Publications HAL

2023

Article dans une revue

titre
A Class of One Dimensional Periodic Microstructures Exhibiting Effective Timoshenko Beam Behavior
auteur
Jean-Jacques Alibert, Emilio Barchiesi, Francesco Dell’isola, Pierre Seppecher
article
ESAIM: Control, Optimisation and Calculus of Variations, 2023, 29, pp.53. ⟨10.1051/cocv/2023048⟩
resume
We study, from a variational viewpoint, the asymptotic behavior of a planar beam with a periodic wavy shape when the amplitude and the wavelength of the shape tend to zero. We assume that the beam behaves, at the microscopic level, as a compressible Euler–Bernoulli beam and that the material properties have the same period as the geometry. We allow for distributed or concentrated bending compliance and for a non-quadratic extensional energy. The macroscopic Γ-limit that we obtain corresponds to a non-linear model of Timoshenko type.
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Article dans une revue
DOI
DOI : 10.1051/cocv/2023048
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https://hal.science/hal-04165376/file/cocv230066.pdf BibTex

Autre publication scientifique

titre
Review of "An energy approach to asymptotic, higher-order, linear homogenization
auteur
Basile Audoly, Claire Lestringant, Pierre Seppecher, Damiano Pasini, Jean-Francois Ganghoffer, Laurence Brassart
article
2023
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Autre publication scientifique
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https://hal.science/hal-04272312/file/JTCAM_11414_review.pdf BibTex

2022

Pré-publication, Document de travail

titre
The obstacle problem in masonry structures and cable nets
auteur
Ada Amendola, Ornella Mattei, Graeme W. Milton, Pierre Seppecher
article
2022
resume
We consider the problem of finding a net that supports prescribed forces applied at prescribed points, yet avoids certain obstacles, with all the elements of the net under compression (strut net) or under tension (cable web). In the case of masonry structures, for instance, this consists in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible due, for instance, to the presence of holes. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region $\Gamma$ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region $\Gamma_1$ available to the obstacle with $\Gamma_1 \subset\Gamma$. The case where some of the forces are reactive, unprescribed but reacting to the other prescribed forces, is also treated. It again reduces to identifying the feasible domain in a linear programming problem. Finally, one may allow a subset of the reactive forces to each act not at a prescribed point, but rather at any point on a prescribed line segment. Then the task reduces to identifying the feasible domain in a quadratic programming problem.
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https://arxiv.org/pdf/2204.00729 BibTex

2021

Article dans une revue

titre
Homogenization of an elastic material reinforced by very strong fibers arranged along a periodic lattice
auteur
Houssam Abdoul-Anziz, Lukáš Jakabčin, Pierre Seppecher
article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2021, ⟨10.1098/rspa.2020.0620⟩
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Article dans une revue
DOI
DOI : 10.1098/rspa.2020.0620
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https://univ-tln.hal.science/hal-02482837/file/Anziz-Jakabcin-Seppecher.pdf BibTex

Pré-publication, Document de travail

titre
Asymptotic comparison of the strain-gradient and micromorphic models when loading forces are widely spread
auteur
Pierre Seppecher, Lukáš Jakabčin
article
2021
resume
In this paper we reconciliate different homogenization results which describe the effective behavior of a heterogeneous material either by a strain-gradient model either by a micromorphic one. Indeed we prove that the solutions of both models are asymptotically very close when considering a loading with increasing wavelength. This result is obtained using the Fourier analysis on the tensor spaces and applies to a large class of micromorphic models. However, we provide an example of a micromorphic model that does not belong to this class and thus cannot be approximated by a strain-gradient model.
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https://hal.science/hal-03432329/file/Seppecher-Jakabcin-2021.pdf BibTex

2020

Article dans une revue

titre
On periodic homogenization of highly contrasted elastic structures
auteur
Lukáš Jakabčin, Pierre Seppecher
article
Journal of the Mechanics and Physics of Solids, 2020, ⟨10.1016/j.jmps.2020.104104⟩
resume
While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables us to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure.
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Article dans une revue
DOI
DOI : 10.1016/j.jmps.2020.104104
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https://univ-tln.hal.science/hal-02379572/file/PHHCES.pdf BibTex
titre
Two-dimensional continua capable of large elastic extension in two independent directions: asymptotic homogenization, numerical simulations and experimental evidence
auteur
E. Barchiesi, F Dell'Isola, François Hild, Pierre Seppecher
article
Mechanics Research Communications, 2020, 103, pp.103466. ⟨10.1016/j.mechrescom.2019.103466⟩
resume
The synthesis of a 1D full second gradient continuum was obtained by the design of so-called pantographic beam (see Alibert et al. Mathematics and Mechanics of Solids (2003) [2]) and the problem of the synthesis of planar second gradient continua has been faced in several subsequent papers: in dell'Isola et al. Zeitschrift für angewandte Mathematik und Physik (2015) [5] and dell'Isola et al. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2016). [4] it is considered a three-length-scale mi-crostructure in which two initially orthogonal families of long Euler beams (i.e. beams much longer than the size of the homogenization cell but much slenderer than it) are interconnected by perfect or elastic pivots (hinges). The corresponding homogenized two-dimensional continuum (which was called pantographic sheet) has a D4 orthotropic symmetry. It has been proven to have a deformation energy depending on the second gradient of in-plane displacements and to allow for large elon-gations in some specic directions while remaining in the elastic regime. However, in pantographic sheets, the deformation energy only depends on the geodesic bending of the actual conguration of its symmetry directions (see for more details Steigmann et al. Acta Mechanica Sinica (2015) [3] and Placidi et al. Journal of Engineering Mathematics (2017) [6]). On the other hand, in Seppecher et al. J. of Physics: Conference Series vol. 319 (2011) [1], it was designed a bi-pantographic architectured sheet where the previously considered Euler beams were replaced by pantographic beams to form a more complex three-length-scale mi-crostructure and it was proven that, once homogenized, such a bi-pantographic sheet, in planar and linearized deformation states, produces a more complete second gradient two-dimensional continuum. Derivatives of elongations along the two symmetry directions now appear in the deformation energy. The aim of the present paper is the experimental validation of the second gradient behavior of such bi-pantographic sheets. As their intrinsic mechanical structure produces a geometrically non-linear behavior for relatively small total deformation , we rst need to extend the homogenization result to the regime of large deformations. Subsequently we compare the predictions obtained using such second gradient model with experimental evidence, as elaborated by local Digital Image Correlation (DIC) focused on the discrete kinematics of the hinges. 1 Introduction In physical literature the problem of synthesis has been confronted in many contexts. Namely, given a La-grangian potential (describing the conservative part of considered phenomena) and a Rayleigh dissipation potential , specied in terms of suitable kinematic descrip-tors, one has to nd a physical system, belonging to a class specied a priori, whose evolution is governed by the corresponding Hamilton-Rayleigh principle. In the period (1930-1970) in which the prevalence of digital computers was not yet achieved, the problem of synthesis of electric circuits was confronted in order to design suitable, and dedicated, analog computers (see, e.g., Kron [13, 14]). When the a priori class of physical systems is constituted by electric circuits with only passive elements, the previous general problem was par-ticularized as follows: given quadratic Lagrangian and Rayleigh potentials, in terms of a nite number of degrees of freedom, one has to nd the graph of a circuit and the interconnecting (linear and passive) electric elements such that it is governed by the corresponding Hamilton-Rayleigh principle. The available results found in the literature for the synthesis of passive electric circuits were recently used to synthesize piezoelectromechanical metamateri-als, suitably tailored to dampen out mechanical vibrations (see dell'Isola et al. [15, 16, 17] and [18, 25]). The synthesis problem for mechanical metamaterials reads as: given any choice of the continuous elds describing a kinematics, given functionals expressing the deformation energy, the kinetic energy and the dissipa-tion potential in terms of these elds, nd the architec-tured mechanical structure (possibly multi-scale) such that, in the homogenization limit, the obtained continuous model is exactly the one chosen a priori. Therefore , the qualitative behaviour of a metamaterial shall be given by its multi-scale architecture rather than by the constituting base materials [31].
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Article dans une revue
DOI
DOI : 10.1016/j.mechrescom.2019.103466
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https://hal.science/hal-02404383/file/MRC2019-ccsd.pdf BibTex

