The strain gradient elasticity is enriched with fractional derivatives of the strain, contributing for a more accurate description of the non-local stress–strain response, introducing a better description of the influence of microstructure (local and non-local), since in some respects the fractal texture of the material is also introduced. Using that theory, the uniaxial tension problem is discussed. The one-dimensional variational problem is presented including not only the strain but also the strain gradient and fractional derivatives of the strain and the strain gradient. The solution to that uniaxial problem is found by the finite element numerical method, suitably revisited for the fractional strain gradient elastic problems.