Since the first studies dedicated to the mechanics of deformable bodies (Euler, D’Alembert, Lagrange) the Principle of Virtual Work (or Virtual Velocities) has been used to supply a firm guidance to the formulation of novel theories. Gabrio Piola dedicated his scientific life to formulate a continuum theory able to encompass a larger class of deformation phenomena and was the first author considering continua with non-local internal interactions and, as particular case, higher gradient continua. Being more recent followers of Piola, Mindlin and then Richard Toupin recognized the Principle of Virtual Work (and its particular case: the Principle of Least Action) as the (only) firm foundation of Continuum Mechanics. Mindlin and Toupin managed to formulate a conceptual frame for Continuum Mechanics which is able to model effectively the complex behavior of so-called architectured, advanced, multiscale or microstructured (meta)materials. Other postulation schemes, instead, does not seem able to be equally efficient. However Navier-Cauchy format for Continuum Mechanics is based on the concept of contact interaction between subbodies of a given continuous body.
To reconcile D’Alembertian mechanics with Cauchy viewpoint it has to be shown how -by means of the Principle of Virtual Powers- it is possible to generalize Cauchy representation formulas for contact interactions to the case of N-th gradient continua, i.e. continua in which the deformation energy depends on the deformation Green-Saint-Venant tensor and all its N − 1 order gradients. Actually it is needed to derive the explicit representation formulas to be used in N-th gradient continua to determine contact interactions as functions of the shape of Cauchy Cuts. It is shown, in particular, that i) these interactions must include edge (i.e. concentrated on curves) and wedge (i.e. concentrated on points) interactions, and ii) these interactions cannot reduce simply to forces: indeed the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green and Germain) is fundamental and unavoidable in the theory of N-th gradient continua. The main concepts of the theory of embedded manifolds are an essential tool in the presented derivation and will be recalled in the form more relevant in the present context, also indicating how some important generalizations should be developed.