Reductible de certaines equations differentielles dans de espaces de banach de dimension quelconque

A differential equation (or a differential system, or a partial differential equation) is said to be reducible if one can find a change of the variable, which transforms it into a simpler one (e.g.: autonomous or linear).

Reduction, whenever possible, allows one, by using the solution of the reduced equation, to prove the existence of a solution to the original equation, and of certain of its properties, and to find an approximate value of this solution.

Résumé

Une équation différentielle (qui peut être un système différentiel ou une équation aux dérivées partielles) est réductible s'il est possible de trouver un changement de variable qui le transforme en une équation différentielle plus simple à étudier (autonome ou linéaire).

La réduction, quand elle est possible, permet de déterminer l'existence, certaines propriétés et une valeur approchée d'une solutionb de l'équation initiale à partir d'une solution de l'équation réduite.

Zusammenfassung

Eine Differentialgleichung (oder ein Differentialsystem, oder eine partielle Differentialgleichung) wird reduzierbar genannt, falls man sie durch eine Änderung der Variablen in eine einfachere (z.B. autonome oder lineare) Gleichung überführen kann. Wenn eine Reduzierung möglich ist, kann man die Lösung des reduzierten Systems benutzen, um die Existenz und bestimmte Eigenschaften der Lösung des Originalsystems zu beweisen. Weiterhin lassen sich auf diese Weise Näherungswerte für die Originallösung finden.     

Generalisation de la methode de la moyenne

A method to reduce some different systems containing a small parameter is proposed. This method makes it possible to regain, in a simple way, the classical method of averaging and to extend it to higher approximations as well as to wider classes of systems. Some examples of applications are then given.     

Etude du comportement en traction de bitumes en fonction de la température et de la vitesse de sollicitation

Summary The tensile behaviour of four grades of asphalt is investigated at various temperatures over a wide range of strain rates. It is shown that their ultimate tensile stress in the ductile range can be properly described by the formalism of Eyring's theory of activated rate processes.

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Zusammenfassung Das Verhalten von vier Bitumina, die in einem sehr weiten Bereich von Deformationsgeschwindigkeiten und Temperaturen untersucht wurden, zeigt deutlich, daß ihre im duktilen Bereiche liegende Zugfestigkeit mittels eines aus der Eyringschen Theorie der Aktivierungsprozesse abgeleiteten Formalismus beschrieben werden kann.

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Avec 5 figures et 2 tableaux     

Optimisation de renforts composites internes de structures tridimensionnelles

The reinforcement of a 3D structure with composite panels is considered. The behaviour of those panels are to be optimized considering the external loading of the 3D structure. The study is within the framework of anisotropic linear elasticity. The 3D model takes into account the rigidity induced by Kirchhoff–Love plate reinforcement. An optimization of the behavior characteristics of the composite reinforcement (density and orientation of fibers at each point) is presented.     

Identification de la relation de dispersion dans les barres

Dispersion and attenuation of longitudinal waves in elastic or weakly viscoelastic rods are measured by analysing the resonant frequencies present in the strain spectrum due to an unknown loading. The method takes the finite measuring time of the test into account. It is applied to an aluminium bar, in which the dispersion relation is identified very accurately at frequencies up to 60 kHz. To cite this article: R. Othman et al., C. R. Mecanique 330 (2002) 849–855.     

Methodes d'equiblibre appliquees au calcul de l'elargissement libre dans le cas de l'etirage de barres rectangulaires

Many forming processes (forging, constraining, drawing…) produce free surfaces flows. It is not easy to foresee evolution of these surfaces, flows are generaly unsteady and three-dimensional. One has compared two equilibrium methods (slabs and columns methods) in the case of rectangular bars drawing with a two planes die (free surfaces on both sides of bars). Metal is supposed to be rigid, perfecly plastic, follows the HILL's principle of maximal work and the VON MISES's plasticity criterium. Every section of bars are supposed to stay rectangular. The better results were obtained with the columns method which is in better agreement with experiments.     

Suivi de fissuration de matériaux par thermographie

The purpose of this work is to present a novel strategy for real-time monitoring crack growth of materials. The process is based on the use of thermal data extracted along the horizontal axis of symmetry of single edge notch tension (SENT) specimens, during fatigue tests. These data are exploited using an implemented program to detect in situ the growth of fatigue crack, with the critical size and propagation speed of the crack. This technique has the advantage to be applicable to a wide range of materials regardless of their electrical conductivity and their surface texture. To cite this article: D. Ait Aouit, A. Ouahabi, C. R. Mecanique 336 (2008).     

Variation de la vitesse de propagation d'ondes ultrasoniques au cours du passage de l'état solide

Sans résumé     

Nouveaux theoremes de stabilite utilisant la methode directe de Liapounov

Several generalizations are given of the first theorem of Liapounov concerning the stability of a critical point of an ordinary differential equation. One uses successively positive definite auxiliary functions, but whose derivatives are not semi-negative definite, functions having a pole located at the critical point, and at last functions which are not positive definite, although they are positive everywhere. Finally, an optimal stability criterion is determined for a certain class of such criteria.

Zusammenfassung

Es werden mehrere Verallgemeinerungen über den ersten Lehrsatz von Liapounov bezüglich der Beständigkeit eines kritischen Punktes einer gewöhnlichen Differential gleichung gegeben. Man verwendet fortlaufend positiv bestimmte Hilfsfunktionen, deren Differentialquotienten jedoch nicht halbnegativ bestimmt sind, nachdem die Funktionen einen im kritischen Punkt lokalisierten Pol haben, und schliesslich Funktionen die nicht positiv bestimmt sind, obwohl sie überall positiv sind. Abschliessend wird ein optimales Beständigkeits-kriterium festgelegt für eine bestimmte Klasse solcher Kriterien.

Abstract

Пrивoдитsя rяд oбoбщeNий пerвoй тeoreмы ЛяпuNoвa oб usтoйchивosти кrитиchesкoй тochки oбыкNoвeNNoгo диффereNциaльNoгo uraвNeNия. ПrимeNяuтsя пosлeдoвaтeльNo пoлoжитeльNo oпreдeлeNы вsпoмaгaтeльNыe фuNкции, пroизвoдNыe кoтorых Ne являuтsя пoлu-oтrицaтeльNo oпreдeлeNNыми, фuNкции s пoлusoм Naхoдящимsя в кrитиchesкoй тochкe, и NaкoNeц фuNкции, кoтorыe Ne являuтsя пoлoжитeльNo oпreдeлeNNыми, хoтя oNи вeздe пoлoжитeльNы. NaкoNeц oпreдeляeтsя oптимaльNый кrитerий usтoйchивosти для Neкoтoroгo клassa тaких кrитerиeв.     

Conséquences de l'agrégation et de la percolation de charges sur le comportement viscoélastique de nanocomposites

The aim of the present Note is to differentiate the relative contributions of the aggregation and the percolation phenomena to the viscoelastic behaviour of composites. This separation is performed through the development of a micromechanical model, by taking into account the percolation concept. To cite this article: P. Mélé et al., C. R. Mecanique 333 (2005).     

Diagramme de Laguerre

This Note presents a generalization of a known fast and robust algorithm of incremental construction of the Delaunay triangulation to the case of the regular triangulation of points in ${\mathbb{R}}^d$. In particular, the transport formula of simplex circumball centers are naturally extended to the case of the regular triangulation. The associated Laguerre diagram can then be obtained by duality from the regular triangulation. Some numerical examples of Laguerre diagrams in three dimensions are given. To cite this article: H. Borouchaki et al., C. R. Mecanique 333 (2005).