title: The postulationsá la D'Alembertandá la Cauchyfor higher gradient continuum theories are equivalent: a review of existing results.
abstract: In order to found continuum mechanics, two different postulations have been used. The first, introduced by Lagrange and Piola, starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients. Then, by using the divergence theorem, a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body. This method assumes ana priorinotion of internal work, regards stress tensors as dual of virtual displacements and their gradients, deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts. The second method, conceived by Cauchy and based on the celebrated tetrahedron argument, starts by postulating the type of contact interactions which can be exerted on the boundary of every (suitably) regular part of a body. Then it proceeds by proving the existence of stress tensors from a balance-type postulate. In this paper, we review some relevant literature on the subject, discussing how the two postulations can be reconciled in the case of higher gradient theories. Finally, we underline the importance of the concept of contact surface, edge and wedges-order forces.
authors.affiliation: DISG, Università di Roma La Sapienza, Rome, Italy; International Research Center M&MoCS, Cisterna di Latina, Italy.
authors.forename: F
authors.identifier: 0000-0002-3607-1113
authors.initials: F
authors.lastname: dell'Isola
authors.affiliation: Institut de Mathématiques de Toulon, (IMATH) , Université de Toulon et du Var, U.F.R. des Sciences et Techniques , La Garde, Cedex, France.
authors.forename: P
authors.initials: P
authors.lastname: Seppecher
authors.affiliation: DIMA, Università di Roma La Sapienza, Rome, Italy; International Research Center M&MoCS, Cisterna di Latina, Italy.
authors.forename: A Della
authors.identifier: 0000-0002-1782-0270
authors.initials: AD
authors.lastname: Corte
keywords: contact interactions; continuum mechanics; higher gradient continua; principle of virtual work