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Accueil du site > Equipes de recherche > Multi-échelle > Séminaire d’équipe/Team seminar

Séminaire d’équipe/Team seminar

The Multi-scale team is held approximately on a monthly basis. This page gathers the description of past instances.

J. Bluthé Hydromechanical modeling and numerical simulation of self-sealing phenomena in the Callovo-Oxfordian claystone 2018-10-DD
P. Monasse The geometric principles of 3D reconstruction from stereo imagery 2018-09-06
C. Arson Micro-macro modeling of chemo-mechanical damage and healing in rocks 2018-06-08
D. Weisz-Patrault Multiphase transformation induced plasticity and applications to forming processes 2018-05-18
M. Lebihain How to decypher fracture surfaces in disordered brittle materials ? 2018-04-24
B. Pan Towards High-Accuracy Digital Image/Volume Correlation Measurements : A Perspective From Imaging 2018-03-29
S. Naili Hierarchical homogenization of fluid saturated porous solid with multiple porosity scales 2018-03-08
N. Nemati Nonlocal dynamics of metafluids and phononic crystals 2017-07-04
V. Kostikov Unsteady free surface flow above the moving circular cylinder 2017-06-20
M. Peigney Hashin-Shtrikman type bounds for nonlinear conductors 2017-06-02
G. Cumunel Damage detection based on changes in modal parameters in beam-like structures : from methods development to experiments 2017-05-15
I. Maruyama Role of aggregate in concrete 2017-03-10
L. Brochard Physical mechanisms underlying the thermo-mechanical behavior of clays 2017-03-03
T. Honorio Effective properties of cement-based materials at early-age : towards estimations in ageing linear poroviscoelasticity 2017-01-10
K. Ioannidou Mesoscale modelling of Calcium Silicate Hydrates in cement 2016-12-06
P. Podio-Guidugli About energy and entropy inflows 2016-09-08
K. Danas Modeling of porous materials consisting of isotropic and anisotropic matrix and implications on deformation localization 2015-12-16
C. Lestringant Flambement d’une poutre précontrainte : compétition entre instabilités micro et macro 2015-11-30
A.-T. Akono Elucidating the Mechanical Resistance of Advanced Geo-composites at the Molecular Length-scale 2015-11-24
C. Montero Descriptions multi-échelles des comportements hygromécaniques du bois 2015-09-24
S. François Numerical modeling and in situ vibration measurements during the design and construction of low vibration floors at the nanotechnology laboratory Corelab 1B 2015-06-11
M. Bertin Propagation des incertitudes dans un modèle réduit de propagation des infrasons 2015-05-21
R.S. Elliott The Knowledgebase of Interatomic Models : An online resource for standardized testing and long-term warehousing of interatomic models and data 2014-09-19
X. Château Approche par homogénéisation du comportement d’une suspension de bulles dans un fluide à seuil 2014-06-02
G. Hassen Homogenized constitutive law for stone column reinforced soils : Elastoplastic behavior and strength capacities 2014-05-27
V. Ehrlacher Optimization of a structurally graded microstructured material 2014-04-24
V. Marry Modélisation moléculaire des fluides dans les argiles 2014-03-31
L. Chamoin Vérification et validation de modèles pour le calcul de quantités d’intérêt en ingénierie mécanique 2014-02-04
S. Brisard Méthodes d’homogénéisation numériques basées sur la discrétisation de l’équation de Lippmann—Schwinger 2014-01-20
H. Talbot Nouvelles méthodes de reconstruction tomographique par optimisation convexe 2013-11-25
F. Legoll Homogénéisation numérique : méthodes MsFEM, HMM, FE2, ... 2013-11-04
R. Cottereau Numerical strategy for the homogenization of random structural models 2013-10-03
L. Gélébart Méthodes FFT pour la simulation numérique de matériaux hétérogènes : développements et applications 2013-02-06

Hydromechanical modeling and numerical simulation of self-sealing phenomena in the Callovo-Oxfordian claystone

Joffrey Bluthé, Université Paris-Est, Laboratoire Navier, CNRS, ENPC, IFSTTAR, France

Extensive preliminary studies have led the National RadioactiveWaste Management Agency (ANDRA) to the choice of the Callovo-Oxfordian (COx) claystone of the Meuse/Haute-Marne as a host rock for a radioactive waste repository. This choice relies on its very low permeability and adequate mechanical properties, which allow the geological layer to act as a natural barrier against the spreading of radionuclides in the biosphere. However, the concept of underground storage relies on the excavation of a network of wells and drifts, which damages the surrounding rock, leading to the creation of a damaged zone along the drift walls. Therefore, the overall water permeability is likely to increase by several orders of magnitude. This damaged zone could represent a preferential pathway for dissolved radionuclides which could reach prematurely the surrounding more permeable geological layers. Thankfully, existing fractures tend to close upon rehydration of the claystone, mainly because of swelling phenomena and delayed deformations. This is referred to as self-sealing.

We propose here to model the hydromechanical couplings that take place during self-sealing so that the progressive resaturation of a drift may be studied from a theoretical standpoint. The swelling phenomena are first studied in a simplified linear elastic context to analyze the influence of geometry and boundary conditions on self-sealing, first at the scale of the sample using the finite element code Cast3M (CEA) to simulate the progressive resaturation around a set of periodic elliptical cracks. Non-trivial effects are brought to light, and lead to the conclusion that self-sealing needs to be investigated at the level of the structure and not only at the level of the material. Thus, further investigations are performed at the scale of the underground drift. Using micromechanics, the EDZ is represented as a medium composed of a homogeneous matrix in which micro cracks are distributed with preferential orientations. An analytical solution is derived for different possible orientations of the cracks, and the crack closure is determined to discuss the self-sealing potential of the rock in the case of a purely mechanical loading on the one hand, and of a purely hydric loading on the other hand. The non-linear case of several crack families which are able to close entirely during the experiment is investigated in an incremental approach using the results obtained in the linear case.

It should be noted that since the operation phase and the resaturation process take place over a hundred years and a few thousands of years respectively, delayed deformations are bound to develop, leading to convergence of the drift walls. This first model provides insights that may be useful for understanding the response of the EDZ, but also when developing a more elaborate model taking into account the viscoplastic behavior of the rock in relation with the water content. Both aspects indeed appear to have a significant impact on the macroscopic response of the COx claystone subject to swelling phenomena.

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The micromechanical model
Courtesy J. Bluthé

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The geometric principles of 3D reconstruction from stereo imagery

Pascal Monasse IMAGINE/LIGM, École des Ponts ParisTech, Université Paris-Est, France

I will discuss the geometric foundations of the 3D reconstruction of a scene from stereoscopic imagery. Based on the classical pinhole model for a camera, the geometric constraints between two photographs, encoded in the so-called fundamental matrix, will be explained. The automatic extraction of matching feature points via the SIFT method and their usage in estimating the fundamental matrix through the RANSAC algorithm will be presented. Finally, I will discuss the current challenges of multi-view stereo reconstruction.