Chapitre d'ouvrage

titre
Microscopic Interpretation of Strain-Gradient and Generalized Continuum Models
auteur
Pierre Seppecher
article
Mechanics of Strain Gradient Materials, 600, Springer International Publishing, pp.71-99, 2020, CISM International Centre for Mechanical Sciences, ⟨10.1007/978-3-030-43830-2_4⟩
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Chapitre d'ouvrage
DOI
DOI : 10.1007/978-3-030-43830-2_4
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titre
Least action and virtual work principles for the formulation of generalized continuum models
auteur
Pierre Seppecher, Francesco Dell'Isola, Luca Placidi, Emilio Barchiesi, Anil Misra
article
Discrete and Continuum Models for Complex Metamaterials, 1, Cambridge University Press, 2020, ⟨10.1017/9781316104262⟩
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Chapitre d'ouvrage
DOI
DOI : 10.1017/9781316104262
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titre
Pantographic Metamaterial: A (Not So) Particular Case
auteur
Francesco Dell'Isola, Mario Spagnuolo, Emilio Barchiesi, Ivan Giorgio, Pierre Seppecher
article
Discrete and Continuum Models for Complex Metamaterials, 1, Cambridge University Press, 2020, ⟨10.1017/9781316104262⟩
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Chapitre d'ouvrage
DOI
DOI : 10.1017/9781316104262
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titre
Higher Gradient Theories and Their Foundations
auteur
Francesco Dell'Isola, Pierre Seppecher, Alessandro Della Corte
article
Encyclopedia of Continuum Mechanics (2020 EDITION), pp.1090-1099, 2020
resume
Continuum mechanics always supplies approximate models for physical systems, in which a more fundamental (possibly discrete or inhomogeneous) microstructure may be somehow neglected. Indeed, Cauchy (or Cauchy-Navier) continuum theory describes efficiently, at a macroscopic level, the behavior of a mechanical system only when the inhomogeneities which the model does not take into account have a characteristic length scale much smaller than the macroscale where phenomena are observed. Therefore, it is now widely accepted that in some circumstances, it is necessary to add to the placement field some extra kinematical fields, to take into account, at a macroscopic level, some aspects of the mechanical behavior of materials having complex microscopic...
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Chapitre d'ouvrage
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https://hal.science/hal-02918379/file/dellisola_seppecher_dellacorte_new.pdf BibTex

