期刊
MATHEMATICS AND MECHANICS OF SOLIDS
卷 20, 期 4, 页码 375-417出版社
SAGE PUBLICATIONS LTD
DOI: 10.1177/1081286513497616
关键词
Hamilton-Piola least action principle; second gradient continua; capillary fluid; variation principle
资金
- International Research Center MMoCS
- RFBR [12-01-00038]
- project BQR Materiaux Meso et Micro-Heterogenes: Optimisation par Modeles de Second Gradient et Applications en Ingenierie [BQR 2013-0054]
In 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. 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, Mindlin, Green-Rivlin, 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 delta C or on C-1 and delta C-1, where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli's law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to analytical continuum mechanics are also presented.
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