Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity

A two-dimensional solid consisting of a linear elastic isotropic material is considered in this paper. The strain energy is expressed as a function of the strain and of the gradient of strain. The balance equations and the boundary conditions have been derived and numerically simulated for those classical problems for which an analytical solution is available in the literature. Numerical simulations have been developed with a commercial code and a perfect overlap between the results and the analytical solution has been found. The role of external edge double forces and external wedge forces has also been analyzed. We investigate a mesh-size independency of second gradient numerical solutions with respect to the classical first gradient one. The necessity of a second gradient modelling is finally shown. Thus, we analyze a non-linear anisotropic problem, for which experimental evidence of internal boundary layer is shown and we prove that this can be related to the second gradient modelling.