J.-J. Marigo

École Polytechnique · Laboratoire de Mécanique des Solides

Structural lattices with quasi-periodic pattern possess interesting dynamic features that can be exploited to control, localize and redirect propagating waves. In this work we show that the properties of a large class of quasi-periodic locally resonant systems (approximated as periodic, with arbitrarily large period) can be performed defining an eq...

We inspect the dynamic behavior of suspended cables presenting a periodic array of scatter elements, consisting of a discrete set of masses that are hanging by means of elastic or rigid connections. By introducing some approximations, we show that the problem in the continuous domain can be brought back to an equivalent discrete problem, whose solu...

This work analyzes the dynamic behavior of structural elements that can be modelled as taut cables with a discrete array of punctual attached and hanging masses. The propagation of mechanical waves is strongly influenced by the presence of such scatter elements. We found that the problem is governed by a discrete equation, whose solutions depend on...

Local resonant metamaterials are a class of microstructured man-made material which attenuate the propagation of waves in certain frequency ranges, known as band gaps. In this work, we study through asymptotic homogenization the anti-plane shear wave propagation in metamaterial with a stiff matrix and soft inclusions, periodically distributed, whic...

We propose a homogenization method based on a matched asymptotic expansion technique to obtain the effective behavior of a two-dimensional linear viscoelastic periodically stratified slab, which accounts for the finite size of the slab. The problem is investigated for shear waves, and the wave equation in the harmonic regime is considered. The obta...

This paper proposes the homogenization for a stratified viscoelastic media with free edge. We consider the effect of two-dimensional periodically stratified slab over a semi-infinite viscoelastic ground on the propagation of shear waves hitting the interface. Within the harmonic regime, the second order homogenization and matched-asymptotic expansi...

We propose a homogenization method based on a matched asymptotic expansion technique to obtain the effective behavior of a periodic array of linear viscoelastic inclusions embedded in a linear viscoelastic matrix. The problem is considered for shear waves and the wave equation in the harmonic regime is considered. The obtained effective behavior is...

We analyze the band structure of a single-phase metamaterial involving low-frequency flexural resonances by combining asymptotic homogenization and Bloch-Floquet analysis. We provide the closed-form expression of the dispersion relation in the whole Brillouin zone. The dispersion relation involve two effective, frequency-dependent, mass densities a...

In this work, we study the transversal vibration of thin periodic elastic plates through asymptotic homogenization. In particular, we consider soft inclusions and rigid inclusions with soft coatings embedded in a stiff matrix. The method provides a general expression for the dynamic surface density of the plate, which we compute analytically for ci...

In a previous study (Marigo et al. in J. Mech. Phys. Solids 143:104029, 2020) we have studied the effect of a periodic array of subwavelength plates or beams over a semi-infinite elastic ground on the propagation of waves hitting the interface. The study was restricted to the low frequency regime where only flexural resonances take place. Here, we...

Metamaterials are generally known for their waves attenuation capabilities. This behaviour, which is related to the microstructure composing these materials, can be due to a Bragg-type scattering mechanism or to local resonances. The objective here is to exploit the two phenomena for generating a system able to localize the energy carried by propag...

We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472 , 20160062 ( doi:10.1098/rspa.2016.0062 )) different transmission conditio...

We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements...

We propose a model of flexural elastic plates accounting for boundary layer effects due to the most usual boundary conditions or to geometrical defects, constructed via matched asymptotic expansions. In particular, considering a rectangular plate clamped at two opposite edges while the other two are free, we derive the effective boundary conditions...

Our study concerns the propagation of acoustic waves through a thin screen made of a periodic arrangement of air bubbles in water. The bubbles are oscillators of the Minnaert type whose dynamics is modified by the containment. This nonlinear dynamics is obtained in the time domain using asymptotic analysis and a homogenization technique involving t...

The time-domain propagation of scalar waves across a periodic row of inclusions is considered in 2D. As the typical wavelength within the background medium is assumed to be much larger than the spacing between inclusions and the row width, the physical configuration considered is in the low-frequency homogenization regime. Furthermore, a high contr...

Among the attractive properties of metamaterials, the capability of focusing and localizing waves has recently attracted research interest to establish novel energy harvester configurations. In the same frame, in this work, we develop and optimize a system for concentrating mechanical energy carried by elastic anti-plane waves. The system, resembli...

We consider the effect of an array of plates or beams over a semi-infinite elastic ground on the propagation of elastic waves hitting the interface. The plates/beams are slender bodies with flexural resonances at low frequencies able to perturb significantly the propagation of waves in the ground. An effective model is obtained using asymptotic ana...

