On Boundary Layers Observed in Some 1D Second-Gradient Theories

Boundary layers are regions into a material domain where gradients localize. They often arise in non-local theories such as second gradient ones, which introduce various internal length scales. This work aims at exploring the properties linked to boundary layers for a few typologies of one-dimensional continua moving in plane. More particularly, three cases are explored: the deflection of an extensible Euler–Bernoulli beam, and the axial deformation of a pantographic beam with nonlinear first gradient and second gradient effects. It is concluded that the size of boundary layers depends on the internal lengths and, when nonlinearities are considered, on the external load.