Third gradient continua: nonstandard equilibrium equations and selection of work conjugate variables

This paper outlines the variational derivation of the Lagrangian equilibrium equations for the third-gradient materials, stemming from the minimization of the total potential energy functional, and the selection of suitable dual variables to represent the inner work in the Eulerian configuration. Volume, face, edge and wedge contributions were provided through integration by parts of the inner virtual work and by repeated applications of the divergence theorem extended to embedded submanifolds with codimension one and two. Detailed expressions were provided for the contact pressures and the edge loading, revealing the complex dependence on the face normals and on the mean curvature. Relationships were specified among the Lagrangian (hyper-)stress tensors of rank lower or equal to four, and their Eulerian counterparts.