- A reformulation of the Symplectic Brezis-Ekeland-Nayroles principle, (doi), (2024)
- The Symplectic Brezis-Ekeland-Nayroles principle (SBEN) is a dissipative version of hamiltonian mechanics [3], [4], [5]. It is named so because it gives a dynamical version of the principles formulated by Brezis-Ekeland [2] for parabolic equations and Nayroles theorems on dissipative systems [10].In this article we continue [6] and reformulate SBEN in a simpler form, which cover non-associated constitutive laws expressed via bipotential theory. We thus obtain a new, dynamical theory for a large class of models which are usually studied in the quasistatic regime. This is the original version of the article written for the preparation of the project ANR-22-CE51-0034.
- Dissipation and the information content of the deviation from hamiltonian dynamics, (journal pdf) (doi), Ann. Acad. Rom. Sci., Math. Appl. 15, No. 1-2, 366-382 (2023)
- We explain a dissipative version of hamiltonian mechanics, based on the information content of the deviation from hamiltonian dynamics. From this formulation we deduce minimal dissipation principles, dynamical inclusions, or constrained evolution with hamiltonian drift reformulations. Among applications we recover a dynamics generalization of Mielke et al quasistatic rate-independent processes.
This article gives a clear and unitary presentation of the theory of hamiltonian inclusions with convex dissipation or symplectic Brezis-Ekeland-Nayroles principle, presented under various conventions first in arXiv:0810.1419, then in arXiv:1408.3102 and, for the appearance of bipotentials in relation to the symplectic duality, in arXiv:1902.04598v1.
- On the information content of the difference from hamiltonian evolution (2019)
- A dissipative version of hamiltonian mechanics is proposed via a principle of minimal information content of the deviation from hamiltonian evolution. We show that we can cover viscosity, plasticity, damage and unilateral contact. This article continues arXiv:1807.10480
- A stochastic version and a Liouville theorem for
hamiltonian inclusions with convex dissipation (2018)
- The statistical counterpart of the formalism of hamiltonian systems with convex dissipation arXiv:0810.1419, arXiv:1408.3102 is a completely open subject. Here are described a stochastic version of the SBEN principle and a Liouville type theorem which uses a minimal dissipation cost functional.
- A symplectic Brezis-Ekeland-Nayroles principle, with
Géry de Saxcé,(2014), Mathematics and Mechanics of Solids 22, 6, (2017)
- We propose a modification of the hamiltonian formalism which can be used for dissipative systems. This work continues arXiv:0810.1419 and advances by the introduction of a symplectic version of the Brezis-Ekeland-Nayroles principle [Brezis Ekeland 1976] [Nayroles 1976]. As an application we show how standard plasticity can be treated in our formalism.
- A variational formulation for constitutive laws described by bipotentials (with
Géry de Saxcé and Claude Vallée),(2011)
- Inspired by the algorithm of Berga and de Saxce for solving the discretisation in time of the evolution problem for an implicit standard material, we propose a general variational formulation in terms of bipotentials.
- Blurred maximal cyclically
monotone sets and
bipotentials (with
Géry de Saxcé and Claude Vallée), Analysis and Applications 8 (2010), no. 4, 1-14
- Let X be a reflexive Banach space and Y its dual. In this
paper we find necessary and sufficient conditions for the existence
of a bipotential for a blurred maximal cyclically monotone graph.
Equivalently, we find a necessary and sufficient condition for
a subgradient differential inclusion to be expressed as a implicit
differential inclusion with the help of a bipotential.
- Blurred constitutive laws and bipotential convex
covers (with
Géry de Saxcé and Claude Vallée), Mathematics and Mechanics of Solids 16(2), (2011), 161-171 , DOI
- In many practical situations, incertitudes affect the
mechanical behaviour that is given by a family of graphs
instead of a single one. In this paper, we show how the
bipotential method is able to capture such blurred
constitutive laws, using bipotential convex covers.
- Bipotentials for non monotone
multivalued operators: fundamental results and applications (with
Géry de Saxcé and Claude Vallée), Acta Applicandae Mathematicae, 110, 2(2010), 955-972 DOI
- This is a survey of recent results about bipotentials
representing multivalued operators. The notion of bipotential is
based on an extension of Fenchel's inequality, with several
interesting applications related to non associated constitutive
laws in non smooth mechanics, such as Coulomb frictional contact
or non-associated Drucker-Prager model in plasticity. Relations
betweeen bipotentials and Fitzpatrick functions are described.
Selfdual lagrangians, introduced and studied by Ghoussoub, can be
seen as bipotentials representing maximal monotone operators. We
show that bipotentials can represent some monotone but not maximal
operators, as well as non monotone operators. Further we describe
results concerning the construction of a bipotential which
represents a given non monotone operator, by using convex
lagrangian covers or bipotential convex covers.
