A recursive formula to compute Lagrangianactions corresponding to an Eulerian edge H-force in elastic materials with asufficiently high grade

In the theory of the generalized elastic bodies, when the expression of the stored energy depends on the placement field derivatives of order higher than (H+2), H-force densities can be defined, acting along the border edges. Such exotic actions, deduced through an approach à la d’Alembert, do work on the H-th derivative of the virtual displacements along the normal to the concurrent boundary faces, that is sensitive to extremely fine features of the microstructure. In particular, each Eulerian edge force density of this kind corresponds to a set of external actions defined in the Lagrangian configuration. This set includes: (i) edge force densities that do work on the virtual displacements; (ii) edge force densities, that do work on the higher order derivatives of the virtual displacement along the direction normal to the concurrent boundary faces, along the direction tangent to these faces and normal to the border edge tangent, or mixed; (iii) wedge forces. When the order H is increased, the analytical deduction of these irreducible work terms may become prohibitive. In this study, a representation formula is outlined, apt to generate all the contributions appearing in the Lagrangian configuration. This approach extends in a recursive manner the integration by part required in the calculus of variations: starting from a line integral, resulting from the total contraction of dual Lagrangian tensors absolutely symmetric, the tensor product of complementary edge projectors is exploited and combined with the divergence theorem for differential manifolds.