5).

Furthermore, microstructure geometrical and mechanical properties of the square matrix with central circle inclusion (Fig. 5.1b) have a crucial effect on the higher gradient constitute parameters (with and without corrections terms). We integrate the dynamical equations of a beam on each nodes in order to determine the dynamic expression of forces in terms of the nodes displacements and rotation.

The longitudinal force is related to the strain as follows:

We assume the propagation of harmonic plane waves of the form:

Inserting the harmonic wave in (1.A.1) leads to:

The solution of previous differential equation writes:

We assume that the compression wave length c

 is much layer than the beam length, thus the compression coefficient 0 ( l )

 is very small, so that the normal force can be developed in powers of 0 ( l )

 using asymptotic expansion:

360

1.A.2 Dynamical expression of the transverse force and bending moments

Following the same procedure, the expressions of the transversal force can be calculated by integrating the transverse equilibrium equation using the constitutive law: We note that y represent the coordinates in the direction of the transverse motion of the beam v.

After deriving twice the moments respecting to y we get: As a summary we obtain:

- We subsequently determine the expressions of the localization operators of Eq. ( 3) for the case of purely normal macroscopic deformations. In such a deformation mode, the macro displacement field is along the direction X of periodic structure of Fig. 3.2 (left), so that:

The coefficients of the localization tensors for the micro-displacements are denoted i a and i b for the first and second layer respectively, defined as follows:

  .

Eq. (3.A.2) contains a total of 10 unknown parameters i a and i b , the superscript therein defining in which phase the displacement or the strain is evaluated. The micro-stress developed in each inner material phase is modeled as an isotropic elastic material that is computed as a function of the micro-strains within Eq. (3.A.2), as follows: