Transformation of surface H-forces in high grade elasticity

Abstract

The expression of the elastic energy, postulated for a given material, specifies through the principle of virtual work which external loadings turn out to be admissible. For the elastic bodies, the energy of which depends on the gradients of the placement higher than $(H+1)$ (or, equivalently, on the derivatives of the deformation gradient higher than $H$), external force densities can be prescribed over the boundary surface, which do work on the $H$-th derivative of the virtual displacements along the normal direction. Such generalized forces with $H=1,2$ have been dealt with in the literature, referred to as double and triple forces: when $H$ is increased, however, the analytical evaluation of these work terms may become prohibitive. Each Eulerian $H$-force density of this kind, acting on the current boundary surface, turns out to be equivalent to a set of diverse actions defined in the Lagrangian configuration. These actions include: (i) surface terms doing work on the virtual displacement vector; (ii) surface terms doing work on the higher order derivatives of the virtual displacement along the normal direction; (iii) edge terms, doing work on the virtual displacement or on its higher order gradients, which may be further reduced. In this study, a recursive algorithm is utilized to generate all these Lagrangian contributions for any $H$. Such an approach rests on the binomial expansion of the tensor products of complementary surface projectors, symmetrized in the work duality, and on the integration by parts combined with the generalized divergence theorem.