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A 3D rendering of a facade of the Opera Garnier from photographs using the OpenMVG open source software
Courtesy P. Monasse

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Micro-macro modeling of chemo-mechanical damage and healing in rocks

Chloé Arson and Xianda Shen, Georgia Institute of Technology

Damage and healing in rocks refer to variations of mechanical and physical properties induced by pore or crack evolution. The gap between microscopic and macroscopic models makes it infeasible to uniquely characterize the pore- and crack- scale mechanisms that control deformation and flow regimes, predict percolation thresholds coupled to changes of rock stiffness, or relate crack rebonding time to stiffness and permeability healing time. The goal of this research is, therefore, to understand and predict chemo-mechanical damage and healing processes in rocks, by coupling microstructural and poromechanical models.

Oedometer tests and triaxial cyclic loading tests conducted by our collaborators on synthetic salt rock are used to characterize mechanical damage, both microscopically and macroscopically. Fabric is captured by 2D Scanning Electron Microscopy, by taking images to the axis of maximum principal stress. Microstructure image analyses indicate that solidity, local solid volume fraction, branch orientation and grain orientation are representative microstructure descriptors. A global fabric tensor, calculated as the product of moments of probability density functions of these four descriptors, is used as an internal variable to model the reduction of stiffness of damaged salt samples. Chemical damage is studied in granite, in which the expansion of biotite minerals is thought to be the cause of spherical weathering fractures. A time-dependent deformation law is established, based on the kinetics of the predominant chemical reactions. A Mori-Tanaka homogenization scheme is proposed to understand why biotite weathering results in the initiation and accumulation of damage in granite.

A chemo-mechanical homogenization model is formulated to predict long term healing processes induced by pressure solution in crystalline media that contain water films. Hill’s inclusion-matrix interaction law is applied to upscale strains and stresses at the scale of a Representative Elementary Volume (REV). Oedometer tests are simulated for specimens containing spherical inclusions with uniformly distributed contact plane orientations. It is observed that in samples containing inclusions with different initial void sizes, inclusions with larger pores have a negligible healing rate and are slowing down the overall REV healing process. The healing rate is higher in specimens with smaller grain sizes. For uniform void size distributions, the healing rate increases with initial porosity, but the final porosity change does not depend on the initial porosity of the sample.

Finally, we explore ways to couple the micro-model of mechanical healing driven by pressure solution with a thermodynamically consistent Thermo-Hydro-Chemo-Mechanical theory. The expression of Helmholtz free energy is established in terms of strain, porosity, temperature (state variables), and chemical, healing and damage potentials (internal variables). We assume that voids are connected and filled with brine. A modified Mori-Tanaka homogenization scheme is used to update the Biot coefficient, the Biot modulus and the stiffness tensor during the healing process (creep). Pore pressure is assumed to be the eigen stress of each void, and the auxiliary eigen stress of the matrix is introduced to satisfy Levin’s theorem. Simulations under uniaxial stress and isotropic stress conditions are performed under various conditions of temperature, pore pressure, void size and void orientation. The proposed model allows identifying the conditions in which the coupling between chemical reactions and mechanical stress can be beneficial for geological storage.

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Multiphase transformation induced plasticity and applications to forming processes

Daniel Weisz-Patrault (Laboratoire de Mécanique des Solides, École Polytechnique)

Transformation induced plasticity (TRIP) classically refers to plastic strains observed during phase transitions that occur under mechanical loads that can be much lower than the yield stress. A theoretical approach based on analytic homogenization is proposed to deal with multiphase changes and to extend the validity of the well-known and widely used model proposed by Leblond, J.-B et al (1989). International journal of plasticity, 5(6):551–572. Besides the generalization for several phases, one can mention three main improvements in the calculation of the local equivalent plastic strain : the deviatoric part of the phase transformation is taken into account, both parent and product phases are elastic-plastic with linear isotropic hardening and the applied stress is considered.

Applications to forming processes will be presented within the framework of residual stress estimation. In particular, the coiling process consist in winding steel strip on themselves for storage. Coil sagging is a major defect that can occur for recent steel grades undergoing phase transitions when the coils cool down. The detailed mechanisms leading to coil sagging are still not well understood, making this phenomenon very difficult to prevent. Mixed analytic/energetic approaches (accounting for thermal expansion, multiphase transitions, transformation induced plasticity and sliding at the interfaces) are developed to simulate efficiently the coil sagging phenomenon. Computation times are compatible with parametric studies.

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Coil sagging
Courtesy D. Weisz-Patrault
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Shear stress (MPa)
Courtesy D. Weisz-Patrault

Daniel Weisz-Patrault is a tenured researcher at the Solid Mechanics Laboratory in Ecole Polytechnique, France. He obtained his engineer and master degrees from Ecole des Ponts ParisTech in 2010, then did his PhD in the Navier laboratory under the supervision of Alain Ehrlacher. He graduated in 2012. He joined the Solid Mechanics Laboratory in 2014. His works have been awarded with 2 PhD prizes and he recently received the Jean Rist medal from SF2M.

His research interests are :

  • Thermo-metallurgical couplings
  • Homogenization of phase transitions in polycrystals
  • Transformation induced plasticity
  • Forming processes and additive manufacturing
  • Fast numerical simulation by mixed analytic/numeric/energetic approaches
  • Inverse methods

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How to decypher fracture surfaces in disordered brittle materials ?

Mathias Lebihain¹², Jean-Baptiste Leblond¹, Laurent Ponson¹, Michel Bornert²

¹Institut Jean Le Rond ∂’Alembert (UPMC, CNRS), Paris, France

²Laboratoire Navier (ENPC, IFSTTAR, CNRS), Marne-la-Vallée, France

Keywords : crack propagation, brittle fracture, heterogeneous materials, toughness discontinuies

With the recent emergence of new manufacturing techniques (e.g. 3D printing), structured materials will flourish in the next few decades. Thus predicting the role played by small-scale heterogeneities on the macroscopic behavior of solids is a prerequisite to design such improved materials and make reliable predictions both on their resistance and their lifetime.

Fracture surfaces are a persistent signature of crack propagation. In that sense, they may contain detailed information on the failure mechanisms [1]. The goal of this study is to extract failure information from the fracture surfaces statistical properties [2]. To this aim, we developed a theorical formalism which depicts crack propagation in a heterogeneous brittle material displaying toughness discontinuities. Using perturbation methods of LEFM [3, 4, 5], we compute efficiently the propagation of a semi-infinite crack in a omogeneous matrix with tens of thousands randomly distributed tougher inclusions.

In this presentation, we will describe how we handle the interaction between the crack front and toughness discontinuities through a (G-Gc)max criterion. The crack can either go across the inclusion or by-pass it (Fig. 1). By putting at work this criterion on large system of inclusions (Fig. 2), we are able to apply concepts of quantitative fractography to characterize the resulting surface roughness and compare numerical results with experimental data on 3D-printed heterogeneous samples.

