Second-gradient continua: From Lagrangian to Eulerian and back

Francesco Dell'Isola (1) , Simon Eugster (2) , Roberto Fedele (3) , Pierre Seppecher (4)

1 UNIVAQ - Università degli Studi dell'Aquila = University of L'Aquila
2 University of Stuttgart
3 POLIMI - Politecnico di Milano [Milan]
4 UTLN - Université de Toulon

In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description, the expression of surface and edge contact interactions (which include forces and double-forces) for second-gradient continua in terms of the normal and the curvature of contact boundary surfaces and edge shapes.

Mots clés: Continuum mechanics; second-gradient continua; Piola transformation; principle of virtual work; Lagrangian formulation; Eulerian formulation

Domaines: Mécanique des matériaux [physics.class-ph]

Francesco Dell'Isola, Simon Eugster, Roberto Fedele, Pierre Seppecher. Second-gradient continua: From Lagrangian to Eulerian and back. Mathematics and Mechanics of Solids, 2022. ⟨hal-03606801⟩