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

TLDR

Capillary fluids, whose deformation energy depends on the second gradient of placement and are also known as Korteweg–de Vries or Cahn–Allen fluids, are a subclass of second‑gradient continua studied in analytical continuum mechanics, with historical roots in Piola’s work. The paper formulates a material description for second‑gradient continua and proves that a stationary action principle holds for capillary fluids. A Lagrangian action is defined in both material and spatial frames, yielding Euler–Lagrange equations and boundary conditions expressed through an objective deformation‑energy density that depends on either C and ∇C or C⁻¹ and ∇C⁻¹, and which reduces to the capillary‑fluid evolution equations for fluid‑specific energies. The variational framework yields a Bernoulli‑type law for capillary fluids and introduces useful kinematic formulas.

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 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.