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Figure 1 : Crack propagation in a heterogeneous material with crossed (light gray) and by-passed (dark gray) inclusions
Courtesy M. Lebihain
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Figure 2 : Fracture surface from a crack interacting with 100k inclusions (Compute time : 2 hours)
Courtesy M. Lebihain


[1] Ponson, L., ”Statistical aspects in crack growth phenomena : How the fluctuations reveal the failure mechanisms”, Int. J. Frac., 201:11-27 (2016)

[2] Ponson, L., ”Crack propagation in disordered materials : how to decypher fracture surfaces”, Annales de Physique, 32 (2007)

[3] Rice, J.R., ”First-order variation in elastic fields due to variation in location of a planar crack front”, ASME J. Appl. Mech., 52:571–579 (1985)

[4] Movchan, A.B., Gao, H. and Willis, J.R., ”On perturbations of plane cracks”, Int. J. Solids Structures, 35:26-27:3419-3453 (1998)

[5] Leblond, J.-B., ”Crack paths in three-dimensional solids” , Int. J. Solids Structures, 36:79-103 (1999)

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Towards High-Accuracy Digital Image/Volume Correlation Measurements : A Perspective From Imaging (2018-03-29)

B. Pan, Institute of Solid Mechanics, Beihang University, Beijing 100191, People’s Republic of China

In the experimental mechanics community, digital image/volume correlation (DIC/DVC) techniques have been widely accepted as the most popular, practical and versatile tool for full-field surface/internal displacement and deformation measurements. Basically, the implementation of DIC/DVC measurements involves two consecutive stages : namely image acquisiton and image processing. At present, the state-of-the-art DIC/DVC algorithms using inverse compositional Gauss-Newton (IC-GN) algorithm and B-spline interpolation scheme allow subpixel/subvoxel registration with an accuracy higher than 0.005 pixels/voxels for computer simulated speckle pattern. However, for real experimental images recorded by common imaging systems, the accruacy of DIC/DVC measurements can be seriously degraded. Since measurement accuracy is always the most important objective in various experimental mechanics applications, the errors associated with image acquistion must be understood, quantified and minimized.

In this talk, we will first point out that all the common imaging systems (e.g. a single camera for 2D-DIC, synchronized two cameras for stereo-DIC, or a x-ray CT scanner for DVC) are neither perfect nor stable, because of the existence of lens distortion and the continual slight changes in imaging geometry associated with the self-heating effect or ambient temperature variations. Then, we systematically investigate the measurement errors in 2D-DIC, stereo-DIC and DVC from the perspective of the stability of image acquisition devices. Our experimental results show that the maximum temperature-induced artificial strains can reach an magnitude of 200, 150, and 400 microstrains for the specific imaging devices used by the authors. Finally, we discuss several approaches that can be used to mitigate or correct these errors. In particular, we focus on an easy-to-implement but effective reference specimen compensation method, and validate its efficacy and practicality by real experiments. Typical application of DIC techniques to civil engineering will also be demonstrated.

Dr. Bing Pan is a full professor in School of Aerospace Science & Engineering at Beihang University (BUAA), China. He received his Ph.D degree in Mechanical Engineering from Tsinghua University in 2008. After working with Professor Anand Asundi in Nanyang Technological University (Singapore) as a postdoctor, he joined Institute of Solid Mechanics, BUAA in 2009. His current research interests mainly focus on advanced optical techniques and their applications in experimental mechanics, especially the digital image correlation, digital volume correlation techniques for surface or internal deformation measurement of solid materials and structures, as well as new experimental techniques for characterizing thermo-mechanical behavior of hypersonic materials and structures. He has published more than 100 peer-reviewed articles in international journals, and six of these papers were selected as ESI highly cited papers. All his publications have been cited more than 3300 times according to Web of Science and more than 6000 times according to Google Scholar. Dr. Pan was selected for Youth Changjiang Scholars (MOE) in 2016, and won the National Natural Science Funds for Excellent Young Scholar in 2013.

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Hierarchical homogenization of fluid saturated porous solid with multiple porosity scales (2018-03-08)

S. Nailih, Université Paris-Est Créteil, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, Marne la Vallée, France

In this work, we investigate the macroscopic response of an elastic porous skeleton subjected to a mechanical loading. The pores of two different sizes are filled with a compressible fluid which can redistribute at both the microscopic and mesoscopic scales since the pores form one system of connected network. We apply the asymptotic homogenization method to upscale a microscopic fluid–structure interaction problem. The obtained poroelastic model describes the matrix behavior at the mesoscopic level. The homogenization procedure is repeated to give rise a double-porosity model relevant to the macroscopic scale. We discuss relationships obtained with the standard Biot poroelasticity theory. As an advantage, the approach reported here provides a direct procedure to calculate the effective properties for any 3D microstructure without any restrictions concerning the shape or topology of the pore network.

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A REV with orthogonal connected microstructures. The matrix occupying Ym represents the dual porosity,the channels form the 3D cross, domain Yc.
Courtesy S. Naili

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Nonlocal dynamics of metafluids and phononic crystals (2017-07-04)

N. Nemati, Laboratoire MSME, Université Paris-Est, Marne la Vallée, France

We present briefly a nonlocal theory of sound propagation in rigid porous media saturated with a viscothermal fluid. The theory that takes fully into account the temporal and spatial dispersions is formulated based on an analogy with electromagnetism and the thermodynamic concept of acoustic energy flux. Within this theory a non-asymptotic homogenization method has been developed, through which the effective-medium parameters can be obtained in functions of frequency and wave vector. We show how metamaterial and phononic crystal behaviors originate in temporal and spatial dispersions in the medium.

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Unsteady free surface flow above the moving circular cylinder (2017-06-20)

V. Kostikov, Novosibirsk State University

A problem on non-stationary free surface flow of an infinitely deep ideal fluid generated due to the motion of a submerged body is considered. The initial formulation of the problem is reduced to an integral-differential system of equations for the functions defining the free surface shape, the normal and tangential components of velocity on the free boundary. Small- time asymptotic solution is constructed for the case of circular cylinder that moves with a constant acceleration from rest. The role of non-linearity is clarified by the analysis of this solution in the context of formation mechanism of added mass layers, splash jets and finite amplitude surface waves.