2019

Article dans une revue

titre
Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms
auteur
Houssam Abdoul-Anziz, Cédric Bellis, Pierre Seppecher
article
Mathematics and Mechanics of Solids, 2019, ⟨10.1177/ToBeAssigned⟩
resume
We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. In these papers, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present paper, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure which we call asymmetrical pantographic structure.
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Article dans une revue
DOI
DOI : 10.1177/ToBeAssigned
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https://hal.science/hal-02065135/file/coupling-vs5.pdf BibTex
titre
Pantographic metamaterials: an example of mathematically driven design and of its technological challenges
auteur
Francesco Dell'Isola, Pierre Seppecher, Jean-Jacques Alibert, Tomasz Lekszycki, Roman Grygoruk, Marek Pawlikowski, David Steigmann, Ivan Giorgio, Ugo Andreaus, Emilio Turco, Maciej Golaszewski, Nicola Rizzi, Claude Boutin, Victor A. Eremeyev, Anil Misra, Luca Placidi, Emilio Barchiesi, Leopoldo Greco, Massimo Cuomo, Antonio Cazzani, Alessandro Della Corte, Antonio Battista, Daria Scerrato, Inna Zurba Eremeeva, Yosra Rahali, Jean-François Ganghoffer, Wolfang Müller, Gregor Ganzosch, Mario Spagnuolo, Aron Pfaff, Katarzyna Barcz, Klaus Hoschke, Jan Neggers, François Hild
article
Continuum Mechanics and Thermodynamics, 2019, 31 (4), pp.851-884. ⟨10.1007/s00161-018-0689-8⟩
resume
In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.
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Article dans une revue
DOI
DOI : 10.1007/s00161-018-0689-8
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https://hal.science/hal-01829943/file/pantographic_metamaterials_.pdf BibTex
titre
On the forces that cable webs under tension can support and how to design cable webs to channel stresses
auteur
Guy Bouchitté, Ornella Mattei, Graeme Milton, Pierre Seppecher
article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019, 475 (2223), pp.20180781. ⟨10.1098/rspa.2018.0781⟩
resume
In many applications of Structural Engineering the following question arises: given a set of forces f1, f2,. .. , fN applied at prescribed points x1, x2,. .. , xN , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x1, x2,. .. , xN in the two-and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f1, f2,. .. , fN applied at points strictly within the convex hull of x1, x2,. .. , xN. In three-dimensions we show that, by slightly perturbing f1, f2,. .. , fN , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.
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Article dans une revue
DOI
DOI : 10.1098/rspa.2018.0781
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https://hal.science/hal-02188481/file/On_the_forces.pdf BibTex
titre
Large deformations of Timoshenko and Euler beams under distributed load
auteur
A. Della Corte, A. Battista, F. Dell'Isola, Pierre Seppecher
article
Zeitschrift für Angewandte Mathematik und Physik, 2019, 70 (2), pp.52. ⟨10.1007/s00033-019-1098-y⟩
resume
In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established.
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Article dans une revue
DOI
DOI : 10.1007/s00033-019-1098-y
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https://hal.science/hal-02269403/file/LargeDeformationsOfTimoshenkoA.pdf BibTex
titre
Advances in Pantographic Structures: Design, Manufacturing, Models, Experiments and Image Analyses
auteur
Francesco Dell'Isola, Pierre Seppecher, Mario Spagnuolo, Emilio Barchiesi, François Hild, Tomasz Lekszycki, Ivan Giorgio, Luca Placidi, Ugo Andreaus, Massimo Cuomo, Simon R Eugster, Aron Pfaff, Klaus Hoschke, Ralph Langkemper, Emilio Turco, Rizacan Sarikaya, Aviral Misra, Michele de Angelo, Francesco d'Annibale, Amine Bouterf, Xavier Pinelli, Anil Misra, Boris Desmorat, Marek Pawlikowski, Corinne Dupuy, Daria Scerrato, Patrice Peyre, Marco Laudato, Luca Manzari, Peter Göransson, Christian Hesch, Sofia Hesch, Patrick Franciosi, Justin Dirrenberger, Florian Maurin, Zacharias Vangelatos, Costas Grigoropoulos, Vasileia Melissinaki, Maria Farsari, Wolfgang Muller, Emek Abali, Christian Liebold, Gregor Ganzosch, Philip Harrison, Rafal Drobnicki, Leonid A Igumnov, Faris Alzahrani, Tasawar Hayat
article
Continuum Mechanics and Thermodynamics, 2019, 31 (4), pp.1231-1282. ⟨10.1007/s00161-019-00806-x⟩
resume
In the last decade the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discrete or continuous) have been introduced. In a previous publication a large part of the already existing literature about pantographic metamaterials has been presented. In this paper we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.
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Article dans une revue
DOI
DOI : 10.1007/s00161-019-00806-x
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https://hal.science/hal-02156824/file/CMAT-S-19-00030.pdf BibTex

Communication dans un congrès

titre
Homogenization of frame lattices leading to second gradient models coupling classical strain and strain gradient terms
auteur
Houssam Abdoul-Anziz, Cédric Bellis, Pierre Seppecher
article
ICoNSoM 2019, Jun 2019, Rome, Italy
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Ouvrages

titre
The Complete Works of Gabrio Piola: Volume II
auteur
Pierre Seppecher, Francesco Dell'Isola, Ugo Andreaus, Antonio Cazzani, Raffaele Esposito, Luca Placidi, Umberto Perego, Giulio Maier
article
Springer International Publishing, 97, 2019, Advanced Structured Materials, ⟨10.1007/978-3-319-70692-4⟩
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Ouvrages
DOI
DOI : 10.1007/978-3-319-70692-4
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Pré-publication, Document de travail

titre
Lagrange multipliers in infinite dimensional spaces, examples of application
auteur
A. Bersani, F. Dell'Isola, Pierre Seppecher
article
2019
resume
The Lagrange multipliers method is used in Mathematical Analysis, in Mechanics, in Economics and in several other fields, to deal with the search of the global maximum or minimum of a function, in presence of a constraint. The usual technique, applied to the case of finite-dimensional systems, transforms the constrained optimization problem into an unconstrained one, by means of the introduction of one or more multipliers and of a suitable Lagrangian function, to be optimized. In Mechanics, several optimization problems can be applied to infinite-dimensional systems. Lagrange multipliers method can be applied also to these cases.
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https://hal.science/hal-02269765/file/multi_lagrange-v8.pdf BibTex

2018

Article dans une revue

titre
Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams
auteur
A Battista, Alessandro Della Corte, F Dell 'Isola, Pierre Seppecher
article
Zeitschrift für Angewandte Mathematik und Physik, 2018, ⟨10.1007/s00033-018-0946-5⟩
resume
In the present paper we study a natural nonlinear generalization of Timoshenko beam model and show that it can describe the homogenized deformation energy of a 1D continuum with a simple microstructure. We prove the well-posedness of the corresponding variational problem in case of a generic end load, discuss some regularity issues and evaluate the critical load. Moreover , we generalize the model so as to include an additional rotational spring in the microstructure. Finally, some numerical simulations are presented and discussed.
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Article dans une revue
DOI
DOI : 10.1007/s00033-018-0946-5
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https://hal.science/hal-01625160/file/paper.pdf BibTex
titre
Homogenization of periodic graph-based elastic structures
auteur
Houssam Abdoul-Anziz, Pierre Seppecher
article
Journal de l'École polytechnique — Mathématiques, 2018
resume
In the framework of Γ-convergence and periodic homogenization of highly contrasted materials, we study cylindrical structures made of one material and voids. Interest in high contrast homogenization is growing rapidly but assumptions are generally made in order to remain in the framework of classical elasticity. On the contrary, we obtain homogenized energies taking into account second gradient (i.e. strain gradient) effects. We first show that we can reduce the study of the considered structures to discrete systems corresponding to frame lattices. Our study of such lattices differs from the literature in the fact that we must take into account the different orders of magnitude of the extensional and flexural stiffnesses. This allows us to consider structures which would have been floppy when considering only extensional stiffness and completely rigid when considering flexural stiffnesses of the same order of magnitude than the extensional ones. At our knowledge, this paper provides the first rigorous homogenization result with a complete second gradient limit energy.
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https://univ-tln.hal.science/hal-01492589/file/abdoul-anziz-seppecher_epreuves.pdf BibTex
titre
Strain gradient and generalized continua obtained by homogenizing frame lattices
auteur
Houssam Abdoul-Anziz, Pierre Seppecher
article
Mathematics and Mechanics of Complex Systems, 2018, ⟨10.2140/memocs.2018.6.213⟩
resume
We determine the effective behavior of periodic structures made of welded elastic bars. Taking into account the fact that flexural and torsional stiffnesses are much smaller than the extensional one we overpass classical homogenization formula and obtain totally different types of effective energies. We work in the framework of linear elasticity. We give different examples of two dimensional or three dimensional micro-structures which lead to generalized 1D, 2D or 3D continua like Timoshenko beam, Mindlin-Reissner plate, strain gradient, Cosserat, or micromorphic continua.
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DOI
DOI : 10.2140/memocs.2018.6.213
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https://univ-tln.hal.science/hal-01672898/file/memocs-v6-n3-p04-s.pdf BibTex

