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Papers by Christian Licht

Asymptotic Analysis

By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variab... more By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variable Hilbert spaces, we propose an asymptotic modeling for the behavior of a linearly elastic plate in bilateral contact with a rigid body along part of its lateral boundary with Norton or Tresca friction.

The piezoelectric thin plate modeling already derived by the authors is extended to rod-like stru... more The piezoelectric thin plate modeling already derived by the authors is extended to rod-like structures. Two models corresponding to sensor or actuator behavior are obtained. The conditions of existence of non local terms in the limit models are discussed.

Presentation de modeles simplifies mais suffisamment precis de comportements dynamiques de joncti... more Presentation de modeles simplifies mais suffisamment precis de comportements dynamiques de jonctions minces elastiques ou non

Piezoelectric materials are widely used in the design of smart structures. It is thus of major te... more Piezoelectric materials are widely used in the design of smart structures. It is thus of major technological interest to provide efficient modelings of such structures. In the case of thin piezoelectric plates, classical studies generally lead to two different models. These two models can be linked to the distinction between sensors and actuators. Here, we extend these results to the second order piezoelectricity, that is to say piezoelectricity with electric field gradient. We recently showed in [1] that three different models have to be taken into account, which broadens the scope of the sensors and actuators field. Second order piezoelectricity being compatible with isotropy (see the introduction below), we also propose a systematic study of the impact of crystalline symmetries on our models and show that a striking effect named 'structural switch-off' appears for some specific crystal classes. This paper aims at presenting these results in a simplified but accurate way.

Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classi... more Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term— a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings.

Le sujet principal de cet article est une approche numerique de la theorie de l'homogeneisati... more Le sujet principal de cet article est une approche numerique de la theorie de l'homogeneisation en mecanique non-lineaire. Une etude theorique menee par Stefan Muller precise que pour une densite d'energie non-convexe, la densite d'energie homogeneisee est donnee par une famille de problemes de minimisation sur tous les multiples de la cellule de base du milieu considere. Cette etude est tout d'abord realisee sur un materiau modele : un composite periodiquement stratifie conformement a l'etude menee par celui-ci. Les premiers resultats numeriques sont interessants et sont similaires a ceux obtenus par cette theorie. En effet, les solutions different en fonction du nombre de cellules de bases pris en compte pour une etude d'homogeneisation.

East-West Journal of Mathematics, 2009

Most of the structures in Civil Engineering consists in assemblies of deformable bodies, thus it ... more Most of the structures in Civil Engineering consists in assemblies of deformable bodies, thus it is of interest to dispose of efficient models of junctions between de- formable solids. The classical schemes of Continuum Mechanics lead to boundary value problems involving several parameters, one being essential: the (low) thick- ness of the layer filled by the adhesive. For usual behaviors of the adherents and the adhesive, it is not difficult to prove existence of solutions, but their numerical approximations may be difficult due to the rather low thickness of the adhesive im- plying a too fine mesh. We propose a simplified but accurate mathematical modeling by a rigorous study of the asymptotic behavior of the three-dimensional adhesive when its thickness goes to zero. Depending on the stiffness of the adhesive, the limit model will replace the thin adhesive layer by either a mechanical constraint along the surface the layer shrinks toward or a material surface; the structure of th...

We study the transient response of a thermoelastic structure made of two tridimensional bodies co... more We study the transient response of a thermoelastic structure made of two tridimensional bodies connected by a thin adhesive layer. Once more we highlight the powerful flexibility of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: considering the geometrical and physical characteristics of the thin layer as parameters, we are able to show in a unitary way that this situation leads to a huge variety of limit models the properties of which are detailed. In particular, according to the relative behaviors of the different parameters involved, new features are evidenced such as the apparition of an added specific heat coefficient for the interface or of additional thermomechanical state variables defined not only on the limit geometric interface but on its cartesian product by any interval of real numbers.

A mathematical analysis, based on the theory of semi-groups of operators on Hilbert space, of lin... more A mathematical analysis, based on the theory of semi-groups of operators on Hilbert space, of linearized problems involving Saint-Venant equations governing shallow water flows is carried out. The first problem concerns the classical linearized Saint-Venant problem, while the second deals with a water quality problem: the evolution of a concentration of products in a flow solution of the first problem. The method seems capable to take into account thermal and multi-phase effects.