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Horizontal motion of circular cylinder of radius r = 0.7 with λ = 10. Solid thick line – higher-order asymptotic solution, dashed line – leading-order solution, solid thin line – exact linear solution.
Courtesy V. Kostikov
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Vertical rising (θ = p/2) of circular cylinder of radius r = 0.5 with λ = 5 : (a) Visualisation of free surface deformation at the dimensionless time t = 0.8 and (b) the thickness of inertial liquid layer about the cylinder at x = 0 predicted by the higher-order asymptotic solution (solid line) and by the leading-order asymptotic (dashed line).
Courtesy V.Kostikov

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Hashin-Shtrikman type bounds for nonlinear conductors (2017-06-02)

M. Peigney, Laboratoire Navier

In a seminal work, Hashin and Shtrikman (1962) have obtained optimal bounds on the effective conductivity of linear composite conductors with statistically isotropic microstructures. Those bounds are explicit functions of the volume fractions and conductivities of each constitutive phase. Several methods have been proposed to extend the results of Hashin and Shtrikman to nonlinear composites. A first method, proposed by Talbot and Willis (1985), makes uses of a homogeneous linear comparison medium and generalizes the variational approach introduced by Hashin and Shtrikman. A second method, due to Ponte Castaneda (1991), employs a heterogeneous linear comparison medium. Both those methods can generate nonlinear bounds of the Hashin-Shtrikman type, i.e. bounds that hold for the whole class of isotropic composites with prescribed volume fractions and conduction properties of the individual phases. In this talk, we focus on a third method, known as the translation method, which has been proposed in the linear setting by Lurie and Cherkaev (1984) and independently by Murat and Tartar (1985). The main idea is to embed the original problem in a problem of higher dimension by considering several loading orientations simultaneously. For two-dimensional conductors, we show that the translation method can produce nonlinear Hashin-Shtrikman type bounds that are strictly stronger than those provided by the Talbot-Willis and the Ponte Castaneda methods.

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Damage detection based on changes in modal parameters in beam-like structures : from methods development to experiments (2017-05-15)

G. Cumunel, Laboratoire Navier

The demand for enhanced performance and reliability of structures in terms of safety, noise-level and durability is ever increasing. Over the past few decades significant research have been conducted on structural health monitoring using many different techniques applied in various fields like aeronautics, industry, or civil engineering.

Given boundary conditions, a linear dynamic system can be defined either by its physical parameters (density, stiffness and damping) or by its modal parameters (eigenfrequencies, mode shapes, and modal damping ratios). Structural parameter modifications result in a change in modal parameters. This observation that changes in structural properties induce changes in modal parameters is the key to using modal methods for damage identification and structural health monitoring.

Two methods that have been developed at Navier are introduced as well as some developments on sensors and experimental results. The first one is a procedure for localization and quantification of modifications in the case of transverse vibrations of an Euler-Bernoulli beam with or without axial force. Two states are considered : an initial or ``healthy’’ one and a modified or ``damaged’’ one with simultaneous modifications which can concern axial force, mass density and/or bending stiffness. The proposed procedure gives first an estimation of the localization of the modification using the mode shapes of the initial state and then an estimation of the relative variations of axial force and/or mass density and/or bending stiffness through a linear optimization technique.

The second one is based on the single value decomposition (SVD). The SVD of an m × n real or complex matrix M is a factorization of the form Um×m∙Σm×n∙Vn×n. The columns of U and V are called the left-singular vectors and right-singular vectors of M, respectively. We applied the SVD on matrices of modal parameters evolution to take advantage of the property of the SVD to determine the main trends, in a statistical point of view, in a data set. We can thus access to the detection of damage by observing right-singular vectors of the matrix of frequencies evolution as well as of the matrix of mode shape evolution, which can moreover localize the damage with left-singular vectors.

Finally, some developments on instrumentation to enhance damage detection using modal curvature methods, as fiber optic or piezoelectric long gage sensors or high speed camera(s) (digital image correlation), are discussed.

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Beams instrumented with small and long gage strain gauges
Courtesy G. Cumunel
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Accelerometers and high-speed camera for damage detection experiments
Courtesy G.Cumunel

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Role of aggregate in concrete (2017-03-10)

I. Maruyama, Concrete Engineering Lab., Architectural Structure and Construction System, Graduate School of Environmental Studies, Nagoya University, Japan

Aggregate generally occupied about 70% of the volume of normal strength concrete. Consequently, the property of aggregate has a large impact on the mechanical performance of concrete, while their impacts are not fully understood. In the presentation, a part of roles of aggregates in concrete is clarified from a viewpoint of volumetric stability. Drying shrinkage, shrinkage-induced cracking, and physical properties of concrete under drying are addressed. In addition, as a special case of volumetric stability problem, concrete under irradiation is briefly introduced based on the advanced knowledge from Japanese national project sponsored by nuclear regulatory authority.

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Cracking behavior of concrete containing non-shrinking limestone aggregate under restraint condition.
Courtesy I. Maruyama
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Cracking behavior of concrete containing shrinking sandstone aggregate under restraint condition.
Courtesy I. Maruyama

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Physical mechanisms underlying the thermo-mechanical behavior of clays (2017-03-03)

L. Brochard, Laboratoire Navier

The thermo-mechanical behavior of clays is known to be especially unusual. Indeed, drained heating experiments show irreversible thermal contractions of normally consolidated clays. In contrast, over-consolidated clays exhibit a reversible expansion at low temperatures followed by irreversible contraction at higher temperatures. Moreover, the temperature at the expansion-contraction transition increases with the degree of over-consolidation. The amplitudes of expansion and contraction are one to two orders of magnitude larger than what is measured for the solid clay minerals. Accordingly, adsorbed water seems to be at the heart of this phenomenon, but geomechanical models do not relate adsorption and thermo-mechanics of clays. We propose in this work a multi-scale approach that does relate adsorption and thermo-mechanics, the result of which prove consistent with experimental observation, thus offering a possible explanation of the physical origin of the thermo-mechanics of clays. This work is conducted in the framework of the ANR project TEAM2ClayDesicc which aims at studying the thermally-induced desiccation of clays.

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Effective properties of cement-based materials at early-age : towards estimations in ageing linear poroviscoelasticity (2017-01-10)

T. Honorio, Laboratoire Navier

At early age, the properties of cement-based materials (CBM) are ageing due (mainly) to hydration processes. The evolutions of mechanical and thermal properties at early age are of interest in industrial applications involving, notably, massive concrete structures in which thermal cracking may harm the performance of the structure.

In this presentation, estimations of the evolution of the effective thermal (heat capacity, thermal conductivity and coefficient of thermal expansion) and mechanical (elastic moduli, creep and relaxation functions) properties are presented in a multiscale framework representing the micro- and mesostructures of CBM.

Estimations of the viscoelastic properties are highlighted. As mentioned above, at early-age, these properties are ageing, i.e. evolves independently with respect to time and loading time. The homogenization tools used here follows the approach introduced by Sanahuja (2013), which combined Mandel’s (1958) correspondence principle in Ageing Linear Viscoelasticity (ALV) in terms of Volterra integral operators with a numerical solution for integrals. Inspired by Sanahuja (2013b), the argument of solidification theory is applied to obtain effective relaxation and creep compliance functions at (hydrating) cement paste, mortar and concrete scales in an undrained saturated scenario (Honorio et al., 2016).

Further, to cope with transformation fields, as the ones found in thermo- and poromechanics, a general formulation of the micromechanics of presstressed or prestrained composites in ALV is presented. Because of the non-commutativity of Volterra operators, two estimations of effective transformations fields are possible. However, these estimations can be proven identical by cause of the consistency condition. Additionally, the results are extended to the case of locally transforming materials due to non-coupled dissolution and/or precipitation of a given (elastic or viscoelastic) phase.

