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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2019

Article dans une revue

- titre
- A new class of cost for optimal transport planning
- auteur
- Jean-Jacques Alibert, Guy Bouchitté, Thierry Champion
- article
*European Journal of Applied Mathematics*, 2019, 30 (6), pp.1229-1263. ⟨10.1017/s0956792518000669⟩- resume
- We study a class of optimal transport planning problems where the reference cost involves a non linear function G(x, p) representing the transport cost between the Dirac mesure x and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case (G(x, p) = R c(x, y) dp) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich-Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with-convergence theory.
- typdoc
- Article dans une revue
- DOI
- DOI : 10.1017/s0956792518000669
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- 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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- titre
- Homogenization of nonlinear inextensible pantographic structures by Γ-convergence
- auteur
- Jean-Jacques Alibert, Alessandro Della Corte
- article
*Mathematics and Mechanics of Complex Systems*, 2019, 7 (1), pp.1-24. ⟨10.2140/memocs.2019.7.1⟩- typdoc
- Article dans une revue
- DOI
- DOI : 10.2140/memocs.2019.7.1
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2018

Pré-publication, Document de travail

- titre
- A new class of cost for optimal transport planning
- auteur
- Jean-Jacques Alibert, Guy Bouchitté, Thierry Champion
- article
- 2018
- resume
- We study a class of optimal transport planning problems where the reference cost involves a non linear function G(x, p) representing the transport cost between the Dirac mesure x and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case (G(x, p) = R c(x, y) dp) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich-Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with-convergence theory.
- typdoc
- Pré-publication, Document de travail
- Accès au texte intégral et bibtex

2017

Article dans une revue

- titre
- Extensional Elastica in large deformation as $$\Gamma $$ Γ -limit of a discrete 1D mechanical system
- auteur
- Jean-Jacques Alibert, Alessandro Della Corte, Ivan Giorgio, Antonio Battista
- article
*Zeitschrift für Angewandte Mathematik und Physik*, 2017, 68 (2), ⟨10.1007/s00033-017-0785-9⟩- typdoc
- Article dans une revue
- DOI
- DOI : 10.1007/s00033-017-0785-9
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Chapitre d'ouvrage

- 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- typdoc
- Chapitre d'ouvrage
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2015

Article dans une revue

- titre
- Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof
- auteur
- Jean-Jacques Alibert, Alessandro Della Corte
- article
*Zeitschrift für Angewandte Mathematik und Physik*, 2015, 66 (5), pp 2855-2870. ⟨10.1007/s00033-015-0526-x⟩- resume
- Since the works by Gabrio Piola, it has been debated the relevance of higher-gradient continuum models in mechanics. Some authors even questioned the logical consistency of higher-gradient theories, and the applicability of generalized continuum theories seems still open. The present paper considers a pantographic plate constituted by Euler beams suitably interconnected and proves that Piola’s heuristic homogenization method does produce an approximating continuum in which deformation energy depends only on second gradients of displacements. The Gamma-convergence argument presented herein shows indeed that Piola’s conjecture can be rigorously proven in a Banach space whose norm is physically dictated by energetic considerations.
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- Article dans une revue
- DOI
- DOI : 10.1007/s00033-015-0526-x
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2013

Article dans une revue

- titre
- A nonstandard free boundary problem arising in the shape optimization of thin torsion rods
- auteur
- Jean-Jacques Alibert, Guy Bouchitté, Ilaria Fragalà, Ilaria Lucardesi
- article
*Interfaces Free Boundaries*, 2013, 15 (1), pp.95-119. ⟨10.4171/IFB/296⟩- resume
- We study a 2d-variational problem, in which the cost functional is an integral depending on the gradient through a convex but not strictly convex integrand, and the admissible functions have zero gradient on the complement of a given domain D. We are interested in establishing whether solutions exist whose gradient "avoids" the region of non-strict convexity. Actually, the answer to this question is related to establishing whether homogenization phenomena occur in optimal thin torsion rods. We provide some existence results for different geometries of D, and we study the nonstandard free boundary problem with a gradient obstacle, which is obtained through the optimality conditions.
- typdoc
- Article dans une revue
- DOI
- DOI : 10.4171/IFB/296
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2011

Article dans une revue

- 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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2008

Article dans une revue

- 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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- Article dans une revue
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2003

Article dans une revue

- 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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2001

Article dans une revue

- titre
- Genericity in Deterministic and Stochastic Differential Equations
- auteur
- Jean-Jacques Alibert, Khaled Bahlali
- article
*Séminaire de Probabilités XXXV*, 2001, 1755, pp.220-240- resume
- We prove that the convergence of the approximation with time delay, as well as pathwise uniqueness, are generic properties in ordinary differential equations as well as in stochastic differential equations. This is done in the case where the coefficients are neither bounded nor time continuous. The approximation with time delay is used to obtain existence of weak solutions for SDE. We also prove L^2-convergence of this approximation when only pathwise uniqueness is assumed.
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- Article dans une revue
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1998

Article dans une revue

- titre
- A lagrange multiplier theorem for control problems with state constraints
- auteur
- Jean-Jacques Alibert, Jean-Pierre Raymond
- article
*Numerical Functional Analysis and Optimization*, 1998, 19 (7-8), pp.51-73- resume
- We prove an existence theorem of Lagrange multipliers for an abstract control problem in Banach spaces. This theorem may be applied to obtain optimality conditions for control problems governed by partial differential equations in the presence of pointwise state constraints.
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- Article dans une revue
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1997

Article dans une revue

- titre
- Non-Uniform Integrability and Generalized Young Measures
- auteur
- Jean-Jacques Alibert, Guy Bouchitté
- article
*Journal of Convex Analysis*, 1997, 4 (1), pp.129-147- typdoc
- Article dans une revue
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- titre
- Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls
- auteur
- Jean-Jacques Alibert, Jean-Pierre Raymond
- article
*Numerical Functional Analysis and Optimization*, 1997, 18 (3-4), pp.235-250- resume
- We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.
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- Article dans une revue
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1992

Article dans une revue

- titre
- An example of a quasiconvex function that is not polyconvex in two dimensions
- auteur
- Jean-Jacques Alibert, Bernard Dacorogna
- article
*Archive for Rational Mechanics and Analysis*, 1992, 117 (2), pp.155-166. ⟨10.1007/BF00387763⟩- resume
- We study the different notions of convexity for the function f γ(ξ) = |ξ|2 (|ξ|2 − 2γ det ξ) where ξ ε ℝ2×2, introduced by Dacorogna & Marcellini. We show that f γ is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦γ¦≦ 2/3 √2, 1, 1+ɛ (for some ɛ>0), 2/√3, respectively.
- typdoc
- Article dans une revue
- DOI
- DOI : 10.1007/BF00387763
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