We extend to the linearly piezoelectric case the mathematical derivation of the linearly elastic ... more We extend to the linearly piezoelectric case the mathematical derivation of the linearly elastic behavior of a plate as the limit behavior of a three-dimensional solid whose thickness 2e tends to zero. Due to classical assumptions on the exterior loadings, a suitable scaling is defined by to study the limit behavior as e goes to 0. Note that the assumptions on the forces are those which provide Kirchhoff-Love limit plate theory while those on the electrical loading involve an index p running over 1, 2 that will imply two kinds of limit models according to the nature and the magnitude of the data. We show that the scaled states converge in a suitable topology to the unique solution of the limit problem indexed by p. These limit problems (p = 1 or 2) are connected with the physical situations where the thin plate acts as an actuator or a sensor.

We present another proof of a study of Bellieud and Bouchitte that we expect to be more suitable ... more We present another proof of a study of Bellieud and Bouchitte that we expect to be more suitable to treat more general geometrical and physical cases.

Utilisation de la theorie des semi-groupes (non-lineaires ou non) pour la resolution de problemes... more Utilisation de la theorie des semi-groupes (non-lineaires ou non) pour la resolution de problemes de couplage structure elastique/ fluide parfaits ou visqueux a surface libre infinie

Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classi... more Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term— a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings.

Nous considerons l'assemblage de deux corps hyperelastiques (non-lineaire) lies par un adhesi... more Nous considerons l'assemblage de deux corps hyperelastiques (non-lineaire) lies par un adhesif hyperelastique de faible rigidite (non lineaire) occupant une petite couche d'epaisseur e. Une strategie numerique du probleme n'est pas vraiment envisageable en raison du grand nombre de degres de liberte lie au maillage de la fine couche de colle et du tres mauvais conditionnement du systeme du a la tres faible rigidite de la colle. Il est donc pertinent de proposer un modele variationnel equivalent evitant ces types de comportement.

... Milieu cellulaire et homogénéisation. P. Aubert, Stéphane Pagano 1 , Christian Licht 1 , Jose... more ... Milieu cellulaire et homogénéisation. P. Aubert, Stéphane Pagano 1 , Christian Licht 1 , Joseph Gril 1. (2004). C3. ... Contributeur : Lmgc Aigle <>. Déposé pour le compte de : STÉPHANEPAGANO <>. Soumis le : Vendredi 3 Septembre 2010, 00:23:37. ...

La prise en compte d'une liaison non reduite a l'adhesion pure entre adhesif et adherent ... more La prise en compte d'une liaison non reduite a l'adhesion pure entre adhesif et adherent conduit a une liaison asymptotique globale traduisant la mise en serie de cette liaison avec la liaison classique limite fournie par le tres mince adherent elastique.

We derive a theory of thin linearly quasicrystalline plates by studying the limit behavior of a t... more We derive a theory of thin linearly quasicrystalline plates by studying the limit behavior of a three-dimensional flat body as its thickness tends to zero. We exhibit the existence of a surprisingly high number of models, each of them linked to a specific set of boundary conditions. As such, these results show that quasicrystals behave as smart materials.

The multi-echo gradient echo (ME-GRE) magnetic resonance signal evolution in white matter has a s... more The multi-echo gradient echo (ME-GRE) magnetic resonance signal evolution in white matter has a strong dependence on the orientation of myelinated axons in respect to the main static field. Although analytical solutions, based on the Hollow Cylinder Model have been able to predict some of the behaviour the hollow cylinder model, it has been shown that realistic models of white matter offer a better description of the signal behaviout observed.In this work, we present a pipeline to (i) generate realistic 2D white matter models with its microstructure based on real axon but with arbitrary fiber volume fraction (FVF) and g-ratio. We (ii) simulate their interaction with the static magnetic field to be able to simulate their MR signal. For the first time, we (iii) demonstrate that realistic 2D models can be used to simulate an MR signal that provides a good approximation of the signal obtained from a real 3D white matter model obtained using electron microscopy. We then (iv) demonstrate ...