Communication dans un congrès

titre
Continua models with strain gradient energy obtained by rigorous homogenization
auteur
Houssam Abdoul-Anziz, Pierre Seppecher
article
ESMC 2018, Jul 2018, Bologne, Italy
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titre
Homogenization results leading to generalized continua models
auteur
Houssam Abdoul-Anziz, Pierre Seppecher
article
CANUM 2018, May 2018, Cap d'Agde, France
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Chapitre d'ouvrage

titre
A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identification and Numerical Results
auteur
Emilio Barchiesi, Francesco Dell'Isola, Marco Laudato, Luca Placidi, Pierre Seppecher
article
Advances in Mechanics of Microstructured Media and Structures , 87, pp 43-74, 2018, Advanced Structured Materials book series (STRUCTMAT), volume 87
resume
In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.
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https://hal.science/hal-01817742/file/pantographic_paper.pdf BibTex

2017

Article dans une revue

titre
Temperature profile in a liquid-vapor interface near the critical point
auteur
Henri Gouin, Pierre Seppecher
article
Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, 2017, 473 (20170229), pp.1-13. ⟨10.1098/rspa.2017.0229⟩
resume
Thanks to an expansion with respect to densities of energy, mass and entropy, we discuss the concept of thermocapillary fluid for inhomogeneous fluids. The non-convex state law valid for homogeneous fluids is modified by adding terms taking account of the gradients of these densities. This seems more realistic than Cahn and Hilliard's model which uses a density expansion in mass-density gradient only. Indeed, through liquid-vapor interfaces, realistic potentials in molecular theories show that entropy density and temperature do not vary with the mass density as it would do in bulk phases. In this paper, we prove using a rescaling process near the critical point that liquid-vapor interfaces behave essentially in the same way as in Cahn and Hilliard's model.
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Article dans une revue
DOI
DOI : 10.1098/rspa.2017.0229
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https://hal.science/hal-01492802/file/Temperature.pdf BibTex

Communication dans un congrès

titre
Homogenization of periodic elastic structures leading to second gradient effects
auteur
Houssam Abdoul-Anziz, Pierre Seppecher
article
EUROMECH-Colloquium 579, Apr 2017, Arpino, Italy
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titre
Convergence of {H}encky-type discrete beam model to {E}uler inextensible elastica in large deformation: rigorous proof
auteur
Jean-Jacques Alibert, Pierre Seppecher, Alessandro Della Corte
article
Mathematical modelling in solid mechanics, 2017
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Ouvrages

titre
Mécanique des milieux continus généralisés
auteur
Samuel Forest, Gianpietro del Piero, Pierre Seppecher
article
Cepadues, 2017, 2364936098
resume
Cet ouvrage traite des milieux continus généralisés. Il s’adresse à un large public de chercheurs en mécanique ou en mathématiques ainsi qu’à des étudiants intéressés par les fondements des modèles utilisés en mécanique. Les méthodes générales pour construire de manière systématique et cohérente des théories de ces milieux sont présentées en détail ainsi que diverses applications. En effet, ces théories se révèlent pertinentes pour décrire un grand nombre de circonstances fréquemment rencontrées en pratique : milieux finement hétérogènes, polycristaux, milieux microfissurés, mais aussi théorie de poutres, des plaques et des coques. Trois points de vue, de portée générale, sont plus particulièrement approfondis?: – La construction des théories de milieux continus généralisés rend indispensable la rationalisation et la systématisation de l’application des grands principes de la physique. Le point de vue des puissances virtuelles assure la cohérence de la représentation des efforts avec les divers raffinements de la description cinématique que l’on peut envisager. Pour la représentation des efforts intérieurs, ce point de vue sera mis en rapport avec l’extension des hypothèses sur lesquelles le théorème de Cauchy est basé. – L’étude des milieux micromorphes dont les milieux de Cosserat constituent un cas particulier important. Les théories de poutres et de plaques en constituent une illustration classique. Le comportement élastoplastique de ces milieux continus généralisés est discuté en vue d’application à la description des polycristaux. – Les théories dites de gradient d’ordre supérieur (par exemple, celle du second gradient). Il s’agit d’un autre type de généralisation de la mécanique des milieux continus classiques qui s’est révélé fécond autant pour la mécanique des solides que la mécanique des fluides. En particulier, des illustrations basées sur l’homogénéisation des treillis périodiques sont présentées. Le livre regroupe les contributions de trois chercheurs qui travaillent sur les fondements des milieux continus généralisés depuis de nombreuses années. Ils ont des points de vues différents mais complémentaires sur le sujet. L’ouvrage a été décomposé en trois parties, chacune correspond à la contribution de chacun des trois auteurs. Elles peuvent êtres lues dans un premier temps d’une manière indépendante. Les lecteurs qui souhaitent comprendre les fondements de la mécanique aussi bien que ceux qui sont intéressés par la modélisation des matériaux trouveront un grand plaisir dans la lecture de chacun e des trois parties.
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2016

Article dans une revue

titre
Cauchy Tetrahedron Argument applied to Higher Contact Interactions
auteur
F. Dell'Isola, A. Madeo, Pierre Seppecher
article
Archive for Rational Mechanics and Analysis, 2016, 219 (3), pp.1305-1341. ⟨10.1007/s00205-015-0922-6⟩
resume
Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.
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DOI : 10.1007/s00205-015-0922-6
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2015

