Deformation-induced coupling of the generalized external actions in third-gradient materials

In this study, diverse typologies of external actions are outlined, which turn out to be admissible for the third-gradient modeling of elastic materials. It is shown how such loading, when prescribed over the boundary surface, along the border edges and at the wedges of a deformable body in the Eulerian configuration, can be transformed into the Lagrangian description generating multiple interactions, with a surprising deformation-induced coupling. Such a phenomenon becomes more and more important at increasing the order of the β β -forces, specified by duality as covectors expending work on the β β th normal derivative of the virtual displacements, being herein at most β =2 β = 2 . Insights are provided into the true nature of such generalized forces, resting on the differential geometric features of the deformation process.