Applications to the estimations of effective properties of CBM in the presence of transformation fields are proposed. This theoretical framework offers a possible path to investigate effects of humidity coupled with viscoelastic behavior. In this context, applications coping with other phenomena leading to ageing in CBM (for instance, degradation processes at late ages as leaching) can also be envisioned.

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From composition to the estimation of properties of interest in early-age analysis
Courtesy T. Honorio


Honorio, T., Bary, B., Benboudjema, F., 2016. Multiscale estimation of ageing viscoelastic properties of cement-based materials : A combined analytical and numerical approach to estimate the behaviour at early age. Cem. Concr. Res. 85, 137–155. doi:10.1016/j.cemconres.2016.03.010

Mandel, J., 1958. Sur les corps viscoélastiques linéaires dont les propriétés dépendent de l’âge. Comptes Rendus L’Académie Sci. 247, 175–178.

Sanahuja, J., 2013a. Effective behaviour of ageing linear viscoelastic composites : Homogenization approach. Int. J. Solids Struct. 50, 2846–2856. doi:10.1016/j.ijsolstr.2013.04.023

Sanahuja, J., 2013b. Efficient Homogenization of Ageing Creep of Random Media : Application to Solidifying Cementitious Materials. American Society of Civil Engineers, pp. 201–210. doi:10.1061/9780784413111.023

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Mesoscale modelling of Calcium Silicate Hydrates in cement (2016-12-06)

K. Ioannidou, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology (MIT), MultiScale Material Science for Energy and Environment, MIT-CNRS Joint Laboratory @ MIT

Organized jointly with the Geotechnics team

Calcium-silicate hydrate (C-S-H) is the main binder in a cement paste. It starts forming from the early stages of cement hydration and it progressively densifies as cement sets. C-S-H nano-scale building blocks form a cohesive gel. Here I present a statistical physics approach recently developed, which allows to investigate the C-S-H gel formation under the out-of-equilibrium conditions typical of cement hydration. Our bottom-up approach is based on colloidal particles, precipitating in the pore solution and interacting with effective forces associated to the ionic environment backed-up from atomistic simulations and experiments. I present the evolution of the space filling of C-S-H at early and later stages of hydration with different particle interactions and compare them with experimental data at different lime concentrations. Moreover, I discuss the comparison of our model of C-S-H with different experimental investigations such as scattering intensity, pore size distributions, specific surface area, local densities and hardness of the material. Our results provide a quantitative insight into how the heterogeneities developed during the early stages of hydration persist in the structure of C-S-H and impact the mechanical performance of the hardened cement paste.

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Mesoscale model of CSH
Courtesy K. Ioannidou

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About energy and entropy inflows (2016-09-08)

P. Podio-Guidugli, Università di Roma TorVergata et Accademia Nazionale dei Lincei, Italie

It will be shown that energy and entropy inflows, contrary to what is usually done, should not be always taken proportional ; and that the proportionality factor need not always be temperature, as is the case when modelling phase segregation at constant temperature, with and without diffusion.

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Modeling of porous materials consisting of isotropic and anisotropic matrix and implications on deformation localization (2015-12-16)

K. Danas, Laboratoire de Mécanique des Solides, École Polytechnique

This work presents rate-dependent constitutive models for porous materials with a matrix phase described by either J_2-flow theory or crystal plasticity with arbitrary number of slip systems and orientations. The material comprises ellipsoidal voids at arbitrary orientations and is subjected to general three-dimensional loadings. The proposed modified variational models (MVAR), are based on the nonlinear variational homogenization method, which makes use of a linear comparison porous material to estimate the response of the nonlinear porous material.

The MVAR models are validated by periodic finite element simulations for a large number of parameters including general void shapes and orientations, various creep exponents (i.e., nonlinearity) and general loading conditions. The MVAR models are found to be in good agreement with the finite element results for all cases considered.

The MVAR model with the isotropic matrix is then implemented in a UMAT and used to simulate 3D geometries of real experimental setups. Strain localization and damage as this is measured by the evolution of the underlying porosity and void shapes is analyzed and discussed. The findings are in good agreement with the experimental observations.

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Flambement d’une poutre précontrainte : compétition entre instabilités micro et macro (2015-11-30)

C. Lestringant, Laboratoire d’Alembert, Université Pierre et Marie Curie

Des observations récentes [1, 2] sur un système expérimental constitué de deux rubans d’élastomères assemblés avec pré-contrainte font état d’un motif de flambement composé de plusieurs hélices de chiralité opposée séparées par des défauts appelés perversions. Un nombre variable de perversions a été observé en faisant varier les valeurs de la précontrainte et des caractéristiques géométriques de la section des rubans. L’étude de la stabilité d’une poutre avec courbure naturelle ne permet pas d’expliquer cette sélection du nombre de perversions.

Nous présentons un modèle 1D faisant un apparaître un motif de flambement de longueur d’onde finie, ce qui permet de rendre compte qualitativement du comportement observé dans [1, 2]. Ce modèle prédit en effet à la fois des instabilités "macroscopiques" (dont la longueur d’onde est sélectionnée par la longueur du système) et des instabilités "microscopiques" (dont la longueur d’onde est sélectionnée par les caractéristiques de la section). Ce modèle est fondé sur une cinématique de poutre mince mais étend le modèle de poutre classique d’Euler-Bernoulli, prenant en compte la flexion dans la section, responsable de l’apparition du mode microscopique.

Nous discutons ensuite la généralisation de ce modèle à un modèle de plaque mince non linéaire ainsi qu’à l’élasticité finie 3D.

[1] J. Huang, J. Liu, B. Kroll, K. Bertoldi, and D. R. Clarke, Spontaneous and deterministic three-dimensional curling of pre-strained elastomeric bi-strips, Soft Matter, 8 (2012), 6291–6300

[2] J. Liu, J. Huang, T. Su, K. Bertoldi, and D. R. Clarke, Structural Transition from Helices to Hemihelices, PLOS, 9, 4 (2014)

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Elucidating the Mechanical Resistance of Advanced Geo-composites at the Molecular Length-scale (2015-11-24)

A.-T. Akono, University of Illinois at Urbana-Champaign

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Descriptions multi-échelles des comportements hygromécaniques du bois (2015-09-24)

C. Montero, Laboratoire de Mécanique et Génie Civil (LMGC), IUT de Génie Civil de Nîmes, Université de Montpellier

Parmi les matériaux de construction, une des particularité du bois réside dans l’origine polymérique naturelle de ses constituants : cellulose, hemicellulose et lignine. Leurs arrangements à différentes échelles d’observation confèrent aux bois des comportements mécaniques particuliers en jeu aux échelles macroscopique d’emploi en structures. Les interactions entre l’environnement hygrothermique des éléments en bois mettent en évidence des comportements mécaniques couplés difficiles à prédire par la diversité biologique des bois, le débit ou encore les procédures de séchage.

