Jean LERBET | Evry University (original) (raw)

Papers by Jean LERBET

Research paper thumbnail of Cohomology of the vector fields Lie algebras on R acting on trilinear differential operators, vanishing on sl(2)

Communications in Algebra, Mar 11, 2021

Abstract The main topic of this paper is to compute the first relative cohomology group of the Li... more Abstract The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields with coefficients in the space of trilinear differential operators that acts on tensor densities, vanishing on the Lie algebra where and

Research paper thumbnail of Explicit analysis of catasrophe on a Timoshenko beam

HAL (Le Centre pour la Communication Scientifique Directe), Mar 24, 2021

Research paper thumbnail of Intrinsic incremental evolution of hypoelastic discrete mechanical systems

Journal of Applied Mathematics and Mechanics, Jun 4, 2023

This paper provides an explicit geometric and coordinate‐free formulation of incremental discrete... more This paper provides an explicit geometric and coordinate‐free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle . Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle is fully used in order to get the geometric counterpart κ of the so‐called tangent stiffness matrix. Subject to specific conditions in , the incremental evolution is then a well‐founded question. An hypoelastic four‐grains granular system illustrates in detail these general results.

Research paper thumbnail of Singularities and Stability of a Work Function

HAL (Le Centre pour la Communication Scientifique Directe), Oct 30, 2009

International audienc

Research paper thumbnail of Dynamics

Research paper thumbnail of Exact Solutions for the Vibration of Finite Granular Beam Using Discrete and Gradient Elasticity Cosserat Models

HAL (Le Centre pour la Communication Scientifique Directe), Nov 20, 2020

The present study theoretically investigates the free vibration problem of a discrete granular sy... more The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Research paper thumbnail of Static bending of granular beam: exact discrete and nonlocal solutions

Research paper thumbnail of High-Voltage Spindles detection from EEG signals using recursive synchrosqueezing transform

This paper proposes a new detection technique applied on electroencephalogram (EEG) signals which... more This paper proposes a new detection technique applied on electroencephalogram (EEG) signals which were measured on Parkinsonian rats during an experiment. Our technique uses a recursive filter-bank-based implementation of the short-time Fourier transform that is sharpened using several time-frequency reassignment methods. A detector is then applied to the obtained representation to allow an identification of the HVS signals which are specific to subjects with neurodegeneratives diseases. Our results show an improvement of the state of the art while paving the way of a real-time implementation.

Research paper thumbnail of Shear vibration modes of granular structures: Continuous and discrete approaches

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Research paper thumbnail of Modelling and stabilization of a load suspended by cable from an airship

Multibody System Dynamics

Research paper thumbnail of Stabilité des systèmes non conservatifs et algèbre linéaire

HAL (Le Centre pour la Communication Scientifique Directe), Dec 20, 2013

L'étude de la stabilité linéaire des systèmes non conservatifsà matrices de rigidité non symétriq... more L'étude de la stabilité linéaire des systèmes non conservatifsà matrices de rigidité non symétriques met en oeuvre des résultats d'algèbre linéaire peu classiques en mécanique. Elle peut même mener a des développements originaux d'algèbre linéaire. Nous illustrerons ces faits sur des problèmes de stabilité sous ajout de contraintes cinématiques dans un système non conservatif, problème qui mènera au concept original de matrice m-définie positive. Le résultat central sera présenté et quelques problèmes mathématiques ouverts suggérés.

Research paper thumbnail of Kinematic Structural Stability

HAL (Le Centre pour la Communication Scientifique Directe), Aug 21, 2016

International audienc

Research paper thumbnail of Dynamic Analysis of Granular Chain Using Cosserat Discrete Modeling

HAL (Le Centre pour la Communication Scientifique Directe), Aug 23, 2021

This paper deals with the free vibration of a beam which is discretized by finite rigid granular ... more This paper deals with the free vibration of a beam which is discretized by finite rigid granular elements and compared its exact solution to continuum local elasticity. This problem, which can be considered as a simple model to rigorously study the effect of the microstructure on the dynamic behavior of a equivalent continuum structural model, can be referred to Cosserat discrete chain or a lattice elastic model with shear interaction. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. First the critical frequencies of the system which change the nature of the results, are obtained. Next, the natural frequencies of such as granular model are analytically calculated whatever the considered modes for the granular beam resting on two supports, starting from the resolution of the linear difference eigenvalue problem. It is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (such a Bresse-Timoshenko beam can be also classified as a continuous Cosserat beam model). A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudodifferential operators, stemming from the continuualization process. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. Scale effects of the granular chain are clearly captured by the continuous gradient elasticity model. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Research paper thumbnail of Kinematics of Open Chains, Singularities

Multi-Body Kinematics and Dynamics with Lie Groups, 2018

In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tr... more In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tree structured rigid body system were exposed. In the present chapter we consider a rigid body system which is a chain of bodies linking up two bodies a and b . Then, with the notation.

Research paper thumbnail of Timoshenko beam under Winkler foundation

We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (... more We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (Winkler foundation) and subjected to a longitudinal force. We use analytical methods to determine buckling load and mode shape rather than numerical methods. Haringx and Engesser models are compared. We show that the rigidity of the wall solely gouverns the phenomena and that the two models are equivalent whatever the parameters of the problem.

