Topology optimization of periodic microstructures with a penalization of highly localized buckling modes

The problem of determining highly localized buckling modes in perfectly periodic cellular microstructures of in nite extent is addressed. A double scale asymptotic technique is applied to the linearized stability problem for a periodic structure built from linearly elastic microstructures. The obtained stability condition for the microscale level is then used to establish a comparative analysis between di erent material distributions in the base cell subjected to the same strain eld at the macroscale level. The idea is illustrated by some two-dimensional nite element examples and used to design materials with optimal elastic properties that are less prone to localized instability in the form of local buckling modes at the scale of the microstructure