Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola's point of view

As Piola would have surely conjectured, the stationary action principle holds also for capillary fluids, i.e. those fluids for which the deformation energy depends on spatial derivative of mass density (a modelling necessity which has been already remarked by Cahn and Hilliard [15, 16]). For capillary fluids it is indeed possible to define a Lagrangian density function whose corresponding Euler-Lagrange stationarity conditions once transported on the actual configuration, via a Piola’s transformation, are exactly those obtained, with different methods, in the literature. We recall that some particulat classes of second gradient fluids are sometimes also called Korteweg-de Vries or Cahn- Allen fluids. More generally those continua (which may be solid or fluid) 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 work, following closely the procedure first conceived by Piola and carefully presented in his works translated in the present volume, a material (Lagragian) description for second gradient continua is formulated. Subsequently a Lagrangian action is introduced and by means of Piola’s transformations this action is calculated in both the material and spatial descriptions. Then the corresponding Euler- Lagrange equations and boundary conditions are calculated by using some kinematical relationships suitably established. Once an objective deformation energy volume density is assumed to depend on either C and ∇C or on C−1 and (where C is the Cauchy-Green deformation tensor) the particular form of aforementioned Euler-Lagrange conditions and boundary conditions are established. When further particularizing the treatment to those energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal [25] or Seppecher [142, 145] for an alternative deduction based on thermodynamic arguments) are recovered. Also a version of Bernoulli’s law which is valid for capillary fluids is found and, in Appendix B, all the kinematic formulas which we have found useful for the present variational formulation are gathered. Many historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented when it has been considered useful. In this context the reader is also referred to Capecchi and Ruta [17].