Pour caractériser les réponses mécaniques à long terme des expérimentations ont été menées sous chargement constant à long terme en environnement régulé. Des modèles rhéologiques permettent de prédire les comportement viscoélastiques (sous climats constants) et mécanosorptifs (sous climats variables) et de les comparer aux coefficients prédictifs établis dans l’Eurocode 5.

Pour comprendre l’origine de ces comportements, des études menées sous rayonnement X sur équipement de laboratoire et en synchrotron à l’ESRF (Grenoble, France) ont permit de quantifier la contribution de la cellulose dans la réponse macroscopique pour différents modes de chargements et conditions climatiques.

Cette présentation est l’opportunité d’apporter une vision synthétique sur la connaissances des comportements hygro-mécaniques couplés avec un regard spécifique sur leurs descriptions multi-échelles. Les travaux présentés combinent une approche phénoménologique à l’échelle macroscopique à la fois expérimentale et numérique pour prédire les comportements des bois en structure et une approche réductionniste à l’échelle des constituants pour comprendre les mécanismes moléculaires à l’origine de la variabilité de la réponse différée en vue d’établir des lois de comportements favorisant la fiabilité des structures bois.

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Numerical modeling and in situ vibration measurements during the design and construction of low vibration floors at the nanotechnology laboratory Corelab 1B (2015-06-11)

S. François, KU Leuven (formerly visiting scholar at Laboratoire Navier)

At KU Leuven, the nanotechnology laboratory Corelab 1B has recently been constructed, consisting of low vibration floors, clean rooms and offices. The top 8 m of soil on the site consist of very soft clay and peat, necessitating the use of a piled foundation. Nearby road traffic imposes a challenge to achieve low vibration levels required for sensitive equipment, especially during rush hours. Therefore, vibration control has been of prime concern from an early design stage.

The dynamic response of several foundation designs have been analysed using state of the art 3D dynamic soil-structure interaction models. A finite element (FE) model of a single module of the structure supported by 4 foundation piles has been coupled to a boundary element (BE) model of the layered soil, accounting for dynamic soil-structure interaction. The transfer function between a unit harmonic vertical point load on a nearby road and the foundation has been computed. It was concluded that a sufficiently high bending stiffness of the foundation piles is needed to reduce vibration levels.

Predicted vibrations are subject to a large level of uncertainty. Therefore, a hybrid methodology has been elaborated where structural models are combined with in situ vibration measurements. The measured pile impedance has been coupled to a finite element model of the low vibration floor, resulting in an updated estimate of the transfer function between the nearby road and the structure. This intermediate verification of the design confirmed the performance of the constructed pile foundation, avoiding late structural adjustments.

Numerical predictions obtained with the coupled FE-BE model and the hybrid methodology are finally compared to vibration measurements in the finalized structure, demonstrating that the actual performance of the low vibration floor. This is due to the fact that only a single module of the structure has been considered in the coupled FE-BE and hybrid analysis. Rigid connections between different modules result in a stiffer and heavier ensemble that is less vibration sensitive than a single module. The presence of a large group of piles also causes a screening effect that reduces vibration levels.

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Propagation des incertitudes dans un modèle réduit de propagation des infrasons (2015-05-21)

M. Bertin, formerly post-doc at Laboratoire Navier

La perturbation d’un système peut donner lieu à de la propagation d’onde. Une façon classique d’appréhender ce phénomène est de rechercher les modes propres de vibration du milieu. Mathématiquement, trouver ces modes consiste à rechercher les valeurs et fonctions propres de l’opérateur de propagation. Cependant, d’un point de vue numérique, l’opération peut s’avérer coûteuse car les matrices peuvent avoir de très grandes tailles. En outre, dans la plupart des applications, des incertitudes sont inévitablement associées à notre modèle. Cela est d’autant plus contraignant que certaines échelles du milieu peuvent avoir une influence majeure sur l’onde. La question se pose alors de savoir s’il faut attribuer d’importantes ressources de calcul pour une simulation dont la précision du résultat n’est pas assurée.

Nous proposons une démarche qui permet à la fois de mieux comprendre l’influence des incertitudes sur la propagation et de réduire considérablement les coûts de calcul pour la propagation des infrasons dans l’atmosphère. L’idée principale est que tous les modes n’ont pas la même importance et souvent, seule une poignée d’entre eux suffit à décrire le phénomène sans perte notable de précision. Ces modes s’avèrent être ceux qui sont les plus sensibles aux perturbations atmosphériques. Plus précisément, l’analyse de sensibilité permet d’identifier les structures de l’atmosphère les plus influentes, les groupes de modes qui leur sont associés et les parties du signal infrasonore qui leur correspondent. Ces groupes de modes peuvent être spécifiquement ciblés dans un calcul de spectre au moyen de techniques de projection sur des sous-espace de Krylov, ce qui implique un gain important en coût de calcul. Cette méthode de réduction de modèle peut être appliquée dans un cadre statistique et l’estimation de l’espérance et de la variance du résultat s’effectue là aussi sans perte notable de précision et avec un coût réduit.

Nous illustrons notamment cette méthode avec l’étude du cas de l’explosion du réacteur de Fukushima (Japon), le 12 mars 2011, pour lequel une onde infrasonore a été détectée 240 km plus au sud, et avec l’expérience de calibration Sayarim 2011 (Israël) où l’explosion de quelques charges a donné lieu à des détections infrasonores sur de multiples stations de mesures, proches comme très éloignées.

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The Knowledgebase of Interatomic Models : An online resource for standardized testing and long-term warehousing of interatomic models and data (2014-09-19)

R.S. Elliott, University of Minnesota

Atomistic simulations using empirical interatomic potentials play a key role in realistic scientific and industrial applications. This talk describes an NSF-funded effort to develop an open-source online tool for promoting the use and reliability of interatomic models. The Knowledge-base of Interatomic Models allows to compare model predictions with reference data, to generate new predictions by uploading simulation test codes, and to download models conforming to an application programming interface (API) standard that has been developed in collaboration with the atomistic simulation community. An overview will be given of the KIM project and its main components which include the KIM API, the KIM data structure for representing arbitrary material properties, the KIM processing pipeline, and the KIM visualization framework.

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Approche par homogénéisation du comportement d’une suspension de bulles dans un fluide à seuil (2014-06-02)

X. Château, Laboratoire Navier

Collaboration avec Thy Thuy Linh Nguyen, Lucie Ducloué, Guillaume Ovarlez et Olivier Pitois.

Dans de nombreuses situations d’intérêt pratique des bulles d’air sont incorporées à une pâte ou un fluide à seuil (élaboration de plaques de plâtre ou de bétons allégés, fabrication de mortiers isolants, mousses alimentaires, ...).

Même si l’ajout d’air à un matériau vise en général à modifier ses propriétés à l’état durci, les propriétés rhéologiques à l’état frais sont également affectées. La compréhension et la modélisation de l’effet de l’ajout d’air sont donc essentielles pour une bonne maîtrise de l’élaboration et la mise en oeuvre des matériaux aérés.