Asymptotic Analysis

By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variab... more By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variable Hilbert spaces, we propose an asymptotic modeling for the behavior of a linearly elastic plate in bilateral contact with a rigid body along part of its lateral boundary with Norton or Tresca friction.

The piezoelectric thin plate modeling already derived by the authors is extended to rod-like stru... more The piezoelectric thin plate modeling already derived by the authors is extended to rod-like structures. Two models corresponding to sensor or actuator behavior are obtained. The conditions of existence of non local terms in the limit models are discussed.

Presentation de modeles simplifies mais suffisamment precis de comportements dynamiques de joncti... more Presentation de modeles simplifies mais suffisamment precis de comportements dynamiques de jonctions minces elastiques ou non

Piezoelectric materials are widely used in the design of smart structures. It is thus of major te... more Piezoelectric materials are widely used in the design of smart structures. It is thus of major technological interest to provide efficient modelings of such structures. In the case of thin piezoelectric plates, classical studies generally lead to two different models. These two models can be linked to the distinction between sensors and actuators. Here, we extend these results to the second order piezoelectricity, that is to say piezoelectricity with electric field gradient. We recently showed in [1] that three different models have to be taken into account, which broadens the scope of the sensors and actuators field. Second order piezoelectricity being compatible with isotropy (see the introduction below), we also propose a systematic study of the impact of crystalline symmetries on our models and show that a striking effect named 'structural switch-off' appears for some specific crystal classes. This paper aims at presenting these results in a simplified but accurate way.

Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classi... more Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term— a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings.

Le sujet principal de cet article est une approche numerique de la theorie de l'homogeneisati... more Le sujet principal de cet article est une approche numerique de la theorie de l'homogeneisation en mecanique non-lineaire. Une etude theorique menee par Stefan Muller precise que pour une densite d'energie non-convexe, la densite d'energie homogeneisee est donnee par une famille de problemes de minimisation sur tous les multiples de la cellule de base du milieu considere. Cette etude est tout d'abord realisee sur un materiau modele : un composite periodiquement stratifie conformement a l'etude menee par celui-ci. Les premiers resultats numeriques sont interessants et sont similaires a ceux obtenus par cette theorie. En effet, les solutions different en fonction du nombre de cellules de bases pris en compte pour une etude d'homogeneisation.

East-West Journal of Mathematics, 2009

Most of the structures in Civil Engineering consists in assemblies of deformable bodies, thus it ... more Most of the structures in Civil Engineering consists in assemblies of deformable bodies, thus it is of interest to dispose of efficient models of junctions between de- formable solids. The classical schemes of Continuum Mechanics lead to boundary value problems involving several parameters, one being essential: the (low) thick- ness of the layer filled by the adhesive. For usual behaviors of the adherents and the adhesive, it is not difficult to prove existence of solutions, but their numerical approximations may be difficult due to the rather low thickness of the adhesive im- plying a too fine mesh. We propose a simplified but accurate mathematical modeling by a rigorous study of the asymptotic behavior of the three-dimensional adhesive when its thickness goes to zero. Depending on the stiffness of the adhesive, the limit model will replace the thin adhesive layer by either a mechanical constraint along the surface the layer shrinks toward or a material surface; the structure of th...

We study the transient response of a thermoelastic structure made of two tridimensional bodies co... more We study the transient response of a thermoelastic structure made of two tridimensional bodies connected by a thin adhesive layer. Once more we highlight the powerful flexibility of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: considering the geometrical and physical characteristics of the thin layer as parameters, we are able to show in a unitary way that this situation leads to a huge variety of limit models the properties of which are detailed. In particular, according to the relative behaviors of the different parameters involved, new features are evidenced such as the apparition of an added specific heat coefficient for the interface or of additional thermomechanical state variables defined not only on the limit geometric interface but on its cartesian product by any interval of real numbers.

A mathematical analysis, based on the theory of semi-groups of operators on Hilbert space, of lin... more A mathematical analysis, based on the theory of semi-groups of operators on Hilbert space, of linearized problems involving Saint-Venant equations governing shallow water flows is carried out. The first problem concerns the classical linearized Saint-Venant problem, while the second deals with a water quality problem: the evolution of a concentration of products in a flow solution of the first problem. The method seems capable to take into account thermal and multi-phase effects.

