Piola transformations in second-gradient continua

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