Research paper thumbnail of Non-linear pre and post buckling behaviour of a Timoshenko beam

HAL (Le Centre pour la Communication Scientifique Directe), Aug 22, 2021

Research paper thumbnail of Hyperelastic or Hypoelastic Granular Circular Chain Instability in a Geometrically Exact Framework

Journal of Engineering Mechanics-asce, Sep 1, 2022

This paper investigates several granular interaction laws used in the modeling of discrete granul... more This paper investigates several granular interaction laws used in the modeling of discrete granular media. In the considered model, each grain interacts with its neighbors with a coupled shear-normal interaction law. The analysis is performed in a geometrically exact framework allowing large rotation and displacement evolutions, without any geometrical approximations. It is shown that most of the granular interaction laws available in the literature are classified as hypoelastic interaction laws, and we precise the requirements to build some hyperelastic interaction laws that avoid artificial dissipation. We also show that the uncoupled granular interaction law is hyperelastic for all the studied models. The analysis is applied to a paradigmatic elementary system of a granular loop with a diamond pattern (a four-grain cyclic granular chain) loaded by concentrated forces. Instabilities are observed for large displacement of the diamond chain for all the classified models. It is observed that the discrepancies between each model may grow during the deformation process. The instability phenomenon is associated with the appearance of a limit load for this granular structural problem due to large nonlinear geometrical effects. Blocking phenomena may also appear for such granular structural systems due to secondary granular contacts.

Research paper thumbnail of A New Result in Linear Stability

Research paper thumbnail of Discrete and gradient Elasticity Cosserat Modeling of Granular Chains

International audienceThe present study theoretically investigates the free vibration problem of ... more International audienceThe present study theoretically investigates the free vibration problem of a discrete granular system. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. Such a granular structural system is confined by discrete elastic interactions, to take into account the lateral granular contributions. This discrete repetitive system could be considered as a discrete Cosserat chain or a lattice elastic model with shear interaction. First, from the vibration equation governing the model, the natural frequencies are exactly calculated for the simply supports boundary conditions. Then it is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. The natural freque...

Research paper thumbnail of The “Transference Principle”

Where a licence is displayed above, please note the terms and conditions of the licence govern yo... more Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

Research paper thumbnail of Cohomology of the vector fields Lie algebras on R acting on trilinear differential operators, vanishing on sl(2)

Communications in Algebra, Mar 11, 2021

Abstract The main topic of this paper is to compute the first relative cohomology group of the Li... more Abstract The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields with coefficients in the space of trilinear differential operators that acts on tensor densities, vanishing on the Lie algebra where and

Research paper thumbnail of Explicit analysis of catasrophe on a Timoshenko beam

HAL (Le Centre pour la Communication Scientifique Directe), Mar 24, 2021

Research paper thumbnail of Intrinsic incremental evolution of hypoelastic discrete mechanical systems

Journal of Applied Mathematics and Mechanics, Jun 4, 2023

This paper provides an explicit geometric and coordinate‐free formulation of incremental discrete... more This paper provides an explicit geometric and coordinate‐free formulation of incremental discrete mechanics in the framework of potentially non integrable hypoelasticity. First, the general framework is developed in order to tackle hypoelasticity as an Ehresmann connection on the cotangent bundle . Two types of incremental evolutions may be distinguished, the weak or integrable incremental evolutions and the strong or non integrable incremental evolutions, according to the nature of the hypoelastic constitutive law. The geometric structure of the double tangent bundle is fully used in order to get the geometric counterpart κ of the so‐called tangent stiffness matrix. Subject to specific conditions in , the incremental evolution is then a well‐founded question. An hypoelastic four‐grains granular system illustrates in detail these general results.

Research paper thumbnail of Singularities and Stability of a Work Function

HAL (Le Centre pour la Communication Scientifique Directe), Oct 30, 2009

International audienc

Research paper thumbnail of Dynamics

Research paper thumbnail of Exact Solutions for the Vibration of Finite Granular Beam Using Discrete and Gradient Elasticity Cosserat Models

HAL (Le Centre pour la Communication Scientifique Directe), Nov 20, 2020

The present study theoretically investigates the free vibration problem of a discrete granular sy... more The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Research paper thumbnail of Static bending of granular beam: exact discrete and nonlocal solutions

Research paper thumbnail of High-Voltage Spindles detection from EEG signals using recursive synchrosqueezing transform

This paper proposes a new detection technique applied on electroencephalogram (EEG) signals which... more This paper proposes a new detection technique applied on electroencephalogram (EEG) signals which were measured on Parkinsonian rats during an experiment. Our technique uses a recursive filter-bank-based implementation of the short-time Fourier transform that is sharpened using several time-frequency reassignment methods. A detector is then applied to the obtained representation to allow an identification of the HVS signals which are specific to subjects with neurodegeneratives diseases. Our results show an improvement of the state of the art while paving the way of a real-time implementation.