We extend to the linearly piezoelectric case the mathematical derivation of the linearly elastic ... more We extend to the linearly piezoelectric case the mathematical derivation of the linearly elastic behavior of a plate as the limit behavior of a three-dimensional solid whose thickness 2e tends to zero. Due to classical assumptions on the exterior loadings, a suitable scaling is defined by to study the limit behavior as e goes to 0. Note that the assumptions on the forces are those which provide Kirchhoff-Love limit plate theory while those on the electrical loading involve an index p running over 1, 2 that will imply two kinds of limit models according to the nature and the magnitude of the data. We show that the scaled states converge in a suitable topology to the unique solution of the limit problem indexed by p. These limit problems (p = 1 or 2) are connected with the physical situations where the thin plate acts as an actuator or a sensor.

We present another proof of a study of Bellieud and Bouchitte that we expect to be more suitable ... more We present another proof of a study of Bellieud and Bouchitte that we expect to be more suitable to treat more general geometrical and physical cases.

Utilisation de la theorie des semi-groupes (non-lineaires ou non) pour la resolution de problemes... more Utilisation de la theorie des semi-groupes (non-lineaires ou non) pour la resolution de problemes de couplage structure elastique/ fluide parfaits ou visqueux a surface libre infinie

Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classi... more Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term— a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings.

Nous considerons l'assemblage de deux corps hyperelastiques (non-lineaire) lies par un adhesi... more Nous considerons l'assemblage de deux corps hyperelastiques (non-lineaire) lies par un adhesif hyperelastique de faible rigidite (non lineaire) occupant une petite couche d'epaisseur e. Une strategie numerique du probleme n'est pas vraiment envisageable en raison du grand nombre de degres de liberte lie au maillage de la fine couche de colle et du tres mauvais conditionnement du systeme du a la tres faible rigidite de la colle. Il est donc pertinent de proposer un modele variationnel equivalent evitant ces types de comportement.

... Milieu cellulaire et homogénéisation. P. Aubert, Stéphane Pagano 1 , Christian Licht 1 , Jose... more ... Milieu cellulaire et homogénéisation. P. Aubert, Stéphane Pagano 1 , Christian Licht 1 , Joseph Gril 1. (2004). C3. ... Contributeur : Lmgc Aigle <>. Déposé pour le compte de : STÉPHANEPAGANO <>. Soumis le : Vendredi 3 Septembre 2010, 00:23:37. ...

La prise en compte d'une liaison non reduite a l'adhesion pure entre adhesif et adherent ... more La prise en compte d'une liaison non reduite a l'adhesion pure entre adhesif et adherent conduit a une liaison asymptotique globale traduisant la mise en serie de cette liaison avec la liaison classique limite fournie par le tres mince adherent elastique.

We derive a theory of thin linearly quasicrystalline plates by studying the limit behavior of a t... more We derive a theory of thin linearly quasicrystalline plates by studying the limit behavior of a three-dimensional flat body as its thickness tends to zero. We exhibit the existence of a surprisingly high number of models, each of them linked to a specific set of boundary conditions. As such, these results show that quasicrystals behave as smart materials.

The multi-echo gradient echo (ME-GRE) magnetic resonance signal evolution in white matter has a s... more The multi-echo gradient echo (ME-GRE) magnetic resonance signal evolution in white matter has a strong dependence on the orientation of myelinated axons in respect to the main static field. Although analytical solutions, based on the Hollow Cylinder Model have been able to predict some of the behaviour the hollow cylinder model, it has been shown that realistic models of white matter offer a better description of the signal behaviout observed.In this work, we present a pipeline to (i) generate realistic 2D white matter models with its microstructure based on real axon but with arbitrary fiber volume fraction (FVF) and g-ratio. We (ii) simulate their interaction with the static magnetic field to be able to simulate their MR signal. For the first time, we (iii) demonstrate that realistic 2D models can be used to simulate an MR signal that provides a good approximation of the signal obtained from a real 3D white matter model obtained using electron microscopy. We then (iv) demonstrate ...