The time-domain propagation of scalar waves across a periodic row of inclusions is considered in 2D. As the typical wavelength within the background medium is assumed to be much larger than the spacing between inclusions and the row width, the physical configuration considered is in the low-frequency homogenization regime. Furthermore, a high contr...

In this work, we exploit the two-scale homogenization approach to compute explicitly the band gaps for out-of-plane wave propagation in ternary locally resonant metamaterials (LRM) with two-dimensional periodicity. The homogenization approach leads to the definition of the dynamic effective mass density, depending on the frequency, that becomes neg...

This paper provides a detailed comparison of the two-scale homogenization method and of the Bloch-Floquet theory for the determination of band-gaps in locally resonant metamaterials. A medium composed by a stiff matrix with soft inclusions with 2D periodicity is considered and the equivalent mass density of the homogenized medium is explicitly obta...

This paper aims to report and analyze the possibility of focusing the mechanical energy carried by anti-plane shear waves, localizing it inside a resonant cavity. The transmission through a pair of identical barriers, constituted by two-dimensional (2-D), two-phase Locally Resonant Metamaterials (LRMs) and separated by the cavity itself, is here co...

Effective model for elastic waves propagating in a substrate supporting a dense array of plates/beams with flexural resonances. Abstract We consider the effect of an array of plates or beams over a semi-infinite elastic ground on the propagation of elastic waves hitting the interface. The plates/beams are slender bodies with fle-xural resonances at...

We present a three-dimensional model describing the propagation of elastic waves in a soil sub- strate supporting an array of cylindrical beams experiencing flexural and compressional resonances. The resulting surface waves are of two types. In the sagittal plane, hybridized Rayleigh waves can propagate except within bandgaps resulting from a compl...

This paper is devoted to the use of gradient damage models in a dynamical context. After the setting of the general dynamical problem using a variational approach, one focuses on its application to the fragmentation of a brittle ring under expansion. Although the 1D problem admits a solution where the damage field remains uniform in space, numerica...

This chapter presents a two‐scale homogenization to encapsulate the effect of rigid inclusions in the vicinity of a rigid wall or forming a structured film in effective conditions. The chapter sets the different ingredients needed to conduct the asymptotic analysis. It provides the definition of a small parameter, which is used to differentiate bet...

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured rid...

The influence of the spacing on the resonance of a periodic arrangement of Helmholtz resonators is inspected. An effective problem is used which accurately captures the properties of the resonant array within a large range of frequencies, and whose simplified version leaves an impedance condition. It is shown that the strength of the resonance is e...

This chapter explains the development of the so‐called "gradient damage models" and their extension to ductile materials and dynamic loading. It presents the construction of gradient damage models for brittle softening materials based on the principle of minimum energy. The chapter discusses the main hypothesis and the need for regularization. It t...

The aim of the present work is to study the stabilizing effect of the non-uniformity of the stress field on the cohesive cracks evolution in two-dimensional elastic structures. The crack evolution is governed by Dugdale's or Barenblatt's cohesive force models. We distinguish two stages in the crack evolution: the first one where all the crack is su...

The paper develops a general framework to derive the effective properties of quasi-periodic elastic medium. By using the asymptotic expansion method, the solution is expanded to the second order by solving a sequence of minimization problems. The effective stiffness tensors fields entering in the expression of the macroscopic energy are obtained by...

We present a homogenization method to treat the problem of the reflection of waves at the free boundary of an elastic body, the edge being structured periodically at the subwavelength scale. The problem is considered for shear waves and the wave equation in the time domain is considered. In the homogenized problem, a boundary condition at an equiva...

We propose an elasto-plastic model coupled with damage for the behavior of geomaterials in compression. The model is based on the properties, shown in [S. Andrieux, et al., Un modèle de matériau microfissuré pour les bétons et les roches, J. Mécanique Théorique Appliquée 5 (1986) 471?513], of microcracked materials when the microcracks are closed w...

Local damage models with softening needs localization limiters to preserve the mathematical and physical consistency. In this paper we compare the properties of strain-gradient and damage-gradient regularizations. Gradient-damage models introduce a quadratic dependency of the dissipated energy on the gradient of the damage field and are nowadays ex...

We inspect the unusual scattering properties reported recently for structures alternating dielectric layers of subwavelength thicknesses near the critical angle for total reflection. In TE polarization, the unusual scattering properties are captured by an effective model with an accuracy less than 1% up to kd∼0.1. It is shown that the propagation i...