Article dans une revue

titre
The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results
auteur
F. Dell'Isola, Pierre Seppecher, A. Della Corte
article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015, 471 (2183), 25 p
resume
In order to found continuum mechanics, two different postulations have been used. The first, introduced by Lagrange and Piola, starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients. Then, by using the divergence theorem, a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body. This method assumes an a priori notion of internal work, regards stress tensors as dual of virtual displacements and their gradients, deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts. The second method, conceived by Cauchy and based on the celebrated tetrahedron argument, starts by postulating the type of contact interactions which can be exerted on the boundary of every (suitably) regular part of a body. Then it proceeds by proving the existence of stress tensors from a balance-type postulate. In this paper, we review some relevant literature on the subject, discussing how the two postulations can be reconciled in the case of higher gradient theories. Finally, we underline the importance of the concept of contact surface, edge and wedge s-order forces.
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https://hal.science/hal-01226235/file/dellisola_seppecher_dellacorte.pdf BibTex

2013

Chapitre d'ouvrage

titre
Transformation elastodynamics and active exterior acoustic cloaking
auteur
Graeme W. Milton, D. Onofrei, Pierre Seppecher, Fernando Guevara Vasquez
article
Richard V. Craster, Sébastien Guenneau. Acoustic Metamaterials, 166, Springer Netherlands, pp.289-319, 2013, Springer Series in Materials Science, 9400748132, 9789400748132
resume
This chapter consists of three parts. In the first part we recall the elastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a mass-spring network model. The transformed networks can be realized with "torque springs", which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. In the second and third parts we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. In the second part, the exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of analytic functions. An explicit solution is given that allows to cloak larger objects at a fixed distance from the cloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloaking problem for the Helmholtz equation in 3D. Our method uses the Green's formula and an addition theorem for spherical outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green's formula.
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2012

Article dans une revue

titre
How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach "à la D'Alembert
auteur
Francesco Dell'Isola, Pierre Seppecher, Angela Madeo
article
Zeitschrift für Angewandte Mathematik und Physik, 2012, 63 (6), pp.1119-1141
resume
Navier-Cauchy format for Continuum Mechanics is based on the concept of contact interaction between subbodies of a given continuous body. In this paper it is shown how -by means of the Principle of Virtual Powers- it is possible to generalize Cauchy representation formulas for contact interactions to the case of N-th gradient continua, i.e. continua in which the deformation energy depends on the deformation Green-Saint-Venant tensor and all its N-1 order gradients. In particular, in this paper the explicit representation formulas to be used in N-th gradient continua to determine contact interactions as functions of the shape of Cauchy Cuts are derived. It is therefore shown that i) these interactions must include edge (i.e. concentrated on curves) and wedge (i.e. concentrated on points) interactions, and ii) these interactions cannot reduce simply to forces: indeed the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green and Germain) is fundamental and unavoidable in the theory of N-th gradient continua.
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https://hal.science/hal-00662376/file/dellisola_seppecher_madeo.pdf BibTex
titre
Optimal thin torsion rods and Cheeger sets
auteur
Guy Bouchitté, Ilaria Fragalà, Ilaria Lucardesi, Pierre Seppecher
article
SIAM Journal on Mathematical Analysis, 2012, 44 (1), pp.483--512. ⟨10.1137/110828538⟩
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no abstract
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DOI : 10.1137/110828538
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titre
A metamaterial having a frequency dependent elasticity tensor and a zero effective mass density
auteur
Graeme Milton, Pierre Seppecher
article
physica status solidi (b), 2012, 249 (7), pp.1412--1414
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no abstract
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Chapitre d'ouvrage

titre
Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: a generalization of the Cauchy tetrahedron argument.
auteur
Francesco Dell'Isola, Pierre Seppecher, Angela Madeo
article
Variational Models and Methods in Solid and Fluid Mechanics, Volume 535 of the series CISM Courses and Lectures, pp.17-106, 2012
resume
The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D'Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salencon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an e ective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defi ned by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)-(1965) , Toupin (1962), Mindlin (1964)-(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell'Isola and Seppecher (1995)-(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.
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https://hal.science/hal-00601624/file/chapt2_dellIsola.pdf BibTex
titre
Fluid Shock Wave Generation at Solid-Material Discontinuity Surfaces in Porous Media
auteur
Francesco Dell'Isola, Pierre Seppecher, Angela Madeo
article
Variational Models and Methods in Solid and Fluid Mechanics, Volume 535 of the series CISM Courses and Lectures , pp. 315-358, 2012
resume
A general set of boundary conditions at the interface between dissimilar fuid-filled porous matrices is established starting from an extended Hamilton-Rayleigh principle. These conditions do include inertial effects. Once linearized, they encompass boundary conditions relative to volume Darcy-Brinkman and to surface Saffman-Beavers-Joseph-Deresiewicz dissipation effects.
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https://hal.science/hal-00601626/file/chapt7_dellIsola.pdf BibTex

2011

Article dans une revue

titre
Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach
auteur
Guy Bouchitté, Ilaria Fragalà, Pierre Seppecher
article
Archive for Rational Mechanics and Analysis, 2011, 202 (3), pp.829-874. ⟨10.1007/s00205-011-0435-x⟩
resume
The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.
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DOI : 10.1007/s00205-011-0435-x
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titre
Linear elastic trusses leading to continua with exotic mechanical interactions
auteur
Pierre Seppecher, Jean-Jacques Alibert, Francesco Dell'Isola
article
Journal of Physics: Conference Series, 2011, 319 (1), 13 p
resume
We study the statics of some trusses, i.e. networks of nodes linked by linear springs. The trusses are designed in such a way that a few number of floppy modes are present and remain when considering the homogenized limit of the truss. We then obtain linear elastic materials with exotic mechanical interactions which cannot be described in the classical framework of Cauchy stress theory. For aim of simplicity, the structures described here are two-dimensional. The extension to the 3D case does not present any difficulty.
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titre
Commentary about the paper " Hypertractions and hyperstresses convey the same mechanical information Continuum Mech. Thermodyn. (2010) 22:163176 " by Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories
auteur
Francesco Dell'Isola, Pierre Seppecher
article
Continuum Mechanics and Thermodynamics, 2011, 23 (5), pp.473-478. ⟨10.1007/s00161-010-0176-3⟩
resume
In this commentary we try to make clearer the state of the art concerning the relation between mechanical contact interactions and the different notions of stresses. We emphasize the importance of the concept of virtual displacements. Its role has been recognized in Mechanics and in Continuum Mechanics long ago (see e.g. [59], [45], or [12], [13]) and it is central as well when starting with an expression of the power expended by internal stresses and deducing the form of contact interactions as when starting with some form of the contact interactions and developping a representa- tion theorem for these contact interactions based on the Cauchy tetrahedron construction.
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DOI : 10.1007/s00161-010-0176-3
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2010

