Eulerian–Lagrangian transformation of H-forces in generalized elasticity by recursive projection and integration

In the class of high-grade materials, the elastic energy is assumed to depend not only on the deformation gradient but also on the higher order gradients of the placement function. This circumstance implies, consistently with the principle of virtual work, an enlarged set of admissible external actions. For deformable bodies with a sufficiently high grade, external H-force densities can be prescribed over the boundary surface, which do work on the H-th derivative of the virtual displacements along the direction of the normal to the face. Analogously, external H-force densities can be prescribed along the curved border edges: these actions do work on the H-th derivative of the virtual displacements along the direction of the normal, of the tangent normal, or a variously ordered sequence of both. Each Eulerian H-force density of the above typologies turns out to be equivalent to a set of different actions defined in the Lagrangian configuration, acting on the differential borders of order higher or equal and working on the gradients of the virtual displacements of order lower or equal. When the order H is increased, the analytical evaluation of these generalized actions may become prohibitive. In this study, a recursive procedure is utilized to generate all these irreducible Lagrangian contributions for any H, starting from the Eulerian actions mentioned above. Such an approach rests on complementary edge and surface projectors: their multiple tensor product generates polynomial expressions, which, after symmetrization, can be dealt with as binomial expansions. The proposed method generalizes the integration by parts required in a variational framework, combining it with the divergence theorem extended to differentiable manifolds with codimension one and two. A meaningful exercise is provided, concerning the transformation of an Eulerian surface action with .