We consider a domain made of a linear elastic material which contains an angular point. A small defect, like a cavity or a crack, is located in the neighborhood of the tip of the wedge. In order to study its influence both on the local and global responses of the body, we use a matched asymptotic expansion method. After the general construction of...

In this supplementary material, we present a validation and a discussion of the effective model in the light of comparison with direct numerical calculations. Specifically, we address (i) the capacity of the model at the second order to describe the deviation from a slot resonator (which corresponds to the limit problem-at the dominant order-) to a...

We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with t...

We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of the same height atop a guiding layer on a soil substrate. An asymptotic analysis shows that the forest behaves like an infinitely anisotropic wedge with effective boundary conditions. We discover that the foliage of trees brings a radical cha...

We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of same height atop a guiding layer on a soil substrate. We discover that the foliage of trees { brings a radical change in} the nature of the dispersion relation of these surface waves, which behave like spoof plasmons in the limit of a vanishi...

The resonance of a Helmholtz resonator is studied with a focus on the influence of the neck shape. This is done using a homogenization approach developed for an array of resonators, and the resonance of an array is discussed when compared to that of a single resonator. The homogenization makes a parameter B appear which determines unambiguously the...

We present an interface model based on two-scale homogenization to predict the coherent scattering of light by a periodic rough interface between air and a dielectric. Contrary to previous approaches where the roughnesses are replaced by a layer filled with an equivalent medium, our modeling yields effective jump conditions applying across the regi...

We establish, both theoretically and experimentally, that metamaterials for water waves reach a much higher degree of anisotropy than the one predicted using the analogy between water waves and their electromagnetic or acoustic counterparts. This is due to the fact that this analogy, based on the two-dimensional shallow water approximation, is unab...

Phase-field models, sometimes referred to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith' criterion. For a one-dimensional p...

We present the homogenization of a periodic array of elastic inclusions embedded in an elastic matrix. We consider shear elastic waves with a typical wavelength \(1/k\) much larger than the array spacing \(h\) and thickness \(e\). Owing to the small parameter \(\eta=kh\), with \(e/h=O(1)\), a matched asymptotic expansion technique is applied to the...

Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [5] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimil...

We present a homogenization model for a single row of locally resonant inclusions. The resonances, of the Mie type, result from a high contrast in the shear modulus between the inclusions and the elastic matrix. The presented homogenization model is based on a matched asymptotic expansion technique; it slightly differs from the classical homogeniza...

The wave propagation in structures involving metamaterials can be described owing to homogenization approaches which allow to replace the material structured at the subwavelength scale by an equivalent and simpler, effective medium. In its simplest form, homogenization predicts that the equivalent medium is homogeneous and aniso-tropic and it is as...

The wave propagation in structures involving metamaterials can be described owing to homogenization approaches which allow to replace the material structured at the subwavelength scale by an equivalent and simpler, effective medium. In its simplest form, homogenization predicts that the equivalent medium is homogeneous and aniso-tropic and it is as...

The wave propagation in structures involving metamaterials can be described owing to homogenization approaches which allow to replace the material structured at the subwavelength scale by an equivalent and simpler, effective medium. In its simplest form, homogenization predicts that the equivalent medium is homogeneous and aniso-tropic and it is as...

We present a homogenization method to find the effective behavior of a periodically stratified slab which accounts for the finite size of the slab. The effective behavior is that of a homogeneous anisotropic slab associated with discontinuity conditions, or jump conditions, for the displacement and for the normal stress at the boundaries of the sla...

We propose in this contribution to investigate the link between the dynamic gradient damage model and the classical Griffith's theory of dynamic fracture during the crack propagation phase. To achieve this main objective, we first rigorously reformulate two-dimensional linear elastic dynamic fracture problems using variational methods and shape der...

In this contribution, we propose a dynamic gradient damage model as a phase-field approach for studying brutal fracture phenomena in quasi-brittle materials under impact-type loading conditions. Several existing approaches to account for the tension-compression asymmetry of fracture behavior of materials are reviewed. A better understanding of thes...

The paper is devoted to gradient damage models which allow us to describe all the process of degradation of a body including the nucleation of cracks and their propagation. The construction of such model follows the variational approach to fracture and proceeds into two stages: (1) definition of the energy; (2) formulation of the damage evolution p...

Research Cite this article: Marigo J-J, Maurel A. 2016 Two-scale homogenization to determine effective parameters of thin metallic-structured films. Proc. R. Soc. A 472: 20160068. http://dx. We present a homogenization method based on matched asymptotic expansion technique to derive effective transmission conditions of thin structured films. The me...

