An effective strategy to transform second-gradient equilibrium equations from the Eulerian to the Lagrangian configuration

In this communication the problem of transforming the equilibrium equations from the Eulerian to the Lagrangian form is discussed with reference to materials governed by secondgradient energy densities. In particular, novel theoretical achievements are outlined, which represent intermediate steps to attain the purpose: the transformation of edge vectors and of complementary orthogonal projectors over the boundary surface; a novel formula based on the divergence theorem for curved surfaces with boundary, relating material and spatial expressions;a remarkable relationship between Lagrangian and Eulerian (hyper-)stress tensors of differentn orders.