Une méthode inverse d'ordre un pour les problèmes de complétion de données

An iterative resolution method for inverse Cauchy problems is presented. The successive iterations satisfy the equilibrium equations exactly. Numerical simulations prove the accuracy of the method and its ability to solve Cauchy problems when the domain boundary is not regular. To cite this article: A. Cimetière et al., C. R. Mecanique 333 (2005).     

Espaces d'écoulements dits « universels », suite 2Spaces of universal flows,part 2

At small velocities, every universal motion can be performed in a fluid whenever the constitutive equation approaches a most general linear model. Both the celebrate theorems of Appell, Cauchy, Helmholtz, Kelvin hold true when dealing with the universal motions. Here the theorem of Bernoulli admits four generalisations. At last, the Helmholtz–Rayleigh theorem about dissipation may be regarded as defining the universal motions. To cite this article: M. Bouthier, C. R. Mecanique 332 (2004).     

Problème spectral pour la propagation conique des ondes élastiques dans un réseau de galeriesA spectral problem for conically propagating elastic waves through an array of cylindrical channels

This Note is devoted to the analysis of elastic waves conically propagating through a doubly periodic array of cylindrical channels. A new method, based on a multiple scattering approach, has been proposed to reduce the problem to an algebraic system of the Rayleigh type. We obtain an eigenvalue problem formulation that enables us to construct the high-order dispersion curves and to study phononic band gap structures in oblique propagation. We note an effect of singular perturbation associated with a small angle of conical propagation. To cite this article: S. Guenneau et al., C. R. Mecanique 330 (2002) 491–497.     

Asymptotic analysis for micropolar fluidsAnalyse asymptotique des fluides micropolaires

The flow of a micropolar fluid through a wavy constricted channel which depends on a small parameter ε?1 is considered. The asymptotic solution is built and justified thanks to a study of the boundary layers terms. The Stokes and Navier–Stokes problems set in a tube structure were previously considered. The method of partial asymptotic decomposition of domain (MAPPD) is also applied and justified for the micropolar flow problem. This method reduces the initial problem to the problem set in the boundary layers domain. To cite this article: D. Dupuy et al., C. R. Mecanique 332 (2004).     

Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaireDetermination of the macroscopic strength criterion of a porous medium by nonlinear homogenization

A porous medium, which matrix is a perfectly plastic solid, is considered. This paper proposes a method to determine the macroscopic admissible stress states. The method is based on a homogenization technique which takes advantage of the equivalence, under certain conditions, between a problem of limit analysis and a ficticious nonlinear elastic problem. The particular case of a Drucker–Prager solid matrix is considered. The method provides an analytical expression for the complete macroscopic strength criterion. To cite this article: J.-F. Barthélémy, L. Dormieux, C. R. Mecanique 331 (2003).     

Nulle contrôlabilité régionale pour des équations de la chaleur dégénéréesRegional null controllability for degenerate heat equations

We are interested in a null controllability problem for a class of strongly degenerate heat equations.First for all T>0, we prove a regional null controllability result at time T at least in the region where the equation is not degenerate. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by application of Carleman estimates combined with the introduction of cut-off functions.Then we improve this result: for all T′>T, we obtain a result of persistent regional null controllability during the time interval [T,T′]. Finally we give similar results for the (non degenerate) heat equation in unbounded domain. To cite this article: P. Cannarsa et al., C. R. Mecanique 330 (2002) 397–401.     

Kinematic shakedown by the Norton–Hoff–Friaa regularising method and augmented LagrangianAdaptation cinématique par la methode de régularisation de Norton–Hoff–Friaa et du Lagrangien augmenté

The shakedown analysis of elastic perfectly plastic structures is formulated as a discrete nonlinear mathematical programming problem by means of the finite element technique. The kinematical problem is regularized through the introduction of the Norton–Hoff viscoplastic material to overcome the non-differentiability of the objective function, and can be solved numerically by the augmented Lagrangian technique. To cite this article: M.A. Hamadouche, C. R. Mecanique 330 (2002) 305–311.     

Une technique de lissage de données basée sur la théorie de la régularisationA filter technique based on regularization theory

A simple filter technique based on the regularization theory is presented. We consider the problem as an optimization one. The regularization theory gives us a suitable theoretical framework to define a functional to minimize. We make a numerical comparation between this method and a classical Fourier technique. To cite this article: J.-M. Fullana, C. R. Mecanique 330 (2002) 647–652.     

Transfert de champs plastiquement admissibles

This paper presents a new approach to interpolate the mechanical fields associated to a given mesh of the computational domain which satisfy the equilibrium equations together with the mechanical criteria which are quadratical in terms of these fields. The method is based on the diffuse approximation techniques. These allow us to construct a field of globally arbitrary order of continuity which approximates accurately the initial discrete mechanical fields. Indeed, the construction is based locally on the resolution of a quadratical optimisation problem under degenerate quadratical constraints for which we propose an analytical solution. The method is applied, in particular, to an equilibrium problem of elastoplastic solid with non linear hardening. To cite this article: P. Villon et al., C. R. Mecanique 330 (2002) 313–318.     