Article dans une revue

titre
Numerical modelling of interfacial soil erosion with viscous incompressible flows
auteur
Frederic Golay, Damien Lachouette, Stéphane Bonelli, Pierre Seppecher
article
Computer Methods in Applied Mechanics and Engineering, 2010, 200 (1-4), pp.383-391. ⟨10.1016/j.cma.2010.09.002⟩
resume
The aim of this study is to develop a numerical model for simulating surface erosion occurring at a fluid/soil interface subject to a flow process. Balance equations with jump relations are used. A penalization procedure including a fictitious domain method is used to compute the Stokes flow around obstacles, in order to avoid body-fitted unstructured meshes and instead use fast and efficient finite volume approximations on Cartesian meshes. The evolution of the water/soil interface is described by using a level set function. The ability of the model to predict the interfacial erosion of soils is confirmed by several numerical simulations.
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DOI : 10.1016/j.cma.2010.09.002
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titre
Interfacial erosion: A three-dimensional numerical model
auteur
Frederic Golay, Damien Lachouette, Stéphane Bonelli, Pierre Seppecher
article
Comptes Rendus Mécanique, 2010, 338 (6), pp.333-337
resume
The aim of this study was to develop a numerical model to simulate the surface erosion occurring at a fluid/soil interface subject to a flow process. We used a penalization procedure to compute flow around obstacles. A fictitious domain method allowed the use of fast and efficient finite volumes approximations on Cartesian meshes and avoided unstructured body-fitted meshes. The water/soil interface evolution was described with a Level Set function. Several numerical simulations confirmed the model's ability to predict the interfacial erosion of soils.
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https://hal.science/hal-00528972/file/golay-lachouette-bonelli-seppecher.pdf BibTex
titre
The optimal compliance problem for thin torsion rods: a 3D-1D analysis leading to Cheeger-type solutions
auteur
Guy Bouchitté, Ilaria Fragalà, Pierre Seppecher
article
Comptes Rendus. Mathématique, 2010, 348 (7-8), pp.467--471. ⟨10.1016/j.crma.2010.01.006⟩
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no abstract
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DOI : 10.1016/j.crma.2010.01.006
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Communication dans un congrès

titre
Interfacial erosion: a three-dimensional numerical model
auteur
Frederic Golay, Damien Lachouette, Stéphane Bonelli, Pierre Seppecher, Cedric Galusinski
article
Euro Mediterranean Symposium On Advances in Geomaterial and Structures , May 2010, Djerba, Tunisia
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Pré-publication, Document de travail

titre
Structural optimization of thin elastic plates: the three dimensional approach
auteur
Guy Bouchitté, Ilaria Fragalà, Pierre Seppecher
article
2010
resume
The natural way to find the most compliant design of an elastic plate, is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotic of such compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in [18]. Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in [16, 17]. Finally, some explicit computation of optimal plates are given.
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https://hal.science/hal-00528976/file/bouchitte-fragala-seppecher_1.pdf BibTex
titre
Hybrid electromagnetic circuits
auteur
Graeme W. Milton, Pierre Seppecher
article
2010
resume
Electromagnetic circuits are the electromagnetic analog at fixed frequency of mass-spring networks in elastodynamics. By interchanging the roles of ε and μ in electromagnetic circuits one obtains magnetoelectric circuits. Here we show that by introducing tetrahedral connectors having ε = μ = 0 one can join electromagnetic and magnetoelectric circuits to obtain hybrid circuits. Their response is governed by a symmetric matrix with negative semidefinite imaginary part. Conversely given any such matrix a recipe is given for constructing a hybrid circuit which has that matrix as its response matrix.
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https://hal.science/hal-00523086/file/miltonseppecher_1.pdf BibTex
titre
Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency
auteur
Graeme W. Milton, Pierre Seppecher, Guy Bouchitté
article
2010
resume
The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done by building upon ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current the primary electromagnetic minimization variational principles have a minimum which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have application to electromagnetic tomography. We also derive saddle point variational principles which correspond to variational principles of Gurtin, Willis, and Borcea.
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https://hal.science/hal-00523225/file/miltonseppecherbouchittA_1.pdf BibTex

2009

Article dans une revue

titre
Boundary Conditions at Fluid-Permeable Interfaces in Porous Media: a Variational Approach
auteur
Francesco Dell'Isola, Angela Madeo, Pierre Seppecher
article
International Journal of Solids and Structures, 2009, pp.28
resume
A general set of boundary conditions at fluid-permeable interfaces between dissimilar fluid-filled porous matrices is established starting from an extended Hamilton-Rayleigh principle. These conditions do include friction and inertial effects. Once linearized, they encompass boundary conditions relative to volume Darcy-Brinkman and to surface Saffman-Beavers-Joseph dissipation effects.
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https://hal.science/hal-00504035/file/Dell_IsolaMadeoSeppecher.pdf BibTex

2008

Article dans une revue

titre
Michell Trusses and Lines of Principal Action
auteur
Guy Bouchitté, Wilfrid Gangbo, Pierre Seppecher
article
Mathematical Models and Methods in Applied Sciences, 2008, 18 (9), pp.1571-1603
resume
We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutually orthogonal families of curves. The method is illustrated on a specific example where uniqueness can be proved by studying an unusual system of hyperbolic PDEs. The questions we address here are of interest in elasticity theory, optimal designs, as well as in functional analysis.
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https://hal.science/hal-00528973/file/bouchitte-gangbo-seppecher.pdf BibTex
titre
Closure of the Set of Diffusion Functionals - the One Dimensional Case
auteur
Jean-Jacques Alibert, Pierre Seppecher
article
Potential Analysis, 2008, 28 (4), pp.335-356
resume
We characterize the closure with respect to Mosco or Gamma-convergence of the set of diffusion functionals in the one dimension case. As commonly accepted we find this closure is a set of local Dirichlet forms. The difficulty is to identify the right notion of locality. We compare different possible definitions. We give a representation theorem for the elements of the considered closure.
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https://hal.science/hal-00527074/file/alibert-seppecher.pdf BibTex