Background: Gradient damage models can be acknowledged as unified framework of dynamic brittle fracture. As a phase-field approach to fracture, they are gaining popularity over the last few years in the computational mechanics community. This paper concentrates on a better understanding of these models. We will highlight their properties during the...

A homogenization method for thin microstructured surfaces and films is presented. In both cases, sound hard materials are considered, associated with Neumann boundary conditions and the wave equation in the time domain is examined. For a structured surface, a boundary condition is obtained on an equivalent flat wall, which links the acoustic veloci...

We present a method of homogenization of thin metallo-dielectric structures as used in the design of artificial surfaces, or metasurfaces. The approach is based on a so-called matched asymptotic expansion technique, leading to parameters being characteristic of an equivalent interface associated to jump conditions. It is applied to an array of meta...

The aim of the present work is to study the nucleation and propagation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale’s cohesive force model. Specifically, we investigate the stabilizing effect of the stress field non-uniformity by introducing a length l which characterizes the stress gradient i...

We present a method of homogenization for metamaterials with ultrathin thickness e, so-called metafilms. The method relies on a separation of scales, a macro scale being associated to the wavelength 1/k in the far field (outer region) and a micro scale associated to the subwavelength size h of the periodic cells in the metamaterial being relevant i...

Pierre Marie Suquet's pioneering work on plasticity paved the way for the mathematical theory of plasticity as we know it today. In this contribution we propose to review the most recent advances on that front and to illustrate how those impact classical problems of quasi-static elasto-plastic evolutions. Most notably, we exhibit new flow rules and...

The uniqueness of the solution to a quasi-static problem in perfect elasto-plasticity is established in the case of a bi-axial test. It is the first example known to us of uniqueness in the context of multi-dimensional elasto-plasticity.

In this paper we present a family of gradient-enhanced continuum damage models whichcan be viewed as a regularization of the variational approach to fracture capable of predicting in aunified framework the onset and space-time dynamic propagation (growth, kinking, branching, arrest)of complex cracks in quasi-brittle materials under severe dynamic l...

Griffith's theory of fracture [1] based on the concept of critical energy release rate G c remains the most used in fracture mechanics thanks to its simplicity in terms of material behavior. However, this theory contains many major drawbacks. In particular, the stress singularity is present at crack tips and it is impossible to initiate crack from...

L'article a pour but d'étudier l'initiation et la propagation de la fissure cohésive au sein d'une structure élastique bidimensionnelle infinie en prenant en compte l'effet essentiel de la non-uniformité du champ de contrainte. L'évolution de la longueur et l'ouverture de la fissure seront mise en évidence. Cette évolution est montrée régulière en...

The aim of the present work is to study the nucleation and prop- agation of cohesive cracks in two-dimensional elastic structures. The crack evolution is governed by Dugdale's cohesive force model. Specifically, we in- vestigate the stabilizing effect of the stress field non-uniformity by introducing a length which characterizes the stress gradient...

G-theta method with appropriate virtual crack extension (θ field) is proposed for the accurate evaluation of energy release rate along a crack edge which is non-orthogonal to the free surface. This method is implemented in the framework of finite element procedure as well as extended finite element one. This numerical procedure is then applied to i...

We derive sufficient conditions that prevent the formation of plastic slips in three-dimensional small strain Prandtl–Reuss elasto-plasticity when the yield criterion is of the Von Mises type. 2010 Mathematics Subject Classification: Primary 74C05, 35Q74, 47J35.

In the framework of rate-independent systems, a family of elastic-plastic-damage models is proposed through a variational formulation. Since the goal is to account for softening behaviors until the total failure, the dissipated energy contains a gradient damage term in order to limit localization effects. The resulting model owns a great flexibilit...

By adopting the ideas of Dugdale [1] and Barenblatt [2] on the surface crack energy and by seeking local mimimum energy [4, 5], the cohesive zone model extends the drawbacks of Griffith's theory [3] related to stress singularity at crack tip and impossible crack initiation from sound structure. Using the variational approach and the complex analysi...

Griffith’s theory of fracture based on the concept of critical energy release rate Gc remains the most used in fracture mechanics thanks to its simplicity in terms of material behavior. However, this theory contains many major drawbacks. On the one hand, it is in general impossible to initiate crack from a sound body because of the stress singulari...

We use matching asymptotic expansions to treat the antiplane elastic problem associated with a small defect located at the tip of a notch. In a first part, we develop the asymptotic method for any type of defect and present the sequential procedure which allows us to calculate the different terms of the inner and outer expansions at any order. This...