Research paper thumbnail of Shear vibration modes of granular structures: Continuous and discrete approaches

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Research paper thumbnail of Modelling and stabilization of a load suspended by cable from an airship

Multibody System Dynamics

Research paper thumbnail of Stabilité des systèmes non conservatifs et algèbre linéaire

HAL (Le Centre pour la Communication Scientifique Directe), Dec 20, 2013

L'étude de la stabilité linéaire des systèmes non conservatifsà matrices de rigidité non symétriq... more L'étude de la stabilité linéaire des systèmes non conservatifsà matrices de rigidité non symétriques met en oeuvre des résultats d'algèbre linéaire peu classiques en mécanique. Elle peut même mener a des développements originaux d'algèbre linéaire. Nous illustrerons ces faits sur des problèmes de stabilité sous ajout de contraintes cinématiques dans un système non conservatif, problème qui mènera au concept original de matrice m-définie positive. Le résultat central sera présenté et quelques problèmes mathématiques ouverts suggérés.

Research paper thumbnail of Kinematic Structural Stability

HAL (Le Centre pour la Communication Scientifique Directe), Aug 21, 2016

International audienc

Research paper thumbnail of Dynamic Analysis of Granular Chain Using Cosserat Discrete Modeling

HAL (Le Centre pour la Communication Scientifique Directe), Aug 23, 2021

This paper deals with the free vibration of a beam which is discretized by finite rigid granular ... more This paper deals with the free vibration of a beam which is discretized by finite rigid granular elements and compared its exact solution to continuum local elasticity. This problem, which can be considered as a simple model to rigorously study the effect of the microstructure on the dynamic behavior of a equivalent continuum structural model, can be referred to Cosserat discrete chain or a lattice elastic model with shear interaction. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. First the critical frequencies of the system which change the nature of the results, are obtained. Next, the natural frequencies of such as granular model are analytically calculated whatever the considered modes for the granular beam resting on two supports, starting from the resolution of the linear difference eigenvalue problem. It is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (such a Bresse-Timoshenko beam can be also classified as a continuous Cosserat beam model). A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudodifferential operators, stemming from the continuualization process. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. Scale effects of the granular chain are clearly captured by the continuous gradient elasticity model. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

Research paper thumbnail of Kinematics of Open Chains, Singularities

Multi-Body Kinematics and Dynamics with Lie Groups, 2018

In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tr... more In the previous chapters all the necessary formula for expanding kinematics or dynamics of any tree structured rigid body system were exposed. In the present chapter we consider a rigid body system which is a chain of bodies linking up two bodies a and b . Then, with the notation.

Research paper thumbnail of Timoshenko beam under Winkler foundation

We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (... more We are interested in buckling for Timoshenko beam supported along its length by an elastic wall (Winkler foundation) and subjected to a longitudinal force. We use analytical methods to determine buckling load and mode shape rather than numerical methods. Haringx and Engesser models are compared. We show that the rigidity of the wall solely gouverns the phenomena and that the two models are equivalent whatever the parameters of the problem.

Research paper thumbnail of Non-linear pre and post buckling behaviour of a Timoshenko beam

HAL (Le Centre pour la Communication Scientifique Directe), Aug 22, 2021

Research paper thumbnail of Hyperelastic or Hypoelastic Granular Circular Chain Instability in a Geometrically Exact Framework

Journal of Engineering Mechanics-asce, Sep 1, 2022

This paper investigates several granular interaction laws used in the modeling of discrete granul... more This paper investigates several granular interaction laws used in the modeling of discrete granular media. In the considered model, each grain interacts with its neighbors with a coupled shear-normal interaction law. The analysis is performed in a geometrically exact framework allowing large rotation and displacement evolutions, without any geometrical approximations. It is shown that most of the granular interaction laws available in the literature are classified as hypoelastic interaction laws, and we precise the requirements to build some hyperelastic interaction laws that avoid artificial dissipation. We also show that the uncoupled granular interaction law is hyperelastic for all the studied models. The analysis is applied to a paradigmatic elementary system of a granular loop with a diamond pattern (a four-grain cyclic granular chain) loaded by concentrated forces. Instabilities are observed for large displacement of the diamond chain for all the classified models. It is observed that the discrepancies between each model may grow during the deformation process. The instability phenomenon is associated with the appearance of a limit load for this granular structural problem due to large nonlinear geometrical effects. Blocking phenomena may also appear for such granular structural systems due to secondary granular contacts.

Research paper thumbnail of A New Result in Linear Stability

Research paper thumbnail of Discrete and gradient Elasticity Cosserat Modeling of Granular Chains

International audienceThe present study theoretically investigates the free vibration problem of ... more International audienceThe present study theoretically investigates the free vibration problem of a discrete granular system. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. Such a granular structural system is confined by discrete elastic interactions, to take into account the lateral granular contributions. This discrete repetitive system could be considered as a discrete Cosserat chain or a lattice elastic model with shear interaction. First, from the vibration equation governing the model, the natural frequencies are exactly calculated for the simply supports boundary conditions. Then it is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. The natural freque...

Research paper thumbnail of The “Transference Principle”

Where a licence is displayed above, please note the terms and conditions of the licence govern yo... more Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.