2007

Article dans une revue

titre
3D-2D analysis for the optimal elastic compliance problem
auteur
Guy Bouchitté, Ilaria Fragalà, Pierre Seppecher
article
Comptes Rendus. Mathématique, 2007, 345 (12), pp.713--718. ⟨10.1016/j.crma.2007.10.039⟩
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no abstract
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DOI : 10.1016/j.crma.2007.10.039
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2006

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titre
Asymptotics of a non-planar rod in non-linear elasticity
auteur
Catherine Pideri, Pierre Seppecher
article
Asymptotic Analysis, 2006, 48 (1-2), pp.33-54
resume
We study the asymptotic behavior of a non-linear elastic material lying in a thin neighborhood of a non-planar line when the diameter of the section tends to zero. We first estimate the rigidity constant in such a domain then we prove the convergence of the three-dimensional model to a one-dimensional model. This convergence is established in the framework of Γ-convergence. The resulting model is the one classically used in mechanics. It corresponds to a non-extensional line subjected to flexion and torsion. The torsion is an internal parameter which can eventually by eliminated but this elimination leads to a non-local energy. Indeed the non-planar geometry of the line couples the flexion and torsion terms.
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2003

Article dans une revue

titre
Determination of the Closure of the Set of Elasticity Functionals
auteur
M. Camar-Eddine, Pierre Seppecher
article
Archive for Rational Mechanics and Analysis, 2003, 170 (3), pp.211 - 245. ⟨10.1007/s00205-003-0272-7⟩
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DOI : 10.1007/s00205-003-0272-7
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titre
Truss modular Beams with Deformation Energy depending on Higher Displacement Gradients
auteur
Jean-Jacques Alibert, Pierre Seppecher, Francesco Dell'Isola
article
Mathematics and Mechanics of Solids, 2003, pp.23
resume
Until now, no third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure. In this paper, we prove that this is possible using pantographic-type structures. Their deformation energies involve combinations of nodal displacements havin the form of second-order or third-order finite differences. We establish the Gamma convergence of these energies to second and third gradient functionals. Some mechanical examples are provided so as to illustrate the special features of these homogenized models.
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https://hal.science/hal-00497327/file/alibert_seppecher_dellisola.pdf BibTex

2002

Article dans une revue

titre
CLOSURE OF THE SET OF DIFFUSION FUNCTIONALS WITH RESPECT TO THE MOSCO-CONVERGENCE
auteur
M. Camar-Eddine, Pierre Seppecher
article
Mathematical Models and Methods in Applied Sciences, 2002, 12 (08), pp.1153 - 1176. ⟨10.1142/S0218202502002069⟩
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DOI : 10.1142/S0218202502002069
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2001

Article dans une revue

titre
Non-local interactions resulting from the homogenization of a linear diffusive medium
auteur
Mohamed Camar-Eddine, Pierre Seppecher
article
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2001, 332 (5), pp.485 - 490. ⟨10.1016/S0764-4442(01)01853-5⟩
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DOI : 10.1016/S0764-4442(01)01853-5
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titre
Line Tension Effect Upon Static Wetting
auteur
Pierre Seppecher
article
Oil & Gas Science and Technology - Revue d'IFP Energies nouvelles, 2001, 56 (1), pp.77-81. ⟨10.2516/ogst:2001009⟩
resume
Adding simply, in the classical capillary model, a constant line density of energy along the contact line leads to ill-posed equilibrium problems. Then, when line tension is present, the equilibrium configuration minimizes a different energy: the relaxedenergy, which explicitly depends on the presence of surface phases (i. e. infinitesimal films) on the boundary of the container. This formulation enables us to describe the modifications of the Young's law and then of equilibrium configurations which are due to line tension.
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DOI
DOI : 10.2516/ogst:2001009
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https://hal.science/hal-02053881/file/seppecher_v56n1.pdf BibTex
titre
Locking materials and the topology of optimal shapes
auteur
Frederic Golay, Pierre Seppecher
article
European Journal of Mechanics - A/Solids, 2001, 20 (4), pp.631-644
resume
We study a problem of structural optimisation using the fictitious material approach. This is connected with the equilibrium of locking materials, which can be approximated by strongly non linear materials. A finite element simulation allows us to experiment with some conjectures about the topology of the optimal solutions.
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2000

Article dans une revue

titre
Regular approximation of free-discontinuity problems
auteur
Guy Bouchitté, C. Dubs, Pierre Seppecher
article
Mathematical Models and Methods in Applied Sciences, 2000, 10 (7), pp.1073--1097. ⟨10.1142/S0218202500000549⟩
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no abstract
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DOI : 10.1142/S0218202500000549
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1998

Article dans une revue

titre
Phase transition with the line-tension effect
auteur
Giovanni Alberti, Guy Bouchitté, Pierre Seppecher
article
Archive for Rational Mechanics and Analysis, 1998, 144 (1), pp.1--46. ⟨10.1007/s002050050111⟩
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no abstract
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DOI
DOI : 10.1007/s002050050111
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1997

