Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids

Abstract

On ne trouvera point de Figures dans cet Ouvrage. Les méthodes que j'y expose ne demandent ni constructions, ni raisonnemens géométriques ou méchaniques, mais seulement des opérations algébriques, assujetties à une marche réguliere et uniforme. Ceux qui aiment l'Analyse, verront avec plaisir la Méchanique en divenir une nouvelle branche, et me sauront gré d'en avoir étendu ansi le domaine." From the Avertissement of the Méchanique Analitique by Lagrange [87]1 AbstractIn this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard [15,16]. We remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. In general continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler-Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C −1 and ∇C −1 , where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal [25] or Seppecher [140,143] for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli's law valid for capillary fluids is found and, in Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to analytical continuum mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta [17]. Part Ithe author obtained the evolution equations for capillary fluids by combining the principle of virtual works in the Eulerian description with the first principle of thermodynamics (also in the case of isothermal motions). This shows that it can be sometimes useful to use an heuristic procedure in which the principle of virtual powers is reinforced by additionally requiring also the validity of the balance of mechanical energy. Also interesting in this context are the results presented in Casal [25], Gavrilyuk and Gouin [68].In the opinion of the present authors the methods of analytical continuum mechanics are the most effective ones (see also [100]), at least when formulating models for mechanical phenomena involving multiple time and length scales. The reader is invited to consider, with respect to this particular class of phenomena, the difficulties which are to be confronted when using continuum thermodynamics, for instance, to describe interfacial phenomena in phase transition (see e.g. d...

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

“…Secondly: it has to be recalled that boundary conditions producing well-posed problems in the case of second gradient continua are more general than when dealing with first gradient continua (see for more details about generalized boundary conditions, e.g., [33,34,46,59] and the historic works by Piola). The procedure which is used in the aforementioned papers can be summarized as follows (see [5,19]): i) one postulates the principle of virtual work, i.e., the equality between internal and external work expended on virtual displacements; ii) one determines a class of internal work functionals involving second gradients of virtual displacement; iii) one determines, by means of an integration by parts, the class of external work functionals which are compatible with the determined class of internal work functionals.…”

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