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

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