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

TLDR: In this article, a Lagrangian action is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments.

Abstract: 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 ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materia...