Generalized continuum media confronted to long and short wavelength instabilities in architected materials

In the context of architected materials, it has been observed that both long-wavelength instabilities leading possibly to localization and short-wavelength commensurate to cell-size instabilities leading to the apparition of a deformation pattern could occur. This work compares the ability of two families of higher order equivalent media, namely strain-gradient and micromorphic media, to capture both mesoscale cell-commensurate and long-wavelength macroscopic instabilities in those materials. The studied architected material consists in a very simple one-dimensional arrangement of non-linear springs, thus allowing for analytical or nearly analytical treatment of the problem, dismissing any uncertainties or imprecisions coming from a numerical method. A numerical solving of the problem is then used to compare the post-buckling predictiion of both models. The study concludes that, even on a very simple case, it is impossible for a strain-gradient Taylor-series expansion type of homogenization method to capture the cell-commensurate instabilities while the micromorphic medium can capture both instabilities but fails to converge properly in the post-buckling regime when localization appears. Micromorphic media are thus the family of equivalent continuum model that are to prefer when dealing with the possibility of patterning inside a structured medium, but if localization is to consider, it would be interesting to combine both strategies into a micromorphic, gradient enhanced equivalent medium