Piola transformations in second-gradient continua

Second-gradient continua are defined as those continua whose internal virtual work functionals depend on the first and second-gradient of the virtual displacement. These functionals can be represented either in Lagrangian (referential) or Eulerian (spatial) description thus defining respectively the Piola-Lagrange as well as the Cauchy-Euler stress and double-stress. In this paper, we deduce the Piola transformation formulae, i.e., those relationships between all Lagrangian and Eulerian fields relevant for the formulation of the Principle of Virtual Work. In particular, we derive the Piola transformations of stress and double-stress as well as the Piola transformations for external virtual work functionals compatible with second-gradient internal work functionals. The latter transformations contain in fact the Piola transformations of the contact surface and line forces as well as the contact surface double-forces.