Article dans une revue

titre
A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium
auteur
Catherine Pideri, Pierre Seppecher
article
Continuum Mechanics and Thermodynamics, 1997, 9 (5), pp.241-257
resume
Homogenization may change fundamentally the constitutive laws of materials. We show how a heterogeneous Cauchy continuum may lead to a non Cauchy continuum. We study the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger linear elastic medium and we prove that, when the Lamé coefficients in the fibers and the radius of the fibers have appropriate order of magnitude, the effective material is a second gradient material, i.e. a material whose energy depends on the second gradient of the displacement.
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titre
Un résultat d'homogénéisation pour un matériau élastique renforcé périodiquement par des fibres élastiques de très grande rigidité
auteur
Catherine Pideri, Pierre Seppecher
article
Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics-Physics-Chemistry-Astronomy, 1997, pp.475-481. ⟨10.1016/S1251-8069(97)80185-6⟩
resume
Nous étudions l'homogénéisation d'un milieu élastique renforcé périodiquement par des fibres de beaucoup plus grande rigidité que la matrice. Nous supposons que les fibres sont parallèles, cylindriques, de section circulaire, et de rayon beaucoup plus petit que la distance de séparation des fibres. Ces hypothèses diffèrent de celles faites dans le cadre classique de la théorie de l'homogénéisation et conduisent à un résultat très différent: nous montrons que, lorsque le rayon des fibres tend vers zéro et que les coefficients de Lamé du matériau les constituant tendent vers l'infini avec des vitesses adaptées, le milieu homogénéisé se comporte comme un milieu de second gradient, c'est-à-dire un milieu dont l'énergie dépend du second gradient du déplacement.
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DOI : 10.1016/S1251-8069(97)80185-6
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titre
Energies with respect to a measure and applications to low-dimensional structures
auteur
Guy Bouchitté, Giuseppe Buttazzo, Pierre Seppecher
article
Calculus of Variations and Partial Differential Equations, 1997, 5 (1), pp.37--54. ⟨10.1007/s005260050058⟩
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no abstract
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DOI : 10.1007/s005260050058
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titre
Edge Contact Forces and Quasi-Balanced Power
auteur
Francesco Dell'Isola, Pierre Seppecher
article
Meccanica, 1997, pp.20
resume
We consider continuous media in which contact edge forces are present. Introducing the notion of quasi-balanced contact force distribution, we are able to prove the conjectures by Noll and Virga [1] concerning the representation of contact edge forces. We generalize the Hamel–Noll theorem on the Cauchy postulate. Then we adapt the celebrated tetrahedron construction of Cauchy in order to obtain a representation theorem for stress states. In fact, we show that two stress tensors of order two and three are necessary for such a representation. Moreover we find the relationship between the notion of interstitial working introduced by Dunn and Serrin [2] and the notion of contact edge force
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titre
Shape optimization solutions via Monge-Kantorovich equation
auteur
Guy Bouchitté, Giuseppe Buttazzo, Pierre Seppecher
article
Comptes rendus de l'Académie des sciences. Série I, Mathématique, 1997, 324 (10), pp.1185--1191. ⟨10.1016/S0764-4442(97)87909-8⟩
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DOI : 10.1016/S0764-4442(97)87909-8
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1996

Article dans une revue

titre
Moving contact lines in the Cahn-Hilliard theory
auteur
Pierre Seppecher
article
International Journal of Engineering Science, 1996, 34 (9), pp.977-992
resume
We establish the equations of motion of an isothermal viscous Cahn-Hilliard fluid and we investigate the dynamics of fluids having moving contact lines under this theory. The force singularity arising in the classical model of capillarity is no longer present. This removal is due to a mass transfer across the interface combined with a finite thickness of the interface. A numerical simulation of the flow in the immediate vicinity of the contact line shows the connection between the static contact angle, the dynamic angle and points out the influence of the velocity
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titre
Transitions de phases avec un potentiel dégénéré à l'infini, application à l'équilibre de petites gouttes
auteur
Guy Bouchitté, Christophe Dubs, Pierre Seppecher
article
Comptes rendus de l'Académie des sciences. Série I, Mathématique, 1996, 323 (9), pp.1103--1108
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HDR

titre
Les fluides de Cahn-Hilliard
auteur
Pierre Seppecher
article
Mécanique [physics.med-ph]. Université du Sud Toulon Var, 1996
resume
There is no abstract
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HDR
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1995

Article dans une revue

titre
Radius and surface tension of microscopic bubbles by second gradient theory
auteur
Francesco Dell'Isola, Henri Gouin, Pierre Seppecher
article
Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, 1995, 320, pp.5
resume
The classical theory of Laplace is not suitable for describing the behavior of microscopic bubbles. The theory of second gradient fluids (which are able to exert shear stresses in equilibrium conditions) allows us to obtain a new expression for surface tension and radius of these bubbles in terms of functionals of the chemical potential. This relationship allows us to generalize the results of Cahn-Hilliard and Tolman.
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titre
The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power
auteur
Francesco Dell'Isola, Pierre Seppecher
article
Comptes rendus de l’Académie des sciences. Série IIb, Mécanique, physique, astronomie, 1995, pp.7
resume
We consider continuous media in which contact edge forces are present. Introducing the notion of quasi- balanced contact force distribution, we are able to prove the conjectures made in [Noll, 1990] concerning the representation of contact edge forces. We first generalise the Noll theorem on Cauchy postulate. Then we adapt the celebrated Cauchy tetrahedron construction to find a representation theorem for stress states. Moreover we find the relationship between double forces Germain [Germain, 1973a,b], interstitial working [Dunn, 1985] and contact edge forces.
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Chapitre d'ouvrage

titre
Boundary effects in phase transitions
auteur
Giovanni Alberti, Guy Bouchitté, Pierre Seppecher
article
Curvature flows and related topics (Levico, 1994), 5, Gakkōtosho, Tokyo, pp.1--11, 1995, GAKUTO Internat. Ser. Math. Sci. Appl
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1994

Article dans une revue

titre
Un résultat de perturbations singulières avec la norme H^1/2
auteur
Giovanni Alberti, Guy Bouchitté, Pierre Seppecher
article
Comptes rendus de l'Académie des sciences. Série I, Mathématique, 1994, 319 (4), pp.333--338
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titre
Un résultat de perturbations singulieres avec la norm H
auteur
Giovanni Alberti, Guy Bouchitté, Pierre Seppecher
article
CR Acad. Sci. Paris, Ser. I, 1994, 319, pp.333--338
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no abstract
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Chapitre d'ouvrage

titre
Cahn and Hilliard fluid on an oscillating boundary
auteur
Guy Bouchitté, Pierre Seppecher
article
Motion by mean curvature and related topics (Trento, 1992), de Gruyter, Berlin, pp.23--42, 1994
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1987

Thèse

titre
Etude d'une modelisation des zones capillaires fluides: Interfaces et lignes de contact
auteur
Pierre Seppecher
article
Mécanique [physics.med-ph]. Ecole Nationale Superieure de Techniques Avancees, 1987. Français. ⟨NNT : ⟩
resume
Nous donnons une description fine des zones capillaires à l'aide de la théorie du second gradient ce qui permet de clarifier les phénomènes de tension superficielle et de tension de ligne. Nous montrons comment par un processus d'intégration à travers les zones capillaires cellesci peuvent être considérées comme des surfaces ou des lignes géométriques porteuses de propriétés matérielles. Pour obtenir des conditions de bilan physiquement acceptables nous faisons une analyse des ordres de grandeur des termes interfaciaux. Enfin nous utilisons la thermodynamique linéaire des phénomènes irréversibles pour écrire les lois de comportement des interfaces et des lignes de contact
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Thèse
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