Which kind of external loading second gradient materials can sustain?

In this communication, materials are considered whose deformation energy depends on both first and second gradient of placement map. When deriving the equilibrium equations by a variational approach, through integration by parts and the reiterated application of the divergence theorem, the class of external loading which are admissible for second gradient materials can be easily identified. Discrepancies with respect to Cauchy’s first gradient continua are highlighted, with special reference to the double forces acting over the boundary faces and the edge loading