Dans une première partie on s’intéresse au problème d’une seule bulle sphérique suspendue dans une matrice élastique incompressible cisaillée à l’infini. On montre que la bulle se comporte comme une particule élastique dont les caractéristiques (module de compressibilité et module de cisaillement) dépendent du module de compressibilité du gaz constituant la bulle, du module de cisaillement de la matrice et d’un nombre capillaire défini comme le rapport du produit du module de cisaillement de la matrice par le rayon de la bulle sur la tension de surface de l’interface matrice air. Ce nombre capillaire quantifie la rigidité de la bulle par rapport à celle de la matrice.

A utilisant cette solution on peut mettre en oeuvre des schémas d’homogénéisation visant à prédire les caractéristiques globales d’’une suspension de bulles dans un matériau linéaire (sous le seuil). Suivant les valeurs du nombre capillaire les bulles peuvent avoir un effet renforçant ou au contraire adoucissant dans le régime linéaire. On s’intéresse ensuite au comportement non linéaire (au dessus du seuil) de la pâte tout d’abord dans le cas d’un comportement plastique parfait puis dans le cas d’un comportement du type Herschel-Bulkley. On donne en particulier des estimations simples des caractéristiques rhéologiques globales du matériau (module de cisaillement, seuil de contrainte, consistance, ...) avant de s’intéresser à la réponse d’un ver de matériau soumis à une histoire simple de déformation.

Ces résultats théoriques sont comparés aux résultats expérimentaux obtenus sur des matériaux modèles au laboratoire.

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Homogenized constitutive law for stone column reinforced soils : Elastoplastic behavior and strength capacities (2014-05-27)

G. Hassen, Laboratoire Navier

The design of soil structures reinforced by columnar inclusions, such as the "stone column" improvement technique, represents a difficult challenge for geotechnical engineers, owing to the composite nature of the reinforced soil and the relatively large number of inclusions involved in the reinforcement process. Apart from the particular situation of a rigid foundation of unlimited horizontal extension, resting upon a soft clay reinforced by regularly distributed inclusions, where simplified calculation procedures can be applied, simulating the behaviour of stone column reinforced soil structures requires the use of more sophisticated numerical procedures.

Referring for instance to a finite element simulation of this kind of structures, where the cylindrical shape of the inclusions should be taken into account, a fully three-dimensional analysis would be required, with a locally refined mesh discretization in order to capture with sufficient accuracy the complex interactions prevailing between the inclusions and the surrounding soil. This would ultimately lead to oversized numerical problems, or at least to the elaboration of a highly complex and sophisticated computational tool, the use of which remains hardly compatible with an engineering design approach.

Taking advantage of the periodic layout of the reinforcing columns into the native soil, the homogenization method offers an attractive alternative, making it possible to overcome the abovementioned difficulties. According to this method, the basic concept of which have already been implemented in a somewhat heuristic manner by [1] or [2], the composite reinforced soil is regarded from a macroscopic point of view (that is at the scale of the whole structure to be designed) as a homogeneous, but anisotropic, medium. Both the stiffness and strength properties of this equivalent medium are derived from solving an elastoplastic boundary value problem attached to the reinforced soil representative volume. Based upon an approximate solution to this problem, in which piecewise constant stress fields are used, the macroscopic constitutive law of the column reinforced soil can be formulated in the context of elastoplasticity in the framework of a finite element formulation as it has been shown in [3].

Besides, as regards the failure or strength capacities of the reinforced soil, the macroscopic criterion is obtained from solving a yield design auxiliary problem attached to the reinforced soil’s unit representative cell by implementing a fem-based numerical approach, making use of semidefinite programming.

The subsequent incorporation of the so-obtained homogenized criteria into global stability analyses of reinforced soil structures appears to be a difficult task, due to the complexity of the corresponding yield surfaces [4]. In order to overcome such a difficulty, a numerical procedure is proposed, based on the use of convex ellipsoidal sets, which provides an accurate approximation to the criterion, involving relatively few parameters, which makes the approximated criterion much easier to handle than the initial one [5].

Finally, the approximation of the macroscopic yield strength criterion is implemented in order to tackle the problem of a classical geotechnical structure, such as evaluating the bearing capacity of a soil foundation resting upon a reinforced soil.


[1] Canetta G., Nova R. (1989). A numerical method for the analysis of ground improved by columnar inclusions. Comp. and Geotech., n°7, pp. 99-114.

[2] Lee J.S., Pande G.N. (1998). Analysis of stone-column reinforced foundations. Int. J. Numer. Anal. Meth. Geomech. n°22, pp. 1001-1020.

[3] Hassen, G., de Buhan, P., & Abdelkrim, M. (2010). Finite element implementation of a homogenized constitutive law for stone column-reinforced foundation soils, with application to the design of structure. Comput. Geotech. 37, 40-49.

[4] Hassen, G., M. Gueguin, & P. de Buhan (2013). A homogenization approach for assessing the yield strength properties of stone column reinforced soils. European Journal of Mechanics-A/Solids 37, 266-280.

[5] Bleyer, J. & P. de Buhan (2013). Yield surface approximation for lower and upper bound yield design of 3d composite frame structures. Computers & Structures 129, 86-98.

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Optimization of a structurally graded microstructured material (2014-04-24)

V. Ehrlacher, CERMICS

(joint work with C. Le Bris, F. Legoll, G. Leugering, M. Stingl, F. Wein)

An approach for the optimization of non-periodic microstructured material through the homogenization method will be presented. The central idea, simsilar to the one used by Pantz and Trabelsi, consists in modeling the material as a macroscopic deformation of an initially periodic material. Following the path of Bensoussan, Lions and Papanicolaou, homogenization formulas can be derived to obtain the expression of the effective stiffness elasticity tensor in the limit when the size of the microcells composing the material tends to zero.

Using reduced-order models obtained via greedy algorithms, the optimization procedure is performed either using the homogenization method. Numerical results obtained on a two-dimensional material will be presented.

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Modélisation moléculaire des fluides dans les argiles (2014-03-31)

V. Marry, Laboratoire PHENIX

Les minéraux argileux sont un des principaux composants de la croûte terrestre. Celles-ci sont très largement étudiées, en particulier dans le cadre du stockage souterrain des déchets radioactifs ou du dioxyde de carbone, ou celui de l’extraction du gaz naturel et du pétrole.

Les modélisations moléculaires, si elles ne peuvent pas prendre en compte la structure globale et complexe du matériau poreux, permettent en revanche d’appréhender les processus au niveau microscopique et d’expliquer le comportement local des fluides en milieu confiné. Les informations ainsi obtenues sur la chimie, la thermodynamique et le transport dans les systèmes argile-fluide sont d’une aide précieuse pour alimenter, compléter, valider ou invalider les modèles théoriques aux échelles macroscopiques.

On se propose ici d’illustrer par quelques exemples les différentes possibilités qu’apportent les simulations moléculaires classiques à la description des systèmes argile-eau et argile-eau-CO2. On montrera aussi l’intérêt d’une démarche multi-échelle pour la description de ces milieux.