- Non maximal cyclically
monotone graphs and construction of a bipotential for the Coulomb's dry
friction law (with
Géry de Saxcé and Claude Vallée), J. of Convex Analysis 17, No 1. (2010), 81-94
- We show a surprising connexion between a property of the
inf convolutions of a family of convex lower semicontinuous
functions and the fact that intersections of maximal cyclically
monotone graphs are the critical set of a bipotential. We then
extend the results from arXiv:math/0608424
to bipotentials convex covers, generalizing the notion of a \
bi-implicitly convex lagrangian cover. As an application we prove
that the bipotential related to Coulomb's friction law is related
to a specific bipotential convex cover with the property that any
graph of the cover is non maximal cyclically monotone.
- Hamiltonian inclusions with convex dissipation
with a view towards applications (2008), Mathematics and its
Applications 1, 2 (2009), 228-251 (journal)
- We propose a generalization of hamiltonian mechanics, as a hamiltonian inclusion with convex
dissipation function. We obtain a dynamical version of the
approach of Mielke
to quasistatic rate-independent processes. Then we show that a class of
models of dynamical brittle damage can be formulated in this setting.
- Four applications
of majorization to convexity in the calculus of variations(2007),
Linear Algebra and its Applications 429, 2008, 1528-1545
- Re-writed and updated version of
math.FA/0105044.
The applications are: (1) simple necessary and sufficient
conditions for an isotropic objective function to be rank one
convex on the set of matrices with positive determinant, (2) a very short
proof of a theorem of Thompson and Freede, Ball, or
Le Dret, concerning the convexity of a class of
isotropic functions which appear in nonlinear elasticity, (3)a lower
semicontinuity result for elastic energy functionals depending on
Hencky's logarithmic strain, (4) a compact proof of
Dacorogna-Marcellini-Tanteri theorem, explaining the
resemblance of this theorem with the Horn-Thompson theorem.
- Construction
of bipotentials and a minimax theorem of Fan (2006) (with
Géry de Saxcé and Claude Vallée)
- In Mechanics, the theory of standard materials is a well-known application
of Convex Analysis. However, the so-called non-associated constitutive laws
cannot be cast in the mould of the standard materials. From the
mathematical viewpoint, a non associated constitutive law is a
multivalued operator which is not supposed to be monotone. A
possible way to study non-associated constitutive laws by using
Convex Analysis, proposed first in [12], consists in constructing
a "bipotential" function of two variables, which physically represents
the dissipation. This is a second paper on the mathematics of the
bipotentials. We prove here another reconstruction theorem for a
bipotential from a convex lagrangian cover, this time using a
convexity notion related to a minimax theorem of Fan.
- Existence
and construction of bipotentials for graphs of multivalued laws(2006)(with
Géry de Saxcé and Claude Vallée), Journal of Convex Analysis, vol. 15, no. 1, 2008, 87-104.
- This is a first paper in convex analysis dedicated to
the bipotential theory, based on an extension of Fenchel's inequality.
Introduced by the second author, bipotentials lead to a succesful new
treatment of the constitutive laws of some dissipative materials:
frictional contact, non-associated Drucker-Prager model, or Lemaitre
plastic ductile damage law. We solve here the problems of existence and
construction of a bipotential for a nonsmooth mechanics constitutive
law.
- The
variational complex of a diffeomorphisms group
- In this paper we propose a variational complex
associated to a
diffeomorphisms group with first order jet in a Lie group. We study the
structure of null lagrangians and we prove some fundamental properties
of them, as well as their connection to differential invariants of the
group action.
- Quasiconvexity
versus group invariance This is the written version of a lecture
held on Feb. 22 at the Mathematical Institute, Oxford, Applied Analysis
and Mechanics Seminars,Hilary Term 1999.
- The lower invariance under a given arbitrary group of
diffeomorphisms
extends the notion of quasiconvexity. The non-commutativity of the
group operation (the function composition) modifies the classical
equivalence between lower semicontinuity and quasiconvexity. In this
context null lagrangians are particular cases of integral invariants of
the group.
- Lower
semi-continuity of integrals with $G$-quasiconvex potential,
Z. Angew. Math. Phys., 53, 6 (2002), pp 949-961
- This paper
introduces the proper notion of variational quasiconvexity associated
to a group of bi-Lipschitz homeomorphisms. We prove a lower
semicontinuity theorem connected to this notion, which improves a
result of Dacorogna and Fusco. In the second part of the paper we apply
this result to a class of functions, introduced in the next paper. Such
functions are multiplicative quasiconvex, hence they induce lower
semicontinuous integrals.
- Majorisation
with applications in Elasticity
- Theorem 5.1 gives simple necessary and sufficient
conditions for an isotropic objective function to be rank one convex on
the set of matrices with positive determinant.
- Theorem 6.2 describes a class of possible
non-polyconvex but multiplicative quasiconvex isotropic functions. This
class is not contained in a well known theorem of Ball (6.3 in this
paper) which gives sufficient conditions for an isotropic and objective
function to be polyconvex.
- I
show that there is a new way to prove directly the quasiconvexity (in
the multiplicative form). Relevance of Schur convexity for the
description of rank one convex hulls is explained.
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