Piola's approach to the equilibrium problem for bodies with second gradient energies. Part I: First gradient theory and differential geometry

In this study, some pioneering contributions, envisaged in the works of Gabrio Piola, were developed through tools of the modern differential geometry and applied to the second gradient continua. Part I introduced the variational approach for the equilibrium problem according to the first gradient theory and exploited differential geometric perspectives for the present scenario. By prescribing the stationarity of the Lagrangian energy functional, the virtual work equations for a Cauchy's medium were recovered. The focus was on the deformation process regarded as a diffeomorphism between Riemannian embedded submanifolds, emphasizing the roles of the pullback metrics and of the covariant differentiation. Novel transport formulae were provided for normal and tangent vectors in the neighborhood of a boundary edge. The divergence theorem for curved surfaces with border was revisited, providing remarkable relationships between Lagrangian and Eulerian expressions involving projectors.