The crust shell and the edge beams of third-gradient continua in current and referential description

Abstract

A former publication has shown that some of the free boundary conditions of a third-gradient continuum can be interpreted as the equilibrium conditions of forces and moments of a crust shell and of edge beams. This was elaborated in the current (Eulerian) description. But it is obvious that a referential (Lagrangian) description of these physical phenomena must look very similar. This is, indeed, demonstrated in the paper at hand and disproves contrary claims in the literature. Moreover, the inductivist idea is omnipresent in the literature that a fictitious cut and a free surface show identical behaviour. But this is only correct with simple materials and not valid with higher gradient ones. We study a cube as a simple example where this wrong approach predicts the following absurd interaction between a subcube and the remainder of the cube: No actions at all appear on the fictitious cuts but there is one single force at the vertex of the subcube. Our findings do not depend on any specific constitutive assumption.