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Vérification et validation de modèles pour le calcul de quantités d’intérêt en ingénierie mécanique (2014-02-04)

L. Chamoin, LMT Cachan

La simulation numérique est à présent devenue un outil de conception indispensable pour l’ingénierie, permettant notamment de prédire le comportement d’objets industriels complexes dans leur environnement. Néanmoins, afin de coller fidèlement à la réalité physique, la simulation nécessite un contrôle permanent des divers modèles mathématiques et outils numériques qu’elle utilise. Cette thématique scientifique, connue sous le nom de Vérification et Validation des modèles (V&V), est une composante de la simulation qui, bien que primordiale, reste souvent fastidieuse et compliquée en pratique car elle nécessite le contrôle de multiples paramètres (de modèle, de discrétisation, etc.) et l’acquisition de nombreuses données expérimentales.

Cependant, dans la très grande majorité des cas, la simulation numérique n’a pas pour objectif de prédire la solution globale du phénomène physique étudié, mais simplement quelques caractéristiques locales de cette solution (contrainte maximale, facteurs d’intensité de contrainte, etc.) appelées quantités d’intérêt et servant directement au dimensionnement. Il est alors cohérent de ne vouloir contrôler que les paramètres de la simulation qui sont influents pour ces quantités d’intérêt, menant ainsi à une démarche de V&V simplifiée.

Au cours de la présentation, nous analysons quelques travaux réalisés ces dernières années en vue de construire des modèles de simulation optimisés en vue du calcul d’une quantité d’intérêt.

Dans un premier temps, nous nous focalisons sur les travaux liés à la vérification « classique », i.e. ceux permettant de construire des discrétisations (maillage EF par exemple) optimales vis-à-vis d’une tolérance d’erreur locale prescrite.

Par la suite, nous nous intéressons au contrôle des résultats de calcul obtenus par réduction de modèle, démarche largement utilisée de nos jours pour simuler les modèles complexes. Dans ce contexte, nous étudions deux cas précis : (i) le couplage de modèles, avec diverses applications (discret/continu, stochastique/déterministe, etc.) ; (ii) l’utilisation de la Proper Generalized Decomposition (PGD) pour représenter la solution de problèmes multi-paramétrés.

Enfin, nous abordons la thématique de validation en étudiant des évolutions récentes dans le recalage des modèles mathématiques, avec pour objectifs d’obtenir une modélisation optimale et un recalage de modèle en temps-réel en vue de la prédiction d’une quantité d’intérêt donnée. Nous verrons que ces évolutions nécessitent notamment un dialogue accru et intelligent entre l’expérience et la simulation.

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Méthodes d’homogénéisation numériques basées sur la discrétisation de l’équation de Lippmann—Schwinger (2014-01-20)

S. Brisard, Laboratoire Navier

La détermination des propriétés élastiques homogénéisées nécessite la résolution d’un problème auxiliaire (équilibre élastique du volume élémentaire statistique). Ce système d’équations aux dérivées partielles peut être réécrit de manière équivalente sous la forme d’une équation intégrale, dite équation de Lippmann—Schwinger, dans laquelle l’inconnue principale n’est plus le champ de déplacement, mais le champ de "polarisation".

Pour résoudre l’équation de Lippmann—Schwinger, il est possible d’utiliser des méthodes basées sur la transformée de Fourier rapide. Initialement proposées par Moulinec et Suquet à partir de 1994, ces méthodes connaissent depuis peu un regain d’intérêt. De même, la méthode de l’inclusion équivalente (Moschovidis et Mura, 1975) peut être vue comme la recherche (dans l’espace réel cette fois-ci) d’une solution approchée à cette équation intégrale.

Ces méthodes correspondent à autant de discrétisations de type Galerkin de l’équation de Lippmann—Schwinger. Ce constat permet de procéder à leur analyse mathématique à l’aide des outils classiques issus de la théorie des éléments finis.

Dans cet exposé, les méthodes basées sur la transformée de Fourier rapide et la méthode de l’inclusion équivalente seront présentées de façon unifiée ; les similarités et les différences entre ces méthodes seront mises en évidence. En particulier, les conditions aux limites associées à la méthode de l’inclusion équivalente seront précisées. Ce dernier point permettra de jeter un éclairage nouveau sur l’approximation classique (due à Willis) de l’opérateur de Green d’un domaine borné.

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Nouvelles méthodes de reconstruction tomographique par optimisation convexe (2013-11-25)

H. Talbot, Université Paris-Est ESIEE/LIGM

La reconstruction tomographique est le plus souvent effectuée par des algorithmes de type rétroprojection, qui statistiquement s’apparentent à des méthodes de type maximum de vraisemblance / moindre-carrés. Comme toutes ces méthodes, elle sont très rapides mais réclament beaucoup de projections et sont sensibles au bruit. De plus des méthodes différentes doivent être adaptées pour tous les cas : faisceau parallèle, conique, trajet hélicoïdal ou non. Des méthodes dites "algébriques" existent depuis longtemps pour corriger certains de ces défauts mais souffrent d’un manque de cohérence.

Depuis quelques temps il est possible de considérer la tomographie comme un opérateur linéaire de projection et de l’intégrer dans des méthodes de résolution de problèmes inverses assez classiques. Depuis environ 2010, des algorithmes efficaces existent permettant de résoudre ces problèmes en considérant uniquement l’opérateur de projection direct, sans nécessiter son inverse, et ce quelque soit la modalité retenue (parallèle, conique, hélicoïdale, etc). De plus il est possible de réaliser des opérations supplémentaires en même temps : débruitage, segmentation, intégration angulaire. Enfin certains problèmes insolubles autrement sont devenus accessibles, par exemple la tomographie locale, où le capteur est plus petit que l’objet à reconstruire.

Dans cet exposé nous présenterons les principes généraux, les algorithmes impliqués et nous montrerons quelques résultats intéressants pour discussion.

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Homogénéisation numérique : méthodes MsFEM, HMM, FE2, ... (2013-11-04)

F. Legoll, Laboratoire Navier

Dans cet exposé, nous décrivons plusieurs méthodes multiéchelles pour le calcul de matériaux présentant des hétérogénéités à une petite échelle (matériaux composites, écoulement de type Stokes dans des milieux perforés, ...). Ces méthodes (dites aussi d’homogénéisation numérique) ont été construites pour traiter les cas (fréquents en pratique) où les hetérogénéités ont une structure géometrique quelconque (ni périodique, ni aléatoire stationnaire), pour lesquelles la théorie classique de l’homogénéisation ne donne pas de formules utilisables en pratique. On s’intéressera en particulier aux méthodes MsFEM, HMM et FE2, introduites depuis une dizaine d’années.

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Numerical strategy for the homogenization of random structural models (2013-10-03)

R. Cottereau, MSSMat

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Méthodes FFT pour la simulation numérique de matériaux hétérogènes : développements et applications (2013-02-06)

L. Gélébart, CEA

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