‘Les fleurs du mal’ – an adaptive wavelet method of arbitrary lines I: convection–diffusion problems« Les fleurs du MAL » – une méthode d'ondelettes adaptive de lignes arbitraires I : problèmes de convection–diffusion

Baudelaire's ‘les fleurs du mal’ refers to various new developments (‘les fleurs’) of the method ofarbitrarylines (mal), since it was first published (in C. R. Acad. Sci. Paris, Sér. I, in 1991). Here we revisit the basic mal (semi-discretization) methodology for stationary convection–diffusion problems and develop an adaptive, wavelet-based solver that is capable of capturing the thin layers that arise in such problems. We show the efficacy and high accuracy of the wavelet-mal solver by applying it to a challenging 2D problem involving both boundary and interior layers. To cite this article: X. Ren, L.S. Xanthis, C. R. Mecanique 332 (2004).     

Material electromagnetic fields and material forces

Electromagnetic fields address configurational forces in a natural way through an energy–stress tensor, which reduces to the Maxwell tensor in the simplest case. This tensor is related to physical forces and to the Cauchy traction in a continuum. Material forces, as opposed to physical forces, are of a different nature as they act upon a site of a continuum where the possible material inhomogeneity is located. A material energy–stress tensor, which is reminiscent of the Maxwell stress, is associated with these forces. Through appropriate balance laws, a material momentum is also associated with material forces. The material momentum is of particular interest in electromagnetic materials as it is intimately related to the pseudomomentum of light [Peierls in Highlights of Condensed Matter Physics, pp. 237–255 (1985) and in Surprises in Theoretical Physics, pp. 91–99 (1979); Thellung in Ann. Phys. 127, 289–301 (1980)]. The balance law for the material momentum can be derived either from the classical physical laws or independently of them. This derivation, which is based on the material electromagnetic potentials and the related gauge transformations, is discussed and commented on for an electromagnetic body.     

Identification of radon transfer velocity coefficient between liquid and gaseous phasesIdentification de la vitesse de transfert de radon entre une phase liquide et une phase gazeuse

Radon transfer between a liquid phase and a gaseous phase is modelled by a Robin's condition (radon flux at the common interface is expressed as function of radon concentrations in the two phases). This condition involves two constants: Ostwald's coefficient (α) and the transfer velocity coefficient (β). Assuming the value of α is known, a method is proposed to determinate the value of β, by studying the radon transfer phenomenon at the laboratory scale. Knowing the initial radon concentrations, the experiment consists in measuring how long the radon flux passes through the common interface. In this stabilisation time radon transport is governed in each phase by diffusion and disintegration. Then, determination of β is equivalent to solving an inverse problem formulated using measured data. A numerical procedure is developed to solve this problem. To cite this article: D.-G. Calugaru, J.-M. Crolet, C. R. Mecanique 330 (2002) 377–382.     

On a class of non-linear elastic materials

This article deals with a family of non-linear hyperelastic materials depending on a parameter varying from 0 to 1; is a masonry-like material and is linear elastic. Some properties of the function delivering the Cauchy stress corresponding to the infinitesimal strain E, are proved; in particular, it is shown that is strongly monotone for >0 and monotone for =0. Moreover, denoting by [u(·;), E(·;), T(·;)] the solution to the equilibrium problem for solids made of a material the convergence of [u(·;), E(·;), T(·;)] for going to 0 and 1, is investigated.     

Homogénéisation d'un milieu élastique fortement hétérogèneHomogenization of a strongly heteregeneous elastic medium

We consider the homogenization of an elastostatic problem in a strongly heterogeneous periodic medium made of two connected components having comparable tensors of elastic moduli, separated by a third medium (soft layer), the thickness of which is of the same order ε than the basic periodicity cell, and such that its elastic moduli tensor becomes infinitely small following a rate ε^r, r>0. If r?2, we identify the homogenized problem. Otherwise, we have to assume moreover that there are no volume forces in the third medium. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 330 (2002) 543–548.     

Finite Amplitude Transverse Waves in Special Incompressible Viscoelastic Solids

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part is a linear combination of A ₁, A ₁ ² and A ₂, where A ₁, A ₂ are the first and second Rivlin–Ericksen tensors. The body is first subject to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that a finite amplitude circularly polarized wave may propagate along either n ⁺ or n ^?, where n ⁺, n ^? are the normals to the planes of the central circular section of the ellipsoid x?B ^?1 x=1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.     

Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ^?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L ² norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.     

Asymptotic decomposition of a singular perturbation problem with unbounded energyDécomposition asymptotique d'une perturbation singulière d'énergie sans limite

We consider a singular perturbation with unbounded energy. We propose here an effective method of finite element computation, fit for accounting for the linear behavior of the solution. The Hilbert space of the variational formulation, H²₀(0,1), is replaced by a simpler subspace containing an asymptotic solution of the initial problem. Error estimates are derived by eliminating some degrees of freedom and a numerical experiment is developped. To cite this article: F. Fontvieille et al., C. R. Mecanique 330 (2002) 507–512.     

On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.     

Direct and inverse inequalities for the isotropic Lamé system with variable coefficients

We consider a Cauchy–Dirichlet problem for the isotropic Lamé system with variable coefficients. We find an estimate for the L² -norm of the surface traction in terms of the initial data and the body force. Then we show that, in absence of body forces, the elastic energy can be controlled by the L² -norm of the surface traction exerted on a suitable sub-boundary, provided that the time interval is sufficiently large. These inequalities are basic for the applicability of the so-called HUM (Hilbert Uniqueness Method) and they can also be used to solve an inverse source problem for the Lamé system.