THE D'ALEMBERT-LAGRANGE PRINCIPLE FOR GRADIENT THEORIES AND BOUNDARY CONDITIONS

Abstract

Asymptotic Methods in Nonlinear Wave Phenomena, pp. 79-95 (2007) No AccessTHE D'ALEMBERT-LAGRANGE PRINCIPLE FOR GRADIENT THEORIES AND BOUNDARY CONDITIONSH. GOUINH. GOUINUniversité d'Aix-Marseille, 13397 Marseille Cedex 20, Francehttps://doi.org/10.1142/9789812708908_0008Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Motions of continuous media presenting singularities are associated with phenomena involving shocks, interfaces or material surfaces. The equations representing evolutions of these media are irregular through geometrical manifolds. A unique continuous medium is conceptually simpler than several media with surfaces of singularity. To avoid the surfaces of discontinuity in the theory, we transform the model by considering a continuous medium taking into account more complete internal energies expressed in gradient developments associated with the variables of state. Nevertheless, resulting equations of motion are of an higher order than those of the classical models: they lead to non-linear models associated with more complex integration processes on the mathematical level as well as on the numerical point of view. In fact, such models allow a precise study of singular zones when they have a non negligible physical thickness. This is typically the case for capillarity phenomena in fluids or mixtures of fluids in which interfacial zones are transition layers between phases or layers between fluids and solid walls. Within the framework of mechanics for continuous media, we propose to deal with the functional point of view considering globally the equations of the media as well as the boundary conditions associated with these equations. For this aim, we revisit the d'Alembert-Lagrange principle of virtual works which is able to consider the expressions of the works of forces applied to a continuous medium as a linear functional value on a space of test functions in the form of virtual displacements. At the end, we analyze examples corresponding to capillary fluids. This analysis brings us to numerical or asymptotic methods avoiding the difficulties due to singularities in simpler -but with singularities- models. Dedication: Dedicated to Prof. Antonio M. Greco. FiguresReferencesRelatedDetailsCited By 4Rankine–Hugoniot conditions obtained by using the space–time Hamilton actionHenri Gouin16 April 2020 | Ricerche di Matematica, Vol. 70, No. 1Dynamic boundary conditions for membranes whose surface energy depends on the mean and Gaussian curvaturesSergey Gavrilyuk and Henri Gouin27 May 2019 | Mathematics and Mechanics of Complex Systems, Vol. 7, No. 2Vesicle Model with Bending Energy RevisitedHenri Gouin17 May 2014 | Acta Applicandae Mathematicae, Vol. 132, No. 1Interfaces endowed with nonconstant surface energies revisited with the d'Alembert–Lagrange principleHenri Gouin1 January 2014 | Mathematics and Mechanics of Complex Systems, Vol. 2, No. 1 Asymptotic Methods in Nonlinear Wave PhenomenaMetrics Downloaded 15 times History PDF download

Locations

• arXiv (Cornell University) - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• DataCite API - View